1 Modeling circuits • Review – Kirchhoff's voltage and current laws – Concept of impedance – Use of Laplace transforms in circuit analysis • Develop – State-variable analysis for circuits
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1
Modeling circuits
• Review – Kirchhoff's voltage and current laws– Concept of impedance– Use of Laplace transforms in circuit analysis
• Develop– State-variable analysis for circuits
2
PI controller example
RiRf Cf
ViVo 0
sC1R
VRV
ff
o
i
i =+
+
Node equations:
3
PI controller example
RiRf Cf
ViVo
⎟⎠⎞
⎜⎝⎛ +−=
sK
KVV i
pi
o
fii
i
fp CR
1K
RR
K ==
0
sC1R
VRV
ff
o
i
i =+
+
4
Example problemR C
v i
L
vc
dtdv
C)t(i
)t(vdtdi
L)t(iR)t(v
c
c
=
++=
5
)t(iC1
dtdv
)t(vL1
)t(vL1
)t(iLR
dtdi
c
c
=
+−−=
)t(v0L
1
)t(v)t(i
0C1
L1
LR
)t(v)t(i
dtd
cc ⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎥
⎦
⎤
⎢⎢
⎣
⎡ −−=⎥
⎦
⎤⎢⎣
⎡
State variables
6
Laplace transforms
)0(Cv)s(sCV)s(I
)s(V)0(Li)s(IsL)s(IR)s(V
cc
c
−=
+−+=
If the initial conditions are zero:
)s(IsC1
sLR)s(V
)s(sCV)s(I
)s(V)s(IsL)s(IR)s(V
c
c
⎥⎦⎤
⎢⎣⎡ ++=
=
++=
or
7
Transfer functionIf the initial conditions are zero, find the ratio of the output I over the input V:
sC1sLR
1)s(V)s(I
)s(H
)s(IsC1
sLR)s(V
++==
⎥⎦⎤
⎢⎣⎡ ++=
8
LC1s
LRs
sL1
1sRC2sLC
sC
sC1sLR
1)s(V)s(I
)s(H
2 ++=
++=
++==
Express transfer functions as the ratio of two polynomials in s
9
Modeling mechanical systems• Review
– Newton's law for translational mechanical systemsdx/dt = v M dv/dt = Σ F
–Newton's law for rotational mechanical systemdθ/dt = ω J dω/dτ = Σ τ
10
Mechanical elements (linear)Mass: f = M a = M dv/dt = M dx2/dt2
Mx
f
fM a
Rigid
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Mechanical elements (linear)Spring: f = K (x1 – x2)
K x1
ff
x2
x1
fK(x1-x2)
12
Mechanical elements (linear)Friction: f = B (v1 – v2)
B x1v1
ff
x2v2
v1
fB(v1-v2)
13
Example
f
B x
K M
)t(f)t(xKdt
)t(dxB
dt
)t(xdM
2
2=++
14
)t(fM
10
)t(v)t(x
MB
MK
10
)t(v)t(x
dtd
or
)t(vdt
)t(dx
)t(fM1
)t(vMB
)t(xMK
dt)t(dv
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−
=⎥⎦
⎤⎢⎣
⎡
=
+−−=
15
MKsM
BsM
1
KsBsM
1)s(F)s(X
)s(F)s(X)KsBsM(
0) cond (initial functionTransfer
)t(f)t(xKdt
)t(dxB
dt
)t(xdM
22
2
2
2
++=
++=
=++
=
=++
16
Analogous rotating elements
torque τ [N m] force f [N]
angle θ [rad] position x [m]
angular speed ω[rad/s] speed v [m/s]
17
Analogous rotating elements
moment of inertia J [kg m2]
mass M [kg]
damping coeff. B [N m s/rad]
damping coeff. B [N s/m]
spring const. K [N m/rad]
spring const. K [N/m]
18
Example
B
KJ
)t()t(Kdt
)t(dB
dt
)t(dJ
2
2τ=θ+
θ+
θ
τ,θ
19
Modeling electromechanical systems
• Models for – DC Generator–DC Motor–Sensors
20
DC Generator• Drive shaft mechanically• Excite the field (sets up air-gap flux)
– Stationary field winding, or– Permanent magnets
• Armature winding rotates though the flux
21
• Commutator works as a rectifier– converts induced ac voltage to dc at
