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Laplace Transforms Circuit Analysis
Passive element equivalents
Review of ECE 221 methods in s domain
Many examples
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Example 1: Circuit AnalysisWe can use the Laplace transform for circuit analysis if we can definethe circuit behavior in terms of a linear ODE.
For example, solve for v(t). Check your answer using the initial and
final value theorems and the methods discussed in Chapter 7.
10 u(t) v(t)
-
+
5 mH
i(0-) = -2 mA
5 k
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Example 1:WorkspaceHint: [r,p,k] = residue([-2e-3 2e3],[1 1e6 0])r = -0.0040, 0.0020,
p = -1000000, 0
k = []
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Example 1:Workspace
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Laplace Transform Circuit Analysis Overview
LPT is useful for circuit analysis because it transforms differentialequations into an algebra problem
Our approach will be similar to the phasor transform
1. Solve for the initial conditions Current flowing through each inductor
Voltage across each capacitor
2. Transform all of the circuit elements to thes domain
3. Solve for the s domain voltages and currents of interest
4. Apply the inverse Laplace transform to find time domainexpressions
How do we know this will work?
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Kirchhoffs LawsNk=1
vk(t) = 0Nk=1
Vk(s) = 0
Mk=1
ik(t) = 0Mk=1
Ik(s) = 0
Kirchhoffs laws are the foundation of circuit analysis
KVL: The sum of voltages around a closed path is zero
KCL: The sum of currents entering a node is equal to the sumof currents leaving a node
If Kirchhoffs laws apply in the s domain, we can use the same
techniques that you learned last term (ECE 221) Apply the LPT to both sides of the time domain expression for
these laws
The laws hold in the s domain
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Defining s Domain Equations: Resistors
R
v(t) -+
i(t) R
V(s) -+
I(s)
v(t) =R i(t) V(s) =R I(s)
Generalization of Ohms Law
As with KCL & KVL, the relationship is the same in the s domainas in the time domain
Note that we used the linearity property of the LPT for bothOhms law and Kirchhoffs laws
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Defining s Domain Equations: Inductors
v(t) -+
i(t) L
V(s) -+
I(s) LsL I
0
V(s) -+
I(s) Ls
I0
s
v(t) =Ldi(t)
dt i(t) =
1
L
t0-v() d+I0
V(s) =L [sI(s) I0] I(s) = 1
sL
V(s) +1
s
I0
V(s) =sLI(s) LI0
Where I0 i(0-)
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Defining s Domain Equations: Capacitors
v(t) -+
i(t)
V(s) -+
I(s)
V(s) -+
I(s)
CV0
C
1sC
V0s 1
sC
i(t) =Cdv(t)dt
v(t) = 1C
t
0-i() d+V0
I(s) =C[sV(s) V0] V(s) = 1
C
1
sI(s)
+
1
sV0
I(s) =sCV(s) CV0 V(s) = 1
sCI(s) +
V0
s
Where V0 v(0-)
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s Domain Impedance and Admittance
Impedance: Z(s) =V(s)
I(s)
Admittance: Y(s) = I(s)V(s)
The s domain impedance of a circuit element is defined for zeroinitial conditions
This is also true for the s domain admittance
We will see that circuit s domain circuit analysis is easier when wecan assume zero initial conditions
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s Domain Circuit Element Summary
Resistor V(s) =RI(s) V =RI R
Inductor V(s) =sLI(s) V =sLI Ls
Capacitor V(s) = 1sC
I(s) V = 1sC
I1
sC
All of these are in the form V(s) =ZI(s)
Note similarity to phasor transform
Identical ifs=j
Will discuss further later Equations only hold for zero initial conditions
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Example 2: Circuit Analysis
10 u(t) v(t)
-
+
5 mH
i(0-) = -2 mA
5 k
Solve for v(t) using s-domain circuit analysis.
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Example 2: Workspace
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Example 3: Circuit Analysis
t = 0
vo
-
+1 k
sin(1000t) 1 F
Given vo(0) = 0, solve for vo(t) for t 0.
