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Modeling
Transfer function form of the model
θn
2
n
Kθ / vω
Gθ / v ( s) =
= 2
2
vin s + 2ζωn s + ωn
The following slides detail a derivation of this analog meter model both as state
space model and transfer function (TF) as shown above.
Why use the TF model? The static experiments provide a measure of the static
gain, so only two more parameters are needed – 3 total. Also, the step response
looks like a classic 2nd order system response.
The state space model needs 5 parameters, so it is not as convenient to use in this
control study.
TF model only needs damping ratio and natural frequency,
which can be readily determined in the laboratory.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin
Analog moving coil meter model
Modeling
• The meter physical model is presented in two different
versions: ‘full model’ and 2nd order.
• The 2nd order model neglects inductance, which is a
good assumption for this application.
• The electromechanical (EM) torque can be modeled
using a gyrator.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin
Recall the models shown before in schematic form:
Electrical
circuit model
Vin
Modeling
im
Meter
movement
Rotational system
needle
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin
Modeling
The EM torque induced on the moving coil is
related to the current – a gyrator model
This slide summarizes the basic force-current relation in a conductor. In a bond graph, this can be
modeled by a gyrator, which gives a net relation between force and current. For a motor or for the case of
the rotational moving coil, this force is resolved into torque.
We find this
Permanent
magnet supplies the
B-field
current
flow direction
F
N
B-field
+
modulation as:
r = Bl sin α
−
copper
rod
v
B
qV
i
F=qV×B
F = r ⋅i
v = r ⋅V
V ≡ xɺ
F
G
•
x
S
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).
dF = dqv × B
It can be shown that the net effect of all charges in the conductor allow us to write:
where dl is an elemental length.
dF = idl × B
For a straight conductor of length l in a uniform magnetic field, you can integrate to find the total force:
F = il × B
With angle α between the vectors, you can arrive at the desired relation:
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
F = ( Bl sin α ) ⋅ i
gyrator modulus
Department of Mechanical Engineering
The University of Texas at Austin
Modeling
Analog moving coil meter bond graph
A bond graph of the meter can take the form shown below. The coil has resistance, Rm,
and inductance, Lm. The needle has moment of inertia, Jn, and there is some damping,
Bn, as well. The spring has stiffness, Ks. 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 (equal to spring
deflection) to input
voltage, vin.
This model can be
derived from the bond
graph, or by application
of Newton’s Laws
(mechanical side) and
KVL (circuit side).
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Electrical
circuit model
EM
conversion
I : Jm
I : Lm
λɺm im
E
vin
iin
vR
rm
vm
1
Rotational system
i i
G
im
iR
hɺn ω n
Tm
Ts
θɺ
1
ωm
C : K s−1
s
TB
ωB
R : Bn
R : RT
RT = Rm + Rs
coil + series
See Appendix B for explanation of
gyrator model for EM transduction.
Department of Mechanical Engineering
The University of Texas at Austin
Full
3rd
Modeling
order model, with inductance
The state-space model for the meter, including the inductance, is 3rd
order.
 hn = J nω n = needle angular momentum

3 States: λ m = Lmim = flux linkage
θ = angular position of needle/spring
 n
ɺ = J ωɺ = T − K θ − Bω

im = λm Lm
h
n
n n
m
s n
n
3
ɺ
State
λ m = Lm ( dim dt ) = vin − (Rm + Rs )im − vin
equations: 
ɺ =ω
θ
n
 n
Note: the needle and the
spring have the same
velocity.
EM gyrator 
Tm = rm im
relations: v = r ω
Also, can choose either the
 m m m
meter flux linkage or
current as the state.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin
3rd
order model state-space equations
Modeling

