LC Circuits and Resonance

Section Overview

You now know two components with opposite personalities: a capacitor whose reactance falls as frequency rises, and an inductor whose reactance rises. Put them together and something remarkable happens. Because one opposition is climbing while the other is dropping, there is exactly one frequency where the two are equal — and there, they cancel. That frequency is called resonance, and it is the single most important idea a capacitor and inductor produce together. This section shows how to find the resonant frequency with f₀ = 1 ÷ (2π√(LC)), how a series and a parallel (tank circuit) arrangement behave oppositely at resonance, the picture of energy sloshing between the two components, and why resonance is the heart of tuning, oscillators, and frequency-selective filters.

Why This Matters

Resonance is how a circuit picks out one frequency from all the others. It's how a radio selects a single station out of the dozens filling the air, how an oscillator generates a clean tone at a chosen frequency, and how a filter passes or rejects a specific band. Every tuned circuit — in radios, transmitters, wireless chargers, metal detectors, and countless signal-processing stages — is an LC resonant circuit at heart. For a repair technician, resonance also explains a distinct failure signature: if the inductor or capacitor in a tuned circuit drifts in value, the resonant frequency shifts, and the circuit tunes to the wrong place — a receiver goes deaf, an oscillator lands off-frequency, a filter passes the wrong band. Knowing that a resonant frequency depends on L and C tells you exactly which components to suspect.

Required Prerequisites

• What Is a Capacitor? — a capacitor stores energy in an electric field; at resonance that energy trades back and forth with the inductor.
• What Is an Inductor? — an inductor stores energy in a magnetic field; it is the capacitor's partner in the energy exchange.
• Capacitor Behavior in AC Circuits — capacitive reactance Xc, which falls with frequency.
• Inductor Behavior in Circuits — inductive reactance XL, which rises with frequency; resonance is where the two reactance curves cross.

Recommended Consumables

No consumables required. The exercises are pen-and-paper resonance calculations; an optional demonstration uses a signal source and an LC pair, but nothing is used up.

Recommended Practice Hardware

• Optional: an inductor and a capacitor to form an LC pair, a signal/function generator with a frequency dial, and a multimeter or oscilloscope to see the response peak or dip near resonance
• The demonstration — sweeping the frequency and watching the LC circuit respond most (or least) strongly at one frequency — makes resonance vivid, but the section stands on the calculations

Everything here is low-voltage signal-level work. No mains, no high-energy sources; the Section 3.1 stored-charge and Section 3.4 inductive-kick cautions still apply to any large component, though the small parts used here are low-energy.

Real-World Applications

The clearest example lives in every radio. An antenna picks up hundreds of stations at once; a tunable LC circuit resonates at just one frequency and selects that station out of the crowd — turning the dial changes L or C to move the resonant frequency. Oscillators use an LC circuit's natural resonance to generate a steady frequency, the timekeeper of transmitters and clocks. Band-pass and band-stop filters use resonance to pass or block a chosen band. Wireless charging and RFID couple energy between two resonant coils. And a switch-mode supply's snubber or an audio crossover uses resonance deliberately. Wherever a circuit needs to care about one frequency, resonance is usually how it does it.

Common Challenges

• Expecting resonance to need a special part. It doesn't — resonance emerges from an ordinary capacitor and an ordinary inductor together, purely because their reactances move in opposite directions with frequency. Nothing is added; the effect is in the pairing.
• Mixing up series and parallel behavior. A series LC circuit is a minimum impedance at resonance (it passes the resonant frequency best); a parallel LC "tank" is a maximum impedance at resonance (it blocks or selects that frequency). They are opposites, and confusing them inverts every filter.
• Forgetting the square root. The resonant frequency depends on 1 ÷ √(LC), not 1 ÷ (LC). Quadrupling the capacitance only halves the frequency, because of the square root — a common surprise.

Safety Notes

Risk Level: Low. This section is analysis and optional low-voltage signal-level demonstration. No mains, no high-energy sources.

Professional Tips Before Starting

• Think of resonance as where the two reactance curves cross. Xc slopes down with frequency, XL slopes up; they meet at exactly one point, and that crossing frequency is f₀. Picturing the crossing makes the whole idea intuitive.
• Remember which arrangement does which: series LC passes its resonant frequency (minimum impedance), parallel LC blocks/selects it (maximum impedance). One mnemonic: in series the signal goes through the low impedance; in a parallel tank the signal is diverted around the high impedance.
• When a tuned circuit is mistuned — a receiver off-station, an oscillator off-frequency — suspect a drifted L or C, because f₀ depends only on those two. Stable, low-drift components exist precisely to hold a resonant frequency in place.

