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Principles and Applications of Electrical Engineering

Giorgio Rizzoni

Chapter 5

Transient Analysis - all with Video Answers

Educators

Chapter Questions

Problem 1

Just before the switch is opened at $t=0$, the current through the inductor is 1.70 mA in the direction shown in Figure P5.1. Did steady-state conditions exist just before the switch was opened?

$$
\begin{array}{rlrl}
L & =0.9 \mathrm{mH} \\
R_1 & =6 \mathrm{k} \Omega \\
R_3 & =3 \mathrm{k} \Omega & & V_5=12 \mathrm{~V} \\
R_2=6 \mathrm{k} \Omega
\end{array}
$$

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Problem 2

At $t<0$, the circuit shown in Figure P5.2 is at steady state. The switch is changed as shown at $t=0$.

$$
\begin{aligned}
V_{S 1} & =35 \mathrm{~V} & & V_{52}=130 \mathrm{~V} \\
C & =11 \mu \mathrm{~F} & & R_1=17 \mathrm{k} \Omega \\
R_2 & =7 \mathrm{k} \Omega & & R_3=23 \mathrm{k} \Omega
\end{aligned}
$$

Determine at $t=0^{+}$the initial current through $R_3$ just after the switch is changed.

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Problem 3

Determine the current through the capacitor just before and just after the switch is closed in Figure P5.3. Assume steady-state conditions for $t \propto 0$.

$$
\begin{array}{ll}
V_1=12 \mathrm{~V} & C=0.5 \mu \mathrm{~F} \\
R_1=0.68 \mathrm{k} \Omega & R_2=1.8 \mathrm{k} \Omega
\end{array}
$$

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Problem 4

Determine the current through the capacitor just before and just after the switch is closed in Figure P5.3. Assume steady-state conditions for $t<0$.

$$
\begin{array}{ll}
V_1=12 \mathrm{~V} & C=150 \mu \mathrm{~F} \\
R_1=400 \mathrm{~m} \Omega & R_2=2.2 \mathrm{k} \Omega
\end{array}
$$

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Problem 5

Just before the switch is opened at $t=0 \mathrm{in}$ Figure P5.1, the current through the inductor is 1.70 mA in the direction shown. Determine the voltage across $R_3$ just after the switch is opened.

$$
\begin{array}{ll}
V_5=12 \mathrm{~V} & L=0.9 \mathrm{mH} \\
R_1=6 \mathrm{k} \Omega & R_2=6 \mathrm{k} \Omega \\
R_3=3 \mathrm{k} \Omega &
\end{array}
$$

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Problem 6

Determine the voltage across the inductor just before and just after the switch is changed in Figure P5.6. Assume steady-state conditions exist for $t<0$.

$$
\begin{array}{ll}
V_5=12 \mathrm{~V} & R_4=0.7 \Omega \\
R_1=22 \mathrm{k} \Omega & L=100 \mathrm{mH}
\end{array}
$$

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Problem 7

Steady-state conditions exist in the circuit shown in Figure P5.7 at $t<0$. The switch is closed at $t=0$.

$$
\begin{aligned}
V_1 & =12 \mathrm{~V} & & R_1=0.68 \mathrm{k} \Omega \\
R_2 & =2.2 \mathrm{k} \Omega & & R_3=1.8 \mathrm{k} \Omega \\
C & =0.47 \mu \mathrm{~F} & &
\end{aligned}
$$

Determine the current through the capacitor at $t=0^{+}$, just after the switch is closed.

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Problem 8

At $t>0$, the circuit shown in Figure P5.2 is at steady state. The switch is changed as shown at $t=0$.

$$
\begin{aligned}
V_{51} & =35 \mathrm{~V} & & V_{S 2}=130 \mathrm{~V} \\
C & =11 \mu \mathrm{~F} & & R_{\mathrm{I}}=17 \mathrm{k} \Omega \\
R_2 & =7 \mathrm{k} \Omega & & R_3=23 \mathrm{k} \Omega
\end{aligned}
$$

Determine the time constant of the circuit for $t>0$.

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Problem 9

At $t<0$, the circuit shown in Figure P5.9 is at steady state. The switch is changed as shown at $t=0$.

$$
\begin{aligned}
V_{51}=13 \mathrm{~V} & V_{52}=13 \mathrm{~V} \\
L=170 \mathrm{mH} & R_1=2.7 \Omega \\
R_2=4.3 \mathrm{k} \Omega & R_3-29 \mathrm{k} \Omega
\end{aligned}
$$

Determine the time constant of the circuit for $t>0$.

