Interview questions & answers
Q1. What is resonance in an electrical circuit and at what frequency does it occur?
Resonance in an electrical circuit occurs when the inductive reactance equals the capacitive reactance (XL = XC), causing them to cancel each other so the circuit behaves as purely resistive. For a series LC circuit, resonance occurs at f0 = 1/(2π√(LC)); for a 1mH inductor and 100nF capacitor, f0 = 1/(2π√(10⁻³ × 10⁻⁷)) ≈ 15.9 kHz. At resonance, the impedance of a series RLC circuit is minimum (equal to R) and current is maximum, while in a parallel RLC circuit impedance is maximum and current from the source is minimum.
Follow-up: Why does a series RLC circuit draw maximum current at resonance?
Q2. What is the Q factor and what does it physically represent?
The Q factor (quality factor) is Q = ω0L/R = 1/(ω0CR) = (1/R)√(L/C), representing the ratio of energy stored in the reactive elements to the energy dissipated per radian — a higher Q means a sharper, more selective resonance. An antenna coil with L=10µH, R=5Ω resonated at 1 MHz has Q = 2π×10⁶×10⁻⁵/5 = 12.57. In a practical LC bandpass filter, Q determines selectivity: a Q of 50 at 10 MHz gives a 3 dB bandwidth of 10MHz/50 = 200 kHz, narrow enough to separate FM stations.
Follow-up: How does the Q factor of a coil change with frequency?
Q3. What is the bandwidth of a resonant circuit and how is it related to Q?
The bandwidth BW is the range of frequencies between the two half-power (-3 dB) points on the resonance curve, and it equals BW = f0/Q for a series or parallel RLC circuit. For a radio receiver IF filter with f0=455 kHz and Q=45, BW = 455/45 ≈ 10.1 kHz — wide enough for an AM audio channel (10 kHz) but rejecting adjacent channels. A higher Q gives a narrower bandwidth and greater selectivity, which is why crystal oscillator circuits (Q > 10,000) are used to select specific frequencies with sub-Hz precision.
Follow-up: What are the upper and lower half-power frequencies of a resonant circuit in terms of f0 and BW?
Q4. What happens to the voltages across L and C individually at series resonance?
At series resonance, the voltages across the inductor and capacitor are each equal to Q × Vs, which can be far larger than the source voltage — this is the voltage magnification effect. For a series RLC circuit with Vs=10V, Q=20, at resonance VL = VC = 20 × 10 = 200V across each reactive element, even though the net reactive voltage (VL - VC) is zero. This Q-times voltage magnification is both useful (in high-Q filter inductors) and dangerous (in power systems where switching transients can generate large resonant overvoltages across capacitor banks).
Follow-up: Why does the voltage across L and C cancel at resonance if each is Q times the source voltage?
Q5. What is the difference between series and parallel resonance?
In series resonance, impedance is minimum at f0 and current is maximum — the circuit acts as a current amplifier from the source perspective; in parallel resonance (anti-resonance), impedance is maximum at f0 and source current is minimum — the circulating tank current can be Q times the source current. A series LC is used in bandpass filters where you want maximum current (and hence maximum voltage across the load resistor) at the resonant frequency. A parallel LC tank circuit in an RF oscillator stores energy and sustains oscillation because the high impedance at resonance minimizes loading on the active gain element.
Follow-up: At what frequency does a parallel LC with lossy inductance actually resonate, compared to the ideal formula?
Q6. What is the impedance of an ideal series LC circuit at resonance?
An ideal series LC circuit (zero resistance) has zero impedance at resonance because XL and XC are equal in magnitude and opposite in sign, resulting in a short circuit at f0. Adding even a small series resistance R = 1Ω to a 10mH-10nF circuit resonating at 159 kHz gives Z = 1Ω at resonance — practical circuits always have some resistance limiting the minimum impedance. An ideal LC circuit at exact resonance with an AC source would allow infinite current to flow, which is physically impossible and why real resonant circuits always include parasitic resistance.
