energetic processes in follow-up electrical control systems

INTERNATIONAL SERIES OF MONOGRAPHS ON

E L E C T R O N I C S A N D I N S T R U M E N T A T I O N

GENERAL EDITORS: D . W . FRY AND W . HIOINBOTHAM

Volume 28

ENERGETIC PROCESSES IN FOLLOW-UP ELECTRICAL CONTROL SYSTEMS

ENERGETIC PROCESSES IN

FOLLOW-UP ELECTRICAL

CONTROL SYSTEMS

by

A. A. BULGAKOV

Translated by

J. B . A R T H U R

Translation edited by

D . K . G H O S H

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This is a translation of the original Russian 3nepzemu-necme npoifeccbi cAednujeao 3Λβκτηροηρμβοόα β zapMonu-uecKOM peoKUMe (Energeticheskiye protsessy sledyashchego elektroprivoda ν garmonicheskom rezhime), published in 1 9 6 0 by the Publishing House of the Academy of

Sciences of the U.S.S.R., Moscow

INTRODUCTION

BY AN electrical control system, we mean any system com-prising electrical machines, mechanical linkages and electrical control units for the purpose of regulating the motion of an industrial object.

All electrical control systems may be classified as (1) those designed to perform some electrical function — such as the defor-mation or shaping of material, shifting loads, etc. or (2) those which perform some specific control function. The latter, in the majority of cases, boils down to the movement of various elements of control and regulation (valves, dampers, rudders, lead screws and similar mechanisms).

An important fundamental type of the second class of control systems consists of systems which convert electrical signals into kinematic motion of the working parts of mechanisms.

An automatic feed control for machine tools is a typical example of a control system.

For brevity, we shall call control systems of the first class, power control systems and those of the second class, servo-

control systems.

Power control systems are usually considered to be the objects

of automation, servo-systems to be the tools of automation. Apart from the general classification, there are important

differences in both classes of systems. The basic requirement of a power control system is the possi-

bility of higher efficiency in the use of available energy, e.g. in adequate stabilization of speed of rotation.

The basic requirement of a servo-system is accurate repro-duction of movements by way of control signals. This require-ment is closely connected with rapid response of the system.

vii

viii INTRODUCTION

As regards efficiency, it is of secondary importance and is often very low.

Servo-control systems may be classified according to their internal design as (1) open-ended servo-systems or (2) closed-loop (i.e. feedback) servo-systems. The latter might be called "follow-up" control systems.

The characteristic feature of servomechanisms is power ampli-fication (the output power to the shaft of a servomechanism may, in principle, be infinitely greater than the power of the control signal at the input to the servomechanism).

Servomechanisms may be classified according to their regime of operation, i.e. (1) the regime of continuous "smooth" dis-placements, e.g. in radar aerial control systems, copying ma-chines, etc. or (2) the regime of finite "stepped" displacements in systems with so-called coordinate control, e.g. in rolling mill pressure equipment, feed mechanisms on drilling machines, etc.

In servomechanisms working in the first regime the require-ment is one of minimum dynamic error which is the same thing as maximum response rate. In servomechanisms working in the second regime the need is for the shifting of the appropri-ate member into the given position with the minimum static error and in the minimum time.

The theory of follow-up systems has been developed mainly as a theory of feedback amplifiers and is most thoroughly worked out on the basis of frequency methods. A large contri-bution has been made in this field by Soviet scientists—Mi-khailov, Tsypkin, Solodovnikov, GoPdfarb [1] et al. Of prime importance in the theory of follow-up systems (after the gen-eral question of stability) are problems connected with the basic requirement of servomechanisms—accuracy of reproduc-tion. As regards the energetic processes in follow-up systems, they have not been well enough studied. But energetic questions cannot be ignored in the design and study of high-quality follow-up systems, if only to increase the aforementioned dyn-amic accuracy.

INTRODUCTION ix

The energetics of follow-up control is concerned with the

systematization, analysis and generalization of load diagrams,

the choice of type of control systems, the determination of the

rated power and other parameters of the motor, the choice of

optimum reduction gear ratio, the study of the effect of dyn-

amic and static loads on the accuracy of reproduction of

command signals and methods of reducing the reproduction

error, etc.

In the present book we consider the fundamental energetical

problems of the theory of follow-up systems as they occur in

one regime of operation, namely the harmonic regime, in which

the command signal varies sinusoidally.

The harmonic regime in follow-up control systems is of

paramount importance for the following reasons:

(1) it corresponds closely to the actual working conditions of certain important machines and mechanisms, e.g. ships' gyroplatforms ;

(2) it can be studied as a regime equivalent to certain dyn-amic regimes;

(3) it is the most convenient stationary regime for studying and assessing the dynamic properties of all kinds of follow-up systems;

(4) experimental frequency characteristics are taken in this regime.

Specific points in the theory and energetics of the harmonic regime in follow-up electrical control systems have been dealt with in the literature.

Lebedev and Dashevskii [2] (1949) examine the motion of an inertia axle, loaded by the torque of dry friction and acted on by a sinusoidal moment of motion. The conditions of contin-uous motion are indicated and the equation of velocity for continuous motion is given. The present author [3] (1952) studies harmonic motion in control systems under idling con-ditions. He introduces the analysis of a phase trajectory (ellipse) on a plane of mechanical characteristics—angular velocity-

χ INTRODUCTION

momentum of the motor. From the geometrical conditions for contact of the ellipse with the mechanical characteristic, a relationship is established between the amplitude and fre-quency of oscillation and the saturation non-linearity and power of the motor in the form of a "limiting" amplitude-frequency characteristic. Equations are given for the power and mean-square moment. The conclusions are extended to a motor with curvilinear mechanical characteristics.

In Tsypkin's monograph [4] (1955), as an example illustrating the application of his theory of relay regulation systems, an-other approach to the problems studied in [2] is given.

In the book by Gille, Pelegrin and Decaulne [6] (1956), the construction of phase trajectories on the mechanical char-acteristics plane in the harmonic regime is used to determine the parameters of follow-up control systems by a trial method. Besekerskii, Orlov, Polonskii and Fedorov, [7] (1958), exam-ine approximate methods for calculating the effect of load-ing on the error. Findeizen [8] gives a general method for finding the motor power in a follow-up system, taking account of the harmonic regime.

Using the harmonic balance method Iosifyan and Kagan [9] study the behaviour of a motor under the action of a sinusoidal torque in the presence of dry friction.

In addition to the above references, there are reports deal-ing with questions which are similar or related to those studied here.

An attempt is made in the present monograph to deal sys-tematically with the fundamental energetic problems of the harmonic regime in follow-up electrical control systems.

The author's main task has been to elucidate the effect of loading on the dynamic characteristics of follow-up control systems and on the choice of rational design parameters. Much of the material is published for the first time.

TABLE OF SYMBOLS

a — saturation level at the input of the non-linear element GB — factor of proportionality between the e.m.f. and the angular

velocity of a d.c. machine with separate excitation GM — factor of proportionality between the torque and the armature

current for the same conditions Ctg — tachogenerator constant — the ratio of the e.m.f. to the angular

velocity D — figure of merit of follow-up control system E, e — e.m.f . Fy G — special symbols for transfer functions /, i — current (general symbol) J — moment of inertia K(p) — transfer function for open circuit and general symbol Κ — static gain (general symbol) K=K(jv) — complex gain (transfer vector) Km= \K(jv)\— modulus of the vector Κ of an open circuit and general

symbol KM — coefficient of elasticity of the mechanical characteristic and

vector of the power element with respect to the torque L — inductance Ρ — power R — active resistance r — loading factor Τ — time constant (general symbol) U — armature control circuit voltage and general symbol V — control signal at input to link (general symbol) w — kinetic energy ζ — gear ratio of reduction gear a — general symbol for angle ß — general symbol for angle y — angle, taking into account the static torque, proportional

to velocity Δ — "statism" δ — argument of the "statism*' vector Δ ε — angular acceleration of motor shaft V — argument of the vector KM

θ — angular error — argument of the error vector

xi

xii TABLE OF SYMBOLS

ι — relat ive angu lar velocity of m o t o r

κ — a r g u m e n t of the gain v e c t o r Κ

λ — durat ion of ve loc i ty impulse μ — re la t ive s t a t i c torque ν — c ircu lar frequency of forced oscil lations of control sys tem ρ — radius v e c t o r of t rac ing point (τ — a r g u m e n t of t h e v e c t o r of t h e short -c ircui ted sys tem ν — ampl i tude e r r o r of "worked off" angle φ — angle of o u t p u t shaft of reduct ion g e a r and general symbol

for a n angle

χ — a r g u m e n t of t h e v e c t o r of the power e lement Κω in p a r t i c u l a r the angle of mechanica l iner t ia = t a n

- 1 vTM

ψ — a r g u m e n t of t h e v e c t o r of the open-circuited sys tem Κ and general symbol for phase

ω — a n g u l a r veloci ty of m o t o r a n d general symbol for a n g u l a r veloci ty

S U B S C R I P T S

a ampl i tude i ind ica tor

a m p amplifier l im l imiting

a m p . p power amlifier M m o m e n t , t o r q u e

a m p . v vo l tage amplifier nom nominal

b l imiting (angu lar ve loc i ty) opt o p t i m u m

c c o m m a n d (angle) Ρ power

c s tar t ing ( torque) Ρ p e a k

c r cr i t ica l s .c shor t c i rcu i t

/ feedback st s t a t i c ( t orque )

9 g e a r b o x υ v o l t a g e

0) a n g u l a r ve loc i ty

I N I T I A L A S S U M P T I O N S

A FOLLOW-UP electrical control system consists of a power unit,

comprising the power elements of the system (the electric motor

and power amplifier), a gear train and an automatic control

unit. The latter consists of a feedback indicator unit which

displays a signal proportional to the angular difference be-

tween the command axle and the output shaft of the gear

train; a voltage amplifier to amplify the difference signal, and

correcting circuits at the input of the voltage amplifier to assure

stability.

Such a system can be represented by a single-loop block dia-

gram (Fig. 1). Some multi-loop systems (those with internal

Φζ Vi V

ω Kg V Kg

Kf

1

Φ

Kf

F I G . 1. Block schematic diagram.

tachometric feedback, for instance) can be reduced to a single-loop circuit by introducing into the main circuit a feedback loop with a transfer function Kf(p). For a single-loop system, Kf(p) = 1.

For clarity, the input signal is taken as the angle of rotation of the command axle, but the results can be extended to sys-tems with electrical inputs by simply calculating the command angle proportional to the controlling signal.

ι

2 ENERGETIC PROCESSES

The output is represented by the angle of rotation of the

output shaft of the gear train.

The power amplifier and the motor should be considered as

a single section of the block diagram, since the output circuit

of the amplifier and the input circuit of the motor, which con-

trols its speed, are in a single loop which transmits energy to

the motor. In other words, the power amplifier and the motor

form a system with mutual parameters which we shall call the

power unit of the follow-up control system.

The power elements in a servo-system have non-linear charac-

teristics: "saturation" of the power amplifier and motor, non-

linear relationship between the angular speed of the motor and

its torque and between the static torque and the angular speed,

backlash in the gears. The following assumptions and re-

strictions will apply herein.

We are considering the established harmonic regime in

follow-up electrical controllers with continuous control with

respect to the angle. Backlash in the gears is not taken into

account, since it has little effect on the processes with which

we are concerned. We take the mechanical characteristics of

the motor to be linear, which is valid, for instance, for constant-

current motors with separate excitation and partly so for

two-phase induction motors. The saturation non-linearity of

the power amplifier-cum-motor is allowed for in an inertialess

non-linear circuit in the input of the power amplifier, and we

assume that the output voltage of the latter varies propor-

tionally to the input signal in the region between the limits of

the voltage (Fig. 2). This we take as the nominal control wind-

ing voltage of the motor.

The amplifier characteristics in the linear region may be given

analytically, e.g. by the transfer function

^amp.v(P) = (1)

or graphically — by its amplitude-phase characteristic

tfamp.v(7>) = tf.mp.YO0e', (2)

http://tf.mp.YO0e'

INITIAL ASSUMPTIONS 3

where V(p) are the Laplace transforms of the input and output

signals V(t); ν is the angular frequency of the signal.

At constant mains voltage, a follow-up controller is acted

on by two external forces —the control action at the input

and the force on the gearbox output shaft due to the static

torque.

F I G . 2. Control and mechanical characteristics.

In the majority of cases, the static load of a control system is caused by frictional forces in its associated transmission and mechanical parts.

Under practical conditions the frictional forces of the me-chanical parts are complex, unstable functions of the velocities of the various members and the forces acting on them, which depend on the external conditions, notably the temperature. Statistical methods would, therefore, have to be invoked to study them.

In this book we consider, for simplicity, three idealized types of loading.

1. Purely dynamic loading or "idling", i.e. the case of a control system with negligibly small static loading.


and

Msi(œ) = Mst (sign as ω) for ω ^ 0 (4)

-M8t < MBt(a>)

<+Mst fo

r ω = 0. (5)

In the first case the static torque is allowed for in the differ-ential equations of motion by a constant "loading factor" r on the velocity and does not appear in the right-hand side as an independent force. Thus, this case becomes formally equiva-lent to perfect idling.

The second case is essentially non-linear in character and, for established harmonic motion, may be reduced, as will be shown below, to the linear case of an active torque given in the form of an independent function of time.

In this more general case, the behaviour of the control sys-tem in the linear region of the power amplifier characteristics may be described by the Laplace transformation equation:

co(p) = K0(p)Vp(p)-KM(p)Mst(p), (6) where

2. Loading by the static torque of "viscous friction", which is proportional to the speed of rotation of the output shaft of the servo.

3. Loading by the static torque of "dry friction", whose magnitude is constant, the sign depending on the sign of the velocity.

The torque due to viscous friction, in the simplest case which we consider, is proportional to the angular velocity

Mst = τω. (3)

The torque due to dry friction, in the usual piecewise-linear approximation, has a constant magnitude, but changes sign according to the sign of the velocity


is the transfer function of the power unit with respect to the angular speed of the motor and

is the transfer function of the power unit with respect to the static torque.

The system of equations describing the power unit of the control system may be solved with respect to any of the vari-ables which characterize it and they will all depend on the static and dynamic loadings. In particular, the internal voltage of the control circuit of the motor will be

U(p) = Fv(p)Vp(p)-FM(p)Mst(p), (9)

where Fv(p) and FM(p) are operators which depend on all the parameters of the power unit, i.e. on the motor as well as the power amplifier.

We shall assume all torques and moments of inertia to be reduced to the motor shaft, designating them without suffixes, like the angular speed of the motor. We shall regard the angle of the output shaft of the gearbox as the output quantity of a servo-system. It is convenient to consider the gearbox as an integrating unit with transfer function

where ζ is the gear ratio from the motor shaft to the gearbox output shaft.

In the stationary regime (p = 0) equation (6) takes the form

ω = Κωνρ-ΚΜΜ, (11)

where

Mst =

M.

This equation expresses the mechanical characteristics of a control system for constant values of the control signal Vp

ω = œb —

KMM, (12

)

where œb = KaVp (13)


is the "limiting" speed of the motor, i.e. the idling speed;

is the coefficient of elasticity of the mechanical characteristic;

( 1 . )

is the angular coefficient of the control characteristic of the

power unit ω = f(Vp) at M = 0.

Figure 2 shows the control and mechanical characteristics

connecting all three output quantities of the power unit (angu-

lar velocity, torque and control signal).

I f the inertia of the control and power amplifier circuits is

negligibly small compared with the mechanical inertia of the

motor, then, under idling conditions (ikfst = 0), from equations

(6) and (11) and the equation of motion of the control system

M = Mit + J ^ (16)

we obtain

^ ' W P - ( 1 7 )

where J is the moment of inertia of the rotating mass of the

system, referred to the motor shaft;

TM —

JKM (18

)

is the electromechanical time constant of the control system. The voltage amplifier is assumed to be linear, taking into

account the fact that the effective limits of non-linearity of its saturation characteristic are wider than the saturation limits of the power amplifier.

The characteristics of the voltage amplifier together with the correcting sections may be expressed analytically by the trans-


fer function

# a „ , p . v ( P ) = - ^ P (19)

where V{ is the transform of the signal at the input of the am-plifier, and graphically by the amplitude-phase characteristic o f

^amp.v(^)-

The feedback indicator unit, including the command appa-ratus, which gives the difference angle (selsyns, rotary trans-formers, etc.), is assumed to be inertialess with transfer co-efficient (constant over the working range)

K _dVi(P) _dv{

where θ is the difference angle between the command axle and the driven axle

Θ(Ρ) = ΨΟ(Ρ)-Ψ(Ρ)- (2 i )

For a single-loop system the transfer function of the feedback section becomes unity

K,(p) = 1. (22)

The transfer function of the complete open-circuit control system will be

K(p) = KiK^WK^pWMHP) ^ · (23)

The angle of rotation of the output shaft of the gearbox, on the basis of equations (21) and (23), can be expressed in the following form

φ(ρ) = <pc(p)Ks.c(P)> (24)

where ΚΛΛ(ρ) = χ ™ y ) (25)

is the closed loop transfer function. From equations (21), (24) and (25) the difference angle can

be expressed by the formula

/λ ι \ / \ 1 / η η \


In the particular case of a control system with internal ta-chometric feedback, the feedback section has transfer function

Kf(p) = (27)

W where Ctg = —^ (28)

is the tachogenerator constant The product of the static amplification factors of all the sec-

tions of the circuit is called the figure of merit of the control system.

D = pK(p)p_+Q. (29)

The figure of merit expresses the ratio of the velocity of the gearbox output shaft to the angular error in the estabilished tracking regime with constant velocity, coc = const.

Since all the factors become zero in the established regime (i.e. with i - - o o ) , the figure of merit may be defined in terms of the transfer functions of the angular velocity at the gearbox output shaft co(p)/z and the error θ(ρ), as p-+ 0.

