E5A: Resonance and Q

E5A covers the behavior of series and parallel RLC circuits at resonance, the mathematical definition and practical effects of the quality factor Q, and the relationship between Q and the half-power bandwidth of a resonant circuit. These concepts underpin filter design, impedance matching networks, and oscillator circuits throughout amateur radio engineering.

The Extra exam draws one question from E5A. Questions require applying the resonant frequency formula, understanding impedance and current behavior at resonance in both series and parallel configurations, calculating Q and bandwidth, and recognizing the consequences of increasing Q.

Resonant Frequency

Every RLC circuit has a resonant frequency at which the inductive reactance XL equals the capacitive reactance XC. At this frequency the two reactances cancel each other, leaving only the resistance in the circuit. The resonant frequency is determined solely by L and C:

Applying this formula to specific values:

• L = 50 μH, C = 40 pF: f₀ = 1 / (2π√(50×10⁻⁶ × 40×10⁻¹²)) = 3.56 MHz
• L = 50 μH, C = 10 pF: f₀ = 1 / (2π√(50×10⁻⁶ × 10×10⁻¹²)) = 7.12 MHz

Note that R does not appear in the formula. Changing resistance shifts the Q and bandwidth but not the resonant frequency.

Series RLC at Resonance

In a series RLC circuit, all three components share the same current. At resonance XL = XC, so the two reactances cancel in series. The net impedance is:

Although the net impedance is just R, the voltages across the individual reactive components can be much larger than the applied voltage. The inductor and capacitor each develop a voltage equal to Q times the applied voltage. This voltage magnification is the defining characteristic of a high-Q series resonant circuit. Increasing Q in a series resonant circuit causes these internal voltages to increase — a critical design consideration in transmitter tank circuits and RF filters.

Parallel RLC at Resonance

A parallel RLC circuit behaves in the opposite way from a series circuit at resonance. Because L and C are in parallel, they exchange energy back and forth between the magnetic and electric fields without drawing significant current from the source.

In a parallel tank circuit, the circulating current flowing between L and C is at a maximum at resonance — often many times the current drawn from the source. The input current from the source is at a minimum because the high parallel impedance limits it. The tank circuit acts as a flywheel for RF energy.

Quality Factor Q

The quality factor Q quantifies how lossy or lossless a resonant circuit is. A higher Q means lower loss and sharper frequency selectivity. Q is defined differently for series and parallel circuits:

The practical consequences of Q value:

• Higher Q → narrower bandwidth — the circuit is more selective
• Higher Q → higher internal voltages in series resonant circuits (voltage magnification = Q)
• Higher Q in an impedance-matching circuit → decreased matching bandwidth — there is a fundamental tradeoff between Q and the frequency range over which a match is maintained

Half-Power Bandwidth

The half-power bandwidth (also called the 3 dB bandwidth) is the frequency range over which the circuit's response is within 3 dB of its peak. Outside this range the power delivered to the load drops below half the peak value.

Applying the formula to specific circuits:

• f₀ = 7.1 MHz, Q = 150: BW = 7,100,000 / 150 = 47.3 kHz
• f₀ = 3.7 MHz, Q = 118: BW = 3,700,000 / 118 = 31.4 kHz

These relationships explain why filter designers choose Q carefully: a crystal filter with Q of 50,000 can resolve a 100 Hz bandwidth at 5 MHz, while a typical LC tank with Q of 100 produces a bandwidth of 50 kHz at the same frequency.

E5A Practice Questions