Deck 14: Application of the Laplace Transform to Circuit Analysis

Question 1

Use Thévenin's theorem to determine v o ( t ) for t 0 in Fig. E14.2.

Figure E14.2

Answer 1

Refer to Figure E14.2 in the textbook.
The Laplace transformed circuit is,
Figure 1
The output voltage is the voltage across the
resistance.
The Thevenin's equivalent for the circuit can be calculated by removing the
resistance.
The resulting circuit is,
Figure 2
From Figure 2 the current in the first loop is,
Apply Kirchhoff's Voltage Law to loop 2.
The Thevenin's equivalent voltage can be calculated as,
The Thevenin equivalent resistance is calculated by replacing the sources in the network with their internal impedances.
Figure 3
The equivalent impedance is,
The Thevenin equivalent circuit is,
Figure 4
Apply voltage division rule at output.
Simplify the quadratic equation '
'we get,
The equation
is,
Express
in a partial fraction expansion.
The value of
is,
The value of
is,
The partial fraction expansion of
is then
Apply inverse Laplace transform.
Thus, the value of the output voltage is,
.

Question 2

Find v o ( t ), t 0, in the network shown in Fig. P14.12 using nodal analysis.

Figure P14.12

Answer 2

Refer to Figure P14.11 in the textbook.
Convert elements into s-domain:
Draw the circuit diagram in s-domain.
Write the following equations using Figure 1.
Apply nodal analysis at node
.
Substitute
and
expressions.
Resolve into partial fractions.
Determine the value of
.
Determine the value of
.
Determine the value of
.
The partial fraction expansion of
is,
Apply inverse Laplace transform.
Therefore, the output voltage is
.

Question 3

Use mesh analysis to find v o ( t ) , t 0, in the network in Fig. P14.25.

Figure P14.25

Answer 3

Refer to Figure P14.16 in the textbook.
The Laplace transformed circuit is,
Figure 1
From Figure 1,
Apply Kirchhoff's Voltage Law to second mesh.
The output voltage is expressed as,
Resolve into partial fractions.
The value of
is,
The value of
is,
The partial fraction expansion is,
Apply inverse Laplace transform.
Therefore, the output voltage is
.

Question 4

Use Thévenin's theorem to find v o ( t ), t 0, in the network in Fig. P14.41.

Figure P14.41

Question 5

Find v o ( t ), for t 0, in the network in Fig. P14.57.

Figure P14.57

Question 6

The voltage response of the network to a unit step input is
Is the response overdamped

Question 7

Find the steady-state response v o ( t ), t 0, in the network in Fig. P14.89.

Figure P14.89

Question 8

Assuming that the initial inductor current is zero in the circuit in Fig. 14PFE-5, find the transfer function V o ( s )/ V s ( s ).

Figure 14PFE-5
a.

b.

c.

d.

Question 9

The transfer function for a network is

Determine the pole-zero plot of H ( s ), the type of damping exhibited by the network, and the unit step response of the network.

Figure E14.13

Question 10

Use superposition to solve Problem 14.11.
Problem 14.11
Use nodal analysis to find i o ( t ) in the network in Fig. P14.11.

Figure P14.11

Question 11

Use Thévenin's theorem to find v o ( t ), t 0, in the network shown in Fig. P14.42.

Figure P14.42

Question 12

Find v o ( t ) for t 0 in the network in Fig. P14.58.

Figure P14.58

Question 13

Find the output voltage, v o ( t ), t 0, in the network in Fig. P14.74a if the input is represented by the waveform shown in Fig. P14.74b.

Figure P14.74

Question 14

Find v o ( t ), t 0, in the network in Fig. P14.5 using node equations.

Figure P14.5

Question 15

Find v o ( t ), t 0, in the network in Fig. P14.13.

Figure P14.13

Question 16

Use superposition to find v o ( t ) , t 0, in the network shown in Fig. P14.27.

Figure P14.27

Question 17

Use Thévenin's theorem to find i o ( t ), t 0, in the network shown in Fig. P14.43.

Figure P14.43

Question 18

Find v o ( t ) for t 0 in the network in Fig. P14.59.

