10.1 R-L-C Resonant Circuits

10.1 R-L-C Resonant Circuits

Quick Answer: f0 = 1/(2π√(LC)). Series resonance: XL = XC, |Z| minimum (= R), current maximum. Parallel (tank) resonance: |Z| maximum, line current minimum. Q measures sharpness; BW = f0/Q between the half-power points. High-Q tanks and IF filters are narrow; low-Q networks are wide.

Topic 3-D (practical circuits) starts where pure impedance formulas left off. Element 3 already taught you that series RLC magnitude is √[R² + (XL − XC)²] and that XL and XC cancel at one frequency. This section turns that cancelation into resonant-circuit behavior—the language of RF tanks, traps, IF transformers, and tuned amplifiers that GROL maintainers align and repair every day.

Why resonance matters on a radio chassis

Almost every transmitter final, receiver RF/IF stage, crystal filter skirt, and antenna matching network exploits resonance: a frequency where inductive and capacitive reactances balance so the network stops looking “reactive” and starts looking like a pure resistance—either very small (series) or very large (parallel). Select the wrong L or C, load the tank too heavily, or mistune the slug, and you lose power, gain, or adjacent-channel rejection.

Resonant frequency — one formula, two topologies

For ideal inductance L and capacitance C (series or parallel):

[ f_0 = \frac{1}{2\pi \sqrt{LC}} ]

Pool-style check: If L doubles, f0 falls by √2 ≈ 0.707. If C is cut to one-fourth, f0 doubles. Powers-of-ten errors (µH vs mH, pF vs nF) create the classic wrong answers of 1.59 MHz vs 5.03 MHz vs 15.9 MHz on numeric stems.

Worked example — IF-style tank

L = 250 µH, C = 100 pF (rough 1 MHz IF ballpark teaching set):

[ \sqrt{LC} = \sqrt{(250 \times 10^{-6})(100 \times 10^{-12})} = \sqrt{2.5 \times 10^{-14}} = 1.581 \times 10^{-7} ]

[ f_0 = \frac{1}{2\pi \times 1.581 \times 10^{-7}} \approx 1.01,\mathrm{MHz} ]

Trim C or L slightly (slug-tuned coil, ceramic trimmer) and the stage tracks the assigned IF channel.

Worked example — HF transmitter tank scale

L = 5 µH, C = 200 pF:

[ f_0 = \frac{1}{2\pi \sqrt{5 \times 10^{-6} \times 200 \times 10^{-12}}} = \frac{1}{2\pi \sqrt{10^{-15}}} \approx 5.03,\mathrm{MHz} ]

Same √(LC) math as earlier impedance chapters—Element 3 loves recycling the 10 µH / 100 pF → ~5 MHz family of numbers with small L/C swaps.

Series RLC resonance — minimum Z, maximum current

In a series R–L–C loop driven by a voltage source:

[ |Z| = \sqrt{R^2 + (X_L - X_C)^2} ]

At resonance XL = XC, so:

[ |Z|_{\mathrm{min}} = R ]

Exam hook you must not miss: impedance of a series RLC circuit at resonance is equal to the resistance only—not zero (unless R = 0), not XL, not XC.

Current and “Q magnification”

If source voltage is V and R is the only remaining impedance, I = V/R is maximum. That large current through XL (or XC) produces a reactive voltage I × XL that can be Q times the source voltage across L or C alone. Series circuits can therefore develop high voltages on L and C even when the generator is modest—useful for traps and dangerous for insulation if Q is high and R is tiny.

Series applications in radio practice

• Series traps across a line: near-zero Z at the trap frequency shorts that frequency to ground (or across a path), notching an interferer.
• Series arms in L-match networks: cancel antenna reactance so the transmitter sees resistance.
• Some crystal and ceramic filter models include series-resonant paths that pass the desired band.

Parallel RLC (tank) resonance — maximum Z, minimum line current

A parallel combination of L and C (with loss resistance modeled in series with L or as a parallel R) is the classic tank circuit. At resonance the circulating current between L and C is large, but the net current drawn from the source is small—the tank looks like a high impedance.

Why RF amplifiers love tanks: a transistor wants a high load impedance at the operating frequency for voltage gain, and a low impedance off-frequency to reject harmonics and out-of-band noise. A parallel tank delivers exactly that peak.