D = ω { ρ) = ρ ψ ( ρ) ram * β ( Ρ ) Ρ- ο θ ( ρ ) ρ^ „ '

(™>

Example. The equations and equivalent circuit of a follow-up control system are worked out with respect to the complete schematic diagram. Let us consider a typical constant current follow-up system with amplidyne having quadrature field and selsyn indicator (Fig. 3). We represent the schematic diagram by an equivalent circuit made up of loops with directional coupl-ings. A directional coupling is a connexion between two cir-cuits, or elements in general, for which a variable quantity in one circuit affects a variable quantity in the second circuit, but a variable quantity in the second circuit does not directly affect a variable quantity in the first circuit. The first circuit is called the controlling circuit and its variable the controlling force or signal, and the second circuit and its variable are said to be controlled. A typical example of the first (controlling


F I G . 3 . Schematic diagram of follow-up control system.

force) is the current in the excitation circuit of a d.c. generator; an example of the second (controlled variable) is the e.m.f. in the armature circuit of such a generator.

A very important property of these directional couplings is the absence of energy transfer from the controlled circuit to the controlling circuit. The controlled armature circuit of a d.c. generator acquires energy from the generator shaft, i.e. from the motor which turns it, not from the excitation circuit. In general, the latter may not consume energy, for example, in machines with permanent magnet excitation and in negatively-biassed grid circuits in valves. I t is this that enables us to control energetic processes with negligible expenditure of power, using various power amplifiers.

I f the schematic diagram is put in the form of an equivalent circuit, there is no need to replace every individual component of the actual system by a corresponding element in the equiv-alent circuit, since the output of one section and the input of the succeeding section are usually combined into one circuit, and the output and input of one section belong to different circuits which can have directional couplings. A section is


described by a set of differential equations and each loop by one equation, and so it is simpler to consider loops as elements of the equivalent circuit.

An equivalent circuit using this "loop" representation will be in the form of a chain of separate loops, some (or all) of which will be connected by directional couplers.

Using this approach, the control system of Fig. 3 can be represented by the simplified equivalent circuit of Fig. 4.

' o m p . v ' o m p . p i ^ ω ω

*

F I G . 4. Equivalent circuit.

The selsyn indicator circuit may be represented by a loop with two sources of e.m.f. One e.m.f. is proportional to the angle of rotation of the command axis φ0Κί and the other to the angle of rotation of the gearbox output shaft <pK{. The difference e.m.f. Vit is applied to the input of the phase sensitive amplifier.

Tachometric feedback may be fed into this loop by means of an e.m.f. proportional to the speed of the motor.

The electronic phase-sensitive amplifier is conveniently replaced by the usual equivalent circuit for a triode, a loop with an e.m.f. 7 υ = μ 7 { , an equivalent active resistance Rv

and the inductance Lv of the amplidyne. The anode current in the circuit (which is the excitation current of the amplidyne) will be

'amp.v — τ ,m r> 1

' * ' ~i~ amp.v F

where

m _ ^amp.v / o o \ ± amp.v — jj> \

Ο Δ)

is the time constant of the amplifier.


The transfer function of this circuit will be

(ρ) _ μΙΞυ

*amp.v(P) - y 7») ~ f T y " > K**)

K amp.v\r/

AT

1a m p . v /

/

where μ is the static amplification factor of the valve.

There is a directional coupling between the phase sensitive

amplifier and the indicator circuit, which is implicit in the

equation relating the e.m.f. of the valve circuit to the output

voltage of the indicator section F a mp v t: μνν The effective

direction is indicated by the arrowheads in the equality sign.

The equation itself does not reflect the directional character of

the coupling.

The second loop corresponds to the transverse amplidyne

circuit. I t has at its input the e.m.f. induced on the brushes

of the amplidyne by the flux due to the current in the preceding

"voltage amplifier*' circuit,

âmp.p âmp.pâmp.v · (^4)

This is the equation of the magnetization characteristic and

it expresses the directional coupling between the two circuits :

the e.m.f. depends on the excitation current but the excitation

current is independent of the e.m.f.

The transverse circuit has an active resistance B p and an

inductance Lp.

The current will be

• _ Iföp y âmp.p/^p · /οκ\ 'amp.p — ι , 7 7 Τ,

v amp.p ~~ ι , 77 ^ 'amp.v > \

0 % J) 1

"t"-i amp.p r 1

τ -i amp.p ^ where

âmp.p =

TT^ (36) JXp

is the time constant of the transverse amplidyne circuit.

The transfer function of this section will be

* amp.p(P) = • (37) 1 ·* amp.p Ρ

The next three interlinked sections represent the main power unit circuit.

1 2 ENERGETIC PROCESSES

The first combines the transverse amplidyne circuit and the armature circuit of the motor. The e.m.f. of the transverse circuit is expressed by the directional equation

ν ρ ~ # p<.mp.p . (38)

The loop consists of the active resistances and inductances of the transverse circuit and the armature circuit of the motor and two generalized voltages: the e.m.f. of the amplidyne Vp and the e.m.f. due to rotation of the motor

Em = CEa>. (39)

The second loop represents the electrical equivalent of the mechanical circuit of the motor. I t consists of two generalized forces: the torque of the motor at the input

Mm = CMi (40)

and the static resistance torque, referred to the motor shaft, at the input

Mst=±Mg (41)

and one parameter — the moment of inertia, J m , of the rotating masses, referred to the motor shaft.

The coefficients of mutual impedance CE and C M , which express the conversion of the electrical energy of the first section into the mechanical energy of the second and vice versa, are equivalent to the coefficient of mutual inductance of an electric circuit and have the same magnitude if the electrical and mechanical powers are given in the same units.

For constant flux Φ

_pN ΙΟ-» Γ V 1 E ~ α 2π [ l / s e c j '

_pN IQ-« k g. m

Λ 2π9.80 A '

where ρ is the number of pairs of poles in the motor;

Ν is the number of conductors in the armature; a is the number of parallel branches in its winding.


It is obvious that

kg-m

= 9.81 sec CE Pel iE J

The third loop gives a simplified representation of the gearbox. I t would be more accurate to represent the gearbox by a

chain of interconnected loops. In the discussion below of the optimum gear ratio we shall use the simplified model of the gearbox, neglecting losses, backlash and elasticity in the inter-mediate gears and assuming that the flywheel masses of the gearbox are concentrated, like a belt drive, in the two axles — the motor axle Jm and the output axle of the gearbox Jg.

Thus, to the gearbox there corresponds a single third loop which has two generalized forces —the torque of the motor, referred to the output shaft of the gearbox, Mg — Mz and the static torque of the load applied to the output shaft of the gearbox, M%u.

The generalized velocity of the loop is the angular speed of the output shaft, cog. The loop contains one parameter — the moment of inertia at the output shaft, Jg. The gear ratio is equivalent to the transformer ratio of an electric circuit.

Correspondingly, the moment of inertia of the load, reduced to the motor shaft, will be J'g — Jgjz

2 and the total reduced moment of inertia

The tachogenerator may be shown in the diagram as a loop having a directional coupling with the second loop of the power section, Et ^ Ctco.

The output circuit of the gearbox has a directional coupling with the loop representing the phase indicator, via an inte-grating circuit, shown conditionally in the diagram.

The triple-loop circuit of the power section, after elimination of the gear ratio of gear box, is described by the two differentia]

7 - 7 4- 9

(42)


equations of electrical and mechanical equilibrium

^ dt

Mst = CMi-J^m. (43)

Using the Laplace transformation and solving with respect to the transform of the angular velocity, we obtain for zero initial conditions

ω = CE(l + TMp+TMTEp*) VP -

- κ " ι + τ1ΙΙτΡτΊϊΜ*- ( 4 4)

In these equations

ΤE = - J K (45)

is the electromagnetic time constant of the power section;

Τ M = JKM (46) is its electromechanical time constant;

KM = j h - (47)

is the coefficient of elasticity of the mechanical characteristic. In fact, multiplying numerator and denominator of RpjCECM

by iy we obtain

Rpi 1 ω,-ω

since RpijCE = cob — ω is the decrease in speed under the action of the torque M=CMi.

Comparing expressions (44) and (6), we obtain for the given controller

KM - ι+τΜ

κ;ί%τ^ <49> and

K m ( p ) = K m 1 + τΙϊ+ΤΡ

ΕΤΜρ> * (50)


The internal e.m.f. of the power section, i.e. the voltage at the terminals between the amplidyne and the motor, can be obtained by subtracting the voltage drop across the active resistance and inductance of the transverse circuit (or adding these voltage drops to the rotary e.m.f. of the motor)

d/

U = Vp~iRA-LA-^.

(51

) Determining the current from equation (43)

l + TMp + TETuf ( '

and substituting in equation (51), we obtain

rj = v l + yTMp + yTmTMp* _

*> l + TMp+TETMp*

where γ = *ψ = (54)

is the ratio of the active resistance of the motor circuit Rm to the total resistance of the armature circuit of the power section Rp =

Bm +

Ra>

Ττη=^'> Τ

α = ^ (85)

are the electromagnetic time constants of the motor and ampli-dyne loops.

I t follows from equations (53) and (9) that

1 - M M P +

1 Ε1MP

Rv l+Tap FM(P) =7T'\4.Τ r)-i-T Τ tf' ( 5 7)

A +

1 MP-T

1 E

1 MP

Note that in the particular case of idling, the power section can be represented as two consecutively connected sections.


The output of the first section and the input to the second will be the voltage U. The transfer function of the first section will be

Λ Μ - ^ . (58)

and the transfer function of the second will be

U(p) - F^p) - 1 + γΤΜρ + γΤΜΤΕρ*· { O V)

These sections will not, however, be directional, as is often stated in the literature. Furthermore, both transfer functions depend on the parameters of all three loops of the power section and cannot be regarded as self-contained units.

The overall transfer function of the above system, from the phase-sensitive amplifier to the motor, will be

K ' i p ) = V ω ( Ρ \ ^ = K^p,(P)Km(p)Ka(p) (60)

or

K P) (1 + ^amp.y P) U + ^amp.p P){\ + TMp + TET M p

2) '

(61)

If we include in this transfer function the inertialess coefficient of the indicator section and the integrating section of the gearbox, we obtain the transfer function of the complete open-circuit system

K'M = W) = KiK'ip)Tp- <62)

C H A P T E R I

T H E E N E R G E T I C S O F C O N T R O L

S Y S T E M S U N D E R N O - L O A D C O N D I T I O N S

Vector diagrams

The loading pn the electric motor in a follow-up control

system in the harmonic regime is the sum of two components —

the dynamic, proportional to the angular acceleration of the

motor, and the static, due to the forces of useful and wasteful

resistance of viscous and dry friction.

Let us consider the dynamics of a control system under no-load

conditions, i.e. in the absence of static torque on the motor

shaft and, therefore, for purely dynamic loading. Such an

idealization is not only of theoretical interest but is also

permissible in practical calculations if the static torque is

sufficiently small.

For a sinusoidal variation in the input, command signal

<Pc = <Pc.a s in vt

(6 3

)

all variable quantities in the control system in the linear region of its characteristic are, in the established regime, harmonic functions of the frequency of the command signal <pc. We can thus use the ordinary symbolic method and consider them as vectors whose projections on the time axis give the corresponding time functions.

The set of vectors which characterizes the behaviour of the control system — the command angle <pc, the angular velocity of the motor ω, its acceleration ε, torque M, difference angle, etc. —gives the vector diagram of the controller for a given constant value of frequency. The loci of the above vectors

17


with frequency as a parameter represent the amplitude-phase characteristics (APC) of the corresponding quantities. The vector diagrams and APC's enable us to obtain a picture of the behaviour of the control system in the harmonic regime which is sufficiently complete and graphic for practical purposes.

The vector diagram of a follow-up system is constructed on the basis of the transfer functions or amplitude-phase charac-teristics of the open-circuit system and the individual sections. Substituting the angular frequency jv for ρ in the transfer function gives the transfer vector

Κ =K{jv) =Ka&, (64)

whose modulus Ka = K(v) and phase ξ = ξ(ν). The hodograph of the tip of this vector for a frequency variation from — oo to -f 0 0 gives the amplitude-phase characteristic of the given system or section.

Let us introduce the following symbols for transfer vectors. The amplifier vector—the general symbol and two stages

of amplification (voltage and power)

*amp = #amp.oe-J*. (65)

A voltage amplifier is indicated by the addition of the sub-script "v" and a power amplifier by a "p".

The argument of the amplifier vector, like those of most of the other sections of the control system, has a negative sign, since due to inertia parameters in its circuit, the output signal usually lags the input, the lag increasing with frequency.

The power section vector, with respect to angular velocity

Κω =KOMe-ù (66)

The angle χ takes account of the phase lag between the angular velocity and the control signal at the input to the power section. When the electromagnetic inertia of the ele-ments of this section can be neglected, the angle X represents only the mechanical inertia of the system.

NO-LOAD CONDITION 19

In this particular case

* · = y r § m e~ix = K a c os

1 e~jx ( 6 7

)

where χ = t a n ' 1 vT M. (68)

As will be seen later, the mechanical inertia angle χ is an important physical parameter, as many dynamic properties and processes are determined by parameters such as signal frequency y , coefficient of elasticity of the mechanical charac-teristic KM, gear ratio 2, moments of inertia of the motor and the load, not considered separately but invariably in the combination vTM = vKM(Jm-^Jl/z

2) which is expressed phys-ically by the angle X.

In addition, the mechanical inertia, in most cases, is the dominant feature in a short-circuited control system and determines the dynamics of the system.

The dimensionless product vTM is often considered as the relative time.

The force vector with respect to moment

The gearbox vector

Kg=-e-jl2rn , (70)

where ζ is the gear ratio. The indicator section vector

Ki - h\. (71)

The feedback loop vector

K, =K,.ae-iK (72)

The transfer vector of the open single-loop circuit will be equal to the product of the vectors of all the sections.


k = kik^kJekf = j r a e - * . ( 7 3 )

The modulus of this vector

Ka =

Ki.a Κ ^ ν ^ Κ ω αΚ α αΚ ι a ( 7 4 )

and the argument

in = v. - 4 - ν 4 -

2

1 ψ = κ+χ + - π+Α; ( 7 5 )

and has a negative sign since it is determined by the arguments of sections with phase lags.

The vector for a closed loop, according to (25), will bo

£ s . c =

=

Ks,cae-K (76)

1 +K

The hodograph of the tip of this vector as the frequency varies from - ~ to 4 -

0 0 describes the amplitude-phase cha-

racteristic in the linear region of a follow-up control system.

The frequency variation of the modulus Ksc a and phase σ give, respectively, the amplitude and phase frequency characteristics of the system

Another useful vector is the "statism" vector

Λ = — • — = Δ > * (78)

which characterizes the accuracy of the control system in the

established harmonic regime, just as the statism factor charac-

terizes the accuracy of an automatic regulation system in the

established regime with constant value of the command signal.

The transfer vectors Ksc and Δ and the unit vector 1 = ej 0

form a feedback triangle

k8.c+À = 1. (79)

NO-LOAD CONDITIONS 21

The moduli and arguments of the vectors of the ampli-

fication factor of the closed loop system and of the statism

are simply expressed in terms of the modulus and argument

of the open loop vector by formulae which are easily obtained

from (79) and (76) by substituting in them Κ from equation

(73) and putting e" j v = cos ψ — j sin ψ.

The modulus of the statism vector is

Δα = 1 , (80) 1 / 1 + 2 / ί α cos y + # g

and the phase

δ = t a n " 1 — — , (81) 1 + Ka cos ψ

ν

where Κα cos ψ and Κα sin ψ are the real imaginary parts of the APC vector for an open loop.

For the amplification vector of a closed loop system, remem-bering that Ksc = AÊ, we obtain the modulus

Ksc = ΚαΔα = Ka (82) V i + 2 j r e o o 8 V + t f *

and the argument

a = ψ-δ = t a n - 1

T jr

8 1 P ψ — (83)

Like ψ, it has a negative sign since the vector of the output angle lags the vector of the command angle.

The transfer vectors of the whole system, if, Κs c and Â,

and of the separate parts, are independent of external forces acting on the system, including the modulus of the input command signal, and characterize the physical and structural properties of a control system for a given command signal frequency. They are analogous to the impedance vectors, or, more strictly, to the admittance vectors of the a.c. theory.

Using the transfer vectors of a system and its separate parts, for any given value of the command signal vector it is easy to construct the vector diagram which characterizes


quantities involved in the system (the output angle and the error angle, the angular velocity and acceleration, the torque of the motor, the voltages and currents in the individual sections and so on).

The vector of the command angle is analogous to the voltage applied to a four-terminal network in circuit theory, the output angle is like the voltage at the output of the four-terminal network and the vectors of the other parameters of the control system are like the internal currents and voltages of the four-terminal network. Due to the presence of the amplifier, this four-terminal network is, essentially, an active one.

This analogy has allowed us to use, in the analysis of control systems, frequency methods which were first evolved in the theory of electronic amplifiers.

In drawing the vector diagram of the parameters of an electric control system, we take the vectors as fixed and the axis of time rotating clockwise and coinciding with the ordinate axis at zero time. The moduli of the vectors are taken as the amplitude values of the corresponding quantities. The command angle vector is directed along the positive direction of the real axis, i.e. we take q?c = (pc = (pca.

The output angle vector will be

φ = <pcÊs.c = Λα6' ] σ · (84)

I t lags by an amount a with respect to the command angle. The error angle vector, according to equations (78), (79) and

(84), is Ô = q>c-<p = cpcÀ = <pcAae>*. (85)

I t leads the command angle cpc by an amount δ. I t follows from equations (84) and (85) that the product of

the command angle vector and the transfer feed-back triangle of equation (79) gives a triangular relationship between the angles

<pc = ψ +6. (86)

The error angle vector Ö and the phase indicator transfer

NO-LOAD CONDITIONS 2 3

vector determine the signal vector at the input to the voltage amplifier

famp = 6*i = ΨοΔΚ^*. (87)

This vector has the same phase as the error angle vector, since we are neglecting the inertia in the phase indicator and, therefore, it leads the command angle vector by an amount δ. Multiplying it by the overall transfer vector of the voltage and power amplifiers, we obtain the control vector influencing the power section

Κ = ^ampfamp = M # i # a m p ^ ~χ )

. (88)

I t lags in phase with respect to the vector of the error angle and the input signal by an amount κ due to the inertia para-meters in the amplifiers.

The product of this vector and the velocity transfer vector of the power section gives the angular velocity of the motor

ώ = Ϋρ£ω = M t f ^ p Z . e « » — * > . (89)

The angular velocity vector lags the control signal vector by an amount χ, accounted for by the inertia in the power section.

