# Laplace Transforms Circuit Analysis • Passive element equivalents

Laplace Transforms Circuit Analysis

Passive element equivalents

Review of ECE 221 methods in

s

Many examples domain

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

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Example 1: Circuit Analysis

We can use the Laplace transform for circuit analysis if we can deﬁne the 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.

i(0-) = -2 mA

10 u(t)

5 k Ω

5 mH

+

v(t)

-

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

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Example 1:Workspace

Hint:

[r,p,k] = residue([-2e-3 2e3],[1 1e6 0]) r = -0.0040, 0.0020, p = -1000000, 0 k = []

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

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Example 1:Workspace

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

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Laplace Transform Circuit Analysis Overview

LPT is useful for circuit analysis because it transforms diﬀerential equations 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 the

3. Solve for the

s s

domain domain voltages and currents of interest

4. Apply the inverse Laplace transform to ﬁnd time domain expressions

How do we know this will work?

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

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Kirchhoﬀ’s Laws

N v k

(

t

) = 0

k

=1

M k

=1

i k

(

t

) = 0

N

V k

(

s

) = 0

k

=1

M k

=1

I k

(

s

) = 0

Kirchhoﬀ’s 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 sum of currents leaving a node

If Kirchhoﬀ’s 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

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

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Deﬁning

s

Domain Equations: Resistors

i(t)

R

+

v(t)

-

I(s)

R

+

V(s)

-

v

(

t

) =

R i

(

t

)

V

(

s

) =

R I

(

s

)

Generalization of Ohm’s Law

As with KCL & KVL, the relationship is the same in the as in the time domain

s

domain

Note that we used the linearity property of the LPT for both

Ohm’s law and Kirchhoﬀ’s laws

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

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v

(

t

) =

L

d

i

(

t

) d

t

V

(

s

) =

L

[

sI

(

s

)

− I

0

]

V

(

s

) =

sLI

(

s

)

− LI

0

Where

I

0

Deﬁning

s

Domain Equations: Inductors

i(t)

L

+

v(t)

-

I

0 s

L I

0

I(s)

+

Ls

V(s)

-

I(s)

+

Ls

V(s)

-

i

(

t

) =

I

(

s

) =

1

L

1

sL

V

0

-

t v

(

τ

) d

(

s

) +

1

s

τ

I

0

+

I

0

i

(0

-

)

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

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Deﬁning

s

Domain Equations: Capacitors

i(t)

C

I(s)

+

v(t)

1

sC

-

V

0 s

+

V(s)

i

(

t

) =

C

d

v

(

t

) d

t

I

(

s

) =

C

[

sV

(

s

)

− V

0

]

-

Where

I

(

s

) =

sCV

(

s

)

− CV

0

V

0

v

(0

-

)

CV

0

1

sC

I(s)

+

V(s)

-

V

V v

(

t

) =

(

(

s s

) =

) =

1

C

1

C

1

sC

I t i

(

τ

) d

τ

0

-

1

s

I

(

s

) +

(

s

) +

V

0

s

+

V

1

s

V

0

0

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

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s

Impedance:

Z

(

s

) =

Y

(

s

) =

V

(

s

)

I

(

s

)

I

(

s

)

V

(

s

)

The

s

domain

impedance

initial conditions of a circuit element is deﬁned for zero

This is also true for the

s

We will see that circuit can assume

s

domain circuit analysis is easier when we

zero initial conditions

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

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Resistor

Inductor

s

Domain Circuit Element Summary

V

(

s

) =

RI

(

s

)

V

(

s

) =

sLI

(

s

)

V

V

=

=

RI sLI

Capacitor

V

(

s

) =

1

sC

I

(

s

)

V

=

1

sC

I

All of these are in the form

V

(

s

) =

ZI

(

s

)

Note similarity to phasor transform

Identical if

s

=

Will discuss further later

Equations only hold for zero initial conditions

R

Ls

1

sC

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

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10 u(t)

i(0-) = -2 mA

5 k Ω

5 mH

Example 2: Circuit Analysis

+

v(t)

-

Solve for

v

(

t

) using

s

-domain circuit analysis.

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

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Example 2: Workspace

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Example 3: Circuit Analysis

sin(1000

t

)

t = 0

1 k Ω

1

μ

F

+

v o

-

Given

v o

(0) = 0 , solve for

v o

(

t

) for

t ≥

0 .