Idealized parallel vs real tanks

Ideal lossless parallel LC would present infinite Z at f0. Real coils have series resistance; solid-state stages load the tank; coupling loops steal energy. Finite loaded Q keeps peak Z finite—still “high,” still the right mental model for Element 3: parallel resonance → high impedance.

| Topology | Z at f0 | Line current at f0 | Typical RF job | |---|---|---| | Series RLC | Min (= R) | Max | Trap, series match | | Parallel RLC tank | Max | Min | Oscillator/amp load, IF transformer primary |

Quality factor Q — sharpness of resonance

Q is the dimensionless figure of merit for how “peaky” the resonance is.

Series RLC definitions (exam-useful forms)

[ Q = \frac{X_L}{R} = \frac{X_C}{R} = \frac{2\pi f_0 L}{R} = \frac{1}{2\pi f_0 C R} ]

(at resonance, XL = XC, so either reactance over R works).

Parallel tank (loaded) intuition

For a parallel tank with parallel resistance R_p representing losses and loading:

[ Q \approx \frac{R_p}{X_L} = \frac{R_p}{X_C} ]

Heavy loading (small R_p) destroys Q. That is why an IF can of a marine receiver is carefully coupled: too much load flattens the response and adjacent-channel junk comes through; too little coupling starves the next stage of signal.

Bandwidth — BW = f0 / Q

The half-power bandwidth (also called the −3 dB bandwidth) of a single tuned circuit is:

[ BW = \frac{f_0}{Q} ]

where BW = f_H − f_L, and f_H and f_L are the upper and lower frequencies at which power in the load is half the resonant value (voltage ≈ 0.707 of peak for a constant-resistance load model).

Worked example — IF filter bandwidth

f0 = 455 kHz, Q = 50:

[ BW = \frac{455,\mathrm{kHz}}{50} = 9.1,\mathrm{kHz} ]

That order of magnitude suits AM voice-style IF widths. Raise Q to 100 and BW halves to 4.55 kHz—tighter, more selective, less audio “room” if overdone.

Worked example — VHF channel-style thinking

f0 = 156.8 MHz (marine VHF Ch-16 ballpark), desired BW ≈ 25 kHz for a simple single-circuit teaching model:

[ Q = \frac{f_0}{BW} = \frac{156.8 \times 10^6}{25 \times 10^3} \approx 6272 ]

A single LC tank rarely holds that Q under real loading—modern radios use multiple poles, crystals, ceramics, or DSP filters. Element 3 still wants the BW = f0/Q relationship cold even when real hardware uses multi-stage filters.

Half-power points (mental picture)

        response
           ^
           |        *
           |       * *
    0.707 -|------*---*------  half-power
           |     *     *
           |    *       *
           +---+----+----+--> f
              fL   f0   fH
                 <─ BW ─>

Energy outside fL–fH is rejected more and more; that is selectivity.

Voltage and current vs frequency — what you measure on the bench

When aligning an IF transformer with a sweep generator and scope, you are literally watching Q and BW: a tall skinny blob is high Q; a short wide blob is overcoupled or low Q.

Tank circuits in RF amplifiers and oscillators

Parallel tanks appear as:

1. Collector/drain loads in Class C or linear RF stages — peak impedance at the carrier; harmonics see lower Z and are attenuated.
2. Oscillator resonators (with feedback) — frequency set by LC (or crystal’s series/parallel modes).
3. IF transformers — double-tuned transformers set bandpass shape; coupling coefficient interacts with Q to produce single-peak or double-hump responses.

Design/maintain hooks for GROL techs:

Series vs parallel — one-page comparison

Exam-day checklist for R-L-C resonance (3-D)

1. f0 = 1/(2π√(LC)) — same for ideal series and parallel LC.
2. Series: Z min = R; current max.
3. Parallel tank: Z max; line current min.
4. Q high → sharp, selective; BW = f0/Q.
5. Half-power points define BW; voltage ~0.707 of peak there in the standard model.
6. Loading a tank lowers Q, widens BW, lowers peak Z.
7. RF amp tanks are parallel; do not answer “series minimum Z” for a collector tank question.

Master these extremes and the BW equation, and Element 3 resonant-circuit items stop being abstract AC math and become the same story you use when you dip a grid-dip meter, peak a plate current dip (legacy), or sweep an IF. Next section moves from passive tanks into op amps, PLLs, VCOs, and mixers—the active practical circuits that generate, lock, and translate those RF frequencies.