model of moving coil meter - mechanical engineering · model of moving coil meter prof. r.g....

ME 144L – Prof. R.G. LongoriaDynamic Systems and Controls Laboratory

Department of Mechanical EngineeringThe University of Texas at Austin

Experimental and Theoretical

Model of Moving Coil Meter

Prof. R.G. Longoria

Updated Summer 2012



System: Moving Coil MeterFRONT VIEW

REAR VIEW

Electrical circuit model

Mechanical model

Meter

movement

Series resistor

‘needle’



Moving Coil Meter Movement

From Figliola and Beasley,

“Theory and Design for

Mechanical Measurements”,

John Wiley and Sons, 1995.

This D’Arsonval meter movement is a basic EM

device that responds to electrical voltage or current

signals.

In the particular meter being used, the

coil pivots such that the conductors

are always perpendicular to the

magnetic field generated by a

permanent magnet, as shown here.



Physical Modeling of System

• We focus on a 2nd order model that neglects inductance.

• The appendix shows a 3rd order model with inductance, and a frequency response comparison of the 2nd order and 3rd order model shows that the simplification is reasonable.

• Methods for deriving transfer functions are reviewed.

• Frequency response derivations are summarized and examples provided in Matlab and LabVIEW.

• These tools should allow comparison of model and experimental data directly on the same or similar plots.



Meter Movement Bond GraphA bond graph of the meter is shown below. The coil has resistance, Rm, and inductance,

Lm. The needle has moment of inertia, Jm, and there is some damping, Bm, as well. The

spring has stiffness, Km. These are parameters for linear constitutive relations for each

of the elements shown in this model. Note, the meter also has an external series resistor

that is not shown here, but the value of that resistance can be added to Rm.

We seek a mathematical

model that relates needle

position, θ, to input

voltage, v.

This model can be

derived from the bond

graph, or by application

of Newton’s Laws

(mechanical side) and

KVL (circuit side).

Causality

assignment shows

this is a 2nd order

system.

Ignore

inductance

(very

small)

See appendix for 3rd order system model with inductance included.

: mJI

ωhɺ

Gmv

mi

mT

mω

mr••

1

BωB

T

:m

BR

KT

Kθɺ

1

:m

LI

:m

RR

Ri

λɺ mi

Rv

v

i:1

mKCE



Simplified Model Equations

( )

angular momentum

angular position of needle/spring

where,

( )

m m

m s m m

m

m m m

m m m

m

m

c s

h J

h T K B

T r i

v r

v vi

R R

ω

θ

θ ω

θ ω

ω

= =

=

= − −

=

=

=

−=

+

ɺ

ɺ

The mathematical model for the meter, neglecting inductance,

States:

State

equations:

EM gyrator

relations: If you know how to derive the

equations from a bond graph, you

see how current is here

determined by the voltage drop.



State Equations – 2nd order

�

21

100

mm m m

m m

m

m

rB K r

hR JhR v

J

θθ

− + −

= + B

A

ɺ

ɺ

��

In state space form:

[ ]�

[ ]�

0 1 0h

y vθθ

= = +

DC

State equations:

Output equation:



Transfer Functions (TF)

• To derive a transfer function (TF), you have several options:

1. Use G(s)=C(sI-A)-1B+D (state-space to TF)

2. Change the state space equations into an ODE of nth order, and use Laplace transform

3. Use a bond graph, and apply impedance methods

• It is likely that you may have learned one or two of these approaches for deriving the TF.

• Method #1 is available in LabVIEW and Matlab, but only if numerical parameters are available.



2nd order SS Model to TF

G s( ) 0 1( )

s Bm

rm2

Rm

+

1

Jm

⋅+

1−

Jm

Km

s

1−

⋅

rm

Rm

0

⋅ 0+

G s( )1

Rm s Bm⋅⋅ Rm Km⋅+ rm2

s⋅+ Rm s2

Jm⋅⋅+

rm⋅

G s( )1

Rm s2

Jm⋅⋅ Rm Bm⋅ rm2

+

s⋅+ Rm Km⋅+

rm⋅

numrm

Rm Jm⋅

den s2

Bm

rm2

Rm

+

1

Jm

⋅ s⋅+Km

Jm

+

In a similar manner as in the

previous case, the G(s)

function is derived here

symbolically.

Keep in mind, this G(s)

function has been defined by

the A,B,C,D system as:

So ( )G sv

θ⇒ =

( )y

G su

=



TF to Frequency Response

• Given G(s), you can determine the frequency response

function (FRF), by setting s = jω, and determining the

amplitude and phase functions of G(s).

• Use:

• Example (1st order system):

( ) ( ) jG j G j e

φω ω=

�1 1

( ) ( )1 1s j

G s G js jω

ωτ τω=

= ⇒ =+ +

( )

( )

2

Im( )1 1

Re( )

1( )

1

0 tan tan ( ) ( )den

num den den

G jωτω

φ φ φ τω φ ω− −

=+

= − = − = − =

Amplitude function

Phase function

Both of these functions are

plotted versus frequency, ω.

