module 2 new partb

8/4/2019 Module 2 New PartB

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CONVERTERS IN ELECTRIC DRIVE SYSTEMS (Part B)

MODELING OF SWITCH-MODE CONVERTERS FOR ELECTRIC DRIVE SYSTEMS

IntroductionModeling is a simplified representation of a physical system. In electrical engineering, physical systems arenormally modeled using mathematical equations. In this module, the models for the DC-DC converters applied toelectrical drives are developed and these will be used for two reasons: (i) to increase our understanding on thedynamic behaviour of the converters (through simulation) and, (ii) to use them in the controller design of the closedloop control systemThe complexity of the developed model of power electronic converters will depend on the applications of the model.For instance, a model of a switching device used to analyze its switching characteristic or switching behaviour isdifferent from a model of the switching device used to study the fundamental behavior of a converter containing thatparticular switching device. The former will include the detailed characteristics of the devices while the latternormally adopts the ideal characteristic of the models. This module will discuss on how to obtain the averaged andlinearised model of the converter intended to be used in the linear controller design. Our goal is to establish alinearised relationship between the control input signals of the converters and the averaged output voltage of theconverters.

Switch-mode converters used in DC drives can be classified as 1-quadrant, 2-quadrant or 4-quadrant. Thequadrants of the converters are defined based on the capability of these converters to synthesize positive ornegative average voltages and also on their capability to conduct either positive or negative currents. The quadrants

of operations of these converters are typically depicted using the V-I graphs as shown in Figure 2.

1-quadrant converterA single-quadrant converter is only capable of producing positive average output voltage and positive current, asshown in Figure 2(a). When the transistor is on, the output voltage v a equals Vdc and current ia builds up. When theswitch is turned off, because of the inductance in the armature, the current has to be continuous therefore will flowthrough the diode (see Figure 1). When the diode conducts, ideally va = 0. The duty cycle of the power transistordetermines the average output voltage of the converter, Va, which is fed to the armature of the DC motor. If thecurrent reaches zero before the next turn on cycle of the transistor, then the current is said to be discontinuous. Thistypically happens when the back EMF of motor is large, the turn off period is too long (small duty cycle) or theinductance of the armature circuit is too low. Assuming a continuous current, the average output voltage is given by

(1)

Where d = ton/T, and ton is the time when the switch is on within the period T. We will not go into details of the 1-quadrant converter this is discussed in many textbooks on power electronics and drives.

Figure 1 1-quadrant converter : va, ia, id

dc

dT

0 dca

dVdtVT

1V

s

==

va

ia

id

Va

T

ton


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2-quadrant converterA 2-quadrant converter operates in two quadrants, normally in positive average voltage with either directions of thecurrent. Figure 2(b) shows a typical 2-quadrant converter used in DC drives. As in all other converters, the status ofthe upper and lower switches in a leg, must always complement, i.e. if the upper switch is on, the lower switch mustbe off or vice versa. The instantaneous output voltage, va, can be either Vdc or zero; if the upper switch is on, theoutput voltage va equals Vdc and if the lower switch is on va = 0. The average value of the output voltage over acycle of the waveform depends on how long the upper (or lower) switch is on, i.e. the duty cycle of the outputvoltage waveform.

Modeling of the 2-quadrant converterThere are several ways in which this duty cycle can be varied. In this module, we will assume a method which is

based on a comparison between the control signal and a triangular wave, as illustrated in Figures 3 and 4.

Referring to Figure 3, the output of the comparator is obtained as follows:

when vc > vtri, upper switch on, va = Vdc(2)

when vc < vtri, lower switch on, va = 0

12

3 4 I

V

(c) Four-quadrant

(b) Two-quadrant

Figure 2 2-quadrant and 4 quadrant DC-DC converter

(a) Single-quadrant

1Q2

3 Q4 I

V

I

Q12

Q3 Q4

V

+

va

-

ia

id

=0

1q

+

va

-

ia

+ va

ia

+

Vdc

+

Vdc

+

Vdc


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Obviously, the instantaneous waveform of va will follow that of q and it is given by: va = q(Vdc) The average value ofva will depend on upper switch is turn on and this in turn depends on the control signal vc. Let us define the averagevalue of q over a cycle (T tri) as d, known as the duty ratio given by (3):

(3)

tri

on

t

0tri

Tt

ttri T

tdt1

T

1dtq

T

1d

ontri

=== +

vc

vtri

+

Vdc

Vdc

q

Figure 3 Control signal for 2-quadrant converter

Figure 4 Controlling the output voltage based on triangular wave comparison

vc

q

Ttri

d

va

0

Vtri

-Vtri

Vdc

0

+

va


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where ton is the duration in which q remains as 1 (i.e. upper switch is on)

We can obtain the relation between the average voltage, Va and the duty ratio, d, by calculating the average valueof va in terms of d,

(4)

If the triangular frequency is assumed very high as compared to the variations of the control signal v c, d in this case,can be assumed continuous. However, when selecting the bandwidth of the closed-loop system containing thiselement, the discrete values of d must be taken into account; that is, the bandwidth must be limited to one or twoorder lower than the triangular frequency. This is to ensure that the control signal does not vary faster than thecapability in which the average voltage can change.

