Q: In Problem 8.48, the voltage $$v_o(t)$$ for an underdamped RLC circuit is given by $$v_o(t) = \frac{V_g}{RC\omega_d} e$$^{-\alpha t}$$ \sin(\omega_d t). $$What is the time $$t at $$which $$|v_o| is $$maximum?

$$t = \frac{1}{\omega_d} \tan$$^{-1}$$\left(\frac{\omega_d}{\alpha}\right)$$

Question 1

In an RLC circuit, the characteristic equation for the natural response is given by $$s^2 + 2$$\alpha s + \$$omega_0^2 = 0$. If $$\alpha \u003c \$$omega_0$$, what type of response does the circuit exhibit?

Accepted Answer

Underdamped

Question 2

A series RLC circuit has $$R = 2\,k\Omega$$, $$L = 2\,H$$, and $$C = 0.5\,\mu F. $$Calculate the undamped natural frequency $$\$$omega_0$$.

Accepted Answer

$$\$$omega_0$$ = 1000\,rad/s$$

Question 3

For the circuit in Problem 8.25, if the initial current in the inductor is $$-30\,mA$$ and the initial voltage on the capacitor is $$60\,V$$, what is the expression for $$i_L(t)$$ for $$t \geq 0$$ when $$R = 8000\,\Omega$$?

Accepted Answer

$$i_L(t) = 15 - 40e$$^{-80t}$$ - 5e$$^{-600t}$$\,mA$$

Question 4

In Problem 8.41, the resistor is adjusted for critical damping. Which of the following is the correct value of $$R$$ for critical damping if $$L = 250\,mH$$ and $$C = 10\,\mu F$$?

Accepted Answer

$$R = 2500\,\Omega$$

Question 5

A switch in an RLC circuit has been open for a long time and closes at $$t=0. $$The initial capacitor voltage is $$0\,V. $$Which of the following best describes the behavior of the inductor current immediately after the switch closes?

Accepted Answer

The inductor current remains at its initial value.

Question 6

Given a series RLC circuit with $$R = 10\,k\Omega$$, $$L = 10\,H$$, and $$C = 1\,\mu F$$, what is the value of $$\alpha ($$the damping factor)?

Accepted Answer

$$\alpha = 1000\,rad/s$$

Question 7

In Problem 8.48, the voltage $$v_o(t)$$ for an underdamped RLC circuit is given by $$v_o(t) = \frac{V_g}{RC\omega_d} e$$^{-\alpha t}$$ \sin(\omega_d t). $$What is the time $$t at $$which $$|v_o| is $$maximum?

Accepted Answer

$$t = \frac{1}{\omega_d} 	an$$^{-1}$$\left(\frac{\omega_d}{\alpha}ight)$$

Question 8

For the circuit in Problem 8.44, the solution for $$v_o(t) is$$ $$v_o(t) = 70e$$^{-5t}$$ \cos 5t - 30e$$^{-5t}$$ \sin 5t\,V$$ for $$t \geq 0. $$What is the initial value $$v_o(0)$$?

Accepted Answer

$$v_o(0) = 70\,V$$

Question 9

In Problem 8.51, the initial value of $$\frac{dv_o}{dt} at$$ $$t=0^+ is $$calculated. If $$C = 12\,\mu F$$ and the current through the capacitor at $$t=0^+ is$$ $$60\,A/s$$, what is $$\frac{dv_o}{dt}(0^+)$$?

Accepted Answer

$$\frac{dv_o}{dt}(0^+) = 855,000\,V/s$$

Question 10

A series RLC circuit is said to be critically damped when $$\alpha = \$$omega_0$$. Which of the following is the correct relationship for R$ in terms of L$ and C$ for critical damping?

Accepted Answer

$$R = 2\sqrt{L/C}$$

Question 11

In Problem 8.28, the switch has been open for a long time before closing at $$t=0. If $$the initial capacitor voltage is $$0\,V$$, what is the value of $$v_o(0)$$ immediately after the switch closes?

Accepted Answer

$$v_o(0) = 0\,V$$

Question 12

For a parallel RLC circuit, which of the following statements is true regarding the steady-state behavior as $$t 	o \infty$$ after a DC source is applied?

Accepted Answer

The inductor acts as a short circuit and the capacitor acts as an open circuit.

Question 13

In Problem 8.16, the equivalent resistance $$R_{eq}$$ for the parallel resistors $$150\,k\Omega$$, $$60\,k\Omega$$, and $$100\,k\Omega is $$calculated. What is the correct value of $$R_{eq}$$?

Accepted Answer

$$R_{eq} = 50\,k\Omega$$

Question 14

A switch in a series RLC circuit is opened at $$t=0$$ after being closed for a long time. If the circuit is underdamped, which of the following best describes the voltage across the capacitor?

Accepted Answer

It oscillates with exponentially decaying amplitude.

Question 15

In Problem 8.42, the initial value of $$v_o is$$ $$300\,V. If$$ $$C = 5\,\mu F$$ and the initial current through the capacitor is $$0$$, what is the initial value of $$\frac{dv_o}{dt}$$?

Accepted Answer

$$\frac{dv_o}{dt}(0^+) = -60\,V/s$$

Question 16

For the circuit in Problem 8.27, with $$R = 10,000\,\Omega$$, $$L = 10\,H$$, and $$C = 1\,\mu F$$, what is the form of the solution for $$i_L(t)$$ for $$t \geq 0$$?

Accepted Answer

$$i_L(t) = A_1$ e$$^{-\alpha t}$$ + A_2$ t e$$^{-\alpha t}$$

Question 17

In Problem 8.51, the solution for $$v_o(t)$$ for $$t \geq 0 is$$ $$v_o(t) = -20e$$^{-25,000t}$$ + 15\cos 15,000t + 41e$$^{-20,000t}$$ \sin 15,000t. $$What is the steady-state value of $$v_o(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$v_o(\infty) = 15\,V$$

Second-Order Differential Equations in RLC Circuits: Calculus Applications

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