 R
r
 − T − m
0 
Jn

 λɺ   Lm
 
λ

m

1
 m   r

Bn
m
 h  +  0 v

ɺ


=
−
−
K
h
State equations:
s
in
n
n




J
J



 
n
n
ɺ
0 
  θn  
 θ n  


1

 0
B
0


Jn


A
Output equation:
 λ 
 m 
y = θ n =  0 0 1   hn  + 0  vin

 C
 θ n  D
In state space form:
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin
Modeling
Analog moving coil meter bond graph – neglect
inductance
If we neglect the inductance, we see that the model reduces to second order.
Note the change in causality (if you understand bond graphs).
This assumption is reasonable given that we observe a step response in the
experiments that looks 2nd order, underdamped.
Now the only states of
interest are the needle
angle (related to spring
deflection) and the
needle rotational
momentum.
I : Jm
I : Lc
λɺm im
E
vin
vm
1
iin
vR
rm
i i
G
im
iR
hɺn ω n
Tm
Ts
θɺ
1
ωm
C : K s−1
s
TB
ωB
R : Bn
R : RT
RT = Rm + Rs
coil + series
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
See Appendix B for explanation of
gyrator model for EM transduction.
Department of Mechanical Engineering
The University of Texas at Austin
2nd
order model, neglecting inductance
Modeling
The mathematical model for the meter, neglecting inductance,
 hn = J nω n = angular momentum
States: 
θ n = angular position of needle/spring
 hɺn = Tm − K sθ n − Bnω n
State
equations: θɺ = ω
 n
n
Tm = rm im
EM gyrator 

relations:
vm = rmω m
where,
iin = iR
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
v
(
=
in
− vm )
(Rm + Rs )
The meter current is now
determined by the voltage
drop, not by the inductor
state.
Department of Mechanical Engineering
The University of Texas at Austin
2nd
Modeling
order state-space equations
In state space form:
State equations:
 

rm2  1


 hɺ   −  Bn + R  J − K s   h
n
T
n
 n =


ɺ
 θn  
 θn
1




0


Jn

A
 h
n
Output equation: y = θ n =  0 1  
 θ
 n
C
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab


  rm 
 +  R  vin
  T 

0 

B

 + 0  vin
  D
Department of Mechanical Engineering
The University of Texas at Austin
Let’s convert this into a
2nd
order ODE equation
Modeling
First, consider just the first equation and write it in terms of the angle:
hn = J nω n = J nθɺn ⇒ hɺn = J nθɺɺn
Substitute into the momentum equation:
2

 1
r
rm
m
ɺɺ
ɺ
⇒ J nθ n +  Bn +   J nθ n  + K sθ n =
vin
RT  J n
RT

Remember that the angle and angular velocity are related, so write
in terms of the angle. This gives the 2nd order ODE we want:
2

 1
 Ks 
 K s   rm

r
r
m
m
⇒ θɺɺn +  Bn +  θɺn +   θ n =
vin =   
vin 
RT  J n
J n RT
 Jn 
 J n   K s RT 

ω n2
ω n2
u(t)
2ζω n
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin
Modeling
Relate to the static gain measured in lab
2

 1
 Ks 
 K s   rm

r
m
θɺɺn +  Bn +  θɺn +   θ n =   
vin 
RT  J n
 Jn 
 J n   K s RT 

ω n2
ω n2
u(t)
2ζω n
For a constant input voltage, the steady-state angle (equilibrium) is
found by making the derivative terms zero,
Recall:

r2  1
K
K r
θɺɺn +  Bn +

=0
⇒ θ nss
m
θɺn +
RT  J n =0
 rm
=
 K s RT
s
Jn
θ nss =

 vin = Kθ /v vin

s
m
J n K s RT
vin
 90 ⋅ deg 
θ =
⋅ vin

15
⋅
V


static gain
model
So, if you want a certain angle, you simply apply, vin = K v /θ θ desired
Works well if parameters are known, remain constant, and
dynamic effects are not significant.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin
Now write in transfer function form
Start in the standard form: θɺɺn + 2ζω nθɺn + ω n2θ n = ω n2u(t)
Modeling
u(t) = Kθ /v vin
Transform to s-domain, and solve for angle-to-voltage relation:
Kθ / vωn2
= 2
vin s + 2ζωn s + ωn2
θn
You can see this model returns the static gain relation when you
make s go to zero (i.e., steady-state).
So, all we need is the damping ratio and the natural frequency to
parameterize this dynamic model. These can be found either from
the physical parameters or by experimentally determining the
values in the lab. We’ll do the latter.
ME 144L – Prof. R.G. Longoria
Dynamic Systems and Controls Lab
Department of Mechanical Engineering
The University of Texas at Austin