These functions are also referred to as ‘Bode plots’, and are commonly found functions in Matlab

(Control Toolbox) and LabVIEW (Control Design Toolkit). If these packages are not available,

you must derive the functions analytically so you can plot in Excel, Matlab, or LabVIEW.



Meter: TF to FRF (2nd order)( ) ( ) j

G j G j eφω ω=

�

( )2 2

22

( ) ( )1 1

m m

m m m m

s jm m m mm m

m m m m m m

r r

R J R JG s G j

r K r Ks B s j B j

R J J R J J

ω

ω

ω ω=

= ⇒ =

+ + + + + +

( )

222

2

2

Im( )1 1

Re( )2

( )

0 tan tan ( )

m

m m

m mm

m m m

mm

den m m

num den denm

m

r

R JG j

K rB

J R J

rB

R J

K

J

ω

ωω

ω

φ φ φ φ ωω

− −

=

− + +

+

= − = − = − =

−

Amplitude function

Phase function

( )2 2Note: 1jω ω= −



Meter: dc gain

222

2

1( 0)

00

m m

m m m m m

m m mm m

m m

m m m

r r

R J R J rG j j

K R KK r

B JJ R J

ω

= ⋅ = = =

− + +

The dc gain is the value of the TF or FRF when ‘s’ or ‘ω’ go to zero,

respectively. So, from the amplitude function,

The dc gain for the moving coil meter is,

0

1( 0) m

m m

rG j j

v R Kω

θω

=

= ⋅ = =



DC Gain from this TFG s( ) 0 1( )

s Bm

rm2

Rm

+

1

Jm

⋅+

1−

Jm

Km

s

1−

⋅

rm

Rm

0

⋅ 0+

G s( )1

Rm s Bm⋅⋅ Rm Km⋅+ rm2

s⋅+ Rm s2

Jm⋅⋅+

rm⋅

G s( )1

Rm s2

Jm⋅⋅ Rm Bm⋅ rm2

+

s⋅+ Rm Km⋅+

rm⋅

numrm

Rm Jm⋅

den s2

Bm

rm2

Rm

+

1

Jm

⋅ s⋅+Km

Jm

+

The DC gain can be found from the G(s)

function by making s=jω go to zero.

( )2 2( ) m m m

m m m m m m

r R JG s

v s B r R s J K J

θ= =

+ + +

So, the DC gain is,

0

1( 0) m

s m m

rG s

v R K

θ

→

→ = = ⋅

�

,

dc current

m dcm dc m dcdc

m m m m

Tr v r i

K R K Kθ

= ⋅ = =

Other relations:



Plotting the FRF

• The FRFs are commonly plotted as amplitude and phase functions (sometimes called Bode plots, although strictly speaking a Bode plot is an approximated ‘sketch’ of the FRF plots).

• LabVIEW and Matlab may have build-in packages:

– In Matlab, if Control Toolbox is available, use: bode().

– In LabVIEW, the Control Design Toolkit has CD Bode.vi.

– Use the online help if you have or want to use these tools.

• For this course, you should generate these plots without these tools, since it is instructive to develop the code for computing the amplitude and phase functions.

• It is important to notice that the amplitude is often plotted in terms of decibels (dB). In this context, the decibel is defined,

• Also, make note of the frequency axes used (rad, deg, etc.).

10dB 20log ( )G jω=See Matlab

example on next

slide.



Example Script

101

102

103

104

-30

-20

-10

0

101

102

103

104

-100

-80

-60

-40

-20

0

% Basic script for plotting amplitude and phase plots

% Example: 1st order system. RGL, 4-15-06

% Plot at selected frequencies

f = [10.000

20.000

50.000

100.00

200.00

400.00

500.00

1000.0

2000.0

5000.0

10000];

%

w = 2*pi*f;

N = length(w);

R=81.36e3;

C=0.005e-6;

tau = R*C;

j = sqrt(-1);

for i=1:N,

gsys(i) = 1/(j*w(i)*R*C+1);

magsys(i) = abs(gsys(i));

dbm(i) = 20*log10(magsys(i));

angsys(i) = atan(imag(gsys(i))/real(gsys(i)));

end

%

subplot(2,1,1), semilogx(f,dbm,'o')

subplot(2,1,2), semilogx(f,angsys*180/pi,'o')

Note, you can handle the

complex function in Matlab

directly.



Appendix

Details on the physical modeling and

on the 3rd order model



Basic Electromechanics

F

B

q V F = q V × B

N

S

+

−

B-fieldcopperrod

current

flow direction

GFv

i•x

Permanentmagnet supplies the

B-field

The differential force on a differential element of charge, dq, is given by:

where B is the magnetic field density, and i the current (moving charge).

It can be shown that the net effect of all charges in the conductor allow us to write:

where dl is an elemental length.

For a straight conductor of length l in a uniform magnetic field, you can integrate to find the total force:

With angle α between the vectors, you can arrive at the desired relation:

dF dqv B= ×� ��

dF idl B= ×��

F il B= ×��

( )gyrator modulus

sinF Bl iα= ⋅��

sinr Bl α=

We find this

modulation as:

F r i

v r V

V x

= ⋅

= ⋅

≡ ɺ

This slide summarizes the basic force-current relation in each conductor. In a bond graph, this can be

modeled by a gyrator, which gives a net relation between torque and current.