Follow-up: What is the impedance of an ideal parallel LC circuit at resonance?
Q7. What is the frequency response curve of a series RLC circuit?
The frequency response of a series RLC circuit shows impedance minimum at f0 = 1/(2π√LC), rising symmetrically (on a log frequency scale) for frequencies above and below resonance as either XC dominates below f0 or XL dominates above it, with the sharpness of the minimum determined by Q. The current response curve I(f) = Vs/|Z(f)| is the mirror image — maximum at f0 with a bell-shaped peak of width BW = f0/Q at the -3 dB points. On a Bode plot, the phase shifts from +90° (capacitive) at low frequencies through 0° at f0 to -90° (inductive) at high frequencies.
Follow-up: What is the phase angle of the series RLC impedance at the upper half-power frequency?
Q8. How is resonance used in power factor correction?
In a power system, inductive loads like induction motors draw lagging reactive current that increases line losses; connecting a capacitor bank in parallel resonates with the load inductance near the supply frequency, ideally canceling the reactive current so the power factor approaches unity. In a 415V, 50 Hz industrial plant with a 100 kVAR inductive load, a 100 kVAR capacitor bank brings the reactive component to zero and reduces the current drawn from the utility by the reactive component. Exact resonance at 50 Hz with the supply network is avoided because it would cause dangerously high circulating reactive currents — the capacitor is sized for partial compensation to a target power factor of 0.95 lagging.
Follow-up: Why is exact resonance at the supply frequency dangerous in a power factor correction installation?
Q9. What is the resonant frequency of a parallel RLC circuit with a lossy inductor?
For a parallel circuit where the inductor has series resistance RL, the resonant frequency is slightly lower than the ideal formula: f_res = (1/2π√(LC))×√(1 - RL²C/L), shifting below f0 = 1/(2π√LC) when RL is significant. For L=1mH with RL=5Ω and C=100nF, the correction factor √(1 - 25×10⁻⁷/10⁻³) = √(1 - 0.0025) ≈ 0.99875, giving less than 0.1% shift — negligible for most circuits. For high-loss inductors with RL comparable to √(L/C), the shift is significant and the circuit may not reach true resonance at all if RL > √(L/C).
Follow-up: At what value of inductor series resistance does the parallel RLC circuit fail to resonate?
Q10. What is dynamic impedance (dynamic resistance) of a parallel resonant circuit?
Dynamic impedance Rd = L/(CR) is the equivalent resistive impedance of a parallel tuned circuit at resonance, and it equals Q × (ωL) = Q/ωC — it can be many times larger than the physical resistor value. A parallel tank circuit with L=100µH, C=1nF, and R=2Ω at 500 kHz resonance has Rd = 10⁻⁴/(10⁻⁹×2) = 50kΩ. The dynamic impedance is what makes the parallel tank circuit useful as a high-impedance load for an RF amplifier: the collector load of a BJT RF stage is typically a tank circuit with dynamic impedance matching the transistor's output impedance for maximum gain.
Follow-up: How does the dynamic impedance of a tank circuit change as R (coil resistance) increases?
Q11. What is a notch filter and how does resonance achieve it?
A notch filter (band-stop filter) provides maximum attenuation at a specific frequency by placing a series resonant LC (nearly zero impedance at resonance) in shunt across the signal path or a parallel resonant LC (maximum impedance at resonance) in series with the signal path. A parallel LC with Q=30 placed in series with a 600Ω load at 50 Hz acts as a 50 Hz notch filter that eliminates power line hum from an audio signal. The depth of the notch is limited by inductor Q — a practical Q of 100 gives about 40 dB attenuation, while an ideal lossless LC would give infinite attenuation at exact resonance.
Follow-up: How does inductor Q affect the depth of attenuation in a notch filter?
Q12. How does loading (adding a parallel resistance) affect the Q of a parallel resonant circuit?