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Problem 10

Steady-state conditions exist in the circuit shown in Figure P5.7 for $t<0$. The switch is closed at $t=0$.

$$
\begin{array}{ll}
V_1=12 \mathrm{~V} & C=0.47 \mu \mathrm{~F} \\
R_1=680 \Omega & R_2=2.2 \mathrm{k} \Omega \\
R_3=1.8 \mathrm{k} \Omega &
\end{array}
$$

Determine the time constant of the circuit for $t>0$.

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Problem 11

Just before the switch is opened at $t=0$ in Figure P5.1, the current through the inductor is 1.70 mA in the direction shown.

$$
\begin{array}{ll}
V_s=12 \mathrm{~V} & L=0.9 \mathrm{mH} \\
R_1=6 \mathrm{k} \Omega & R_2=6 \mathrm{k} \Omega \\
R_5=3 \mathrm{k} \Omega &
\end{array}
$$

Determine the time constant of the circuit for $t>0$.

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Problem 12

Determine $v_C(t)$ for $t>0$. The volage across the capacitor in Figure P5. 12 just before the switch is changed is given below.

$$
\begin{array}{lll}
v_c\left(0^{-}\right)=-7 \mathrm{~V} & I_s=17 \mathrm{~mA} & C=0.55 \mu \mathrm{~F} \\
R_1=7 \mathrm{k} \Omega & R_2=3.3 \mathrm{k} \Omega &
\end{array}
$$

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Problem 13

Determine $i_{N_1}(t)$ for $t>0$ in Figure P5.9.

$$
\begin{array}{ll}
V_{51}-23 \mathrm{~V} & V_{52}=20 \mathrm{~V} \\
L=23 \mathrm{mH} & R_1=0.7 \Omega \\
R_2=13 \Omega & R_3=330 \mathrm{k} \Omega
\end{array}
$$

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Problem 14

Assume DC steady-state conditions exist in the circuit shown in Figure P5. 14 for $t<0$. The switch is changed at $t=0$ as shown.

$$
\begin{array}{ll}
V_{S 1}=17 \mathrm{~V} & V_{S I}=11 \mathrm{~V} \\
R_1=14 \mathrm{k} \Omega & R_2=13 \mathrm{k} \Omega \\
R_3=14 \mathrm{k} \Omega & C=70 \mathrm{nF}
\end{array}
$$

Determine:
a. $\mathrm{v}(t)$ for $t>0$.
b. The time required, after the switch is operated, for $V(t)$ to change by 98 percent of its total change in voltage.

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00:39

Problem 15

The circuit of Figure P5. 15 is a simple model of an automotive ignition system. The switch models the "points" that switch electrical power to the cylinder when the fuel-air mixture is compressed. $R$ is the resistance between the electrodes (i.e., the "gap") of the spark plug.

$$
\begin{array}{ll}
V_G-12 \mathrm{~V} & R_G-0.37 \Omega \\
R=1.7 \mathrm{k} \Omega &
\end{array}
$$
Determine the value of $L$ and $R_1$ so that the voltage across the spark plug gap just after the switch is changed is 23 kV and so that this voltage will change exponentially with a time constant $\mathrm{r}=13 \mathrm{~ms}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator

Problem 16

The inductor $L$ in the circuit shown in Figure P5. 16 is the coil of a relay. When the current through the coil is equal to or greater than +2 mA the relay functions. Assume steady-state conditions at $t<0$. If

$$
\begin{aligned}
& V_s=12 \mathrm{~V} \\
& L=10.9 \mathrm{mH} \quad R_1=3.1 \mathrm{k} \Omega
\end{aligned}
$$

determine $R_2$ so that the relay functions at $t=2.3 \mathrm{~s}$.

