Resonance
Resonance is the phenomenon behind every tuned circuit. It is the specific condition in an LC circuit where the inductive reactance exactly equals the capacitive reactance — where the inductor's opposition and the capacitor's opposition cancel each other completely, leaving only resistance in the circuit. At this one special frequency, remarkable things happen: in a series circuit, current surges to its maximum; in a parallel circuit, impedance reaches its peak. Oscillators, filters, wireless chargers and metal detectors are all designed around resonance.
Current versus frequency in a series RLC circuit. At the resonant frequency f0, current reaches its maximum value V/R — limited only by resistance. The sharpness of the peak is determined by the Q factor. Higher Q means a narrower, taller peak.
View LargerResonance as Energy Exchange
To understand resonance, think about a child on a swing. The swing oscillates at its natural frequency — determined by the length of the rope, not by how hard you push. If you push at exactly the right moment (matching the swing's natural frequency), small pushes produce large swings. Push at the wrong timing and the swing stops. The swing stores energy in kinetic form (at the bottom, moving fast) and potential form (at the top, momentarily stationary). Energy transfers continuously between these two forms at the natural frequency.
An LC circuit works identically. An inductor stores energy in its magnetic field; a capacitor stores energy in its electric field. At resonance, energy sloshes continuously back and forth between the inductor's magnetic field and the capacitor's electric field, at a natural frequency determined by L and C. The small amount of resistance in the circuit dissipates a tiny fraction of this energy each cycle — without resistance, the oscillation would continue forever.
This energy exchange model explains several important properties of resonant circuits: why large voltages can build up across reactive components even when the supply voltage is small; why resonant circuits can "ring" (oscillate freely after being excited); and why the frequency of oscillation is determined only by L and C, not by resistance.
The Resonance Condition: XL = XC
Resonance occurs at the frequency where the inductive reactance equals the capacitive reactance:
XL = XC
2πf0L = 1 / (2πf0C)
At this frequency, the +jXL and −jXC terms in the impedance expression cancel exactly: Z = R + j(XL − XC) = R + j(0) = R. The impedance becomes purely resistive — just the circuit's resistance, with no reactive component at all.
This is a remarkable result: a series circuit containing an inductor and a capacitor — two reactive components — presents no net reactance at all at resonance. It behaves as if neither the inductor nor the capacitor were present, leaving only resistance.
The Resonant Frequency Formula
Solving 2πf0L = 1/(2πf0C) for f0 gives the resonant frequency formula:
f0 = 1 / (2π√(LC))
where:
f0 = resonant frequency in hertz (Hz)
L = inductance in henrys (H)
C = capacitance in farads (F)
Equivalently, using angular frequency ω0 = 2πf0:
ω0 = 1 / √(LC)
Both L and C appear under the square root, which has important implications:
- Doubling L (with C fixed) reduces f0 by a factor of √2 ≈ 1.414 — only 29% reduction
- Quadrupling L reduces f0 by a factor of 2 — halves the frequency
- The same relationship holds for C — quadrupling C halves the resonant frequency
- To cover a 4:1 frequency range, you need a 16:1 variation in L or C (or 4:1 in each)
A quartz crystal resonates at 7.000 MHz. If the equivalent series inductance of the crystal is 12 mH, what capacitance is the crystal equivalent to?
Rearranging: C = 1 / (4π² × f0² × L)
C = 1 / (4π² × (7 × 106)² × 12 × 10-3)
C = 1 / (4 × 9.8696 × 4.9 × 1013 × 0.012)
C = 1 / (2.313 × 1013 × 0.012) = 1 / (2.776 × 1011) = 3.6 fF (femtofarads)
The crystal equivalent capacitance is tiny — in the femtofarad range — which is why crystal oscillators have very high Q and precise, stable resonant frequencies.
A parallel tank circuit uses a 1.0 µH inductor. What capacitor value is needed to resonate at 14.2 MHz?
C = 1 / (4π² × f0² × L)
C = 1 / (4π² × (14.2 × 106)² × 1.0 × 10-6)
C = 1 / (39.478 × 2.016 × 1014 × 10-6)
C = 1 / (7.961 × 109) = 125.6 pF
A 120 pF capacitor would resonate slightly above 14.2 MHz; a 130 pF would resonate slightly below. In practice, you would use a 100 pF fixed capacitor in parallel with a variable 50 pF trimmer to allow adjustment to exactly 14.2 MHz.
A wireless charger's transmit coil has an inductance of 8 µH. What series capacitance makes it resonate at 140 kHz, a typical wireless charging frequency?
