Q: Given a parallel RLC circuit with $$R = 400\,\Omega$$, $$L = 50\,\text{mH}$$, and $$C = 50\,\text{nF}$$, what is the value of the Neper frequency $$\alpha$$?

$$\alpha = 25,000\,\text{rad/s}$$

Q: A parallel RLC circuit has $$R = 500\,\Omega$$, $$L = 1\,\text{mH}$$, and $$C = 1\,\mu\text{F}. $$Calculate the resonant frequency $$\$$omega_0$$.

$$\$$omega_0$$ = 10,000\,\text{rad/s}$$

Question 1

Which of the following is the correct second-order differential equation describing the natural response of a parallel RLC circuit in terms of the voltage $$v(t)$$ across the elements?

Accepted Answer

$$\frac{d^2$v}{dt^2$} + \frac{1}{RC} \frac{dv}{dt} + \frac{1}{LC} v = 0$$

Question 2

Given a parallel RLC circuit with $$R = 400\,\Omega$$, $$L = 50\,	ext{mH}$$, and $$C = 50\,	ext{nF}$$, what is the value of the Neper frequency $$\alpha$$?

Accepted Answer

$$\alpha = 25,000\,	ext{rad/s}$$

Question 3

For a parallel RLC circuit, if $$\$$omega_0$$^2 \u003c \$$alpha^2$$, what type of voltage response will the circuit exhibit?

Accepted Answer

Overdamped

Question 4

Which of the following expressions correctly gives the characteristic roots for the parallel RLC circuit's natural response?

Accepted Answer

$$s_{1,2} = -\frac{1}{2RC} \pm \sqrt{\left(\frac{1}{2RC}\$$right)^2$$ - \frac{1}{LC}}$$

Question 5

A parallel RLC circuit has $$R = 500\,\Omega$$, $$L = 1\,	ext{mH}$$, and $$C = 1\,\mu	ext{F}. $$Calculate the resonant frequency $$\$$omega_0$$.

Accepted Answer

$$\$$omega_0$$ = 10,000\,	ext{rad/s}$$

Question 6

In the context of RLC circuits, what is the physical significance of the Neper frequency $$\alpha$$?

Accepted Answer

It is the rate at which the oscillations decay in the circuit.

Question 7

For the underdamped case in a parallel RLC circuit, which of the following correctly expresses the voltage $$v(t)$$?

Accepted Answer

$$v(t) = e$$^{-\alpha t}$$ [B_1$ \cos(\omega_d t) + B_2$ \sin(\omega_d t)]$$

Question 8

A parallel RLC circuit is critically damped. Which of the following is true about its characteristic roots?

Accepted Answer

$$s_1$$ and $$s_2$$ are real and equal

Question 9

In a real-world biomedical device, a parallel RLC circuit is used to filter out high-frequency noise from a heart rate sensor. If the circuit is underdamped, what will be the effect on the sensor's voltage output when a sudden change in heart rate occurs?

Accepted Answer

The voltage will oscillate before settling to the new value.

Question 10

Given the step response of a parallel RLC circuit with a DC current source, what is the final value of the inductor current $$i_L(\infty)$$?

Accepted Answer

$$i_L(\infty) = I_f ($$the current source value)

Question 11

Which initial condition is used to solve for the constants $$A_1$$ and $$A_2 in $$the general solution $$v(t) = A_1$ e$$^{s_1 t}$$ + A_2$ e$$^{s_2 t}$$ for a parallel RLC circuit?

Accepted Answer

$$v(0)$$ and $$\frac{dv}{dt}\bigg|_{t=0}$$

Question 12

A parallel RLC circuit has $$R = 250\,\Omega$$, $$C = 2\,\mu	ext{F}$$, and $$L = 0.32\,	ext{H}. $$Calculate $$\alpha$$ and $$\$$omega_0$$.

Accepted Answer

$$\alpha = 1,000\,	ext{rad/s}$$, $$\$$omega_0$$ = 1,250\,	ext{rad/s}$$

Question 13

In the context of the step response of a parallel RLC circuit, what is the main difference between the natural response and the forced response?

Accepted Answer

Natural response is due to the initial conditions, forced response is due to the external source.

Question 14

A parallel RLC circuit is used in a radio receiver to select a particular frequency. Which parameter should be adjusted to change the resonant frequency $$\$$omega_0$$?

Accepted Answer

The capacitance $$C or $$the inductance $$L$$

Question 15

For a parallel RLC circuit, if $$\$$omega_0$$^2 = \$$alpha^2$$, what is the form of the voltage response?

Accepted Answer

$$v(t) = D_1$ e$$^{-\alpha t}$$ + D_2$ t e$$^{-\alpha t}$$

Question 16

In a parallel RLC circuit, which element primarily determines the initial voltage $$v(0)$$?

Accepted Answer

The capacitor

Question 17

A parallel RLC circuit is used in a chemical sensor to detect rapid changes in concentration. If the circuit is overdamped, what is the expected behavior of the sensor's voltage output after a sudden change?

Accepted Answer

The voltage will slowly approach the new value without oscillation.

Question 18

Which of the following best describes the physical meaning of the resonant frequency $$\$$omega_0$$ in a parallel RLC circuit?

Accepted Answer

It is the frequency at which the circuit naturally oscillates in the absence of damping.

Chapter 8: RLC Circuits – Second-Order Differential Equations and Their Applications

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