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Problem 17

Determine the current through the capacitor just before and just after the switch is closed in Figure P5.17. Assume steady-state conditions for $t<0$.

$$
\begin{array}{ll}
V_1=12 \mathrm{~V} & C=150 \mu \mathrm{~F} \\
R_1=400 \mathrm{~m} \Omega & R_2=2.2 \mathrm{k} \Omega
\end{array}
$$

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Problem 18

Determine the voltage across the inductor just before and just after the switch is changed in Figure P5.18. Assume steady-state conditions exist for $t<0$.

$$
\begin{array}{ll}
V_s=12 \mathrm{~V} & R_5=0.24 \Omega \\
R_1=33 \mathrm{k} \Omega & L=100 \mathrm{mH}
\end{array}
$$

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Problem 19

Steady-state conditions exist in the circuit shown in Figure P5.7 for $t<0$. The switch is closed at $t=0$.

$$
\begin{array}{ll}
V_1=12 \mathrm{~V} & C=150 \mu \mathrm{~F} \\
R_1=4 \mathrm{M} \Omega & R_2=80 \mathrm{M} \Omega \\
R_3=6 \mathrm{M} \Omega &
\end{array}
$$

Determine the time constant of the circuit for $t>0$.

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Problem 20

Just before the switch is opened at $t=0$ in Figure P5.1, the current through the inductor is 1.70 mA in the direction shown.

$$
\begin{array}{ll}
V_5=12 \mathrm{~V} & L=100 \mathrm{mH} \\
R_1=400 \Omega & R_2=400 \Omega \\
R_3=600 \Omega &
\end{array}
$$

Determine the time constant of the circuit for $t>0$.

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Problem 21

For the circuit shown in Figure P5.21, assume that switch $S_1$ is always open and that switch $S_2$ closes at $t=0$.
a. Find the capacitor voltage, $v_c(t)$, at $t=0^{+}$.
b. Find the time constant, $\tau$, for $t \geq 0$.
c. Find an expression for $\mathrm{Ec}_{\mathrm{C}}(t)$ and sketch the function.
d. Find $v_c(t)$ for each of the following values of $t$ : $0, \tau, 2 \pi, 5 \tau, 10 \tau$.

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Problem 22

For the circuit shown in Figure P5.21, assume that switch $S_1$ is always open; switch $S_2$ has been closed for a long time, and opens at $f=0$.
a. Find the capacitor voltage, $v_C(t)$, at $t=0^{+}$.
b. Find the time constant, $\tau$, for $t \geq 0$.
c. Find an expression for $v_c(t)$ and sketch the function.
d. Find $v_c(t)$ for each of the following values of $t$ : $0, \tau, 2 \pi, 5 \tau, 10 \tau$.

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Problem 23

For the circuit of Figure P5.21, assume that switch $S_2$ is always open, and that switch $S_1$ has been closed for a long time and opens at $t=0$. At $t=t_1=3 \tau$, switch $S_1$ closes again.
a. Find the capacitor voltage, $v_c(t)$, at $t=0^{+}$.
b. Find an expression for $a_c(t)$ for $t>0$ and sketch the function.

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Problem 24

Assume both switches $S_1$ and $S_2$ in Figure PS. 21 close at $t=0$.
a. Find the capacitor voltage, $v_c(t)$, at $t=0^{+}$.
b. Find the time constant, $\tau$, for $t \geq 0$.
c. Find an expression for $v_c(t)$ and sketch the function.
d. Find $v_c(t)$ for each of the following values of $t$ : $0, \tau, 2 \tau, 5 \tau, 10 \tau$.

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Problem 25

Assume both switches $S_1$ and $S_2$ in Figure P5.21 have been closed for a long time and switch $S_2$ opens at $t=0^{+}$.
a. Find the capacitor voltage, $v_c(t)$, at $t=0^{+}$.
b. Find an expression for $v_c(t)$ and sketch the function.
c. Find $u_c(t)$ for each of the following values of $t$ : $0, \tau, 2 \tau, 5 \tau, 10 \tau$.

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Problem 26

For the circuit of Figure P5.26, determine the time constants $\tau$ and $\tau^{\prime}$ before and after the switch opens, respectively. $R_5=4 \mathrm{k} \Omega, R_1=2 \mathrm{k} \Omega$, $R_2=R_5=6 \mathrm{k} \Omega$, and $C=1 \mu \mathrm{~F}$.

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Problem 27

For the circuit of Figure P5.27, find the initial current through the inductor, the final current through the inductor, and the expression for $i_L(t)$ for $t \geq 0$.