Adding a parallel resistance Rp across a parallel LC tank circuit reduces the effective Q from Q0 = Rd (dynamic impedance without load) divided by (Rd in parallel with Rp) × ω0C, effectively lowering Q because the external resistance provides an additional energy dissipation path. In an RF amplifier where the tank circuit collector load has Rd = 50kΩ and the next stage's input resistance is 5kΩ, the loaded Q drops by a factor of (50||5)/50 = 0.091 — reducing selectivity by 91%. This is why RF amplifiers use impedance transformation (tapped coils or capacitive dividers) to couple between a high-Q tank and a low-impedance load without degrading the tank Q.
Follow-up: What is the loaded Q and how is it calculated when an external load is connected across a tank circuit?
Q13. What is the half-power frequency and how is it derived?
The half-power frequencies f1 and f2 are where the power dissipated in the circuit drops to half of its maximum value at resonance, equivalent to the frequencies where the impedance magnitude is √2 times the minimum (for series RLC) or √2 times less than the maximum (for parallel RLC). For series RLC: |Z|² = R² + (XL - XC)² = 2R² at half-power, giving XL - XC = ±R, from which f2 - f1 = R/(2πL) = f0/Q. At f1 = 455 kHz - 5 kHz = 450 kHz and f2 = 460 kHz for a Q=45.5 IF transformer at 455 kHz, the voltage across the load is 0.707 of its peak value, corresponding to -3 dB.
Follow-up: What is the phase angle of the series RLC circuit's impedance at the half-power frequencies?
Q14. What is the relationship between bandwidth, Q, and component values?
BW = f0/Q = R/(2πL) = 1/(2πRC), showing that bandwidth depends on resistance and the reactive element values: increasing R widens the bandwidth (reduces Q), increasing L or C narrows it (for fixed R). In a crystal filter at 10.7 MHz IF used in FM receivers, the crystal's Q of 50,000 gives BW = 10.7MHz/50000 = 214 Hz — impossibly narrow for any LC circuit with practical Q of 100-500. For a conventional LC IF transformer at 455 kHz with L=10mH and required BW=10kHz: R = 2πL×BW = 2π×0.01×10000 ≈ 628Ω, which sets the required Q-damping resistance.
Follow-up: How does a quartz crystal achieve Q factors of 10,000 to 100,000 compared to typical LC circuits?
Q15. How is resonance used in a crystal oscillator?
A quartz crystal is an electromechanical resonator with a very high Q (10,000–100,000) that oscillates at its series or parallel resonant frequency, determined by its precise mechanical dimensions rather than LC component tolerances. In a Colpitts crystal oscillator in a microcontroller's clock circuit, the 32.768 kHz crystal (f = 2^15 Hz) maintains frequency to within ±20 ppm regardless of temperature or component aging — an LC oscillator would drift by hundreds of ppm. The crystal's piezoelectric effect converts electrical energy to mechanical vibration and back, with the mechanical resonance providing the extreme Q that no practical inductor-capacitor combination can approach.
Follow-up: What is the difference between the series and parallel resonant frequencies of a quartz crystal?
Common misconceptions
Misconception: At resonance the impedance of an RLC circuit is always zero.
Correct: At series resonance impedance equals R (minimum but nonzero due to resistance); only an ideal lossless series LC has zero impedance at resonance.
Misconception: Series and parallel resonance occur at the same frequency in a real circuit.
Correct: In a parallel circuit with a lossy inductor, the true resonant frequency is slightly lower than 1/(2π√LC) due to the series resistance of the coil.
Misconception: A higher Q always means a better circuit.
Correct: Higher Q gives narrower bandwidth and higher selectivity, which is desirable in filters and oscillators but undesirable in power factor correction where exact resonance at supply frequency causes dangerous circulating currents.
Misconception: At series resonance the voltages across L and C are zero because they cancel.
Correct: The net reactive voltage (VL - VC) is zero at series resonance, but individually VL and VC are each Q times the source voltage — they are large and equal, not zero.