In the particular case, when all the electromagnetic inertia parameters of the power section are negligible, the angle χ is due only to mechanical inertia in the system and is expressed by the equation

χ = t a n " 1 vTM. (90)

The angular velocity vector may be obtained in another way, by differentiating the vector of the output angle φ

ώ = = jvcpz = ω αβ J ^ 2 , (91)

where ωα = K*.<2V<Pc.a =

νζΨα (

9 2)

is the amplitude of the angular velocity of the motor. The following equation relating the phases of the vectors

in the control system emerges from equations (89) and (92):

σ = ± π + κ + χ-ο. (93)


The internal e.m.f. of the power section, i.e. the voltage at the terminals between the motor and the power amplifier, from equations (6) and (9) may be expressed by the vector

Û = ώζ?= FvYp, (94)

where

Fv = Fv{v)eX

is the transfer vector of the power section with respect to voltage.

The phase of the voltage vector lies between those of the control signal Ϋρ and the angular velocity ώ.

The amplitude of the output angle, expressed in terms of voltage, is given by the equation

<P=-At-Ü =^Ϋν (95) jvFvz jvz

In the simplest case of a motor control circuit with no inertia, when the transfer function of the power section takes only mechanical inertia into account and is given by equation (17), and fiv = 1 (i.e. Û = Ϋ ) , the amplitude of the output angle is given by the expression

- K° U - K » U — χ (96)

or, putting KJJ = cob (see equations (13) and (94) ),

φα = Λ = ^ cos χ. (97)

The angular acceleration of the motor is given by the vector

ε = jvd> = ν2ζφ> (98)

dco and the dynamic torque M$ = J — by the vector

]Slj = -1ν2ζ<γ. (99)

In the given case of idle running, it is equal to the torque vector of the motor.


The angular acceleration and dynamic torque vectors are in

antiphase with the output angle vector.

Figure 5 shows the complete vector spectrum for the follow-

up electrical control system for one value of frequency.

F I G . 5. Vector spectrum.

Similarly, we can construct the vectors of any intermediate quantities, such as the voltages and currents in the individual amplifier circuits.

Accuracy of reproduction of harmonic oscillations. The optimum gear ratio of the gearbox

In the harmonic regime the accuracy of a control system is determined by the degree of divergence between the oscilla-tions of the input command signal and those of the output shaft of the gearbox (Fig. 6,a). These deviations have different characters in the operational region of the control system where its characteristics can be taken as linear and in the region where non-linearity becomes important and non-linear distortion occurs.

In the linear region the error in reproducing oscillations, as


already shown, is characterized for any value of frequency by

the statism transfer vector.

The statism vector, according to equation (78), is given

by the relative vector error in the reproduction of the angle

A = ^ t = A = _ * ! L e » . (100)

<Pc <Pc <Pc.a

I t depends only on the structural and physical properties

of the system and provides a figure of merit for it. The closer

the modulus to unity and the smaller the argument, the better

the system.

F I G . 6. Error diagrams: a — space; b — vector; c — time.


The absolute vector error in oscillation reproduction is given by the vector of the difference angle (Fig. 6,b)

6 = ö0Ä. (101)

The difference angle vector Ô characterizes the floating error in angle reproduction since its projection on the time axis gives the instantaneous value of the geometric angle between the command axis and the output axis

0 ( 0 = ? > â( 0 - 9 > ( 0 = èá sin (vt-ô). (102)

Its modulus gives the greatest value of the angular error and its argument the phase of this greatest value relative to the amplitude of the command angle.

The accuracy of a control system in the harmonic regime is characterized also by the amplitude and phase errors. The amplitude error represents the difference in amplitude of the command and output angles and is given (Fig. 6,c) by

øÄ.á = <Pc.a-<Pa- (X° 3 )

Its relative magnitude is

õ = Øï.á~Øá =^á < ( 1 0 4)

<Pc.a <Pc.a

The phase error or angular error is the phase of the output angle with respect to the command angle

ó = arg <p c-arg ö. (105)

The relation between the amplitude and phase vector of the difference angle and the amplitude and phase errors is readily established from the feedback triangle [equation (86)].

The relative amplitude error (Fig. 7) is

v= - V l + z J 2 - 2 z l a c o s o + l. (106)

Assuming that, in practice, Ä\<^\, and approximating the square root, we find that v = z l a c o s ô , or

(107)

: ENERGETIC PROCESSES

The phase error is given by equation (83)

sin ψ a = t a n - 1

Ka + cos ψ (108)

These equations show that the amplitude error and, especially, the phase error are significantly less than the amplitude and

.κ" U,) I + K ' V , )

-1 \ -0-5 / \ l l 1 / 1

/ 0 5 1 f ι t

\ 1 ^

YKV2) \ ι+κ' w

F I G . 7. The sign of the amplitude error.

phase (respectively) of the difference angle. Their magnitude depends on the structural parameters of the system and the frequency of the oscillations by way of Ka (74) and ψ (75).

The amplitude error can be either positive (φα<φ0,α) or negative (<paxpCM), or, for certain values of frequency, equal to zero (φα = φ0Μ).

The condition governing the sign of the amplitude error and when it becomes zero is easily established from the feedback vector triangle [equation (86)] with sides cpca, φα and θ or

* NO-LOAD CONDITIONS 2 9

from the similar triangle [equation (79)] with sides 1, K s c a

and Äá.

The amplitude error passes through zero when ö0 = ö0.á or Ksc = 1, i.e. when the vectors i f s c and 1 form an isosceles triangle with base Äá. But, from equation (76), Ksc = 1 means that I Ê J = 11 + K\, and this is possible (Fig. 7) only if the real and imaginary parts of the transfer vector of the open loop system are, respectively,

RcK = Ka cos y) = - ' (109)

and

Im Ê = f(nsmy> = ±lj4KfL-l.

and its argument

Ø0Ã = t an" 1 yj4Kl-l. (110)

In the ])articular case, Ka(v1) = Ka(v2), the argument %pCT may have two values in the range

3ð ð ~ T ^ ø < ~~2'

In the frequency band in which the APC of the open loop system lies to the right of the line R e i f = — - , the amplitude error is positive ; if to the left it is negative ; and at the points of intersection it is equal to zero.

The sinusoid of the output angle thus has the same amplitude as that of the command angle but lags it by the phase error angle ó.

In the limiting case when the amplitude error is zero, the phase error is not, of course, zero due to the presence in the system of inertia parameters. Its limiting value (at v = 0) is found from equation (83), substituting ipcr from equation (HO),

( I l l )


In the particular case, when Ka = — \ , tan ψοτ = 0, y>CT = —π, Κ = - | -e~J Î\ £ s c = e~jn and the angular error σ = —π.

The critical value of the phase error decreases as the amplifi-cation factor of the open loop system increases and, for large enough values, aCT & t a n " 1 (1 jKa).

The modulus of the statism vector in the absence of an amplitude error has the value

since | l £ | = | 1 - f È. |. I t could become zero i f the amplitude and phase errors simultaneously became zero.

An important structural parameter of a control system is the gear ratio of the gearbox.

There is a definite optimum value of the gear ratio for which the control system responds to a given displacement in the minimum time. This optimum value of the gear ratio gives the minimum starting-up and braking time for systems with an open-circuit control loop on switching the full voltage.

Under idling conditions ( J f g t = 0) the optimum gear ratio, in the above sense, is given by the formula

where Jm and Jl are the moments of inertia of the motor and the load, respectively (the inertia of the intermediate parts of the gearbox is ignored here).

For this value of the gear ratio (which corresponds to the minimum of the product Jz = JmZ + Jl\z) the kinetic energy of the motor is equal to the kinetic energy of the load

(112)

(113)

Jmω2 _Jlüif

~~2 ~ΊΓ (114)


For a control system working in the harmonic regime, the

gear ratio of the gearbox affects the accuracy of reproduction

of the oscillations of the output shaft of the gearbox and their

limiting amplitude. The motor power necessary to cope with a

given value of angle at a given frequency also depends on the

gearbox ratio.

The effect of the gear ratio on the accuracy of the controller

is revealed through the amplification factor of the open circuit

loop JS", on which depends the statism vector À> see equation

(78).

We select from equation (73) of the transfer vector of the

amplification factor of the open circuit loop that part Kz

which depends on the gear ratio,

È = &'£z. (115)

I t combines two vectors —the vector of the power section

with respect to angular velocity (66) and the gearbox vector (70)

(116)

Neglecting, for simplicity, the electromagnetic inertia of

the power section, we obtain, from equations (67) and (70),

1 vzjl + v2Th ( Ð 7 )

In this vector, the gear ratio affects both the modulus and

the argument.

The argument depends on the gear ratio because the angle

is determined by equation (68), in which ÔM = JKM. For

simplicity we assume that the flywheel masses of the system

are concentrated, as in a belt drive, in two axes only— in the

motor shaft J m and in the gearbox output shaft J g . Thus, the

total moment of inertia, referred to the motor shaft, is expressed

by equation (42)

J = J

m+Jg-2 ·


The angle χ will be

χ = tan"-1 vKM m + Jg -4r . (118)

Angle # will decrease monotonically with increasing gear ratio.

The gear ratio of a step-down gearbox may vary from the minimum value of 2 = 1, for which the load is connected directly to the motor shaft, up to values which may be in the hundreds or even thousands.

Consequently, angle χ may vary between limits no greater than

tan" 1 vKM{Jm + Jg) - χ > tan" 1 vKMJm. (119)

The argument of the vector Kv and, indeed, of the transfer vector Κ of the whole circuit [i.e. the angle ψ in equation (75) which contains χ] vary between even narrower limits.

Using equations (42) and (46), the modulus of the vector Kz is given by the formula

Kz= , K

: (l 2

° )

As the gear ratio increases, Kz first of all increases (away from the minimum value with 2=1), and then decreases, varying between the limits

0 < Kz R<Z K

<* (121) y/l + *KUJm+Je)

2

By examining expression (120) for a maximum (or the expression under the square root in the denominator for a minimum) we can find the critical gear ratio for which Kz tends to a maximum

- O p t . , - ^ ! + v 2 K h Jl - yJl + V2T2M- W

Here and below

T m = K M J m ; Tg = K M J g . (123)


The maximum value of the amplification factor Kz for z o pt is

Kz = K

» (124)

I f the frequency is increased, the critical gear ratio rises monoton-

ically and tends asymptotically, asv o o , to a limit

This limiting value of the gear ratio to give maximum amplifi-

cation factor, is equal to its value in equation (113), optimized

with respect to the starting and stopping conditions by switch-

ing a constant voltage.

I t is convenient to write formula (122) in the form

where

f° P t' V " V l + ^ m

W 0 P t'

;0 Pt . l - J l + V2 7 2 - ^ O p t . v

f O r J η = J„

(125)

(126)

Figure 8 is a graph of z o p t el as a function of vTg in a semi-

logarithmic plot. Using this graph it is easy to find the optimum

o L 0-1 a i - O

FIG. 8. Optimum gear ratio as a function of frequency.


gear ratio of the gearbox for any frequency and given control system parameters Jgf KM and Jm.

The critical value of the gear ratio zopt v gives maximum accuracy of reproduction of oscillations for the given frequency, since the accuracy of a follow-up control system increases with increasing gain of the open-circuit loop containing Kz M.

The effect of the gear ratio on the accuracy coincides with its effect on rapid response.

In the harmonic regime the phase error a serves as an indica-tion of rapidity of operation. The smaller it is, the smaller the inertia of the system and, therefore, the quicker the response.

But, according to equations (108) and (111), the greater the gain, the smaller the phase error.

Thus, in the harmonic regime the optimum gear ratio depends on the frequency. This gear ratio is optimum in terms of both accuracy and rapid response. In addition, the magnitude of the angle of the output shaft of the gearbox has a maximum at the optimum value of the gear ratio. In fact, as the gear ratio increases, this value first rises and then falls (Fig. 9) . I t is clear from a comparison of equations (97), (116) and (120) that the critical gear ratio for which the amplitude of the angle is a maximum is given by formula (122).

The maximum value for this gear ratio is

The aforementioned connexion between the critical gear ratio (122) for maximum gain at a given frequency and the optimum gear ratio (113) for minimum start/stop time is fully understood.

Starting and stopping by direct switching correspond to an infinite command signal frequency for which equation (122) reduces to equation (113).

The effect of the gear ratio in the harmonic regime on the gain (and, therefore, the dynamic accuracy and speed of re-sponse), the value of the output angle of the gearbox and also,

Ψ = wb.lim (127)

vj2vTMy/l + v*T*l'

NO-LOAD CONDITIONS 3 5 •(0

01 =

V/V

—S

9UH

n

93[OJq

ß

èÀ

=

V

/V

—seu

l] Pîios) eje^

eurB

J'Bd

uia^S

iCs no oi^vi J-B

ÖS eq

cj jo

qoejje

eqx

*6 -oi^j


as will be shown, the overall power of the motor, depends on

one and the same function of the gear ratio. This function can

be obtained, for example, from equation (120) and written

in the form

It depends on the mechanical inertia parameter # m=tan~ 1i>7 7

m, where Tm = JmKw and on the ratio of the moment of inertia of the load to the moment of inertia of the motor. In order to show quantitatively the effect of the gear ratio on the above quantities, we give in Fig. 9 a semilogarithmic plot of a family of curves of f{z) as a function of ζ for various values of the mechanical inertia parameter vTm and two values of the ratio of moment of inertia of the load to moment of inertia of the motor rotor.

These curves show how great an influence the gear ratio has on the dynamics of the system.

We can draw the following general conclusions, based on the above discussion.

There is no single optimum value of the gear ratio for all dynamic regimes. Therefore, the choice of gear ratio must be coordinated with the predominant frequency characteristic of the actual working part of the control system which must have an essentially statistical character.

Since the operational frequency band of follow-up control systems is limited in the majority of cases to comparatively low values, formula (113) gives an over-estimate of the value.

I t is also necessary to bear two facts in mind.

1. The effect of the gear ratio in a closed loop follow-up system is felt mainly through the overall gain which may be varied by the amplifier.

2. The optimum gear ratio depends on the load.

1 (128)


The energetics of electrical control systems

The effect of the load on the behaviour of an electrical control system is conveniently studied with the help of the curve of speed of the motor as a function of its torque.

For second order systems the curve co=f(M) is the phase trajectory of the representative point and uniquely charac-terizes the state of the system at any moment of time. For higher order systems this trajectory is the projection of the image point on a plane of two independent variables—the angular velocity ω and the torque M.

Unlike the system of coordinates usually used for construct-ing phase trajectories—velocity as a function of coordinates and system of coordinates—force phase trajectories on a mechanical characteristics plane have an important advantage in the analysis of the dynamics, especially the energetics of control systems, since the product of these coordinates gives the power.

A phase trajectory drawn on the mechanical characteristics plane may be considered as a load diagram for the control system, expressed in an obvious form, and not as a function of the time parameter.

I f the relation between the curve œ=f(M) and time is estab-lished by means of the appropriate parameter scale, it gives a complete picture of the behaviour of the control system in the dynamic regimes, just as the mechanical characteristics give a picture of the behaviour of the control system in the stationary regime.

The phase trajectory ω, M shows the relationship between the dynamic and static loads and the mechanical characteristics of the motor, the role of saturation non-linearity of the power amplifier, the static torque and mechanical characteristics and so on.

For periodic motion'; the phase trajectory .ω, M has the form of a closed curve, which, in the harmonic regime, is an ellipse (Fig. 10).


Let us find the equation of the load ellipse and establish a relation between it and the mechanical characteristics and vector diagram of a follow-up control system. We shall consider the plane of the mechanical characteristics as a complex plane, taking the torque axis as the real axis and the angular velocity

as the imaginary axis. The equation of the ellipse is described by the radius vector

ρ = tâ+jœ. (129)

taken from the origin of coordinates in the mechanical charac-teristic plane.

The instantaneous value of the torque M and angular velocity ω of the motor are expressed by the vectors J&L and ώ of the vector diagram (Fig. 5).

To do this, we rotate them anticlockwise, multiplying by the unit vector e?9i.


The rotating vectors are expressed in terms of the initial vectors of the fixed vector diagram by the equations

Mt = = -Μαβ><«-σ); (130)

ώί = ώ&ν1

= 7 ω α β^ - σ > . (131)

The instantaneous values of the torque and velocity are given by the projections of the rotating vectors on the real axis

M = -Ma cos (vt-a) = \(ä% + ä u ) \ (132)

ω = -œasm(vt-a) = y (ώ, + ώ^) . (133)

Here the suffix c indicates the conjugate of the fundamental vector

Mu = -Μαβ-*νί-σ); œtc = - ; ω α e"*"'-^. (134)

Substituting M and ω from equations (132), (133), (134) in equation (129) and collecting terms, we obtain

ρ = \ { Ü ^ t ) + \(Mix+jcbtx). (135)

Eliminating the unit vectors e*(9t~a) and e ~ j ( W _ c r) from the brackets we can put ρ in the form

ρ = Ρ + Q (136)

where

Ρ = -λ(Μα+ωα), (137)

Q = -\(Μα-ωα) (138)

and, therefore

Q = -\(Μα + ωα)4<«-'>-\(Μα-ωα)*-*«-'Κ (139)


Putting vt — <r = 0, we find one semi-axis of the ellipse

èi = -Ma, (140)

directed along the axis of abscissae, i.e. along the torque axis, and putting vt — σ = π/2 we find the other semi-axis

ρ2 = / ω α, (141)

directed along the ordinate axis, i.e. the angular velocity axis.

Thus, the semi-axes of the load ellipse lie along the axes of coordinates and are equal in amplitude to the torque and angular velocity of the motor [equations (92) and (99)]:

Ma = Jv2z<pa = Jv2zKStCa<pCM (142) and

ω α = νζφα = vzK8tCa(pca (143)

The angular argument of the vector ρ will be

arg ρ = t a n " 1 - ^ = t a n " 1 t a n ^ — — , (144)

and its modulus

ρ = y/a>2 + M2 = yfco* sin2 (vt - a) + M\ cos2 (vt-σ). (145)

The equation of the ellipse in rectangular coordinates will be

ω 2 M2

It is evident from equations (142) and (143) that for a given frequency the load ellipse of a given control system increases in proportion to the amplitude of the command signal equally along both axes of coordinates. For a given amplitude of output angle the ordinates of the ellipse are extended in proportion to the frequency and the abscissae in proportion to the square of the frequency (Fig. 10).