armature terminals
egea
iaRaLa
Thevenin equivalent of armature for PM generator
RL
22
)t(iR)t(e
)t(K)t(e
)t(edt
)t(diL)t(iR)t(e
aLa
g
aa
aaag
=
ωΦ=
=−−
–Assume flux Φ is constant, load inductance is negligible, speed ω is the input:
23
sLRR
KR
)s()s(E
sLRRK
)s()s(I
)s(IsL)s(I)RR()s(K
aLa
La
aLa
a
aaaLa
++
Φ=
Ω
++Φ
=Ω
++=ΩΦ
– Current Ia is one possible output, and voltage Ea is another:
24
– Compare results to that of text, where speed is constant and field voltage is the input
–The same physical system will have many different models each valid under some set of assumptions
25
DC Motor• Apply a dc source to the armature• Excite the field (sets up air-gap flux)
–Stationary field winding, or– Permanent magnets
26
• Commutator works as an inverter– converts dc terminal voltage to ac
voltage on rotating armature winding
emes
ia Ra
Thevenin equivalent of armature for PM motor
τ,θJ
B
ea
RsLa
27
)t(iK)t(
)t(dt
)t(dB
dt
)t(dJ
)t(edt
)t(diL)t(i)RR()t(e
dt)t(d
Kdt
)t(dK)t(e
am
2
2
ma
aaass
mm
=τ
τ=θ
+θ
+++=
θ=
θΦ=
28
)s(IK)s(sBsJ
)s()s(
sLRR)s(E)s(E
)s(I
)s(sK)s(E
am
2
aas
msa
mm
=τ+
Τ=Θ
++−
=
Θ=
29
Es
Em
G1(s)Ia Km
τG2(s)
Θ
Km s
sBsJ
1)s(G
RRLs1
)s(G
22
asa1
+=
++=
30
Es T(s)Θ
)s(G)s(GKs1
)s(G)s(GK)s(E)s(
)s(T21
2m
21m
s +=
Θ=
assa
2msasaa
2a
3m
RRR where
)KBR(s)JRBL(sJLs
K
+=
++++=
31
La is often a small value, then a simpler transfer function is found:
assa
2msasa
2m
RRR where
)KBR(sRJs
K)s(T
+=
++=
32
Sensors• Text describes several sensors:
– Position sensor is optical encoder or potentiometer
– Speed sensor is a tachometer: a dc generator or encoder (if the speed is non-zero)
– Accelerometers based on displacement of an inertia
33
Optical encoder
Reference window LED and light sensors
count angle increments
34
Speed measurement• Count encoder pulses per unit time or
measure time between pulses– Inaccurate at small speeds when very few
pulses occur• PM DC generator eg = K ω
35
Accelerometer• Ideally:
– Isolated mass (known value) with force measurement a = f/m
• Example in text:– Piezoelectric crystal e = Ka f = Ka M a or a = e/(Ka M)
36
Modeling some other systems
• Models for –Transformers–Gears
• Analogs
37
• Ideal transformer– Lossless two-winding device
e1 e2
1
2
2
1
2
1
2
1
NN
ii
NN
ee
==
38
• Ideal gear train– Lossless two shaft rotating device
τ1,θ1τ2,θ2
2
1
2
1
1
2
2
1
2
1
rr
rr
=ττ
=θθ
=ωω
r2
r1
39
One axis robot arm• Motor with gear train on output shaft
driving one-axis robot arm
MotorGearTrain Arm
θL
40
Ea
Em
Ia Kt
τ ωm
Km
Model of one axis robot arm
n
θm
θL
mm RsL1+ BsJ
1+
ωm
s1
θmModel of MotorGear
41
Analog systems• Systems with equations of the same
type are analogs of each other– Example: compare the differential
equations for a mass-spring-friction mechanical system to an RLC series circuit
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