LaplaceCircuits.pdf

7/25/2019 LaplaceCircuits.pdf

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Laplace Transforms Circuit Analysis

Passive element equivalents

Review of ECE 221 methods in s domain

Many examples

J. McNames Portland State University ECE 222 Laplace Circuits Ver. 1.63 1


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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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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: 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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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 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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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: 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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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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Hint: [r,p,k] = residue([320e3],[1 5e3 0])r = [-64, 64]

p = [-5000,0]

k = []



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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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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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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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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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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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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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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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LaplaceCircuits.pdf