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Chapter 7 Quiz — Second-Order RLC Circuits

Test your understanding of RLC circuit behavior, the characteristic equation, damping conditions, resonant frequency, and the quality factor. Each question includes a detailed explanation.

Question 1

A series RLC circuit has \(R = 100 \text{ }\Omega\), \(L = 10 \text{ mH}\), and \(C = 1 \text{ µF}\). What is the undamped natural frequency \(\omega_0\)?

  1. \(1{,}000 \text{ rad/s}\)
  2. \(10{,}000 \text{ rad/s}\)
  3. \(100{,}000 \text{ rad/s}\)
  4. \(31{,}623 \text{ rad/s}\)
Show Answer

Correct Answer: B

\(\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{0.01 \times 10^{-6}}} = \frac{1}{\sqrt{10^{-8}}} = \frac{1}{10^{-4}} = 10{,}000 \text{ rad/s}\). The natural frequency depends only on L and C, not on R. R controls damping (how quickly oscillations die out) but not the fundamental oscillation frequency. In Hz: \(f_0 = 10{,}000/(2\pi) \approx 1{,}592 \text{ Hz}\).

Concept Tested: Natural Frequency \(\omega_0 = 1/\sqrt{LC}\)

Question 2

For the same series RLC circuit (\(R = 100 \text{ }\Omega\), \(L = 10 \text{ mH}\), \(C = 1 \text{ µF}\)), calculate the damping coefficient \(\alpha\) and damping ratio \(\zeta\). What type of response will the circuit exhibit?

  1. \(\alpha = 5{,}000 \text{ rad/s}\), \(\zeta = 0.5\) — underdamped (oscillatory)
  2. \(\alpha = 10{,}000 \text{ rad/s}\), \(\zeta = 1.0\) — critically damped
  3. \(\alpha = 5{,}000 \text{ rad/s}\), \(\zeta = 0.5\) — overdamped
  4. \(\alpha = 20{,}000 \text{ rad/s}\), \(\zeta = 2.0\) — overdamped
Show Answer

Correct Answer: A

Series RLC: \(\alpha = R/(2L) = 100/(2 \times 0.01) = 5{,}000 \text{ rad/s}\). Damping ratio: \(\zeta = \alpha/\omega_0 = 5{,}000/10{,}000 = 0.5\). Since \(\zeta < 1\), the circuit is underdamped — it will oscillate around its final value with decaying amplitude. The damped frequency is \(\omega_d = \omega_0\sqrt{1-\zeta^2} = 10{,}000\sqrt{1-0.25} = 8{,}660 \text{ rad/s}\).

Concept Tested: Damping Coefficient \(\alpha = R/2L\) and Damping Ratio \(\zeta = \alpha/\omega_0\)

Question 3

For a series RLC circuit to be critically damped, what condition must hold between the damping coefficient \(\alpha\) and the natural frequency \(\omega_0\)?

  1. \(\alpha > \omega_0\)
  2. \(\alpha < \omega_0\)
  3. \(\alpha = \omega_0\)
  4. \(\alpha = 2\omega_0\)
Show Answer

Correct Answer: C

Critical damping occurs when \(\alpha = \omega_0\) (equivalently, \(\zeta = 1\)). This is the boundary between oscillatory (\(\alpha < \omega_0\), underdamped) and non-oscillatory (\(\alpha > \omega_0\), overdamped) behavior. At critical damping, the characteristic equation has a repeated real root \(s = -\omega_0\), giving the fastest return to equilibrium without any overshoot — the "Goldilocks" condition for applications like analog meters, camera stabilization, and automotive suspension design.

Concept Tested: Critically Damped Condition \(\alpha = \omega_0\), \(\zeta = 1\)

Question 4

The resonant frequency of a series RLC circuit is \(f_0 = 1{,}592 \text{ Hz}\) (approximately). The inductance is \(L = 10 \text{ mH}\). What is the capacitance?

  1. 100 nF
  2. 10 µF
  3. 1 µF
  4. 100 µF
Show Answer

Correct Answer: C

Using \(f_0 = 1/(2\pi\sqrt{LC})\): \(\omega_0 = 2\pi \times 1592 \approx 10{,}000 \text{ rad/s}\). Solving for C: \(C = 1/(\omega_0^2 L) = 1/((10{,}000)^2 \times 0.01) = 1/(10^8 \times 0.01) = 1/10^6 = 1 \times 10^{-6} \text{ F} = 1 \text{ µF}\). This confirms the LC values from earlier questions — 10 mH and 1 µF resonate at about 1,592 Hz.

