Q: If the RMS voltages across the resistor, capacitor, and inductor in a series RLC circuit are $V_R = 40\ \text{V}$, $V_C = 80\ \text{V}$, and $V_L = 60\ \text{V}$, what is the RMS value of the total voltage?

$V_{\text{rms, total}} = \sqrt{40^2 + (60 - 80)^2}$

Q: Which of the following correctly expresses the phase angle $\phi$ between the total voltage and the current in a driven RLC circuit?

$\tan \phi = \frac{X_L - X_C}{R}$

Q: A circuit has $R = 5\ \Omega$, $L = 0.1\ \text{H}$, $C = 50\ \mu\text{F}$, and is driven at $\omega = 200\ \text{rad/s}$. What is the impedance $Z$?

$Z = \sqrt{5^2 + \left(200 \times 0.1 - \frac{1}{200 \times 50 \times 10^{-6}}\right)^2}$

Q: Which equation gives the steady-state current amplitude $I_{\text{max}}$ in a driven RLC circuit with a peak voltage $V_0$?

$I_{\text{max}} = \frac{V_0}{Z}$

Question 1

In a driven RLC circuit, which of the following best describes the phase relationship between the voltage across the capacitor and the current?

Accepted Answer

The voltage across the capacitor lags the current by $\frac{\pi}{2}$.

Question 2

Given a driven RLC circuit with $R = 10\ \Omega$, $L = 0.2\ 	ext{H}$, and $C = 100\ \mu	ext{F}$, what is the resonant angular frequency $\omega_0$?

Accepted Answer

$\omega_0 = \frac{1}{\sqrt{LC}}$

Question 3

In a driven RLC circuit, the impedance $Z$ is given by:

Accepted Answer

$Z = \sqrt{R^2 + (X_L - X_C)^2}$

Question 4

If the RMS voltages across the resistor, capacitor, and inductor in a series RLC circuit are $V_R = 40\ 	ext{V}$, $V_C = 80\ 	ext{V}$, and $V_L = 60\ 	ext{V}$, what is the RMS value of the total voltage?

Accepted Answer

$V_{	ext{rms, total}} = \sqrt{40^2 + (60 - 80)^2}$

Question 5

Which mnemonic helps remember the phase relationships in AC circuits for capacitors and inductors?

Accepted Answer

CIVIL

Question 6

What is the reactance of a capacitor in an AC circuit with angular frequency $\omega$ and capacitance $C$?

Accepted Answer

$X_C = \frac{1}{\omega C}$

Question 7

A radio receiver uses a resonant RLC circuit to select a frequency. If the inductance is $L = 1\ 	ext{mH}$ and the capacitance is $C = 10\ 	ext{nF}$, what is the resonant frequency $f_0$ in Hz?

Accepted Answer

$f_0 = \frac{1}{2\pi\sqrt{LC}}$

Question 8

In a driven RLC circuit, what happens to the current amplitude as the driving frequency approaches the resonant frequency?

Accepted Answer

The current amplitude reaches a maximum.

Question 9

Which of the following correctly expresses the phase angle $\phi$ between the total voltage and the current in a driven RLC circuit?

Accepted Answer

$	an \phi = \frac{X_L - X_C}{R}$

Question 10

A circuit has $R = 5\ \Omega$, $L = 0.1\ 	ext{H}$, $C = 50\ \mu	ext{F}$, and is driven at $\omega = 200\ 	ext{rad/s}$. What is the impedance $Z$?

Accepted Answer

$Z = \sqrt{5^2 + \left(200 	imes 0.1 - \frac{1}{200 	imes 50 	imes 10^{-6}}ight)^2}$

Question 11

In a driven RLC circuit, if the driving frequency is much lower than the resonant frequency, which component dominates the impedance?

Accepted Answer

The capacitor ($C$)

Question 12

Which equation gives the steady-state current amplitude $I_{	ext{max}}$ in a driven RLC circuit with a peak voltage $V_0$?

Accepted Answer

$I_{	ext{max}} = \frac{V_0}{Z}$

Question 13

A medical device uses an RLC circuit to filter signals. If the device requires a sharp resonance, which parameter should be minimized?

Accepted Answer

Resistance ($R$)

Question 14

Which of the following is true about the voltages across the resistor, capacitor, and inductor in a driven RLC circuit?

Accepted Answer

Their instantaneous values always add up to the total voltage.

Question 15

If the driving frequency is set exactly at the resonant frequency of a series RLC circuit, what is the impedance?

Accepted Answer

$Z = R$

Question 16

In a real-world scenario, why can't the RMS voltages across the resistor, capacitor, and inductor simply be added to find the total RMS voltage in a series RLC circuit?

Accepted Answer

Because the voltages are not all in phase with each other.

Question 17

A cell phone receiver uses a series RLC circuit to tune to a specific frequency. If the circuit is tuned to resonance, what happens to the current for a given applied voltage?

Accepted Answer

The current is maximized.

Question 18

Which formula correctly gives the reactance of an inductor in an AC circuit?

Accepted Answer

$X_L = \omega L$

Question 19

In a driven RLC circuit, if $X_L = X_C$, what is the phase angle $\phi$ between the total voltage and the current?

Accepted Answer

$\phi = 0$

Question 20

A driven RLC circuit is used in a chemical sensor to detect resonance. If the sensor's circuit has $R = 2\ \Omega$, $L = 0.5\ 	ext{H}$, and $C = 20\ \mu	ext{F}$, what is the resonant angular frequency $\omega_0$?

Accepted Answer

$\omega_0 = \frac{1}{\sqrt{0.5 	imes 20 	imes 10^{-6}}}$

Driven RLC Circuits, Phasors, Impedance, and Resonance in AC Circuits

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