Where the Two Reactances Meet

The Resonant Frequency

Recall the two reactances. Capacitive reactance Xc = 1 ÷ (2πfC) falls as frequency rises; inductive reactance XL = 2πfL rises. Plot both against frequency and one curve slopes down while the other slopes up, so they cross at exactly one frequency. At that crossing the two oppositions are equal — XL = Xc — and because they act in opposite senses (an inductor's voltage leads, a capacitor's lags), they cancel. That special frequency is the resonant frequency, f₀, and setting XL equal to Xc and solving gives a clean result:

f₀ = 1 ÷ (2π√(LC))

with L in henries, C in farads, and f₀ in hertz. Notice it depends only on L and C — not on the drive, not on resistance — and it involves the square root of their product, so changes have a softened effect.

A worked example. Take L = 10 mH (0.01 H) and C = 1 µF (0.000001 F):

f₀ = 1 ÷ (2π√(0.01 × 0.000001)) = 1 ÷ (2π√(0.00000001)) = 1 ÷ (2π × 0.0001) ≈ 1592 Hz (about 1.6 kHz)

Now watch the square root at work: quadruple the capacitance to 4 µF and the frequency doesn't drop to a quarter — it drops only to a half, about 796 Hz, because √4 = 2. And a radio-frequency example shows the range: L = 100 µH with C = 100 pF gives f₀ = 1 ÷ (2π√(0.0001 × 0.0000000001)) ≈ 1.59 MHz — right in the AM broadcast band. Small L and C values push resonance up to radio frequencies; large ones bring it down to audio.

Series vs. Parallel: Opposite Behaviors

How an LC circuit acts at resonance depends on whether the two components are in series or in parallel, and the two are exact opposites.

• Series LC — capacitor and inductor in a line. At resonance their reactances are equal and cancel, so the net reactance is zero: the circuit's impedance drops to a minimum (just the small unavoidable resistance). A series-resonant circuit therefore lets the resonant frequency through most easily — it's an acceptor, the basis of a band-pass filter that picks out f₀.
• Parallel LC — capacitor and inductor side by side, the classic tank circuit. At resonance the two branches' currents are equal and opposite and largely cancel in the external circuit, so the tank's impedance rises to a maximum. A parallel-resonant tank therefore blocks or selects the resonant frequency — it's the basis of a band-stop filter, or, seen the other way, the frequency-selecting element in an oscillator or tuner.

Series passes its frequency, parallel blocks its frequency: opposite arrangements, opposite behavior, same resonant frequency.

The Electrical Pendulum

Why does energy behave so dramatically at resonance? Because an LC circuit is an electrical pendulum. A charged capacitor pushes current through the inductor, building the inductor's magnetic field as the capacitor empties; then the collapsing magnetic field pushes current back, recharging the capacitor the other way; and the cycle repeats. Energy sloshes back and forth between the capacitor's electric field and the inductor's magnetic field, just as a pendulum trades between height (potential energy) and motion (kinetic energy). Left alone, this exchange happens naturally at f₀ — the circuit "rings" at its resonant frequency. Drive it at that frequency and you reinforce the swing on every push, which is why the response builds up so strongly there and nowhere else.

Q: How Sharp the Resonance Is

Not every resonance is equally selective. The Q factor describes how sharp the resonance is — how narrow the band of frequencies around f₀ that the circuit strongly responds to. A high-Q circuit resonates in a very narrow band: it is highly selective, ringing hard at f₀ and ignoring nearby frequencies (a radio that separates closely-spaced stations cleanly). A low-Q circuit responds over a broader band: less selective, but sometimes exactly what's wanted when a range of frequencies should pass. Resistance in the circuit lowers Q by dissipating the sloshing energy, which is why high-Q resonant circuits use low-loss inductors and capacitors. You don't need the formula here — the idea is enough: higher Q means a sharper, more selective resonance.

Common Mistakes

• Dropping the square root in f₀. The frequency is 1 ÷ (2π√(LC)); the LC is under a square root. Forgetting it turns a factor-of-four change into a factor-of-two error.
• Reversing series and parallel. Series LC is minimum impedance (passes f₀); parallel tank is maximum impedance (blocks/selects f₀). Getting this backwards inverts the filter's job.
• Thinking resonance depends on the drive or resistance. f₀ is set by L and C alone. Resistance affects the sharpness (Q), not the resonant frequency.
• Assuming a wider resonance is always worse. High Q is sharper and more selective, but low Q — a broader response — is the right choice when a band of frequencies must pass. Sharpness is a design choice, not a universal goal.

Troubleshooting Guidance

Resonance failures have a characteristic signature: a tuned circuit that works but is tuned to the wrong frequency. A receiver that has gone deaf or drifted off-station, an oscillator running off-frequency, or a filter passing the wrong band all point to a shifted resonant frequency — and since f₀ = 1 ÷ (2π√(LC)) depends only on L and C, one of those two has drifted. A capacitor that has lost capacitance or an inductor with shorted turns (lower L) both raise the resonant frequency; a leaky or increased-value part lowers it. Because the relationship runs through a square root, even a modest component drift can move f₀ enough to matter in a sharp, high-Q circuit. When a resonant circuit is mistuned rather than dead, suspect the L and the C before anything else, and remember that this is exactly why such circuits specify stable, low-drift components — holding f₀ in place is the whole point.