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Example 3: WorkspaceHint: [r,p,k] = residue([1e6],conv([1 0 1e6],[1 1e3]))r = [ 0.5000, -0.2500 - 0.2500i, -0.2500 + 0.2500i]
p = 1.0e+003 *[ -1.0000, 0.0000 + 1.0000i, 0.0000 - 1.0000i]
k = []
[abs(r) angle(r)*180/pi]
ans = [ 0.5000 0, 0.3536 -135.0000, 0.3536 135.0000]
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Example 3: Workspace
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Example 3: Plot of Results
0 5 10 15 20 250.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1 TotalTransient
Steady State
Time (ms)
vo
(t)
(V)
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Example 4: Circuit Analysis
40 V
10 mF
10 mH
t = 0
v
-
+
50
50
100
175
175
Solve for v(t).
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Example 4: WorkspaceHint: [r,p,k] = residue([1e-3 20 0],[1 21.25e3 10e3])r = [-1.2496, -0.0004]
p = [-21250,-0.4706]
k = [0.0010]
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Example 4: Workspace
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Example 5: Parallel RLC Circuits
C L R v(t)
-
+
i(t)
Find an expression for V(s). Assume zero initial conditions.
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Example 6: Circuit Analysis
8 H v
-
+
iL
20 k0.125 F
Given v(0) = 0 V and the current through the inductor isiL(0
-) = 12.25 mA, solve for v(t).
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Example 6: WorkspaceHint: [r,p,k] = residue([98e3],[1 400 1e6])r = [ 0 -50.0104i, 0 +50.0104i]
p = 1.0e+002 * [ -2.0000 + 9.7980i, -2.0000 - 9.7980i]
k = [] [abs(r) angle(r)*180/pi]
ans =[ 50.0104 -90.0000, 50.0104 90.0000]
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Example 6: Workspace
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Example 6: Plot ofv(t)
0 5 10 15 20 25 30 35 40
40
20
0
20
40
60
80
Time (ms)
(v
olts)
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Example 6: MATLAB Codet = 0:0.01e-3:40e-3;v = 50*exp(-200*t).*sin(979.8*t);t = t*1000;h = plot(t,v,b);set(h,LineWidth,1.2);
xlim([0 max(t)]);ylim([-23 40]);box off;xlabel(Time (ms));ylabel((volts));title();
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Example 7: Series RLC Circuits
R L
Cv(t)
vR(t) -+ v
L(t) -+
vC(t)
-
+
Find an expression for VR(s), VL(s), and VC(s). Assume zero initialconditions.
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Example 7: Workspace
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Example 8: Circuit Analysis
50 nF
10 nF 2.5 nF
80 V
t = 0
v1(t)
-
+
v2(t)
-
+
20 k
There is no energy stored in the circuit at t= 0. Solve for v2(t).
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Example 8: Workspace
Hint: [r,p,k] = residue([320e3],[1 5e3 0])r = [-64, 64]
p = [-5000,0]
k = []
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Example 8: Workspace
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Example 9: Circuit Analysis
0.25 v1
20 H
0.1 F600 u(t)
v1
v2
10
140
Solve for V2(s). Assume zero initial conditions.
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Example 9: Workspace
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Example 9: Workspace
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Example 10: Circuit Analysis
10 mF
9 i(t)
u(t)
a
b
i(t)
100
Find the Thevenin equivalent of the circuit above. Assume that thecapacitor is initially uncharged.
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Example 10: Workspace
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Example 10: Workspace
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Example 11: Circuit Analysis
v(t)
L
R vo(t)
-
+
iL
Find an expression for vo(t) given that v(t) = etu(t) and
iL(0) =I0 mA.
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Example 11: Workspace
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Example 11: Workspace
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Example 12: Circuit Analysis
100 mH
vs(t)
vo(t)
-
+
200
1 k1 F
Find an expression for Vo(s) in terms ofVs(s). Assume there is noenergy stored in the circuit initially. What is vo(t) ifvs(t) =u(t)?
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Example 12: Workspace
Hint: [r,p,k] = residue([-0.2 0],conv([1 2e3],[1 1e3]))r = [-0.4000, 0.2000]
p = [-2000, -1000]
k = []
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Example 12: Workspace
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Example 13: Circuit Analysis
vo(t)
-
+
vs(t)R
L
RA
CA R
B
CB
Find an expression for Vo(s) in terms ofVs(s). Assume there is noenergy stored in the circuit initially.
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Example 13: Workspace
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Example 13: Workspace
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Example 14: Circuit Analysis
vs(t)
vo(t)
-
+
RL
C
R
Find an expression for Vo(s) in terms ofVs(s). Assume there is no
energy stored in the circuit initially.
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Example 14: Workspace
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