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07:26

Problem 28

Al $t=0$, the switch in the circuit of Figure P5.28 opens. At $t=10 \mathrm{~s}$, the switch closes.
a. What is the time constant for $0<t<10 \mathrm{~s}$ ?
b. What is the time constant for $t>10 \mathrm{~s}$ ?

Sheh Lit Chang
Sheh Lit Chang
University of Washington

Problem 29

The circuit of Figure P5.29 includes a model of a voltage-controlled switch. When the voltage across the capacitor reaches 7 V , the switch is closed. When the capacitor volage reaches 0.5 V , the switch opens. Assume that the capacitor voltage is initially $V_C=0.5$ V and that the switch has just opened.
a. Sketch the capacitor voltage versus time, showing explicitly the periods when the switch is open and when the switch is closed.
b. What is the period of the voltage waveform across the $10-\Omega$ resistor?

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13:34

Problem 30

At $t=0$, the switch in the circuit of Figure P5.30 closes. Assume that $i_L(0)=0 \mathrm{~A}$. For $t \geq 0$,
a. Find $i_L(t)$.
b. Find $v_{L_1}(\mathrm{f})$.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 31

In the circuit shown in Figure P5.31:

$$
\begin{array}{ll}
V_{S 1}=15 \mathrm{~V} & V_{S 2}=9 \mathrm{~V} \\
R_{S 1}=130 \Omega & R_{52}=290 \Omega \\
R_1=1.1 \mathrm{k} \Omega & R_2=700 \Omega \\
L=17 \mathrm{mH} & C=0.35 \mu \mathrm{~F}
\end{array}
$$

Assume that DC steady-state conditions exist for $t<0$. Determine the voltage across the capacitor and the current through the inductor and $R_{52}$ as $t$ approaches infinity.

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Problem 32

In the circuit shown in Figure P5.31:

$$
\begin{array}{ll}
V_{51}=12 \mathrm{~V} & V_{52}=12 \mathrm{~V} \\
R_{51}=50 \Omega & R_{52}=50 \Omega \\
R_1=2.2 \mathrm{k} \Omega & R_2=600 \Omega \\
L=7.8 \mathrm{mH} & C=68 \mu \mathrm{~F}
\end{array}
$$

Assume that DC steady-state conditions exist at $t<0$. Determine the voltage across the capacitor and the current through the inductor as $t$ approaches infinity. Remember to specify the polarity of the voltage and the direction of the current that you assume for your solution.

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Problem 33

If the switch in the circuit shown in Figure P5.33 is closed at $t=0$ and:

$$
\begin{array}{ll}
V_s=170 \mathrm{~V} & R_5=7 \mathrm{k} \Omega \\
R_1=2.3 \mathrm{k} \Omega & R_2=7 \mathrm{k} \Omega \\
L=30 \mathrm{mH} & C=130 \mu \mathrm{~F}
\end{array}
$$

determine, after the circuit has returned to a steady state, the current through the inductor and the voltage across the capacitor and $R_1$.

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Problem 34

If the switch in the circuit shown in Figure P5.34 is closed at $t=0$ and:

$$
\begin{array}{ll}
V_S=12 \mathrm{~V} & C=130 \mu \mathrm{~F} \\
R_1=2.3 \mathrm{k} \Omega & R_2=7 \mathrm{k} \Omega \\
L=30 \mathrm{mH} &
\end{array}
$$
Determine the current through the inductor and the voltage across the capacitor and across $R_1$ after the circuit has returned to a steady state.

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Problem 35

If the switch in the circuit shown in Figure P5.35 is closed at $t=0$ and:

$$
\begin{array}{ll}
V_5=12 \mathrm{~V} & C=0.5 \mu \mathrm{~F} \\
R_1=31 \mathrm{k} \Omega & R_2=22 \mathrm{k} \Omega \\
L=0.9 \mathrm{mH} &
\end{array}
$$

Determine the current through the inductor and the voltage across the capacitor after the circuit has returned to a steady state.

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Problem 36

At $t<0$, the circuit shown in Figure P5.36 is at steady state and the voltage across the capacitor is +7 V . The switch is changed as shown at $t=0$ and:

$$
\begin{array}{ll}
V_S=12 \mathrm{~V} & C=3300 \mu \mathrm{~F} \\
R_1=9.1 \mathrm{k} \Omega & R_2=4.3 \mathrm{k} \Omega \\
R_1=4.3 \mathrm{k} \Omega & L=16 \mathrm{mH}
\end{array}
$$

Determine the initial voltage across $R_2$ just after the switch is changed.