For a given output angle and frequency, the dimensions of the ellipse depend on the moment of inertia of the system and the gear ratio of the gearbox. The ordinates of the ellipse


are proportional to the gear ratio. The abscissae of the ellipse have a minimum at the optimum gear ratio [equation (113)], since in this case the product of the reduced moment of inertia and the gear ratio has a minimum. The semi-axes of the ellipse are then equal to

Ma = 2φαν2 y]JmJg, ωα = <pav

For a fixed gear ratio, the ordinates of the ellipse remain constant while the abscissae are proportional to the reduced moment of inertia of the control system. The value of the input command angle for a given load ellipse is determined by the value of its output angle and the gain of the closed loop system for the given frequency [equation (84)].

The load ellipse takes the form of a circle if it is drawn in normalized units ω ' = ω / ω α along the ordinate axis and M1— = M\Ma along the axis of abscissae.

I t is possible to give on the ellipse the distribution of points of a parameter—time or angle χ = νί with the help of a para-meter scale. I t is simplest of all to construct a scale for one period in the form of a circle with centre at the origin. The distribution of parameter points on the scale will be uniform and may be expressed by the angular scale

m * = ^ " [ r ^ } < 1 4 7)

where Tf is the period of forced oscillations of the system, i.e. the time of one rotation of the vector ρ.

The zero of the scale lies at the point of intersection with the axis of abscissae. Each quadrant corresponds to one quarter of a period. The first and third quadrants correspond to the motor regime and the second and fourth to the generator regime of operation of the motor.

The power developed by the system at a given instant of time is given by the area of the rectangle bounded by the


coordinate axes and the abscissa and ordinate of the appropriate point on the ellipse. As a function of time the instantaneous value of the power is expressed by a sine function of twice the frequency

Ρ = ωΜ = Pa sin 2(vt-a) (148)

and may be represented by the vector

Ρ = Ρ α e-J'2 ( w

-°->, (149) where

Pa = \ ωαΜα = J Jœlv = j Jz\lv\ (150)

The power has a maximum value at time

π/4 + σ T A

-—

Τ ~ ·

The projection of the power vector on a time axis rotating at twice the frequency traces out a closed curve, like a four-petalled rose (Fig. 11).

Since no static resistive forces are applied to the system in the case of idle running under consideration, the mean value of useful power of the motor in a time interval of a number of half periods is equal to zero. A positive value of power corre-sponds to the motor regime, a negative value to the generator regime. The first and third petals of the rose, lying in the corre-sponding quadrants of the ω, M diagram of the mechanical characteristics, represent power consumed by the motor due to the increasing kinetic energy of the rotating masses of the system first in the forward direction and then in the reverse direction of rotation. The second and fourth petals of the rose and quadrants of the diagram represent power delivered by the rotating masses of the system back into the network with the motor working in the generator regime.

The kinetic energy stored or released by the flywheel masses in the system in each quarter period is, according to equation (143),

w = Ja* = J y g v V ( 1 5 1)


F I G . 11. Load "rose".

On the load diagram, the power developed by the motor may be shown as a curve drawn as a function of motor torque (Fig. 10). We have, from equation (146),

ω=±ω^1'{ΐΪ· (152)

The equation for power as a function of torque is

ρ = [ ± ω α 7 ι - ( ^ ) 2 ] · [ ± ^ (i53)

I t will be recalled that the amplitude of the angle <pa through

which the gearbox output shaft turns, is related to the amplitude

of the command angle <pc a by the modulus of the transfer

vector of the closed loop system K8Ca [equation (76)].


π π

Another, somewhat different, value is given by the root-mean-square power of the motor

λ - . - Η ΐ * · * - ^ - iß™*'*' "56>

0

which is V2 times smaller than the peak power. The first formula, which gives a somewhat larger value

(by 16 per cent), should be taken as more reliable. This question is discussed in detail below.

Amplifier non-linearity. The limiting amplitude-frequency

characteristic and power of the motor

In a real system the characteristics of all the sections in which the output quantity is a function of the input are non-linear. From the energetics point of view, the most important is saturation non-linearity.

The curve has symmetrical branches in all four quadrants, the meaning of which we shall discuss. Each branch has a maximum whose coordinates are

MCT = ±^ßMa; ω 0 Γ = ± ^ | ω α · (154)

Hence we can obtain the value already found for the amplitude of the power under idling conditions [equation (150)].

Since the useful power of the motor and its mean power at the axle are zero in the harmonic regime, they cannot be used to estimate the thermal load of the motor which is determined by its losses.

An approximate load characteristic for the motor from the point of view of its losses is provided by the mean power of the motor over one quarter period,

τ T

Pm = Y j = 1 Jz2<PW = *Pn- (155)


The output quantity has an upper limit due to anode current saturation in the valves, magnetic saturation in the amplidyne. etc. These non-linearities are called saturation non-linearities. In idealized form, they are drawn as sections of straight lines (Fig. 2).

The limiting signal level is different for different sections. I t is usually lowest in the power section of the control motor and in the power amplifier, since this limit determines the nominal power of the elements of the power section and the overall dimensions of the whole system. In control systems with electronic amplifiers, the lowest limit is usually found at the output stage of the amplifier.

For practical purposes it is necessary to know the relation between saturation non-linearity, amplitude and frequency of the output signal and the nominal power of the motor, i.e. its overall size.

To simplify the discussion, we assume that the saturation non-linearity takes place in one non-linear section which has no inertia. The remaining sections of the system are assumed to be linear. This allows us to transfer the non-linear section to any position in the single-loop system. I t is most convenient to put it between the indicator and the voltage amplifier. The limiting level of the non-linear section is assumed to be such that the voltage in the control circuit of the motor is within its nominal value.

I f the signal at the output of the non-linear section exceeds the critical value, the system goes beyond the linear region and all sinusoidal quantities will be distorted. An exact analysis of this case can be carried out by, for example, the method of matching, but it leads to rather complicated expressions. We shall, therefore, restrict ourselves to an approximate study based on the method of harmonic linearization.

In general, in a real system a non-sinusoidal periodic signal appears at the input to the non-linear section. Truncation of this signal by the saturation non-linearity produces further distortion in it. The sum of the harmonics of the signal pass


where F(Va) is the signal at the output of the non-linear element

for an input signal Va. For F a < a , it is equal to Va and for

Va>a it has a constant limiting value of b (Fig. 2 ) ; ±a and

±b are the coordinates of the limit point which marks the

junction of the linear and saturation regions.

Figure 12 is a graph of the equivalent gain as a function

of the input signal for saturation non-linearity. In the linear

region it is unity, decreasing hyperbolically in the saturation

region.

K(Va)=l for - α < Γ α

< α ;

K(Va) = -^+-| y- for α < 7 α < 2 α ;

K(Va) = — for Va * 2a. v a

(158)

Formula (157) is also valid for certain other inertialess non-linear elements with single-valued characteristics, partic-ularly for elements with a region of insensitivity.

through the power amplifier and the motor which possess

inertia parameters. These inertia parameters filter the signal,

reducing the amplitude of its upper harmonics. I f this filtering is

sufficiently effective, we can neglect the upper harmonics

and assume that the signal remains sinusoidal in all sections

of the system. Thus, the non-linear inertialess section can be

replaced by a linear one with an equivalent gain whose magni-

tude depends on the amplitude of the input signal.

The expression for the equivalent gain is easily obtained

from the formula for the coefficient of the first term in the

Fourier expansion for the signal at the output of a non-linear

element with a sinusoidal input signal.

I t may be written as a simple approximate formula, due to

Tsypkin [5],


This kind of characteristic occurs in some types of power amplifier in follow-up control systems, for example, the thyra-tron type.

!-0

0-8

0 -6

Κ 0 - 4

0 - 2

0 2 4 6 8 10 Vp ο

F I G . 12. Equivalent gain.

All the relationships we have found for the linear region of operation in servo-systems are valid for the non-linear region if the equivalent transfer coefficient for a non-linear section, K(Va), is used in the transfer function of the voltage amplifier, i.e. if we bear in mind that

^ m p = ^ a m p ^ ( ^ ) and £ ; m p = * Ϊ * ρ . α e ~ ' \ (159)

where i f a mp is the static gain in the linear region and i * r a p is the same, allowing for saturation.

Consequently, all the equations and vector diagrams will be valid only for one value of signal amplitude at the input to the non-linear element.

In order to determine the equivalent gain for the non-linear section, it is necessary to know the magnitude of the amplitude at the input to this section which in turn depends on the equiv-alent gain. The magnitude of this amplitude may be calculated as follows.


From equation (87), the signal vector at the output of the phase indicator is

^amp = Kx6 = φ,Κ,Δ = φ / (160) l+KK(Va)

where Κ is the gain of the open-loop circuit in the linear region (transfer vector).

This equation can be solved graphically with respect to the unknown signal amplitude by finding the point of intersection of the curve

fi(Va) =

<Pca ί (161) \l+KK(Va)\

and the straight line

UVa) =

Va. (162)

Using the signal amplitude at the input of the non-linear element, the equivalent gain of the non-linear section is calcu-lated from equation (157), then the gain and transfer vector of the amplifier, i f amp from equation (159). We can now find the vector of the gain of the open-loop circuit K*, the angular velocity of the servo

and all the other quantities. The expressions for the power obtained in this way will

be less than the actual values, since they take account of only the first harmonic of the signal. This discrepancy might have been allowed for by a correction factor but it would be difficult to determine it. I t depends on the filter action of all inertia elements in the closed-loop circuit and on the actual distribution of saturation non-linearity among the sections. However, the calculation of one first harmonic to a sufficient accuracy answers our problems.

The amplitude of the first harmonic at the output of a non-linear element, according to equation (157), is given by

Va.out = VaK(Va) =\[F{Va) + F{\Va)\ (164)


and has the form of a series of straight lines (see dotted line,

Fig. 13); in the first linear part, for F a < a , V' = Va; in the

second part, for a < Va < 2α, V" = \ b + \V a and in the

third part, for 7 α > 2 α , the value is constant, a'cmt = y

F I G . 1 3 . Amplitude at the output of a non-linear element.

Calculation by the exact formula [1] gives

F = ^ { ^ [ i ~ ^ i n 2 a ] + c o s 2 a } ' (165)

where • - ι

a

a = s i n 1

-ψ- ,

and the graph of the output signal is a monotonie curve tending asymptotically to the limiting value

V = - b ^ 1-276. π


Thus, the amplitude of the first harmonic at the output of a saturation element does not exceed the value corresponding to the limit point Va = a, K(Va) = b by more than 27 per cent for any value of the input signal.

Hence it follows that in a closed-loop follow-up system when the amplitude of the input signal increases above a certain critical value the amplitude of the error vector 6=<pc — φ will increase considerably, since the amplitude of the output angle, due to saturation, does not increase and does not compensate for the increase in the input signal.

Since, in the saturation zone, an infinitesimal increase in the amplitude of the first harmonic causes a large increase in the amount of distortion in the curve of output angle and the dynamic error, it must be concluded that the operational range of a follow-up control system in the harmonic regime is limited in practice to the linear part of the characteristic of the principal element by distortion. I t is advisable to take the limit of this linear region as being defined by the nominal voltage of the motor control circuit. We shall assume in the majority of cases below that this limit is combined with those of the limiting values of the parameters, F a — o o .

In the complex plane of the vector diagram and the APC, it is convenient to represent the limits of the linear region as a circle with centre at the origin and radius equal to the limiting voltage of the motor control circuit, which we have taken to be equal to its nominal voltage. This circle is called the nominal motor voltage circle. I t is constant for all frequencies. The value of the power section control signal corresponding to the nominal motor voltage varies with frequency, and, from equation (94) we get

* V a = ·£*- · (166) r v.a

Figure 14 shows the APC's of the motor voltage and the difference angle 0, and also the nominal voltage circle. The parameter of the APC, i.e. the frequency v, at the point of


intersection of the circle with the voltage APC gives the critical,

i.e. maximum, frequency at which the controller can still

work in the linear region of the power amplifier characteristic

and, therefore, non-linear distortions due to its saturation

non-linearity are excluded.

F I G . 14. Amplitude-phase characteristic.

The voltage vector for this frequency will have a critical value Ûcr, its modulus will be the nominal voltage and its argument will have some fixed value.

The broken line in Fig. 14 is part of the circle which bounds the region of limiting values of the parameters, beyond which the control system cannot operate ( F a = o o ) . The radius of this circle is 27 per cent greater than the radius of the nominal voltage circle.


On the mechanical characteristics plane, the operational regime of the control system is bounded by an ellipse inscribed in the mechanical characteristic for the nominal motor volt-

Actually, the ellipse which is the locus of the point ω, M in the harmonic regime gets bigger as the amplitude and frequency of the command signal increase and for a certain value touches the limiting mechanical characteristic, i.e. the nominal voltage characteristic (Fig. 15).

F I G . 1 5 . Load ellipse and tangent.

This point of contact determines the boundary between the linear region and the saturation region, and the corresponding ratio between the angle and the frequency gives the limiting amplitude-frequency characteristic: the amplitude of the output angle as a function of frequency for the given boundary conditions.

The equation of this characteristic may be obtained from equation (95), putting the voltage equal to the nominal value or the control signal of the power section Vp equal to the limiting value.

Κ k

Φα. lim = T-£ - ^nom = Ϋρ. lim- (1 6 7

)

jvFv jvz


Hence, bearing in mind equation (94), we find the equation of the limiting amplitude-frequency characteristic

Ψα. lim =

v

jg ^nom · ( 1 6 8)

The value of the difference angle vector corresponding to this boundary or limiting regime is easily found from formula (85), remembering that <jp = 0i£:

6iim='r4E*rÜ™- ( 1 6 9)

jvFOKz

For an inertialess control winding of the motor, we find from equation (96)

9 a. l im = K

f ™ = ^5 2 5L c o

« χ. (17°) vzjl +

irTfc vz and from equation (97)

m

œ

b.\\m 0J

b. lim n r k_ „ /171\ Ψα. lim = , a — =

0 08 Χ· (

1 7 1)

vz y/l + v2TM νζ Here ^ö.iim *® the limiting boundary speed of the motor,

i.e. the motor speed under perfect idling conditions (M = 0) and for the nominal voltage;

χ is the phase lag of the angular velocity of the motor behind the control voltage [equation (68)].

The functions (168), (170), (171) express the limiting (more accurately, boundary) amplitude-frequency characteristic of the controller, since they give the maximum possible amplitude which the actual controller can supply at any given frequency. The amplitude of the command signal can thus in principle have any magnitude as long as it is not less than that which determines the boundary value a of the signal at the input to the non-linear element.

Saturation non-linearity does not affect the phase-frequency characteristic, as long as we take the appropriate non-linear section as being single-valued and, therefore, not introducing extra phase shift.


Qc = \Jo>2 + M2 = <pavZyJs\Ti2 (vtc-a)+Jvcos2 (vtc — σ ) . (172)

Let us eliminate the time tc, corresponding to the point of contact, using the equation for the slope of a tangent

d 1 tan ( 1 8 0 - a ) = = J > c ot K : - * ) . (173)

where a is the internal angle of the tangent. Since the tangent coincides with the mechanical charac-

teristic, then t a n a = KM (174)

and from formula (173), using equation (68) we find

cot {vtc-o) = vT = tan χ. (175)

Substituting the value of tc from this equation in formula

(172), we obtain

Qc = <Pa™J i + )2 = <pavzJco82x + Jvsm2x. (176)

In addition, from equations (142) and (143) we can derive an equation for the angular coefficient of the radius vector

tan β = ^- = tan (vL — σ) = (177) r M Jv v 0 9 Jv v

and, eliminating tc,

t an ß = t Ä v = =

^c o t χ

·

( 1 7 8)

To obtain the equation for the limiting amplitude-frequency characteristic (171), it only remains to solve the oblique-angled

An approximate equation for the limiting amplitude-frequency characteristic may also be obtained directly from the geometri-cal relationships of the ellipse inscribed in the limiting mechani-cal characteristic (Fig. 15).

The modulus of the radius vector at the point of contact, from equation (145), is


triangle formed by the radius vector ρ0, part of the mechanical characteristic and the intercept on the axis of abscissae, know-ing the side QC and angles α and /?, as given by formulae (174), (176) and (178).

So far we have been considering a system in which the motor has linear mechanical characteristics. However, the limiting amplitude-frequency characteristic exists for motors of any type. The formula for the limiting amplitude-frequency charac-teristic may be generalized to include motors with curvilinear mechanical characteristics. In this case the quantities cobUm

and T M , which occur in this formula, must be considered as design parameters, the values of which are determined by an imaginary mechanical characteristic coinciding with the tangent through the point of contact of the ellipse and the actual mechanical characteristic (Fig. 15).

To determine the values of the parameters of this imaginary mechanical characteristic, namely the limiting boundary velocity cobl[m and the electromechanical time constant ΤM, i t is necessary first to determine the point of contact and draw the tangent. This problem can be solved by the following method [3].

1. On the basis of the limiting mechanical characteristic, construct a family of curves ω = f(M/Jv) for various constant values of ν as a parameter (Fig. 16).

2. Superimpose on this family of curves a family of arcs of the circles

ω2+(τν-)2= {νψα)2· ( 1 7 9)

These circles correspond to the load ellipse of equation (146) to scales which are irrelevant and which vary along both axes.

3. Draw the family of tangents through the obtained points of contact. The intercepts of the tangents on the ω-axis give the values of the boundary velocity cobAim and the slopes


of the tangents give the product Τ Mv for the fixed value of

frequency corresponding to each curve. This follows directly

from the expression for the slope of the tangent

tan α' = ωο. lim

M, (180)

F I G . 1 6 . Determination of the limiting A . F . C .

In this way, ^5.iim and Τ^ (yT^) are determined for all fixed values of frequency and are substituted in the formula for the limiting amplitude-frequency characteristics.