Figure P14.59

Question 19

The voltage response of a network to a unit step input is
Is the response underdamped

Question 20

Use Thévenin's theorem to determine v o ( t ) for t 0 in Fig. E14.6.

Figure E14.6

Question 21

Determine the steady-state voltage v oss ( t ) in the network in Fig. E14.14 for t 0 if the initial conditions in the network are zero.

Figure E14.14

Question 22

Use superposition to find v o ( t ), t 0, in the network in Fig. P14.28.

Figure P14.28

Question 23

Find i o ( t ), t 0, in the network shown in Fig. P14.44.

Figure P14.44

Question 24

Find v o ( t ) for t 0 in the network in Fig. P14.60.

Figure P14.60

Question 25

The transfer function of a network is given by the expression
Determine the damping ratio, the undamped natural frequency, and the type of response that will be exhibited by the network.

Question 26

Question 27

Use Laplace transforms and nodal analysis to find i 1 ( t ) for t 0 in the network shown in Fig. P14.6. Assume zero initial conditions.

Figure P14.6

Question 28

Use Laplace transforms and mesh analysis to find v o ( t ) for t 0 in the network shown in Fig. P14.14. Assume zero initial conditions.

Figure P14.14

Question 29

Solve Problem 14.14 using Laplace transforms and source transformation.
Problem 14.14
Use Laplace transforms and mesh analysis to find v o ( t ) for t 0 in the network shown in Fig. P14.14. Assume zero initial conditions.

Figure P14.14

Question 30

Find i o ( t ), t 0, in the network shown in Fig. P14.45.

Figure P14.45

Question 31

Determine the initial and final values of the current i ( t ) in the network shown in Fig. P14.61.

Figure P14.61

Question 32

The transfer function of the network is given by the expression

Determine the damping ratio, the undamped natural frequency, and the type of response that will be exhibited by the network.

Question 33

A single-loop, second-order circuit is described by the following differential equation:

Which is the correct form of the total (natural plus forced) response
a. v ( t ) = K 1 + K 2 e t
b. v ( t ) = K 1 cos t + K 2 sin t
c. v ( t ) = K 1 + K 2 te t
d. v ( t ) = K 1 + K 2 e t cos t + K 3 e t sin t

Question 34

Solve Learning Assessment E7.3 on page 261 using Laplace transforms.
Learning Assessment E7.3
In the circuit shown in Fig. E7.3, the switch opens at t = 0. Find i 1 ( t ) for t 0.

Figure E7.3

Question 35

Find the steady-state response v oss ( t ) in Fig. E14.15.

Figure E14.15

Question 36

Use source exchange to solve Problem 14.11.
Problem 14.11
Use nodal analysis to find i o ( t ) in the network in Fig. P14.11.

Figure P14.11

Question 37

Find i o ( t ), t 0, in the network in Fig. P14.46.

Figure P14.46

Question 38

Determine the initial and final values of the voltage v o ( t ) in the network in Fig. P14.62.

Figure P14.62

Question 39

The voltage response of a network to a unit step input is

Is the response critically damped

Question 40

Find the input impedance Z ( s ) in the network in Fig. P14.1.

Figure P14.1

Question 41

Use Laplace transforms to find v ( t ) for t 0 in the network shown in Fig. P14.7. Assume zero initial conditions.

Figure P14.7

Question 42

Solve Problem 14.14 using Laplace transforms and nodal analysis.
Problem 14.14
Use Laplace transforms and mesh analysis to find v o ( t ) for t 0 in the network shown in Fig. P14.14. Assume zero initial conditions.

Figure P14.14

Question 43

Use source transformation to find v o ( t ), t 0, in the circuit in Fig. P14.31.

Figure P14.31

Question 44

Find v o ( t ) for t 0 in the network in Fig. P14.47.

Figure P14.47

Question 45

Find v o ( t ) for t 0 in the network in Fig. P14.63.

Figure P14.63

Question 46

The transfer function of the network is given by the expression

Determine the damping ratio, the undamped natural frequency, and the type of response that will be exhibited by the network.

Question 47

Find v o ( t ) for t 0 in Fig. E14.2 using nodal analysis.

Figure E14.2

Question 48

Solve Learning Assessment E7.6 on page 268 using Laplace transforms.
Learning Assessment E7.6
Consider the network in Fig. E7.6. If the switch opens at t = 0, find the output voltage v o ( t ) for t 0.