Verification & Testing Methods

Check your understanding before moving on:

• [ ] Calculate a resonant frequency with f₀ = 1 ÷ (2π√(LC)) and state what happens to it if the capacitance is quadrupled.
• [ ] State whether a series LC and a parallel LC (tank) present minimum or maximum impedance at resonance, and what each does to a signal at f₀.
• [ ] Explain resonance as energy sloshing between the capacitor's electric field and the inductor's magnetic field.
• [ ] Explain what a higher Q means for a resonant circuit, and why a drifted L or C mistunes it.

Then try the practice exercises below — pen-and-paper calculations, with an optional frequency-sweep demonstration.

Practice Exercises

1. Find the resonant frequency (10 minutes, pen and paper). Calculate f₀ = 1 ÷ (2π√(LC)) for: (a) L = 1 mH, C = 1 µF; (b) L = 250 µH, C = 100 pF. Convert to henries and farads first, and give each answer in Hz or MHz. (Hint: (a) is in the audio range, a few kHz; (b) around 1 MHz.)
2. The square-root effect (5 minutes, pen and paper). An LC circuit resonates at 2000 Hz. Without full recalculation, state its new resonant frequency if you (a) quadruple the capacitance, and (b) quarter the inductance. Explain each using the square root in the formula.
3. Series or parallel? (5 minutes, reasoning). You need a circuit that lets a 455 kHz signal pass while impeding others, and separately one that blocks 455 kHz while passing the rest. State which LC arrangement (series or parallel/tank) you'd use for each, and why, in terms of impedance at resonance.
4. Diagnose a mistuned circuit (10 minutes, reasoning). A receiver's tuned circuit has drifted and now resonates too high in frequency. Using f₀ = 1 ÷ (2π√(LC)), name two component changes that could cause the frequency to rise, and explain why stable components are specified for tuned circuits.

These core ideas — the resonant frequency, series vs. parallel behavior, the energy-exchange picture, and Q/selectivity — are tested in the Chapter Quiz at the end of this chapter, where a score of 80% is required to continue.

Key Takeaways

• Resonance occurs where a capacitor's falling reactance and an inductor's rising reactance become equal (XL = Xc) and cancel, at the resonant frequency f₀ = 1 ÷ (2π√(LC)).
• f₀ depends only on L and C, through a square root: quadrupling either one only halves the frequency.
• A series LC circuit is minimum impedance at resonance and passes its frequency best; a parallel LC tank circuit is maximum impedance and blocks or selects its frequency — opposite behaviors, same f₀.
• Resonance is an electrical pendulum: energy sloshes between the capacitor's electric field and the inductor's magnetic field, so the circuit rings at f₀.
• The Q factor describes how sharp and selective the resonance is; higher Q means a narrower band, and resistance lowers Q by dissipating the sloshing energy.
• Tuning, oscillators, and frequency-selective filters all use resonance; a drifted L or C shifts f₀ and mistunes the circuit, which is why such circuits use stable components.

Skills Learned

• You can now calculate a resonant frequency with f₀ = 1 ÷ (2π√(LC)) and predict how it shifts when L or C changes.
• You can now state whether a series or parallel LC circuit is minimum or maximum impedance at resonance and what each does to a signal.
• You can now explain resonance as energy exchange between the two fields and describe qualitatively what a higher Q means.
• You can now connect resonance to tuning, oscillators, and filters, and explain why a drifted component mistunes a resonant circuit.
• You can now see the capacitor and inductor not just as opposites but as partners that together produce a frequency-selective effect neither has alone.

Glossary Additions

• resonance — the condition in an LC circuit where the inductor's and capacitor's reactances are equal and cancel, at a single frequency where the circuit responds most strongly (series) or presents its highest impedance (parallel); energy exchanges freely between the two components there.
• resonant frequency — the frequency f₀ = 1 ÷ (2π√(LC)) at which an LC circuit resonates, set only by the inductance and capacitance.
• tank circuit — a capacitor and inductor in parallel, which presents a maximum impedance at its resonant frequency and is used to select or reject that frequency (in tuners, oscillators, and band-stop filters).
• Q factor — a measure of how sharp and selective a resonance is: a higher Q means the circuit responds over a narrower band around the resonant frequency; resistance in the circuit lowers Q.

Suggested Next Sections

Must read next:

• Capacitor and Inductor Failure Modes — the chapter's practical close: how these components actually fail on a bench (bulged and dried-out capacitors, open windings, shorted turns) and how to recognize each, including how a drifted value mistunes the resonant circuits from this section.

Recommended:

• Inductor Behavior in Circuits — XL rising with frequency is one of the two reactance curves whose crossing produces resonance.
• Capacitor Behavior in AC Circuits — Xc falling with frequency is the other curve; resonance is where they meet.