In the particular case of a non-monotonic mechanical charac-teristic (Fig. 17) with a maximum (multiphase asynchronous motor), the point of contact is shifted to one of the limiting positions along the co/a>a-axis or the M /ilfa-axis. In the first case, o)b\im = coaf ΤM = 0 and, from equation (171), the limiting amplitude is

™ Ω

6 . lim / 1 Q 1 \

Z»lim

NO-LOAD CONDITIONS

In the second case Ma = McMm and the amplitude is

MCm um

57

Ψα. lim — (182)

The equation of the limiting amplitude-frequency charac-teristic enables us to study an important practical point,

F I G . 17. Determination of the limiting A . F . C for the boundary case.

namely, how the maximum possible amplitude and frequency of oscillation of the output angle are related to the nominal power of the motor.

An expression has been obtained above (150) for the amplitude of the power at the motor shaft, which is valid for the linear region and, therefore, for its boundary with the saturation region.

Let us substitute in it the value of the limiting amplitude of the angle from equation (171)

a. lim

1 J »CO*, lir

2 [1 + *TV\ (183)

Here, according to (13), cobm]lm may be expressed in terms


of the limiting value of the turning moment

ωο . lim = KMMC l i m. (184)

Taking equation (18) into account as well, we obtain

Pa. LIM = ~2œb. Um^c. LIM 1 _|_ V2J^2

= "4

Ω&· HM^C. LIM

S IN 2%·

(185)

In this formula, the product c o ö l lm ikf c l im is related to the limiting value of the maximum mechanical power of the given motor.

0 MC, M C

M

F I G . 1 8 . Mechanical power parabola.

Actually, since the angular velocity of the motor is

ω = œb — KMM, (186)

then the mechanical power at the motor shaft as a function of the torque varies parabolically (Fig. 18)

Ρ = œbM — Κ M M2 (187)

and has a maximum at

Mcr = ±Mc; œcr=\œb. (188)


Hence the maximum mechanical power is

Pp=\<obMc. (189)

I f œb and Mc have the limiting values, i.e. are taken at the

nominal values of motor voltage, the expression

Pp.\im = Τ

œb. l i m ^ c . lim (190)

gives the magnitude of the limiting mechanical power, i.e.

the maximum power the motor can develop at the nominal

voltage.

This quantity depends on the overall dimensions of the

motor and its design features, but does not directly reflect

the state of heating of the motor.

Substituting from equation (190) in (185), we obtain the

relation between the amplitude of the motor power in the

harmonic regime and the maximum mechanical power of the

motor as a function of frequency

vT M

Pa. lim =

%Pp. lim j ^ 2 ^ 2 * (191)

As the frequency is raised, the power amplitude first increases

and then decreases (Fig. 19).

PO.CR PT>ïim

0 I 2 3 4 5 6 7 8 9 10

F I G . 1 9 . Power as a function of frequency.


By inspection we see that for the critical frequency

»cr = J - (192) 1 M

the power amplitude has a maximum equal to the maximum

mechanical power of the motor

Pa = P p . um- (193)

We obtain from equation (171) the amplitude of the output

angle at the critical frequency corresponding to the maximum

power amplitude

Ψ α . ^ =ω

~ ^ φ - . (194)

In expression (191) vTM = tan χ, (195)

where χ is the phase of the oscillations of the angular velocity

of the motor relative to the input signal of the power section,

if the other inertia parameters of this section are neglected

(64). On this basis, using elementary trigonometric identities,

formula (191) may be written in the form

^ a . um = Pp. lim sin 2χ. (196)

The power amplitude varies sinusoidally as a function of the double angle 2χ and has a maximum at % = 45°.

The above formulae do not directly include the nominal power of the motor, but this relation is easily obtained since the heating of the motor is determined by the mean load power (over a quarter period) [equation (155)] and in the harmonic regime this is π/2 times less than the power amplitude.

Pm=yz^l=^JV0>l. (197)

Thus, the mean power as a function of frequency varies like the power amplitude and has a maximum at the same critical frequency [equation (192)], equal to the maximum mechanical power of the motor, divided by π/2.


In accordance with the conditions of permissible heating,

the motor is chosen so that its nominal power is equal to or

greater than the mean power developed by it in the given

regime.

So that the controller can run for a lengthy period over

the entire frequency range without overheating, the nominal

(rated) power of the motor must satisfy the condition

2 Pnom — Ρ m ~ "Z"Pmech. lim · (198) π

The resultant relation

Pmech. lim 71

λ = m

J ;c n

- "m = £ (199)

^nom ^

depends only on the design of the motor.

It can be written in the following more graphic form

where

• = *(l +

K*

M» ™ \ ( l +

ω»οη» \ ( 2 0 0 )

- *nom) 4

\ w

nom J \ KMMnomJ

k t T= ^ B L (201) m nom

is the multiplier of the starting torque;

* „ o m = (202)

is the slip at nominal torque.

We can now draw the following important conclusion.

In the absence of static loading, any motor can operate

at all frequencies without thermal overloading, regardless

of its nominal voltage, provided its design factor λ is less than

π/2 (Fig. 20). Otherwise the frequency band is divided into

three parts. In two of them the motor will be under-rated as

regards heating, whereas in the third, which lies between the

other two, overheating will occur.


F I G . 20. Determining the nominal power.

The boundary values of the frequency can be found from

juation (191), putting

This gives the equation equation (191), putting Ρ α Μ ϊη =

πΙ

2Ρηοτη

a nd ^p.llm =

A^

>nom-

vT M π 4λ'

(203)

the solution of which for ν gives the two values which split

the frequency band into its parts:

1 (2λΜ \W \

In the frequency band

(204)

(205)

the motor will overheat.

From a study of expression (200), it is not difficult to show

that λ has a minimum, equal to 1, at KM =

From the geometrical relationships (Fig. 15) it is possible

to solve the converse problem of finding the parameters which


the motor must have in order to operate over a given amplitude

of oscillation of the angle at a given frequency. To do this,

we can make use of the equations obtained above for the

limiting amplitude-frequency characteristic.

Solving ( 1 7 1 ) for the limiting angular velocity of the motor

(ω at M=0)

ω5. um = φα. imWy/l + ^ n · (2 0 6

)

The starting torque of the motor (M at ω = 0 ) , from equation

( 1 2 ) , is

Mc lim = = V Ï T ^ . ( 2 0 7 )

Hence, the maximum mechanical power of the motor, from

equation ( 1 9 0 ) , is

Pmech.li

m = (1 + „«Tfr). ( 2 0 8 )

and the nominal power of the motor necessary to operate

over an amplitude φα at frequency ν for a long time without

overheating, from equation ( 1 9 9 ) , will be

^ n o m = y l " m " V ( l + v ^ ) . ( 2 0 9 )

Eliminating TMin formulae ( 2 0 8 ) or ( 2 0 9 ) according to equations

( 1 8 ) and ( 1 1 4 ) , we obtain

In the right-hand side of this equation, there are only two unknowns (design parameters of the control system): the coefficient of elasticity of the mechanical characteristic KM

and the gear-ratio of the gearbox z. The motor power has a minimum as either parameter is varied.

Examining equation ( 2 1 0 ) for turning-points in the usual way, using the equation

dPmech.lim = ^ { 2 n)


we find the critical value of the gear-ratio for which the motor power is a minimum. I t can be shown that

zcv — ôpt.v ~ X\~T^~ \ / ι Ι „2772 — ôptôpt.i * (212)

i.e. this value of the gear-ratio agrees with the value in equation (125), which gives the maximum accuracy and maximum amplitude of angle oscillation and is therefore optimum in all the relationships.

Using the equation

^Pmech.lim = 0, (213) dKM

we can find the critical coefficient of elasticity

KM.CT = γν = 7 1 — > (2 1 4

)

for which the motor power is a minimum. The mechanical time constant for this value of the coefficient

of elasticity is the reciprocal of the circular frequency of forced oscillations:

T^ = \ = LT<- <215)

The angle of mechanical lag at the critical value of the coefficient of elasticity is independent of frequency and equal to 45°:

X c r = tan"* v T m c t = 450 ( 2 1 6)

The optimum gear ratio of the gearbox depends on the coefficient of elasticity, and the critical value of the coefficient of elasticity on the gear ratio. The simultaneous solution of equations (212) and (214) for the two unknowns 2 O PT and KMcr would yield the values for which the motor power would have the minimum possible value.


However, these two equations do not have a simultaneous solution in the actual range of the variables. Geometrically, this means that the curve of the optimum value of ζ on the ζ, ΚM plane does not intersect the curve on this plane of the critical value of KM c r, i.e. the minima of the motor power with respect to the two variables do not coincide.

For the optimum value of gear ratio, the motor power will be

PMECHUM = /I+JÎKI*f1 + " 2 ( J m K M + LV

( 2 1 7 )

I t increases with KM) i.e. the motor power decreases as the rigidity of its mechanical characteristic increases.

For the critical coefficient of elasticity [equation (214)] the motor power is

^ech.l im = j < ^ W = 1 ?î^^+/|) . ( 2 1 8 )

I t increases as the gear ratio increases. I t follows from this that, in practice, it is necessary to choose

a motor with a rigid characteristic and determine the optimum gear ratio for it from formula (212). Since the motor power in-creases with frequency, the calculation must be done for a frequency at the top end of the working band.

Of special interest is the case where the gear ratio has the optimum value for starting and braking

«*->/&· < 2 i 9 >

I t has already been shown that this is the value of the gear ratio [equation (212)] in the limiting case, when v-+<*>.

The total moment of inertia, referred to the motor shaft, for this value of gear ratio is twice the moment of inertia of the motor

J = 2Jm, ( 2 2 0 )


and the critical value of the coefficient of elasticity of the motor is

Kmci = JV7^- (221)

The limiting speed of the motor (ω at M — 0) will be

a>b = <Pa.bvJ*3^, (222) and the starting torque

Me = 2ψα.η^

JZJmJi- (223)

The motor has a maximum limiting power

^ c h . i i m = < ) W - (224)

The relationship between the values of the maximum limiting power P\ P" and P"\ calculated from formulae (217), (218) and (224), depends on the choice of values of ΚM in P' and ζ in P

n. I f the optimum value of ζ is chosen, (122) and (212),

then P'" > P', since the chosen value of ζ is less than z o pt

[equation (219)].

When a motor is chosen according to its mean power, by equation (198), and checked for maximum limiting power [equations (217), (218) and (224)], it must be checked when hot (unless it is known that the factor λ (119) has plenty in reserve).

The overheating temperature of the motor is determined by its losses. The losses depend on the control voltage and the loadings on the power to the motor shaft.

In the limiting regime, the motor voltage has a constant nominal value. Due to this, there exists between the losses and the power to the motor shaft a unique relationship which can be obtained from the curve of efficiency as a function of useful power at the nominal voltage.

Since Ρ


F I G . 21. Loss power.

From the graph of loss as a function of useful power for a

given motor we can construct the graph of power loss against

time. I t has the form of a periodic non-sinusoidal curve with a

constant component (Fig. 21). The minimum value of the

losses corresponds to the constant part and the periodic part

represents the alternating losses.

The mean value of power loss is

π ~2~v

Pioss.m = \ P a | ^ s i n 2(r<) dt. (228)

0

the power loss is

Plots = λ

-^Ρ· (

2 2 6)

In the harmonic regime the loss power, from equation (148)

is

^loss = ^ P e | e i n 2 r f | f (227)

where the efficiency η is also a function of time. By definition

it can only have positive values.


If we replace the variable efficiency (a function of the power)

by some equivalent fixed value ηΜ (which depends on the

amplitude of the power), then

π

PiosS.m =^Pa [ sin 2(vt) dt = i ^ ü L P m , (229)

0

where Pm is the mean load power (over a quarter period of

the signal).

This equation provides a basis for selecting the overall

dimensions of the motor according to the mean load power

over a quarter period.

For checking the selected motor, it is necessary to calculate

the mean power loss [equation (228)] and make sure that it

is less than the loss power of the selected motor at the rated

loading with some in reserve, i.e.

Pl0ss.m ^ P i]

~L • (230)

The reserve is necessary in view of the fact that in the harmonic

regime the cooling conditions are worse than when operating

at a constant nominal speed due to the periodic reduction of

speed.

The calculation of the loading for a given motor is considered

in the next chapter.

C H A P T E R 2

L O A D I N G B Y A S T A T I C T O R Q U E

P R O P O R T I O N A L T O V E L O C I T Y

The equation of motion of a control system

The formal description of the motion of a follow-up control system loaded by a static torque proportional to velocity reduces easily to the case we have already considered with purely dynamic loading.

A static torque proportional to velocity may be written in the form

where r is a constant proportionality factor characterizing

the magnitude of the load.

The equation of motion of an electric control system will be

Consequently, the excess torque, which determines the motion of the system, M — rco = J dœ/dt decreases proportionally to velocity (Fig. 22).

The motion is the same as it would have been under idling conditions but on another mechanical characteristic with a lowered limiting speed œh and lower values of the velocity factor KQ9 the coefficient of elasticity KM and, therefore, a reduced electromechanical time constant TM of the system.

M6t = reo (231)

(232)

69


F I G 22. Mechanical characteristics of a control system loaded by the torque of viscous friction.

Substituting the expression for the motor torque from equa-

tion (232) into the mechanical characteristic equation

we obtain ω = Κωνρ-ΚΜΜ,

Κωνρ = ω(1 + ΚΜτ) + ΤΜ^

(233)

(234)

In this expression TM = JKM is the electromechanical time constant. The product KMr also has a completely deter-mined physical interpretation : it can be thought of as a relative parameter characterizing the static torque proportional to velocity. The static torque at the limiting boundary velocity of the motor is denoted by Mr = Mst for ω = ω 5 1 ί π 1. Then (Fig. 22)

^ö.lim M

r Mr KMr = McMm

œbMm

M (235)

c.lim

The parameter μ = ΚΜτ is equal to the ratio of the static

torque at the boundary velocity (ω for M = 0) to the starting

LOADING BY A STATIC TORQUE 71 torque of the motor. Theoretically it can vary from zero to infinity, but in practice μ never exceeds 1. As can be seen from Fig. 22, if μ = 1, the steady speed of the motor is half the limiting value and the steady torque is half the starting torque. I t is known that under these conditions the motor develops maximum useful power [equation (190)] and, therefore, for μ = 1 it has the maximum possible load.

Dividing both sides of equation (234) by l + μ , we obtain

K* ν „ , TM deo

ι D — ω

~f Ϊ · - j T (20b) l + μ p l + μ dt v '

or

KVp = œ'b + T ' M ^ . (237)

where

κ'° = τ τ μ <238>

is the equivalent velocity factor of the motor, taking into account the effect of loading by the static torque of viscous friction :

a* = K'QVp (239)

is the corresponding equivalent limiting velocity of the motor (ω for M = 0) and

Τ'" = ΤΓμ = JK'M ( 2 4 0)

is the equivalent mechanical time constant. The effect of the fictitious lowering of the electromechanical

time constant is explained physically (Fig. 22) by the reduction of the steady speed of the motor which is given by the ordinate of the point of intersection of the mechanical and load charac-teristics œ'b. Thus, for the same value of mechanical time constant the transition process becomes established corre-spondingly more quickly.


The transfer function of the motor of a loaded system is of the same form [equation (17)] as for no-load conditions,

but with different values of the constants KQ and Τ M [see equations (238) and (240)], the equivalent magnitudes of which will now depend on the load factor μ, i.e.

l + μ

Substituting this expression for Κω(ρ) in the transfer function for an open-loop system [equation (23)] and considering it as an equivalent transfer function for a loaded system, we can extend to the latter all the conclusions reached for the case of pure dynamic loading. However, this interpretation is inadequate from the engineers point of view, since it conceals the contribution of a variable energetic factor — loading by the invariant design parameters of the system expressed by the transfer vectors. Our task is to clarify this feature.

Circle diagrams

I t has already been shown that, in general, the behaviour of the power section of a servo-system in the linear part of the characteristics is described by equation (6)

o>(p) = Ka(p)Vp (p) - KM

(p)Mtt (p).

(6)

In the given case, the static torque can be expressed in

the form

^ B t ( P ) = rœ(p). (243)

Substituting this expression in equation (6) and replacing ρ

by jv we obtain ώ = &0Ϋν-μΐμώ. (244)

LOADING B Y A STATIC TORQUE 73

where

^ = 1 ^ · (245)

Hence

. r

ω =

μ~ κΜ

Ϋρ. (246)

This equation gives an expression for the angular velocity of

the motor, taking account of loading and shows that the manner

in which the latter reacts back on the velocity can be formally

Vp

- τ — μ - τ — μ

F I G . 2 3 . Equivalent block diagram.

represented by a negative feedback circuit connected across

the section (Fig. 23) and having a transfer function (Ϊ1

μ/Ε0) proportional to the load factor μ. Thus, the equivalent transfer

function of the power section for a loaded system is given in

terms of the actual transfer function by the equation

Κ = — . (247)

Here and below, a dash refers to a quantity in the loaded system.

Thus, in order to obtain the equivalent transfer vectors of a system loaded by a static torque proportional to velocity, it is sufficient to replace the transfer vector of the power section by the equivalent transfer vector [equation (247)].


For a loaded system we then obtain the following equivalent transfer vectors:

open loop:

closed loop:

* ' = _ * _ ; (248)

statism :

i t ' - K - K - ^ a c • f 24Q)

A> = -JL^= = Δ 1 + f * . (250) 1+K' + 1 + ΑμΚ

As the load parameter μ varies, at constant command signal frequency, all the transfer vectors will describe geometrical loci which, as can be seen from equations (248), (249) and (250), are circular.

The circle for the open loop transfer vector [equation (248)] passes through the origin at the point μ = °°(Fig. 24). The centre is given by the vector

Ρ Ρ m = Ê ^ = Ê £c , (251)

$ μ Χ - Ρ μ 2jP^j

the modulus of which is equal to the radius of the circle r = | m \.

Here and below the suffix c denotes vectors which are conju-gate to the principal vectors, the suffix J denotes the imaginary part and μ the real part of a vector.

The distribution of load parameter points round the circle may be found using a scale, the equation of which is

ρ = &(1+μϊ·μΛ). (252)

The scale passes through the tip of the vector È. at the point μ = 0.