The relation between d and vc is obtained as follows:

When vc = Vtri,p , d = 1 and when vc = -Vtri,p, d = 0. Assuming d is continuous, the relation between d and v c as canbe seen from Figure 5,: is obtained as

(5)

Consequently, the relation between vc and Va can be obtained by substituting (5) into (4),

(6)

Equation (6) is the average value of the output voltage of the converter within the switching frequency, i.e. thetriangular frequency. In other words, the switching waveforms have been removed. To illustrate this, a SIMULINKmodel is developed containing the switching as well as the average models of the 2-quadrant converter as shown in

Figure 6(a). The average model uses (6), whereas the switching model is developed using the converter modelfrom SimPowerSystem toolbox. The control signal vc is a square wave with amplitude of 10 unit which iscompared with the triangular waveform with an amplitudeof 15 unit. The DC voltage, V dc, is set to 200V. Accordingto (6), the average value of the output voltage within the switching frequency will have an offset of 100V that swings

between 33.33V to 166.67V. A simple load consisting of R = 10 and L=10mH are used. The simulation results inFigure 6(b) clearly show the waveform of the switching current following its average waveform.

dc

dT

0

dc

tri

a dVdtVT

1V

tri

==

Vtri,pVtri,pvc

d

1

0

0.5

p,tri

c

V2

v5.0d +=

c

p,tri

dcdca v

V2

VV5.0V +=

Figure 5 Relationship between d and vc


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Figure 6 Simulation of 2-quadrant converter: (a) SIMULINK blocks, (b) Average voltageVa , switching and averagecurrent waveforms

In order to systematically design the linear controller for a closed loop system containing this converter usingclassical control system design methodology such as Bode plot, the small signal model of the converter needs to bedeveloped. The small signal model will be extracted from (6) through linearization. This can be achieved byintroducing small perturbations in Va and vc, separating the ac component and finally taking the Laplace transform ofit as follows:

(7)

Separating the dc and ac components,

( ) ( )ccp,tri

dcdcaa v

~vV2

VV5.0v~V ++=+

Continuous

powergui

g

A

+

-

Universal Bridge

1

0.01s+10

Transfer Fcn1

iave

To Workspace5

iau

To Workspace1

Out1

Subsystem

Signal

Generator

Series RLC Branch

Scope1

Relay1

Relay

-K-

Gain3

DC Voltage Source i+

-

Current Measurement

100

Constant

0.2 0.25 0.3 0.35 0.4 0.450

50

100

150

200

0.2 0.25 0.3 0.35 0.4 0.45

5

10

15

(a)

(b)

Averagedcurrent

Switchingcurrent

Average model

Va

Averaged andswitching current


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DC : (8)

AC : (9)

By taking Laplace transform of equation (9), the small signal transfer function between vc and Va can be obtained asshown in Figure 7

4-quadrant converter

The model developed for the 2-quadrant converter can be used as a building block in developing the model for the4-quadrant converter. As illustrated in the Figure 8, the 4-quadrant converter is composed of two legs, with each legsimilar to that of the 2-quadrant converter. We will consider two switching schemes normally employed: (i) Bipolarswitching scheme, and (ii) unipolar switching scheme.

Modeling of a 4-quadrant converterThe instantaneous voltage va of the 4-quadrant converter can be made either equals Vdc , -Vdc or 0.

Va = Vdc when Q1 and Q2 are ONva = -Vdc when Q3 and Q4 are ONva = 0 when current freewheels through Q and D

The fact that the instantaneous output voltage can become negative (i.e. Vdc), makes it possible for the converterto operate in the 3

rdand 4

thquadrants. There are two possible methods of synthesizing the desired average output

voltage: the output voltage that swings between Vdc and Vdc or the output voltage that swings between Vdc and 0(or Vdc and 0). These two options give rise to two possibilities of switching schemes known as the bipolar andunipolar switching schemes.

c

p,tri

dcdca v

V2

VV5.0V +=

c

p,tri

dca v

~

V2

Vv~ =

p,tri

dc

V2

Vvc(s)va(s)

leg A leg B

+ va

Q1

Q4

Q3

Q2

D1 D3

D2D4

Figure 7 Small signal transfer function of the 2-quadrant converter

Figure 8 4-quadrant converteria


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Bipolar switching

Leg A and Leg B obtained the switching signals from the same control signal as shown in Figure 9. This implies thatswitching of Leg A and Leg B are always complements. The upper switch of leg A is turned on at the same timewith the lower switch of leg B. Similarly, the lower switch of leg A is turned on at the same time with the upper switchof leg B.