Meter Movement Bond GraphA bond graph of the meter is shown below. The coil has resistance, Rm, and inductance,

Lm. The needle has moment of inertia, Jm, and there is some damping, Bm, as well. The

spring has stiffness, Km. These are parameters for linear constitutive relations for each

of the elements shown in this model. Note, the meter also has an external series resistor

that is not shown here, but the value of that resistance can be added to Rm.

We seek a mathematical

model that relates needle

position, θ, to input

voltage, v.

This model can be

derived from the bond

graph, or by application

of Newton’s Laws

(mechanical side) and

KVL (circuit side).

Causality

assignment

shows this

is a 3rd order

system.

:m

JI

ωhɺ

Gm

v

mi

mT

mω

mr••

1

BωB

T

:m

BR

KT

Kθɺ

1

:m

LI

:m

RR

Ri

λɺ mi

Rv

v

i:1

mKCE



Full Model Equations

angular momentum

flux linkage


( )

m m

m m

m s m m

c s m m

m

m m m

m m m

h J

L i

h T K B

v R R i v

T r i

v r

ω

λ

θ

θ ω

λ

θ ω

ω

= =

= = =

= − −

= − + −

=

=

=

ɺ

ɺ

ɺ

The mathematical model for the meter, including all the effects

described is,

States:

State

equations:

EM gyrator

relations:

Note: the needle and the

spring have the same

velocity.



Simplified Forms and Relations

2

2

flux linkage


( )

m m m

m m

m m m s m

mm c s m m

m

h J J

L i

d dh J J T K B

dt dt

diL v R R i v

dt

ω θ

λ

θ

θ θθ θ

λ

θ ω

= =

= = =

= = = − −

= = − + − =

ɺ

ɺ ɺɺ

ɺ

ɺ

We often choose to make use of simplified formulations; the state

space equations may not be suited for answering questions we have.

Note how the state

variables are related

to other useful

variables.

The state equations

are related to both 1st

and 2nd order forms

that we might use.



State Equations

�

0

1

0

01

0 0

m m

m m

m mm

m m

m

R r

L J

r Bh K h v

J J

J

λ λ

θ θ

− − = − − +

B

A

ɺ

ɺ

ɺ

��

In state space form:

[ ] [ ]�

0 0 1 0y h v

λ

θ

θ

= = + DC

��

State equations:

Output equation:



Full Model SS to TF

G s( ) 0 0 1( )

sRm

Lm

+

rm−

Lm

0

rm

Jm

sBm

Jm

+

1−

Jm

0

Km

s

1−

⋅

1

0

0

⋅ 0+

G s( )rm

s3

Lm Jm⋅⋅ s2

Lm Bm⋅⋅+ s Lm Km⋅⋅+ Rm s2

Jm⋅⋅+ Rm s Bm⋅⋅+ Rm Km⋅+ rm2

s⋅+

G s( )rm

s3

Lm Jm⋅⋅ Rm Jm⋅ Lm Bm⋅+( ) s2

⋅+ Rm Bm⋅ Lm Km⋅+ rm2

+

s⋅+ Rm Km⋅+

numrm

Lm Jm⋅

den s3

Rm

Lm

Bm

Jm

+

s2

⋅+Rm Bm⋅

Lm Jm⋅

Km

Jm

+rm

2

Lm Jm⋅+

s⋅+Rm Km⋅

Lm Jm⋅+

Using the G(s) formula,

apply directly to the

derive the form shown

here to the right.

With a symbolic

processor, this is easily

accomplished (e.g.,

Matlab, MathCAD, or

Mathematica).

C

(sI-A)-1

B

D



Example using bode()

clear all

% examples parameters for moving coil

% this example plots both the 3rd and 2nd order system

global Rm rm Jm Km Bm Lm

Rm = 15085;

rm = 0.003;

Jm = 2e-7;

Km = 10e-6;

Bm = 9e-7;

Lm = 0.05;

% 3rd order case

A1 = [-Rm/Lm -rm/Jm 0;rm/Lm -Bm/Jm -Km;0 1/Jm 0];

B1 = [1;0;0];

C1 = [0 0 1];

D1 = [0];

sys1 = ss(A1,B1,C1,D1);

[num1,den1]=ss2tf(A1,B1,C1,D1)

% 2nd order case

A2 = [-(Bm+rm*rm/Rm)/Jm -Km;1/Jm 0];

B2 = [rm/Rm;0];

C2 = [0 1];

D2 = [0];

sys2 = ss(A2,B2,C2,D2);

[num2,den2]=ss2tf(A2,B2,C2,D2);

bode(sys1,sys2)

-400

-300

-200

-100

0

Magnitu

de (

dB

)

100

102

104

106

-270

-180

-90

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

2nd order system3rd order

system

Note how the two models only deviate at very high frequency –

we will never excite the meter at this frequency range!! The 2nd

order system is clearly applicable for all cases of interest.

model of moving coil meter - mechanical engineering · model of moving coil meter prof. r.g....

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