The circle for a closed loop also passes through the origin (Fig. 25). The vector of its centre is

™ = £ - (253)


K'

+

F I G . 2 4 . Circle diagram for open-loop system.

and the equation of the parameter scale is

ρ = £ ( 1 + * β + μ / „ . β ) . (254)

The statism vector circle does not pass through the origin. I t follows from the equation

A = l-K.c (255)

that the statism circle can be found if we transfer the circle for the closed loop system to the point + 1 and change the sign, i.e. turn it through an angle π. The equation of the scale of the parameter μ will be

ρ = - i t ( l + £ c + ^ c ) . (256)


F I G . 2 5 . Circle diagram for closed-loop system.

The above circles completely determine the vectors of the output angle and the difference angle and, therefore, of all the characteristic quantities of a follow-up control system.

Thus, the behaviour of a follow-up control system loaded by a static torque proportional to velocity is described by circle diagrams with parameters related to the static torque μ.

The output angle vector circle is the same as the transfer vector circle for a closed system, multiplied by the command angle vector:

Ψ = <PcK.c = Φ , ) t, . (257) 1 + Αμ¥μ


or, assuming that we put <pc = <pc, simply the K'sc circle,

scaled up by the modulus of the command angle. The vector

of the centre of the circle, the radius and the parameter scale

are all changed correspondingly.

The difference angle circle, too, is the same as the circle

of the statism vector, multiplied by the modulus of the command

angle

Ö' = M ; = Ö - I ± ^ . (258)

The signal vector at the input to the voltage amplifier will

have as its geometric locus the statism circle multiplied by

the factor Ki

Kmv = KP. (259)

The product of this circle and the transfer vector of the

voltage and power amplifiers i f a mp gives the circle for the

power section control signal described by the vector

Κ = ^ a m p ^ m p = P ^ W ' (2 6 0

)

The circle for the angular velocity of the motor can be derived

from the circle for the vector of the angle swept through by

the output shaft of the gear box. To do this we have to take

its derivative multiplied by the gear ratio of the gearbox, i.e.

multiply the circle for the angle φ together with the vector

of its centre and scalar parameter μ by the vector vz and rotate

it through π/2

αϊ

= φ jvz

= ώ \—;— . (261)

The angular velocity circle can also be constructed from the difference signal ( Ϋρ) circle, by multiplying by the equivalent power section transfer vector,

ώ' = K'Jv (262) Differentiating again, we obtain the circle for the acceleration

vector :


This circle becomes the geometric locus of the dynamic torque if its scale is multiplied by the reduced moment of inertia of the system

1 M] = e ' J = Mj

(264)

The angular velocity circle can also be used as the circle for the static torque if its scale is multiplied by the load factor r:

i l / s t = fco = τώ 1

(265)

F I G . 2 6 . Load vector diagram.

Adding the circles for the dynamic and static torques we obtain the circle for the motor torque (Fig. 26), described by the vectors

M' = M'j+Mst. (266)

Since in any given case the static torque is determined by the angular velocity of the motor and the dynamic torque by its derivative, the motor torque can be expressed in terms of the dynamic component of the torque. According to equations

LOADING B Y A STATIC TORQUE 7 9

(264) and (265), we find

ϊί' = J f iT i ( l - / tany) = Α*1"1***7,. (267) 1 f μί^Λ

Here, γ is the angle by which the motor torque lags behind the dynamic torque in phase (Fig. 26). Its magnitude depends on the ratio of the static to the dynamic torque

^ η γ = ζ^ =

7^Μ

=

μ c o t χ· ( 2 6 8)

The vector

l - ; t a n y = / 1 + - ^ - e - ^ = — — e~* . (269)

Let us now find the locus of the motor control circuit voltage. To do this, we must substitute in the power section equation [equation (9)]

Û = FvYp-FMMit

the values of the control signal Ϋρ and static torque üifst = rcü' which are observed under actual load conditions, i.e. those marked with dashes. Then, using equation (246), we find

Substituting in equation (260) À' from formula (250), we obtain

Ϋ'ρ = « M ^ a m p , 1

1μ

\ - (271) l+K+μFμ

and substituting this expression in equation (270), we find

ι+K+μFμ ' Ü1 = φ,Κ^^ / * £ μ \ , (272)

where, for brevity, we put

Fx= (273)


Thus, the circle for the overall state is also the locus of the voltage controlling the motor. The vector of its centre is

m = φ,Κ, i f a m p FvKc-Fx(l + Kc) a m p ( l+tf )*V . e - ( l + * e ) V

the radius

(275)

and the equation for the scale parameter

ρ = I* VEKM&M C + + ^ Β

(276)

I t is possible to obtain from these equations the limiting

values of angle and frequency, i.e. the values at the nominal

voltage. From equation (272), putting | = ^nom> w e^ n ( ^

lim ~~ w

nom (277)

The vertical lines denote the modulus of the expression which can be found from the circle diagram.

From the limiting value of the output angle we obtain

9>lim = «Pc.lim^s.c = U nom , ^ ^ ^1 ^ , · (2 7 8

)

I *ν + * Χ . μ \Ζ ν

In order to study the behaviour of a follow-up electric control system and construct its operational characteristics it is not necessary to draw all the circles. We can confine ourselves to drawing the basic circles and determine the characteristic quantities from these circles using the scale appropriate to each characteristic quantity, the scale being complex, i.e. vector, in the majority of cases.

I t is convenient to take three circles as basic circles, e.g. the closed loop transfer vector circle, the statism circle and the motor torque circle. The values of the vector scales for the circles are easily derived from the above equations.


Figure 27 shows the circle diagram of a follow-up control

system for output angle, torque and voltage with r = 4, and

Fig. 28 the same for ν = 10.

F I G . 2 7 . Circle diagram for ν=4.

Consideration of the transfer vector equations [equations (249), (250) and (257)-(264)] shows that the corresponding characteristic quantities depend on the load in different ways. F o r the vectors of the closed loop gain i f s c , the output angle amplitude, its derivatives — velocity and acceleration, and also the dynamic torque proportional to the latter, this depend-ence is weak. I t is determined by the presence in the denomi-nator of the sum 1 + μϊ

1

μΔ, in which the term μ$μΔ is small compared with unity since Δ« \ and vanishes for no-load conditions.


For the other vectors—those of the open-loop gain [equation (248)], statism [equation (250)], difference angle [equation (258)] and the signals of all the intermediate sections of the

/* = 0 · 5 μ'-\

F I G . 2 8 . Circle diagram for ν = 10.

amplifier circuit, in particular — the dependence on load is strong by equations (259), (260) and (271) . I t is determined, not only by the factor \+μ!*μΑ in the denominator, but also by a factor 1 + μFß in the numerator (except for the È. vector, in which it is in the denominator), where μ$μ is much greater compared with unity. Thus the expressions for the charac-


1+μΡμΑ

Then the closed loop gain vector will be

K« * ^ s . c ( l - ^ Z Î ) ; (280)

the output angle

φ' = φ(1-μΡμΑ); (281)

the angular velocity . /

OJ the acceleration

ε the dynamic torque

M] = Ά^Ι-μί^). (284)

ώ(\-μΐμΑ); (282)

ε(1-μΡμ J ) , ; (283)

Instead of circles, the loci of these quantities will be straight lines — the chords passing through the point μ = 0 of the exact circles.

For the other vectors, which are strongly dependent on the load (except for the open-loop gain), the above approximation leads to the expression

•l±tpL= (1+μΡμ)(1-μΡμΑ) 1+μΡμΑ

* 1 + / ^ ( 1 - Z f ) - Δ μ* Fl * 1+μϊ*^, (285)

To a rougher first approximation for these quantities we can neglect the term μί1

μΑ in the denominator of the expression for the closed loop transfer vector [equation (249)] and the statism transfer vector [equation (250)]. In doing this we are neglecting variations in the closed loop vector with load,

teristics of a control system under load can be simplified if we can tolerate some error.

Without introducing much error, we can, by the usual approximation formula, put

1 1-μΡμΑ. (279)


(287)

To construct the linear loci according to the approximate formula, the angle, velocity, acceleration and no-load dynamic torque vectors have to be multiplied by the vector (1 —μ$μΑ), whose construction is shown in Fig. 29.

F I G . 2 9 . Approximate loeus.

In the simplest case, when only mechanical inertia is taken into account in the power section, the vector

1

taking Ê'sc = Êsc, and assuming that the statism vector varies linearly with load

A «

Λ(1 + ^

μ ) , (286)

which is equivalent to replacing an arc of a circle by its chord. The circles for all the other quantities are also replaced by straight lines. The exception is the open loop gain for which the locus is still a circle.

The smaller the load (i.e. μ) the smaller the error from this assumption.

The locus of the internal voltage vector of the power section will also be a straight line whose equation can be obtained from equation (272) if we neglect the μί1

β term in the denomi-nator. Then


Load diagrams

F o r a follow-up control system loaded by a static torque proportional to angular velocity, the load diagram, i.e. the phase trajectory, described by the point ω, M, is an ellipse, as in the case of no-load. The difference between them lies in the fact that the motor torque leads the angular velocity not by 90°, as in the no-load case, but by an angle Φ = 90°— γ, where

^ t a n - ^ t a n - Ä . ( 2 8 9)

By analogy with the no-load case and taking account of loading by the first approximation formulae [equations (286) — (288)], we can write the formula for the trajectory as a radius vector

Q = Μ +;ω. (290)

The instantaneous values of the torque and velocity in this equation are given in terms of the vectors in the vector diagram

Μ = -Μαβ-ΐ(σ+^ (291)

and

ώ = j(oae-j<T. (292)

Multiplying the vectors M and ώ by the unit time vectors e + J > t, we obtain the rotating vectors:

fundamental-

er ; = J f V w = - M'^-'-y), (293)

ώ\ = ω&νί = 1ω'α&^-σ) (294)

and conjugate —

M'Lc = - M'ae-i<*-"-y), (295)

œLc= - / ω ; θ " ^ - σ ) . (296)


The instantaneous values of torque and velocity can now be written in the form

M\ = -M'acos(vt-a-y) = \ (M[ + Α[Λ) (297)

and

ω\ = -ωα sin (vt-σ) = \ (ώ\ + ώ\Λ). (298)

Substituting M' and ω' from equations (297) and (298) in equation (290) and collecting terms, we obtain

or

Here

(299)

(300)

Ρ = - | ( Λ Τ β β - * + ω β) = -j(Mima+œa-1MsXma) = Pe»>

(301)

and

Q = _ i . ( j | f a e * - ( ü e ) = -λ(Μ1α-ωα+)ΜΒ{.α) = Qe*

(302)

F I G . 3 0 . Calculation of Ρ and Ç .

From the vector diagram (Fig. 30) or analytically from the appropriate equations, we find the moduli of the vectors

Ρ and Q


Ρ = \\/(MjM + (üa)2 + Mlta,

Q =±yJ(MjM-coa)* + M*t.a

and their arguments

ρ = 2π— t a n - 1

(303)

(304)

M st. α M

and

q = 27i-f- tan - ι .

j.a + ω α (305)

Substituting in equation (300) the expressions for Ρ and Q given in the middle parts of equations (301) and (302). The equation for the radius vector [equation (300)] then becomes

or ρ = P e ^ - ^ - f Q^m~a)-M%x>a s i n ( v * - a )

q = Q0-M8tta sin (vt-a), (306)

where ρ0 is the radius vector for the no-load case and Ρ and Q are its parameters [equations (137) and (138)].

If, using equation (298), we replace sin (vt — a) in formula (306) by - ( ω / ω α) , then

Q = °o + M st.α ω = ρ0 + τω. (307)

Hence it follows that the motor torque under load at any instant of t ime or for any value of speed is the sum of the no-load torque and the static torque for that velocity.

Thus, the phase trajectory for a motor under load can be constructed by summing the no-load ellipse and the straight line for the static torque, drawn at an angle y r = tan""1 r to the ordinate axis (Fig. 31) .

However, this construction is valid only to a first approxima-tion, since we have neglected the variation in velocity under load which is allowed for by the exact equation for the velocity


[equation (261)] and the second approximation equation [formula (287)]. The latter equality is linear with respect to a lowering of the load velocity and so the corresponding correction to the no-load ellipse can be made without much difficulty.

F I G . 31. Deformation of the ellipse.

Deformation of the load ellipse is equivalent to rotating it about its centre and stretching it. Let us put

ρ = α + β q = oc-ß. (308)

Then equation (300) can be written in the form

ρ = &α[Ρ 6><*-σ+ν + Q e-M-'+V]. (309)

This is the equation of an ellipse with its axes inclined at an angle to the coordinate axes. Putting vt — σ + β = 0 in equation (309) we find the major semi-axis of the ellipse

gx = e*»(P + Q), (310)


and putting vt — a+ß = j(nl2), the minor semi-axis

62 = *>a?(P-Q)- (311)

Let us find the values of α and /?. Prom equation (308)

« =!(?+?); ß = \{p-q)>

Substituting for ρ and g from equations (305), we find

-XÄ-t1--^' , 3 , 2 )

< 3 1 3>

Thus, for loading by a static torque proportional to velocity, the ellipse is swung round with respect to the coordinate axes by an angle α which increases with the magnitude of the static torque, i.e. of the parameters r and μ (Fig. 31).

The distribution of the parameter (time) points round the ellipse can be found using a scale which is drawn, as for the no-load case, in the form of a circle of arbitrary radius

ρ = ρ β ^ ί +* - σ ) . (314)

Its angular scale is

Τ

1

m9 = —

= y ,

sec/rad. (315

)

The equation of the ellipse can obviously be obtained from the scalar expressions for the instantaneous values of the torque and angular velocity, eliminating the parameter time,

If we use the relative quantities ω'/ωα and M'jM'a and new coordinates m and η which are inclined at a constant angle of 45° to the fundamental ones, the equation of the ellipse becomes simply


The semi-axes of this ellipse are

É?i = yfl+siny; Q2 = yjl-amy. (318)

Under no-load condition, the ellipse in these units becomes a circle which, as the load increases, is stretched out along an axis at an angle of 45° (Fig. 32).

F I G . 3 2 . Deformation of the circle.

Since the torque of the motor is the sum of two components which are 90° out of phase —the dynamic torque proportional to acceleration and the static torque proportional to velocity— the energetic processes in a follow-up control system can be considered as the superposition of two components — dynamic or idling and static or loaded.

It is not difficult to see the complete analogy between the dynamic and static components of power in a follow-up control system in the harmonic regime and the reactive and active components of power in a resistance-capacitance electrical circuit coupled to a sinusoidal voltage. Moment of inertia of the control system corresponds to the dynamic capacity.


The instantaneous power at the motor shaft is

Ρ = Μω = Μαωα sin (vt — σ) cos (vt — σ- γ), (319) or

Ρ = Maœasin (vt-σ) sin (vt-σ+Φ), (320)

where

0 = J - Y (321)

is the phase shift between the torque of the motor and that of its static component which coincides in phase with the angular velocity.

Hence, remembering that

Ma cos Φ = MBif

and Ma sin Φ = Mj9

we obtain

Ρ = Mst.aœasin2(vt--a)+±coaMusm2(vt-a) (322)

or, re-arranging equation (319),

Ρ = - ^ ^ ο ο 8 Φ - ^ ξ ^ ο ο 8 [ 2 ( Λ - σ ) + Φ ] . (323)

The first term (Figs. 33, 34) gives the active power for a static load —in equation (322), the instantaneous value which varies according to a sine-square law, and in equation (323), the mean value

Λ . = J sin* d(vt) = ^ψ^- = \o>lr. (324) 0

The second term [equations (322) and (323)] gives the instan-taneous power for dynamic torque, oscillating at twice the frequency. By analogy to an a.c. circuit, it can be called reactive power, after estimating the imaginary quantity

Pj = Μ α

2

ω * sin Φ = MjmTmmAœTmmM = \o>arJv, (325)

ENERGETIC PROCESSES 9 2

Fie. 38. Instantaneous power.

F I G . 34. Power "rose".


where Mjrm8m and ω Γ m 8 are the root-mean-square values of the dynamic torque and angular velocity of the motor.

This magnitude of reactive power of the motor can be con-sidered as imaginary power of the motor in the harmonic regime in the idling state.

Since the mean value of the power for dynamic torque is zero, the real power actually developed by the motor shaft is equal to the static torque power, i.e. the "active" power. We can also determine, as in an a.c. circuit, the apparent power

p c = VF|PhP3 = Μηωη (326)

To show more clearly the effect of loading, the apparent power can be expressed in terms of the idling power, i.e. the reactive power Pj and the load parameters γ (or Φ) and μ

Ρ, = cosy 2 ψα vTM

(327)

The power at the motor shaft varies with torque according to the graph (Fig. 35), the equation of which may be obtained

0 Me

F I G . 35. Power as a function of torque.

from the equation of the ellipse (326). The latter gives this expression


= »«{wa

si*r±J1-wrs2r) (328)

for the angular velocity. Hence, the required equation for the power is

p = ω« (wa

sin y± J1 - wa

0082 γ) Μ· (329)

The power curve has a maximum at a critical value of torque

Μα = Μα -*ψ?, (330)

corresponding to the critical angular velocity

/ . / l ± s i n y , / l ± s i n y \ cocr = ω α( sin y / -—L±0osyl 2 ~ ~ 7 · ( }

The maximum power will be

_> 1 + s i n y ,^ 1 + s iny . Μ = ω α

+

2

7 = ü f i .««a 2 c os y (332)

or, using equations (89), (91) and (289)

' . - ï ^ l ^ r p * } (333) The above formulae give the useful power at the motor shaft

of a control system with known characteristics as a function of its operating regime: amplitude and frequency of oscillation of the angle and relative static torque in the linear region of its characteristic.

Motor power and control system parameters

Let us now consider how the load affects the choice of motor power and system parameters and also the behaviour of its limiting characteristics.

I t can be seen from Fig. 10 that for a given amplitude and frequency of command signal, the greater the load the sooner


the ellipse touches the limiting mechanical characteristic. Thus, the effect of the load can be explained from the geomet-rical relationships of the ellipse inscribed in the limiting mechan-ical characteristics which bound the linear region of operation.

The instantaneous values of angular velocity and torque of the motor under load, according to equations (293) and (294), will be

ω = —ωα sin (vt — or), (334)

M = -M cos (vt-σ-γ). (335)

The slope of the tangent to the ellipse

tan ( „ - « , = £ = ωα ο ο β Κ -σ )

M M sin (vtc — α— γ)

is equal to the coefficient of elasticity of the mechanical charac-teristic with the sign changed.

t a n a = KM. (337)

Here the subscript c denotes the time at which the motor torque and angular velocity fall into the limiting mechanical characteristic and the ellipse touches it.