In a forward braking mode where the average voltage Va is positive and smaller than the back emf of the armature,current will flow through D1 and D2 when va = Vdc and will flow through Q3 and Q4 when va = -Vdc

Figure 10 shows the key waveforms for the bipolar switching scheme.

From previous analysis, the average voltage for Leg A and Leg B is given by:

VAO = dA(Vdc) and VBO = dB(Vdc)=(1-dA)(Vdc) (10)

Similarly relation between vc and dA and dB can be written as:

For Leg A (11)

For Leg B (12)

Figure 9 4-quadrant converter with bipolar switching scheme

vc

vtri

+

Vdc

q

-Vdc

q

Vdc

+vAO

+vBO

A B

O

Leg A

Leg B

p,tri

cA

V2

v5.0d +=

p,tri

cB

V2

v5.0d =


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We are interested in the voltage across the armature circuit, VAB. Therefore,

VAB = VAO VBO = (dA (1-dA))Vdc = (2dA -1)Vdc (13)

Substituting dA from (11) into (13) gives,

(14)

Equation (14) gives the average value of the output voltage within the switching frequency. Figure 12(a) shows theSIMULINK blocks of the 4-quadrant converter with bipolar switching scheme. The simulation result as shown inFigure 12(b) indicates that the switching output current exactly following its averaged value. The relationshipbetween the output voltage averaged value and the control signal (14) is linear since there is no DC offset present.By taking the Laplace transform of (14), the transfer function between vAB(s) and vc(s) is obtained:

q average value, dA

c

p,tri

dcAB v

V

VV =

Figure 10 Key waveforms of 4-quadrant converter with bipolar scheme

2Vtrivc

q average value, 1-dA

vAO

vBO

Vdc

Vdc

Vdc

Vdc

vAB = vAO - vBO


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(15)

The DC gain for 4-quadrant converter transfer function is therefore twice that of the 2-quadrant converter.

Figure 12 4-quadrant converter: (a) SIMULINK blocks, (b) simulation results: va, switching and averaged currents

)s(vV

V)s(v c

p,tri

dcAB =

p,tri

dc

VV

vc(s) va(s)

Figure 11 Small signal model of the 4-quadrant converter (bipolar)

Continuous

powergui

v+-

Voltage Measurement

g

A

B

+

-

Universal Bridge

1

0.01s+10

Transfer Fcn1

iave

To Workspace5

vau

To Workspace2

iau

To Workspace1

Out1

Subsystem

Signal

Generator

Series RLC Branch

Scope1

Relay1

Relay

-K-

Gain3

DC Voltage Sourcei+-

Current Measurement

0.2 0.25 0.3 0.35 0.4 0.45-150

-100

-50

0

50

100

150

0.2 0.25 0.3 0.35 0.4 0.45-15-10-505

1015

Va

Averaged andswitching current


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Unipolar switching

The switching signals for Leg B is obtained from the inverse of control signal for Leg A as illustrated in Figure 13.Figure 14 shows the key waveforms of the unipolar switching scheme.

According to our previous analysis, the continuous duty ratio for Leg A, dA, is given by:

(16)

Since Leg B uses the inverse control signal, the continuous duty ratio for Leg B is therefore given by:

(17)

This gives and average armature voltage as,

VAB = (dA dB)Vdc = (18)

The transfer function obtained for unipolar switching scheme is therefore the same as the bipolar switching scheme.

The instantaneous output voltage waveforms of the two schemes are however different. For the same triangularfrequency, the switching frequency of the output voltage using the unipolar scheme is doubled when compared tothe bipolar scheme thus effectively reducing the output current ripple. The instantaneous output voltage, which onlyswings between Vdc and zero, also contributes to the reduction in the output current ripple.

p,tri

cA

V2

v5.0d +=

p,tri

cB

V2

v5.0d =

c

p,tri

dc vV

V

Figure 13 4-quadrant converter with unipolar switching scheme

+

Vdc

vc

vtri

qa

Vdc

-vc

vtri

qb

Leg A

Leg B

+vAO

+vBO

A B


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References:

N. Mohan, Power Electronics: Converters, applications and design John Wiley and

Sons, 1995.

N. Mohan, Electric Drives an integrative approach MNPERE, 2000.

W. Leonhard, Control of electrical drives, Springer-Verlag, 1984.

2vtri

vc

qB

qA

Figure 14 Key waveforms of 4-quadrant converter with unipolar scheme

-vc

vAO

vBO

vAB

Vdc

Vdc

Vdc

module 2 new partb

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