Concept Tested: Resonant Frequency \(f_0 = 1/(2\pi\sqrt{LC})\)

Question 5

A series RLC circuit has \(Q = 50\) and resonant frequency \(f_0 = 500 \text{ kHz}\). What is the 3 dB bandwidth?

  1. 25 MHz
  2. 10 kHz
  3. 1 kHz
  4. 25 kHz
Show Answer

Correct Answer: B

Bandwidth and Q are related by \(BW = f_0/Q = 500{,}000/50 = 10{,}000 \text{ Hz} = 10 \text{ kHz}\). A Q of 50 is fairly selective — only frequencies within ±5 kHz of the 500 kHz center pass with less than 3 dB attenuation. Higher Q means narrower bandwidth and better selectivity. This is why radio IF stages and crystal filters have high Q values.

Concept Tested: Q Factor and Bandwidth \(BW = f_0/Q\)

Question 6

The damping ratio of an RLC circuit is \(\zeta = 0.1\). What is the quality factor Q, and how would you classify the response?

  1. \(Q = 5\), overdamped
  2. \(Q = 0.05\), critically damped
  3. \(Q = 5\), underdamped
  4. \(Q = 10\), underdamped
Show Answer

Correct Answer: C

The relationship between Q and \(\zeta\) is: \(Q = 1/(2\zeta) = 1/(2 \times 0.1) = 5\). Since \(\zeta = 0.1 < 1\), the circuit is underdamped — it will oscillate before settling. With \(\zeta = 0.1\), the percent overshoot is \(e^{-\pi \times 0.1/\sqrt{1-0.01}} \times 100 \approx 73\%\) — very oscillatory. High Q (low damping) means sharp resonance, narrow bandwidth, and many oscillation cycles before settling.

Concept Tested: Q Factor and Damping Ratio \(Q = 1/(2\zeta)\)

Question 7

At resonance in a series RLC circuit, what happens to the impedance and what is the phase angle between source voltage and current?

  1. Impedance is maximum and phase angle is 90°
  2. Impedance equals R (minimum) and phase angle is 0°
  3. Impedance is zero and phase angle is 180°
  4. Impedance equals \(X_L + X_C\) and phase angle is 45°
Show Answer

Correct Answer: B

At resonance, \(X_L = X_C\), so they cancel: the total reactance is zero. The circuit impedance reduces to just \(R\) (minimum), and current is maximum. Since voltage and current are in phase (phase angle = 0°), the power factor is 1. Note that individual voltages across L and C can exceed the source voltage at resonance — they cancel each other but are large internally.

Concept Tested: Series RLC Resonance Conditions

Question 8

The characteristic equation of a second-order circuit is \(s^2 + 2\alpha s + \omega_0^2 = 0\). If \(\alpha = 3{,}000 \text{ rad/s}\) and \(\omega_0 = 5{,}000 \text{ rad/s}\), what are the roots \(s_{1,2}\)?

  1. Two distinct real roots: \(-1{,}000\) and \(-5{,}000\)
  2. A repeated real root at \(-3{,}000\)
  3. Complex conjugate roots: \(-3{,}000 \pm j4{,}000\)
  4. Complex conjugate roots: \(-5{,}000 \pm j3{,}000\)
Show Answer

Correct Answer: C

Since \(\alpha < \omega_0\) (3,000 < 5,000), the circuit is underdamped. The roots are complex: \(s_{1,2} = -\alpha \pm j\omega_d\) where \(\omega_d = \sqrt{\omega_0^2 - \alpha^2} = \sqrt{25{,}000{,}000 - 9{,}000{,}000} = \sqrt{16{,}000{,}000} = 4{,}000 \text{ rad/s}\). So \(s_{1,2} = -3{,}000 \pm j4{,}000\). Complex conjugate roots mean oscillatory (underdamped) behavior with oscillation frequency \(\omega_d = 4{,}000 \text{ rad/s}\) and decay rate \(\alpha = 3{,}000\).

Concept Tested: Characteristic Equation Roots \(s_{1,2} = -\alpha \pm j\omega_d\)

Answers Summary

Question Answer Concept Tested
1 B Natural Frequency \(\omega_0 = 1/\sqrt{LC}\)
2 A Damping Coefficient and Ratio
3 C Critically Damped Condition
4 C Resonant Frequency
5 B Q Factor and Bandwidth
6 C Q Factor and Damping Ratio
7 B Series RLC Resonance
8 C Characteristic Equation Roots