Figure E7.6

Question 49

Use nodal analysis to find i o ( t ) in the network in Fig. P14.16.

Figure P14.16

Question 50

Solve Problem 14.14 using Laplace transforms and Thévenin's theorem.
Problem 14.14
Use Laplace transforms and mesh analysis to find v o ( t ) for t 0 in the network shown in Fig. P14.14. Assume zero initial conditions.

Figure P14.14

Question 51

Find v o ( t ) for t 0 in the network shown in Fig. P14.48.

Figure P14.48

Question 52

Find v o ( t ) for t 0 in the network in Fig. P14.64.

Figure P14.64

Question 53

Find the steady-state response i o ( t ) in the network shown in Fig. P14.80.

Figure P14.80

Question 54

If all initial conditions are zero in the network in Fig. 14PFE-2, find the transfer function V o ( s )/ V s ( s ).

Figure 14PFE-2
a.

b.

c.

d.

Question 55

For the network shown in Fig. P14.8, find v o ( t ), t 0.

Figure P14.8

Question 56

Use loop equations to find i 1 ( t ) in the network in Fig. P14.17.

Figure P14.17

Question 57

Use Thévenin's theorem to solve Problem 14.16.
Problem 14.16
Use nodal analysis to find i o ( t ) in the network in Fig. P14.16.

Figure P14.16

Question 58

Find i o ( t ) for t 0 in the network shown in Fig. P14.49.

Figure P14.49

Question 59

For the network shown in Fig. P14.65, determine the value of the output voltage as t .

Figure P14.65

Question 60

Find the steady-state response v o ( t ) in the network shown in Fig. P14.81.

Figure P14.81

Question 61

Find the input impedance Z ( s ) of the network in Fig. P14.2.

Figure P14.2

Question 62

Find i 0 ( t ) for t 0 in Fig. E14.9.

Figure E14.9

Question 63

For the network shown in Fig. P14.18, find v o ( t ), t 0, using mesh equations.

Figure P14.18

Question 64

Use Thévenin's theorem to solve Problem 14.17.
Problem 14.17
Use loop equations to find i 1 ( t ) in the network in Fig. P14.17.

Figure P14.17

Question 65

Find v o ( t ) for t 0 in the network shown in Fig. P14.50.

Figure P14.50

Question 66

Determine the initial and final values of the voltage v o ( t ) in the network in Fig. P14.66.

Figure P14.66

Question 67

Find the steady-state response v o ( t ) in the network shown in Fig. P14.82.

Figure P14.82

Question 68

Find v o ( t ) in the network in Fig. E14.3 using loop equations.

Figure E14.3

Question 69

For the network shown in Fig. P14.9, find i o ( t ), t 0.

Figure P14.9

Question 70

Use mesh equations to find v o ( t ), t 0, in the network in Fig. P14.19.

Figure P14.19

Question 71

Use Thévenin's theorem to find i o ( t ), t 0 , in Fig. P14.35.

Figure P14.35

Question 72

Find v o ( t ) for t 0 in the network shown in Fig. P14.51.

Figure P14.51

Question 73

Given the network in Fig. P14.67, determine the value of the output voltage as t .

Figure P14.67

Question 74

Determine the steady-state response v o ( t ) for the network in Fig. P14.83.

Figure P14.83

Question 75

The initial conditions in the circuit in Fig. 14PFE-3 are zero. Find the transfer function I o ( s )/ I s ( s ).

Figure 14PFE-3
a.

b.

c.

d.

Question 76

Find v o ( t ) for t 0 in Fig. E14.10.

Figure E14.10

Question 77

Use loop analysis to find v o ( t ) for t 0 in the network in Fig. P14.20.

Figure P14.20

Question 78

Use Thévenin's theorem to find v o ( t ), t 0, in the network in Fig. P14.36.

Figure P14.36

Question 79

Find v o ( t ), t 0, in the network shown in Fig. P14.52.

Figure P14.52

Question 80

Determine the output voltage v o ( t ) in the network in Fig. P14.68a if the input is given by the source in Fig. P14.68b.

Figure P14.68