Section 7.3: AC Resonance

Why This Matters for the Exam

Resonance occurs in AC circuits when inductive and capacitive reactances are equal ($X_L = X_C$). In aviation engineering, resonance is a powerful tool used for radio tuning and antenna coupling, but it is also a major safety hazard. Unintended resonance can cause extreme voltages or currents that damage aircraft electronics and wiring insulation. EASA exams test your ability to calculate the resonant frequency, identify the distinct features of series and parallel resonant circuits, and calculate the Q-factor and bandwidth.

Conditions for Resonance

Resonance occurs in a circuit containing both inductance ($L$) and capacitance ($C$) when the inductive reactance ($X_L$) equals the capacitive reactance ($X_C$). Because $X_L$ increases with frequency and $X_C$ decreases with frequency, there is exactly one frequency at which they are equal. This is the resonant frequency ($f_r$). To find the resonant frequency, set $X_L = X_C$: $2\pi f_r L = \frac{1}{2\pi f_r C}$ Solving for $f_r$ yields the resonant frequency formula: $f_r = \frac{1}{2\pi \sqrt{L C}}$ where $f_r$ is in Hertz, $L$ is in Henries, and $C$ is in Farads. This formula applies to both series and parallel resonant circuits.

Series Resonance

In a series RLC circuit at resonance:

1. Reactances Cancel: Because $X_L = X_C$ and they are 180 degrees out of phase, they cancel each other out ($X_L - X_C = 0$).
2. Impedance is Minimum: The total impedance of the circuit is at its absolute minimum and is equal to the resistance: $Z_{series} = R$
3. Current is Maximum: Since impedance is minimum, the current drawn from the supply is at its maximum: $I_{max} = \frac{V_{supply}}{R}$
4. Phase Angle is Zero: The circuit behaves as a pure resistor. The supply voltage and current are in-phase, resulting in a power factor of 1.0 (unity).
5. Voltage Magnification (Safety Hazard): While the total voltage across the reactances is zero, the voltage across the individual components ($V_L$ and $V_C$) can be much higher than the supply voltage: $V_L = V_C = I_{max} \times X_L = Q \times V_{supply}$ This voltage magnification can create voltages thousands of volts high in aircraft systems, causing insulation breakdown and catastrophic damage.

Parallel Resonance (Anti-Resonance)

In a parallel RLC circuit at resonance:

1. Branch Currents Cancel: The current through the inductor ($I_L$) and the current through the capacitor ($I_C$) are equal in magnitude but 180 degrees out of phase. They circulate within the parallel loop (tank circuit) and cancel each other out at the supply line.
2. Impedance is Maximum: Since the reactive branch currents cancel, the line current drawn from the supply is minimized. For an ideal parallel circuit with no resistance in the inductive branch, the impedance at resonance is theoretically infinite. For a practical tank circuit where the inductor has a series resistance $R_{series}$, the resonant impedance is: $Z_{parallel} = \frac{L}{C R_{series}} = Q \times X_L$
3. Current from Supply is Minimum: The supply line current is at its minimum, equal to $V_{supply} / Z_{parallel}$.
4. Current Magnification: High circulating currents flow between the inductor and capacitor within the parallel loop. The circulating current is $Q$ times larger than the line current drawn from the supply.

Quality Factor (Q) & Bandwidth

The Quality Factor ($Q$) is a measure of the selectivity and efficiency of a resonant circuit (stored energy vs. dissipated energy):

• For a series resonant circuit, $Q = \frac{X_L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$. A lower resistance gives a higher, sharper Q.
• For a practical parallel resonant circuit, $Q = \frac{X_L}{R_{series}}$.
• Bandwidth ($BW$) is the frequency range over which the response (current in series, voltage in parallel) is at least $70.7\%$ ($3\text{ dB}$ down) of its maximum value. It is defined as: $BW = f_2 - f_1 = \frac{f_r}{Q}$ where $f_1$ and $f_2$ are the half-power frequencies. High-Q circuits have a narrow bandwidth and high selectivity; low-Q circuits have a wide bandwidth and low selectivity.

Series vs. Parallel Resonance Comparison

Worked Exam Calculations

Example 1: A series circuit has $L = 0.5\text{ H}$, $C = 2\ \mu\text{F}$, and $R = 10\ \Omega$. Calculate the resonant frequency and the Quality Factor ($Q$).

1. Resonant Frequency: $f_r = \frac{1}{2\pi \sqrt{L C}} = \frac{1}{2\pi \sqrt{0.5 \times 2 \times 10^{-6}}} = \frac{1}{2\pi \sqrt{10^{-6}}} = \frac{1000}{2\pi} \approx 159.2\text{ Hz}$
2. Quality Factor: $Q = \frac{1}{R}\sqrt{\frac{L}{C}} = \frac{1}{10}\sqrt{\frac{0.5}{2 \times 10^{-6}}} = 0.1 \times \sqrt{250,000} = 0.1 \times 500 = 50$ If a 10V AC supply at 159.2 Hz is connected, the voltage across the capacitor will be $Q \times V_{supply} = 50 \times 10 = 500\text{ V}$, demonstrating the overvoltage hazard.

Example 2: Using the values from Example 1, calculate the bandwidth ($BW$) of the circuit. $BW = \frac{f_r}{Q} = \frac{159.2\text{ Hz}}{50} \approx 3.18\text{ Hz}$ This narrow bandwidth shows the circuit is highly selective.

Exam Traps & Tips

• Impedance Confusion: The most common EASA trap is confusing series and parallel behavior. Remember: Series is Minimum Impedance (S-M-I) and Parallel is Maximum Impedance (P-M-I). Under pressure, write this down immediately on your scratch paper.
• Inductor Resistance: Real inductors always have internal series resistance ($R_{series}$). In parallel resonance questions, this resistance limits the maximum impedance to a finite value ($L / (C R_{series})$) rather than infinity.