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Problem 37

In the circuit shown in Figure P5.37, assume that DC steady-state conditions exist for $t<0$. Determine at $t=0^{+}$, just after the switch is opened, the current through and volage across the inductor and the capacitor and the current through $R_{52}$ -

$$
\begin{array}{ll}
V_{51}=15 \mathrm{~V} & V_{52}=9 \mathrm{~V} \\
R_{51}=130 \Omega & R_{52}=290 \Omega \\
R_1=1.1 \mathrm{k} \Omega & R_2=700 \Omega \\
L=17 \mathrm{mH} & C=0.35 \mu \mathrm{~F}
\end{array}
$$

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Problem 38

In the circuit shown in Figure P5.37:

$$
\begin{array}{ll}
V_{51}=12 \mathrm{~V} & V_{\Omega 2}=12 \mathrm{~V} \\
R_{S 1}=50 \Omega & R_{\Omega 2}=50 \Omega \\
R_1=2.2 \mathrm{k} \Omega & R_2=600 \Omega \\
L=7.8 \mathrm{mH} & C=68 \mu \mathrm{~F}
\end{array}
$$

Assume that DC steady-state conditions exist for $t<0$. Determine the voltage across the capacitor and the current through the inductor as $t$ approaches infinity. Remember to specify the polarity of the voltage and the direction of the current that you assume for your solution.

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07:28

Problem 39

Assume the switch in the circuit of Figure P5.39 has been closed for a very long time. It is suddenly opened at $t=0$, and then reclosed at $t=5 \mathrm{~s}$.
Determine an expression for the inductor current for $t \geq 0$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 40

Assume the circuit of Figure P5.40 initially stores no energy. The switch is closed at $t=0$, and then reopened at $t=50 \mu \mathrm{~s}$. Determine an expression for the capacitor voltage for $t \geq 0$.

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Problem 41

Assume the circuit of Figure P5.41 initially stores no energy. Switch $S_1$ is open, and $S_2$ is closed. Switch $S_1$ is closed at $t=0$, and switch $S_2$ is opened at $t=5 \mathrm{~s}$. Determine an expression for the capocitor voltage for $t \geq 0$.

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Problem 42

Assume that the circuit shown in Figure P5.42 is underdamped and that the circuit initially has no energy stored. It has been observed that, after the switch is closed at $t=0$, the capacitor voltage reaches an initial peak value of 70 V when $t=5 \pi / 3 \mu s$, a second peak value of 53.2 V when $t=5 \pi \mu \mathrm{~s}$, and eventually approaches a steady-state value of 50 V . If $C=1.6 \mathrm{nF}$, what are the values of $R$ and $L$ ?

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05:12

Problem 43

Given the information provided in Problem 5.42, explain how to modify the circuit so that the first two peaks occur at $5 \pi \mu \mathrm{~s}$ and $15 \pi \mu \mathrm{~s}$. Assume that $C$ cannot be changed.

Vishal Gupta
Vishal Gupta
Numerade Educator

Problem 44

Find $i$ for $t>0$ in the circuit of Figure P5.44 if $i(0)=4 \mathrm{~A}$ and $v(0)=6 \mathrm{~V}$.

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Problem 45

Find $v$ for $t>0$ in the circuit of Figure P5.45 if the circuit is in steady state at $t=0^{-}$.

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02:35

Problem 46

Find $i$ for $t>0$ in the circuit of Figure P5.46 if the circuit is in steady state at $f=0^{-}$.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:35

Problem 47

Find $i$ for $t>0$ in the circuit of Figure P5.47 if the circuit is in steady state at $t=0^{-}$.

Kajal Gautam
Kajal Gautam
Numerade Educator

Problem 48

Find $v$ for $t>0$ in the circuit of Figure P5.48 if the circuit is in steady state at $t=0^{-}$.

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03:29

Problem 49

The circuit of Figure P5.49 is in steady state at $t=0^{-}$. Find $v$ for $t>0$ if $L$ is (a) 2.4 H , (b) 3 H , and (c) 4 H

Keshav Singh
Keshav Singh
Numerade Educator

Problem 50

Find $v$ for $t>0$ in the circuit of Figure P5.50 if the circuit is in steady state at $t=0^{-}$.

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