The ratio of the amplitudes of the velocity and torque in this expression, assuming that Mja = Ma cosy (Fig. 26), may be written as

ΊΓ,-Φ- <338> Using this relationship, equation (336) can be solved with

respect to the contact time tc

M' ~~ vJ tan (vtc-a) cos y + sin γ ~~ ^M ' (339)

giving

tan(v£c — σ) = — f - t a n y = cot χ + tan y. (340)


This expression, which we shall for the moment denote

by a, can be written in the form [see equation (289)]

t a n ( i * c - * ) = i ± ^ = a. (341) V L

M

It shows that the time at which the motor finds itself on

the limiting mechanical characteristic depends on frequency,

load, phase error and control parameters.

We can now obtain an expression for the modulus and

argument of the radius vector of the point of contact gc. The

modulus of the vector is

Qc = y/a>* + M*. (342)

Substituting for ω and M from equations (334) and (335)

and eliminating tc, using equation (351), we obtain

a2 + v V

2[ l + a t a n r ]

2

Qc = ω α ^ ^ . (343)

Using the same equations, we find for the argument of this

vector M

cot β = — = vJ[cot ( v * c - a ) + tan γ] (344)

and after eliminating tc

t a n / ? = _ 4 = 2£+μ)*;Μ , (345)

v J [ l + t a n y ] ν2Τ% + μ(1+μ) or, using formula (68),

tan β = •-== - 1 + μ

xx ί . (346) r vJ[tan χ+(1+μ) tan γ] v

To determine the parameters of the limiting mechanical

characteristic Μ„ ι, , c &.iim> it now only remains to solve

the oblique-angled triangle formed by the radius vector ρ0,

the intercept on the axis of abscissae Mc (or the ordinate axis

col i m) and part of the mechanical characteristic, knowing the

side qc and angles α and β.

LOAD INGBY A STATIC TORQUE 97

Then sin (α + /S) , Ω . 0 , x

^ c . i i m = Qc J„ „ = ec(cos/9 + s in /Scota) sin a

(347)

or, multiplying throughout by ÜTM = tan a,

o>b.um = ^c(sin/3 + Z M cos/S). (348)

Substituting for ρ0 from equation (343) and expressing sin β

and cos β in terms of tan β [equation (345)], this reduces to A +

V YM ( l + a t a n y)

^ö . l im = ω

α 7 = = = » (349)

and after substituting for α and ω α

o>ö.nm = ^ W ^ M + ^ + j " )2-

F o r the starting torque, we obtain

2

dim φαν^η + (1+μ)*.

(350)

(351)

The limiting mechanical power of the motor necessary for the system to develop the required amplitude and frequency under load will be

^mech.lim = ^ [ V^ f f + (1 + μ)2]· (352)

M

This formula can be interpreted geometrically (Fig. 36). The motor power is proportional to the hypotenuse of the

F I G . 3 6 . Characteristic load triangle.


right-angled triangle whose other sides are vTM and l + μ , the angle adjacent to the base vTM being vtc — σ and the angle opposite the base being χ when μ = 0.

All the above formulae reduce to the idling formulae for μ = 0. As in the idling case, the limiting mechanical power of the

motor has minima for certain values of the control system parameters, gear ratio ζ and coefficient of elasticity KM.

Let us write formula (352) in the form

^ m e d U t m = ^ ^ K M ( j m Z + ^ - J + (1+μ^ (353)

From the equation

^-^mech.lim dz

we find the optimum value of the gear ratio

0 (354)

- V(l f μ)2 + ν2Κ^ V a + ^ - r V T S (355)

The optimum gear ratio depends on the load and decreases as it increases.

From the equation

^^mech.lim = 0. (356) dKM

we find the critical coefficient of elasticity

KM.cr = ±±P. (357)

It, too, depends on the load and increases as it increases, i.e. loading requires a damping of the mechanical characteristic of the motor.

The electromechanical time constant of a system with an optimum mechanical characteristic

' Μ ^ - ψ - ^ Τ , (358 ,


is greater than for a system with optimum characteristic

for idling.

The mechanical inertia angle o f a system with a critical

characteristic also increases

X c T = t a n - M i * i"). (359)

The optimum gear ratio, as in the idling case, depends on

the rigidity of the mechanical characteristic and, conversely,

the critical rigidity depends on the gear ratio, but there is

no pair of simultaneous values which will ensure minimum

motor power.

The limiting mechanical power of the motor for optimum

gear ratio, as in the idling case, decreases with increasing

rigidity of mechanical characteristic and increases with in-

creasing gear ratio for the critical mechanical characteristic.

In the latter case it is equal to

^mech.cr = \Μφ>(1+μ). (360)

In the particular case of the gear ratio [equation (219)]

which is optimum as regards starting and stopping time,

the critical value of the coefficient of elasticity is

K m ct = ^ -

( 3 6 1)

The limiting speed of the motor for the indicated values

of 2 O PT and K M XT will be

« M i m = +AO ^ 2 (362)

and the starting torque

.lim = 29>αν

2 y/2JmJnom- (363)

Thus, loading requires an increase in motor speed but does not

affect the starting torque.

The limiting mechanical power of the motor

^mech.lim = ^ ^ n o m (1 + μ) (364)


will be greater than the no-load power by a factor μ, where μ

is the relative load. From equation (350) we can obtain the equation of the

limiting amplitude-frequency characteristic for the loaded motor

<paMm = . (365) vz J (l + μ) 2 + ν

Loading reduces the amplitude in proportion to the ratio of the hypotenuse of the characteristic triangle (Fig. 36) for idling to the hypotenuse for loading.

C H A P T E R 3

L O A D I N G B Y T H E S T A T I C T O R Q U E

O F D R Y F R I C T I O N

The various regimes of motion

A reactive load whose magnitude is practically independent

of velocity only the sign changing with that of velocity, is

called a static torque of dry friction

Mst (ω) = ilif8tsign ω for ω ^ 0;

- Mst < Mst (ω) < Mat for ω = 0.

Such a load is, thus, often called a constant static torque.

As a result of the essential non-linearity of the torque of

dry friction, the motion of the servo-system will not be sinusoi-

dal even for low signal amplitudes which do not arise from

linear parts of the characteristics of sections with saturation.

And for large signal amplitudes, there are errors due to saturation

non-linearity in addition to the errors due to load non-linearity.

There are, therefore, two possible types of motion of a servo-

mechanism, depending on the input signal amplitude:

(1) small amplitude signals which do not go beyond the

linear region and

(2) large amplitude signals when motion extends into the

saturation region.

Depending on the magnitude of the static torque, two essen-

tially different cases are also possible:

(1) the case of small loads for which the velocity is a con-

tinuous function of time, and

101


(2) the case of large loads for which the velocity varies discontinuously with flat spots, i.e. periods when the motor is stationary.

The conditions under which discontinuous motion arises are explained with reference to Fig. 2. The motor does not move until the control voltage reaches a certain critical value at which the starting torque of the motor MC(M for ω = 0) is equal to the static torque.

F I G . 37. Torque oscillograms.

Figure 37a shows an oscillogram of the starting torque Mc,

whose magnitude is proportional to the control voltage Vp9

the angular velocity ω and the static torque M8t for the case of discontinuous motion. Motion commences when Mc = MBt.

Figure 376 shows the case of small loads, when the phase ψ0

of the velocity variation becomes such that the starting torque of the motor at the point ω = 0, dû)/d£>0 is greater than the static torque and the difference torque at that point, Mc — MsV

STATIC TORQUE OF DRY FRICTION 103

Mc-Mst = J — . da)

~dt

The boundary between small and large loads will be estab-lished later. I t is determined by the ratio of the static torque to the starting torque.

The case of small loads and small signals

The motion of a follow-up control system, loaded by the static torque of dry friction, in the linear region of small signals can be considered as motion of a linear system under the action of two external independent forces :

(1) the sinusoidal command signal at the input to the system

and

(2) the square-wave torque applied to the gear-box shaft

of the servo-motor at the output of the system.

Actually, the static torque is a reactive force which depends on the motion of the system but only in so far as the torque changes sign as the velocity passes through zero. In established periodic motion the periodic curve of static torque can be considered as a given function of time. To do this it is only necessary to ascertain the phase between this curve and the other external force —the command signal.

Let us choose as the origin of time the beginning of a period of the angular velocity of the motor, i.e. the point ω = 0, dœ/dt > 0. Then the command angle, the phase of which lags behind the velocity, will have a certain initial phase ψ0

<Pc(0 = <Pc.a sin (vt-tpo). (366)

The initial phase ψ0 depends on the moment of static resist-ance by virtue of the formulated conditions and will be deter-mined later.

is available to increase the acceleration of the inertia masses

of the system

1 0 4 ENERGETIC PROCESSES

The angular velocity of the motor is given by the control system equation (6) relative to the transform

ω(ρ) = KJp)Vp (p) - KM (p)Mst (p) ( 3 6 7 )

where

Mst(p) =

^ s t t a n ^ - ( 3 6 8 )

is the transform of the stepped wave of the moment of static resistance.

The signal at the input to the power section is

Vp (P) = Wc(P) - <PiPWi #amp (Ρ)· ( 3 6 9 )

Substituting this expression in equation (367), transforming to the angle at the gear box shaft by the formula

and assuming equation (23), we obtain

where

Δ ( ρ) = T + T ( p ) " ( 3 7 1)

This expression shows that the angle swept through by the controller for a static dry friction load, and, therefore, all its derivatives (velocity, acceleration and dynamic torque) are comprised of two components: the purely dynamic idling component (ilf8t = 0) and the static resistance load component

<p(t) = ?>o (0 -Pet (0 . ( 3 7 2 )

The dynamic idling component has already been studied. Let us now consider the second component, that due to the load. Its magnitude as a function of time

* t ( 0 = ΨΜ = Κμ{11Δ{Ρ) Μλ (ρ) ( 3 7 3 )


is most easily found, as in the case of a non-sinusoidal voltage applied to an electric circuit, by expanding the square wave of the static torque in a Fourier series

AM . 00

^ s t ( 0 = — - Σ s i n A^ > ( 3 7 4 )

where the A's are all the positive odd numbers, and substituting

in the admittance ^ M ^ — — for ρ the frequencies of the zp

corresponding harmonics. This gives

4 Μ , <~ ι 9>.t(«) = -^£TK„.hAhain

(hvt-Vn + dh), (375)

where KMM= KM(hv); Ah = A(hv)

are the moduli of the vectors i f A i [equation (69)] and À [equa-tion (78)], and r\h—r\(hv) and <5Λ= 6(hv) are the arguments for the frequencies of the corresponding harmonics.

The difference angle 6(t) = cpc(t) — <p(t), using equation (372), is given by

0(0 = e0(t) + Vst(t). (376)

The right-hand term takes account of the component of the error arising from the static load. The control winding voltage of the motor, from equation (9), will be

U(t) = U0(t)-Ust(t), (377) where

4 M , °° ι Ust(t) = — ^ Σ Τ F * (AV> S IN (HV + Vb) (378)

is the voltage reduction due to the static load and ψΕ is the argument of the vector F M(jhv).

The angular velocity of the motor can be obtained by mul-tiplying φ(ρ) by zp from equation (372), giving

œ(t) = ω0(*) + ω Λ( 0 , (379)


where,

u>et(0 = — τ 1 Σ τΚ**Δ* C 0S + (380)

is the velocity reduction due to the static load. The angular acceleration of the motor is obtained by differen-

tiating twice e(t) = 6 o ( * ) ( 3 8 1 )

where 4Λί 4. ν 00

= — z r ~ ~ Σ ^ Μ ^ Λ sin (Arf- i , fc + i f c) (382)

is the reduction in acceleration due to the load. The dynamic torque of the motor M5 = Je will be

Mtf) = Mu(t)-Mm(t), (383) where

M m (t) = 4 JM« ν f hKM h Ah sin (Αvi - 7 ? Λ + ί Λ ) (384)

is the reduction in the dynamic torque of the static load caused by the reduction in the angular velocity and acceleration.

The torque developed by the motor is obtained by adding the dynamic and static components of the load

M(t) = MW + M^t) (385) or

M(t) = MQ(t)+Mx{t)y (386) where

M0(t) = J f , .„(«)

is the motor torque for a purely dynamic load:

MM) = M8t(t)-Mut(t) (387)

is the component of the torque due to the static load, taking into account both the static torque directly and the reduction in the dynamic torque caused by it.

In order to determine the motion of the motor completely, it is necessary to find the value of the initial phase ψ0 of the

STATIC TORQUE OP DRY FRICTION 107

oscillations of the command angle q>c. In our formulae this phase appears in the expressions for all the quantities for purely dynamic loading.

Le t us use equation (379), the equation for angular velocity of the motor. Assuming that ω = 0 at t = 0, we obtain

AM. 00 1

*>o(0) = — s ± Σ TKM.h A cos (ηΗ - dh). (388) π h-i ft

But from equation (91)

- j ( i . - f ) ω = ω α β ^ J m (389)

Hence, assuming that, as in equation (366), the command angle vector has initial phase ipQ in the given case, we find

ω 0(0) = ω α cos (σ+ψ0). (390)

Substituting this expression in equation (388), we obtain

4M 00 1

cos(<r + v>o) = ^ Z 7 7 L Σ TKMMAhcos(vh-ôh), (391)

whence

— Γ 1 Σ Τ^ ΐ ί Λ Λ ο ο β ( ΐ ί Λ - β Λ ) -er . (392)

Using this equation, i t is possible to calculate the phase ψ0

to the required accuracy as a function of the static torque and frequency of oscillations, v.

The obtained expressions make it possible to calculate all the quantities that interest us as a function of time to the required accuracy, i.e. taking account of the desired number of harmonics.

Thus, for a control system loaded by the static torque of dry friction and for limited command signal amplitudes which do not force the system into a saturation zone, the output angle and all the other quantities consist of two components—a sinusoidal idling component and a non-sinusoidal reactive load component.


The amplitudes of the harmonics of this load component decrease rapidly with increasing command signal frequency and number of harmonics due to the filter action of the inertia sections of the system, mainly the moment of inertia at the motor shaft. Harmonics of different quantities are attenuated at different rates since, in the denominator of the amplitude of the harmonics of the output angle, difference angle and voltage, the order of the harmonic appears to the second power; in the denominator o f the angular velocity, i t appears to the first power; and in the acceleration and dynamic torque the harmonic number does not appear at all (the harmonics are attenuated only by the increase in the modulus of the gain J f ) . Consequently, non-linear distortions due to static torque affect the motor torque most of all and the output angle, angular error and motor voltage least of all. Therefore, the application of linear frequency methods to the calculation of the dynamic accuracy of real, non-linear systems gives satisfactory results.

On this basis we can afford, in the majority of cases, to neglect the upper harmonics in the load components of all the variable quantities in a follow-up system and use vector diagrams.

As before, let us assume that the command angle vector coincides in phase with the positive real semi-axis <jPc = ç>c. Then, from the accurate derived functions of time, discarding the upper harmonics and shifting the phase of the fundamental harmonic by an angle ψ0, we obtain the following vector expres-sions.

The output angle vector

Φ = n - ^ M i t & M A , (393)

the difference angle vector

6 = 60+^MBtKMA; (394)


the voltage vector

Ü = U0-±MstFM; (395) 71

the angular velocity vector

ώ = ώΌ + — Mst£M À\ (396) 71

the angular acceleration vector

è = é0-J^MstvÊMÀ; (397) 71

the dynamic torque vector

Mj = M0-^JMstvÊ:MÀ; (398)

the vector of the first harmonic of the static torque

* r t =-MsteßW (399) 71

(which is phase-coincident with the angular velocity vector); the motor torque vector

71

The case of large loads

When the moment of static resistance exceeds a certain critical value, the motion of the system becomes intermittent with intervals when the motor is stationary. The motion is described by more complicated transcendental equations than in the case of small loads. To simplify the discussion, we shall take the gain of the power section with respect to torque as inertialess —KM(p) = KM (in the case of a constant current control system this assumption corresponds to neglecting the inductance of the armature circuit). Then the angular

J 1 0 ENERGETIC PROCESSES

ω α K0Vpa

(409)

velocity of the motor can be expressed by the mechanical characteristic equation

ω = K0Vp-KM M, (401)

where the motor torque is

M = Mst-fJ^. (402)

Let us take the time of starting (ω = 0) as the time origin,

*' = o.

We shall assume that the signal controlling the power section is sinusoidal and that the upper harmonics of velocity are smoothed out in the inertia sections of the amplifier

Vp = Vpasin(vt + C). (403)

We find the initial phase ζ from the critical value of the signal Vp at which the motor moves from rest. From equation (401) for ω = 0, M = Mst (Fig. 37) and

Vp.cr = V

pasinC = ^ M s t (404) ω

we obtain

-*xe

r p.a

± γ± c

From the equalities (401) and (402), using formulae (12) and (18) we obtain

ω(1 + ΤΜρ) = ω α 8 Ϊ η ( ι * + 0 - ω 4 , (406)

where ω α = KJp.a (407)

is the amplitude of the angular velocity;

ωΔ = K»Vv.a (408)

is the reduction in the velocity due to the action of the static torque.

To simplify these equations, we introduce the relative units

ω ω


the relative angular velocity, and

" = t (4,o)

the relative static torque. Equation (406) becomes

+Τ Mp) = Βΐη(νυ + ζ)-μ. (411)

The solution of this equation reduces to

ι = [cos%sin(^ + C - % ) - / u ] - [ c o s C s i n ( C - ^ ) ~ i u ] . (412

Here χ = t a n - * v T^ ( 4 1 3)

From equation (412) we can determine the duration of the "velocity pulse", i.e. the time of continuous rotation of the motor. To do so, we have to put i = 0 and vt — λ in equation (412), where λ is the angular duration of the velocity pulse.