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

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Example 3: Workspace

Hint:

[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]

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

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Example 3: Workspace

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

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Example 3: Plot of Results

0

−0.2

−0.4

−0.6

−0.8

0

1

0.8

0.6

0.4

0.2

Total

Transient

5 10

Time (ms)

15 20 25

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

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50 Ω

10 mF

50 Ω

Example 4: Circuit Analysis

100 Ω

t = 0

175 Ω

175 Ω

40 V

+

v

-

10 mH

Solve for

v

(

t

) .

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

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Example 4: Workspace

Hint:

[r,p,k] = residue([1e-3 20 0],[1 21.25e3 10e3]) r = [-1.2496, -0.0004] p = [-21250,-0.4706] k = [0.0010]

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

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Example 4: Workspace

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

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Example 5: Parallel RLC Circuits

i(t) C L R

+

v(t)

-

Find an expression for

V

(

s

) . Assume zero initial conditions.

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

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Example 6: Circuit Analysis

0.125

μ

F

i

L

8 H

+

20 k Ω

v

-

i

Given

L

(0

-

v

(0) = 0

) =

12

.

25

V and the current through the inductor is mA, solve for

v

(

t

) .

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

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Example 6: Workspace

Hint:

[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]

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

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Example 6: Workspace

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

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Example 6: Plot of

v

(

t

)

80

60

40

20

0

−20

−40

0 5 10 15 20

Time (ms)

25 30 35 40

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

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Example 6: MATLAB Code

t = 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(’’);

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

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v(t)

Example 7: Series RLC Circuits

+

v

R

(t)

+

v

L

(t)

-

R L

C

+

v

C

(t)

-

Find an expression for conditions.

V

R

(

s

) ,

V

L

(

s

) , and

V

C

(

s

) . Assume zero initial

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

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Example 7: Workspace

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

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t = 0

80 V

+

v

2

(t)

-

Example 8: Circuit Analysis

20 k Ω

+

v

1

(t)

-

50 nF

10 nF 2.5 nF

There is no energy stored in the circuit at

t

= 0 . Solve for

v

2

(

t

) .

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

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Example 8: Workspace

Hint:

[r,p,k] = residue([320e3],[1 5e3 0]) r = [-64, 64] p = [-5000,0] k = []

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

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Example 8: Workspace

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

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Example 9: Circuit Analysis

0.25 v

1

600 u(t)

10 Ω

v

1

20 H

0.1 F

v

2

140 Ω

Solve for

V

2

(

s

) . Assume zero initial conditions.

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

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Example 9: Workspace

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

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Example 9: Workspace

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

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Example 10: Circuit Analysis

10 mF

i(t) a

u(t)

9 i(t)

100 Ω b

Find the Thevenin equivalent of the circuit above. Assume that the capacitor is initially uncharged.

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

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Example 10: Workspace

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

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Example 10: Workspace

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

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v(t)

i

L

L

R

Example 11: Circuit Analysis

+

v

o

(t)

-

i

Find an expression for

L

(0) =

I

0 mA.

v o

(

t

) given that

v

(

t

) = e

− αt u

(

t

) and

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

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Example 11: Workspace

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

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Example 11: Workspace

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

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1

μ

F

1 k Ω

v s

(t)

Example 12: Circuit Analysis

200 Ω

100 mH

+

v

o

(t)

-

Find an expression for

V o

(

s

) in terms of

V s

(

s

) . Assume there is no energy stored in the circuit initially. What is

v o

(

t

) if

v s

(

t

) =

u

(

t

) ?

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

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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 = []

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

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Example 12: Workspace

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v s

(t)

C

A

R

A

Example 13: Circuit Analysis

C

B

R

B

R

L

+

v

o

(t)

-

Find an expression for

V o

(

s

) in terms of energy stored in the circuit initially.

V s

(

s

) . Assume there is no

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

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Example 13: Workspace

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

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Example 13: Workspace

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Example 14: Circuit Analysis

C

R

v s

(t) R

L

+

v

o

(t)

-

Find an expression for

V o

(

s

) in terms of energy stored in the circuit initially.

V s

(

s

) . Assume there is no

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

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Example 14: Workspace

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

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