C = 1 / (4π² × (140 × 103)² × 8 × 10-6)
C = 1 / (39.478 × 1.96 × 1010 × 8 × 10-6)
C = 1 / (39.478 × 1.568 × 105)
C = 1 / (6.190 × 106) = 0.162 µF ≈ 162 nF
A 160 nF capacitor in series with the 8 µH coil resonates the circuit close to 140 kHz, so a large current flows in the coil and a strong magnetic field couples power to the phone.
Resonant Frequency Calculator
LC Resonant Frequency Calculator — f = 1/(2π√LC)
Enter two known values to calculate the third. Leave the unknown field blank or enter 0.
What Happens at Resonance
At the resonant frequency, the circuit's behavior changes dramatically depending on whether it is configured as a series or parallel circuit. The following two lessons (the Series Resonant Circuits lesson and the Parallel Resonant Circuits lesson) cover series and parallel resonance in detail, but here is a brief summary:
| Parameter | Series resonance | Parallel resonance |
|---|---|---|
| Impedance | Minimum (= R only) | Maximum (= very high) |
| Current from source | Maximum (= V/R) | Minimum |
| Voltage across L and C | Maximum (Q × Vsupply) | Maximum (= supply voltage) |
| Phase angle | 0° (purely resistive) | 0° (purely resistive) |
| Circuit "looks like" | Short circuit (if R is low) | Open circuit (if Q is high) |
Above and Below Resonance
The behavior of an LC circuit on either side of resonance is predictable and symmetrical in terms of reactance:
Below resonance (f < f0): XC > XL (capacitive reactance dominates). The series circuit appears capacitive (net negative reactance), and impedance rises above R. The parallel circuit appears inductive.
At resonance (f = f0): XC = XL exactly. Net reactance is zero. Series impedance equals R. Parallel impedance is maximum.
Above resonance (f > f0): XL > XC (inductive reactance dominates). The series circuit appears inductive (net positive reactance), and impedance rises above R. The parallel circuit appears capacitive.
A useful memory aid: a series resonant circuit is capacitive below its resonant frequency and inductive above it. A parallel resonant circuit is inductive below its resonant frequency and capacitive above it — exactly the opposite. This is because series and parallel circuits exhibit dual behavior at resonance.
Resonance in Everyday Equipment
Oscillators and clocks. An LC oscillator runs at the resonant frequency of its tank circuit, and a quartz crystal is an extremely precise mechanical resonator with an equivalent LC circuit. The crystal in a watch, a microcontroller board or a computer sets its timing by resonance.
Filters. Band-pass and band-stop filters are built from resonant circuits. A ceramic or crystal resonator filter selects a narrow range of frequencies inside many wireless chips, and a resonant notch removes a single troublesome frequency such as mains hum. These filters cascade several resonant circuits to achieve a flat passband and steep skirts.
Wireless charging and induction. Wireless phone chargers, electric toothbrush bases and induction cooktops drive a coil at resonance so that a large current flows and a strong magnetic field couples energy to the receiving coil or the pan. RFID tags and contactless payment cards are tuned to resonate at 13.56 MHz so they can draw power from the reader's field.
Unwanted resonance. Resonance is not always welcome. Every real capacitor resonates with its own lead inductance; every PCB track has inductance and capacitance; a loudspeaker cone resonates mechanically. Designers check that these resonances fall outside the frequencies the circuit uses, or damp them with resistance.
Frequently Asked Questions
What happens if there is no resistance in an LC circuit?
In a theoretically ideal LC circuit with zero resistance, the resonant oscillation would continue forever with no energy loss — and the current and voltage at resonance would become infinitely large when driven by a source at the resonant frequency. In practice, all real inductors and capacitors have some resistance (the inductor's wire resistance, the capacitor's equivalent series resistance). This limits the Q factor to a finite value and limits the maximum voltages and currents. Very high-Q components can produce voltages across reactive elements that are hundreds of times the supply voltage, which is why resonant circuits must be designed with adequate voltage ratings on their components.
Can you change the resonant frequency without changing L or C?
No — the resonant frequency f0 = 1/(2π√LC) depends only on L and C. To change f0, you must change at least one of them. In practice, variable capacitors (trimmers and varactor diodes) are the most common way to tune resonant circuits because capacitance is easier to vary continuously than inductance. Switched inductors (multiple taps or separate coils) are used for switching between ranges. Varactor diodes — whose capacitance changes with applied DC voltage — allow electronic (voltage-controlled) tuning, used in phase-locked loops and wireless chips.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.