We finally obtain the equation

cos χ sin (ζ-χ)-μ = [cos χ sin (λ + ζ-χ)-μ\βλ001*, (414)

where in the given case of large loads, from equations (405) and (410),

μ = sin ζ. (415)

Thus, we can eliminate one of the variables (ζ or μ) from equation (412). Eliminating ζ and solving for μ, we obtain μ =

cos#[sin (λ—χ) βλcoiχsin χ]

yj { [ 1 - cos χ cos ( λ - χ)] e x c ot * - sin2 χ}2 + {sin ( λ - χ) e x c ot * + sin λ}2. cos2 χ

(416) or, after re-arranging,

cos χ {sin% + sin (λ — χ) β λ c ot *}

yjsin2 χ - 2 sin %(sin χ - cos χ sin λ) e x c o t * + Γ β 2 λ c o t * '

(417)


X = [ l + c o s 2 £ - 2 c o s % c o s ( A - 7 ) ] e 2 X c o t* .

This equation gives the inverse function of the unknown

velocity pulse duration λ from the argument—the relative

load μ for various values of the parameter χ (Fig. 38).

180

160

140

120

100

8 0

'60

4 0

2 0

5

\ \ 1

0 - 2 0 - 8 1-0 0 - 4 0 - 6

M

F I G . 38. The velocity pulse duration function.

As the load μ decreases, the rotation interval λ increases

until it reaches the maximum critical value

(418)

This critical value marks the boundary between continuous

and intermittent rotation.

where


The critical value of the relative load, μ 0 Γ, is found from equation (416), by substituting XCT from formula (418)

Per sin χ cos χ [ 1 + e*c o t*]

V s i n 2 x[ 1 - 2 en c ot *] + [1 + 3 cos2 χ] β 2π c ot * (419)

This equation shows that the critical value of the static torque depends on the starting torque of the motor and the parameter χ, i.e. on the signal frequency and the mechanical time constant TM. This dependence is shown, in relative units, in Fig. 39.

F I G . 3 9 . The critical load function.

For a given μοτ> equation (415) gives us the initial phase of the velocity ζ.

A further reduction in the static torque leads to the case of small loads which we have already considered in more general form. The initial phase ζ will be determined not only by the static torque but also by the acceleration, i.e. the dynamic torque. In this case, putting 2 = 0, l = 0 in equation (411) we find

sin ζ (420)


and

η Απ cot χ 1

sin (χ - ζ) = -Ζ— . — i (422) Α ' cos χ e « c o t* - f l ν '

^ ^ -8 ί η

" ΐ · ^ τ ΐ · ( 4 2 3)

This equation is a particular case of equation ( 4 1 4 ) . We shall proceed in an approximate manner by the method

of harmonic linearization. We expand the periodic impulse curve of relative velocity [equation ( 4 1 2 ) ] into a Fourier series. This series has odd harmonics only. The amplitude of the first harmonic is

and its phase is

V»ln = t a n - i | i , (425)

where

COS Ύ

ax = c o s ( C - x ) - s i n λ cos (ζ-χ + λ)-

- 2 sin χ sin (ζ - χ) [sin χ - sin (χ + λ) e " x c o t *]} -

- 4 ~ ^ {cos 2 χ - cos A + sin χ s in(x + A ) e - X c o t* } . ( 4 2 6 )

COS Y &i = - { A sin ( £ - % ) + sin λ sin (ζ-χ + λ)-

71

- 2 sin χ sin (ζ-χ) [cos χ - c o s (% + A ) e - X c o t* ] } -

l ^ ^ j s i n λ - ^ - s i n 2 χ + 8 ΐ η χ cos (χ + λ)e"Xcotxj . ( 4 2 7 )

As the static torque decreases, the duration of rotation will remain constant (λ = π), but the initial phase ζ will change.

Putting i = 0 and vt = n in the general equation (412), we solve it for f.

[cos χβΐη (£ -%) + μ ] = [μ- cos χ sin (ζ- χ)] e - " c o t* . (421)

Hence


In these equations the velocity pulse duration λ is a function

of the relative torque // = sin~1 ζ and the parameter χ [see

equation (416) and Fig. 38].

I t is evident from equations (424), (425) that if the inter-

mediate parameter λ is eliminated, the relative amplitude

and phase of the first harmonic will depend only on the two

arguments—relative load μ and parameter % = t a n _ 1 vTM.

O 0 - 2 0 - 4 0 - 6 0 - 8 1-0 0 0 - 2 0 - 4 0 - 6 0 - 8 1-0

F I G . 4 0 . Amplitude and phase of the first harmonic of velocity as a function of relative load: a — amplitude; b — phase.

Figure 40 shows graphs of the relative amplitude and phase of the first harmonic as a function of the load μ for various constant values of the parameter χ. The broken line shows the boundary of the region of small loads, i.e. of continuously varying velocity.

The absolute value of the amplitude of the first harmonic ω α 1 will depend on two variables—the amplitude of the control signal Vpa and the static torque Mst. This dependence is expressed not only directly in terms of the relative velocity e, but also indirectly in terms of the relative static torque μ.


From equations (409) and (410), the amplitude of the velocity

is

ωα.ι = 'a.l^a = la.lK.Vp.a (428)

and the static torque is

ΜΛ = μΜ£ = μ ^ - ν ρ Μ . (429) ω

If we neglect the upper harmonics, assuming that they are

filtered out as in the case of saturation non-linearity for large

signals, we can use all the vector equations obtained above

for the no-load condition and also all the previous conclusions.

I t is necessary merely to replace the vector of the gain of the

power section in the linear region, Κω by the equivalent vector

corresponding to the first harmonic of the angular velocity

in the given case of the continuous regime.

The equivalent transfer vector of the power section can be

written in the form

Kl = -=i5==e-fc' . (430)

In the given case, because of the presence in the power section of an inertia parameter as well as non-linearity, it depends not only on the amplitude of the input signal but also on its frequency and time constant. This dependence applies to both the modulus of the vector K*0 and its phase, i.e. the correction for non-linearity is a vector. Let us find it.

In the linear case the amplitude of the velocity is

" a = J±—Vw, (431) V 1 + ν

2Ί%

and in the non-linear case the amplitude of the first harmonic is

»·"' - · " - • - j 3 x r » - ( 4 3 2)

Hence the modulus of the equivalent vector becomes

Κ; = ιΑΛΚ.. (433)


The phase of the equivalent vector, from equation (403), is

χ* = ζ - ψ ο = t a n - Ä - y i . 0 , (434)

since the velocity pulse is displaced relative to the signal vector Vp by an angle £, and the first harmonic of the pulse is displaced relative to the velocity pulse by an angle ψ10.

In order to calculate the vector of the gain of the power section Κ*ω we must first know the signal amplitude at the input to this section ( F p a) , on which depends the vector modulus. This amplitude can be calculated, as before, by the graphical solution of the closed loop equation.

The power section signal vector for a non-linear circuit, according to equations (78) and (88), is

ν*Ρ = φ*κ:κΤ> ( « δ ) 1 + Κ

where

i£* = (436)

is the transfer vector of an open loop with an equivalent power section vector K*0 which itself depends on the signal amplitude

To solve the equation we draw the function

fl(Vp.a) = <Pc.aKi

Lamp

1 + (437)

and the straight line Vpa = Vp. The point of intersection gives the required value of Vpa.

Using this value of Vp<α, μ is calculated from equation (419), whereupon the values of the relative coefficient of the first harmonic c10 and its phase ψί0 are found from the curves (Fig. 40). Finally, the modulus and phase of the equivalent vector Kl are calculated from formulae (433) and (434).


The load diagram and motor power

When a control system is loaded by a static torque of dry friction the load diagram, i.e. the phase trajectory described by the point ω, Μ, can be taken as an ellipse, as in the case of viscous friction, only to a first approximation, when the static torque is allowed for in the first harmonic, and for large loads only the first harmonic of the velocity is taken into account.

In this approximation, the static torque, as in the case of viscous friction, is in phase with the angular velocity but its amplitude is now independent of velocity.

The instantaneous values of the torque and velocity are expressed in the phase trajectory equation

ό = M + jœ (438)

in terms of the vectors in the vector diagram. For simplicity, we shall ignore the reduction of the dynamic torque due to the static load, since its magnitude is small.

The motor torque is

M = Hj + Msi = Mj + j - M9i e-*, (439) 4 71

and the angular velocity

ώ = jcoae~j(r. (440)

Multiplying the vectors ÏÎ and ώ by the unit time vector e j v i, we obtain the rotating vectors

Mt = -Ma&to-'-y* (441) and

wt = ]ωαβ^-σ\ (442) where

γ = t a n - 1 ^ ^ (443)

is the phase difference between the vectors of the motor torque

and its dynamic component.


The instantaneous values of the torque and velocity are expressed in terms of these vectors by the formulae

Μ = \(Mt + Mtx) = -Ma cos (vt-σ-γ) (444)

and

ω = γ (ώι + (btx) = - ωα sin (vt - a). (445)

Putting these expressions into equation (438), we find the equation of the ellipse

ρ = P &{vt-a) + Q e - ^ - ' \ (446) where

P = -j(Mi.a + a>a-j* Mat^=P0 + j^Mst; (447)

Q = - \ ( M U - ω α + j·| M^j = Q 0 - î \ M* . (448)

The ellipse for a load vector ρ, as in the case of viscous friction, can be considered as the geometric sum of the idling ellipse and static torque ellipse.

Substituting equations (447) and (448) in formula (446), we obtain

p = ρ - i ^ 5 l 8 i n ( r t - c r ) . (449) 71

Eliminating sin (vt — σ), using equation (445),

Q = ëo + — r 1 TT · ( 4 5 ° )

Comparison of this formula with equation (307) shows that, to the given approximation, the static load adds to the idling ellipse a component proportional to angular velocity, as in the case of viscous friction. In Fig. 41 it is shown as a straight

line through the origin at an angle tan""1 ( ^ ^ 8 T J to the

ordinate.


Thus, in a given case the deformation of the idling ellipse by the static load leads to a rotation of the ellipse through an angle α which can be calculated by the same formula (312) except that the amplitude of the static torque will be

Mst.a = instead of MsUa = coar.

TC

F I G . 4 1 . Load ellipse.

This first approximation diagram can be made more precise without much extra complication, if as before, neglecting the variation of angular velocity with load, we calculate exactly the static torque and add it to the ordinates of the idling ellipse. This leads to a displacement of the upper half of the idling ellipse relative to the lower half by an amount 2ilfs t.

The instantaneous power at the motor shaft for the case of loading by the static torque of dry friction, as for viscous friction, consists of two components: reactive power—the dynamic idling component, and active power—the static component of the effective load.

The dynamic component varies, as already shown, at double


the frequency and with amplitude proportional to the cube of the frequency [equation (150)] and the square of the amplitude of the angle.

The static component of the power in the given case is proportional to the angular velocity, since the static torque remains constant throughout each half-period of variation of the velocity. I t can only have a positive sign, since the static torque always opposes the velocity and varies according to the sign of the rectified sine wave

• P . U = Pet.a\sin(vt-a)\. (451)

The amplitude of the static power is equal to the product of the amplitude of the angular velocity and the static torque

P*t.a = MBtcoa = Μ8Χζφαν. (452)

The mean value of the static power, i.e. the active power of the load, is ,

ΡΛ = MÉPS- J sin (vt) d(vt) = 1 Mst<oa. (453) 0

The apparent power at the motor shaft is

Pc = y/Pj+Pl = coa J(^iJ + ^ J m a J . (454)

In these formulae the static torque is allowed for accurately by a square wave. I f this is approximated by its first harmonic, the formulae for the power will have a form analogous to the case of viscous friction.

To a first approximation, all the conclusions and considera-tions already discussed apply as regards the motor power and control system parameters. I t is only necessary to replace the static torque of dry friction by the equivalent torque of viscous friction. As shown, the amplitude of the torque of

4M viscous friction, equivalent to dry friction, is equal to

π


and relative load

' « I - = ^

< * « >

Ρ = ί;κ"τγ' ( 4 5 6 )

The angular parameter of the load will be

Using these formulae, all the results obtained for the case

of loading by viscous friction can be easily utilized.

for a value of angular velocity ω α. This corresponds to a coeffi-

cient of viscous friction

REFERENCES

1. L . S . G O L ' D F A R B . On certain non-linearities in regulation systems (Ο nekotorykh nelineinostyakh ν sistemakh regulirovaniya) Avtomatika i telemekhanika, No. 5, 1947.

2. S . A. L E B E D E V , L . N. D A S H E V S K I I . The effect of dry friction on the variation of velocity under a sinusoidal applied torque. (Vliyaniye sukhogo treniya na izmenenie skorosti pri sinusoidarnom prilozhennom momente) Tr. In-ta elektrotekhniki Akad. Nauk. SSSR, 4, 1949.

3. A. A. B U L G A K O V . On the limiting amplitude of sinusoidal variation of output angle in follow-up electric control systems (O predePnoi amplitude sinusoidal'noi otrabotki ugla ν sledyashchikh elektropri-vodakh) Avtomatika i telemekhanika, No. 6, 1962.

4. Ya. Z. T S Y P K I N . The theory of Automatic Control Relay Systems. (Teoriya releinykh sistem avtomaticheskogo regulirovaniya) G.I.T.T.L., 1955.

5. Ya. Z. T S Y P K I N . On the relation between the equivalent gain of an element and its characteristic. (O svyazi mezhdu ekvivalentnym koeffitsientom usileniya elementa i ego kharakteristikoi) Avtomatika i telemekhanika, No. 4. 1956.

6. I . C . G I L L E , M . PELEGRIN, P . D E C A U L N E . Théorie et technique des

asservissements, Paris, 1956. 7. V. A. BESEKERSKII , V. P. ORLOV, L. V. POLONSKII , S. M. F E D O R O V .

The Design of Low Power Follow-up Systems, (Proyektirovanie sledyash-chikh sistem maloi moshchnosti) Sudpromgiz, 1958.

8. W. F I N D E I Z E N . Dobo's elementu wykonawcrego w serwomehanizme, Archivum Automatyki i telemechaniki, 4, p. 1, 1959.

9 . A. G. I O S I F ' Y A N , Β. M. K A G A N . The Fundamentals of Follow-up

Control Systems. (Osnovy sledyashchego privoda) Gosenergoizdat, 1954.

123

I N D E X

Accuracy 25-30, 34 Amplidyne 10, 11, 16, 45 Amplification factor 3J—33 Amplifier

non-linearity of 44 et seq power 1, 2, 15 vector 18 voltage ] , 6, 7

Amplitude Frequency Characteristic (AFC)

44, 52-57 Phase Characteristic (APC) 18,

20, 29, 50 Angle

mechanical inertia 18, 19, 30, 99 output, limiting 80, 100

Circle diagram 72—84 limiting 51 nominal voltage 50

Closed loop vector 20, 21, 74, 83

Design factor (λ) of motor 61, 62 Difference angle (see Error vector) Directional coupling 8, 9, 11 Dry friction 4, 5, 17 101// Dynamic loading 3, 5, 17, 37

Elasticity, coefficient of 63-66, 95, 98-99

Electromechanical time constant 55-60, 69-72, 98

Ellipse, load 38-44, 52, 85, 88, 118// Equivalent

circuit 8—10

gain 47-48 Error vector 22, 27-30, 50, 77,

82, 105, 108

Feedback loop vector 19, 73 transfer function 1 unit 1, 7, 8

Force vector 19 Frequency, critical 61, 80 Friction

dry 4, 5, 17, 101// viscous 4, 5, 17, 69-100

Gain, equivalent 47—48 Gearbox 1, 5, 13

vector 19, 31, 32 Gear ratio 13, 30-36, 40-41, 5 3 -

54, 63-65, 98-100

Harmonic linearization 45, 114

Indicator section vector 19 Inertia, moment of 13, 30-31, 4 0 -

41, 65

Load critical, for continuous rota-tion 112-113 diagram 38, 85-94, 118// ellipse 38-44, 52, 85, 88, 118// rose 42

Loading dynamic 3, 5, 17, 37 static 4, 5, 17, 37, 69//

125

126 INDEX

large loads 109// small loads 103-109

Mechanical inertia angle 18, 19, 36, 99

Merit, figure of 8 Moment of inertia 13, 30-31,

40-41, 65 Motor 1, 2, 12, 15

coefficient of elasticity 63-66, 95, 98-99

design factor (λ) 61-62 power 41-44, 57-68, 94-100,

118// limiting 97-100

saturation 2, 45, 50 speed 38-44, 71-73, 77-79, 83,

85-100, 104-107, 109// limiting 55-63, 69-72, 9 5 -100

Non-linearity of amplifier 44 et seq

Open loop vector 18-21, 74, 82, 117 Output

angle, limiting 80, 100 power 91-94, 120// vector 22, 34-36, 76, 108

Overheating of motor 59—62, 66— 68

Phase trajectory 37, 85-90, 118// Power

loss 66-68 motor 41-44, 57-68, 94-100,

118// output 91-94, 120// section vector 18-20, 23, 31-33,

48, 52-53, 69-73, 116-117

Ratio of gearbox 13, 30-36, 40-41, 53-54, 63-65, 98-100

Rose, load 42

Saturation 2, 45 et seq Speed (see Velocity) Static loading 4, 5, 17, 37, 69// Statism vector, 20, 21, 26, 30, 74-

75, 77, 81-84

Tachogenerator constant 8, 13 Time constant, electromechanica

55-60, 69-72, 98 Torque 38-44, 56-66, 69-72, 78-

81, 83// relative 111—116

Transfer function feedback 1 overall 16

Tsypkin's formula for equivalenl gain 46

Velocity angular

of motor 38-44, 71-73, 77-79. 83, 85-100, 104//

hmiting 55-63, 69-72, 9 5 -100

relative 111-116 factor 18-20, 23, 31-33, 48, 52-

53, 69-73, 116-117 pulse duration 111-116

Vector diagram 17—25 amplifier 18 closed loop 20, 21, 74, 83-84 error 22, 27-30, 50, 77, 82, 105,

108 feedback loop 19, 73 force 19 gearbox 19, 31, 32 indicator section 19 open loop 18-21, 74, 82, 117 output 22, 34-36, 76, 108 power section 18-20, 23, 31 -

33, 48, 52-53, 69-73, 116-117 statism 20-21, 26, 30, 74-77, 81,

82-84 voltage 24, 52-53, 80-81, 84,

109

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I . D . U R U S O V

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by S. A . D O G A N O V S K I I and V. A . I V A N O V

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by M. M. B O T V I N N I K ,

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