Series Resonant Circuits
A series resonant circuit is one of the most important building blocks in electronics. It is a resistor, inductor, and capacitor wired in series — one after another — and driven by an AC signal. At one specific frequency, this simple combination of three components behaves in an almost magical way: it passes that frequency with almost no opposition, while blocking frequencies above and below. Filters that pick out one frequency, crystal oscillators and wireless chargers all rely on exactly this behavior.
In the previous lesson you learned that resonance occurs when XL equals XC and that the impedance of a series circuit drops to just its resistance at resonance. This lesson goes deeper: you will learn exactly how sharp that frequency selectivity is, what controls it, and how voltages inside the circuit can far exceed the supply voltage. These concepts — Q factor, bandwidth, and voltage magnification — are the working vocabulary of every band-pass filter, resonant converter and tuned circuit.
A series RLC circuit. R, L, and C are wired end-to-end with a single current flowing through all three. At the resonant frequency, XL = XC, they cancel, and the circuit impedance drops to just R — current is maximum.
View LargerThe Series Circuit at Resonance
In a series RLC circuit, the same current flows through every component — the resistor, the inductor, and the capacitor all carry identical current at every moment. This is a crucial difference from parallel circuits and it explains much of what makes series resonance useful.
The total impedance of a series RLC circuit is:
Z = √(R² + (XL − XC)²)
where XL = 2πfL and XC = 1/(2πfC)
At frequencies below resonance, XC is large and XL is small, so XC − XL dominates and the circuit looks capacitive. At frequencies above resonance, XL is large and XC is small, so XL − XC dominates and the circuit looks inductive. At exactly the resonant frequency, XL = XC, the two reactive terms cancel completely, and the impedance formula reduces to:
Zres = R
This is the minimum impedance the circuit can have — equal to the series resistance alone.
This minimum-impedance condition at resonance is the defining characteristic of the series resonant circuit. Because current I = V/Z and Z is at its minimum, current is at its maximum at resonance. The circuit is said to be in resonance when it draws the most current from the source.
Minimum Impedance: Why Series Resonance Passes Maximum Current
It seems paradoxical that adding more components to a circuit — an inductor and a capacitor — can make it pass more current than just a resistor alone. But at resonance, this is exactly what happens: the series RLC circuit passes the same current as if only the resistor were present. The inductor and capacitor are "invisible" at the resonant frequency.
The reason is that XL and XC are equal and opposite in their effect on the circuit. Inductive reactance causes voltage to lead current; capacitive reactance causes voltage to lag current. At resonance these effects are equal and cancel, leaving the circuit's behavior determined entirely by R.
Think of it as a tug-of-war where both teams are equally matched. The net effect on the rope is zero — it doesn't move despite enormous forces being applied. In the RLC circuit, the inductor and capacitor exert large reactive forces, but they cancel each other, and the resistor is all that's left controlling current flow.
A series circuit has R = 10 Ω, L = 50 µH, C = 100 pF. Find the resonant frequency and compare impedance at f0, at 10% below f0, and at 10% above f0.
Step 1 — Resonant frequency:
f0 = 1 / (2π√(LC)) = 1 / (2π√(50×10⁻⁶ × 100×10⁻¹²))
= 1 / (2π√(5×10⁻¹⁵)) = 1 / (2π × 7.071×10⁻⁸) = 2.251 MHz
Step 2 — Impedance at resonance:
Z = R = 10 Ω
Step 3 — Impedance at 10% below resonance (f = 2.026 MHz):
XL = 2π × 2.026×10⁶ × 50×10⁻⁶ = 636.6 Ω
XC = 1 / (2π × 2.026×10⁶ × 100×10⁻¹²) = 785.6 Ω
Xnet = XL − XC = 636.6 − 785.6 = −149.0 Ω
Z = √(10² + 149²) = √(100 + 22201) = √22301 = 149.3 Ω
Step 4 — Impedance at 10% above resonance (f = 2.476 MHz):
XL = 2π × 2.476×10⁶ × 50×10⁻⁶ = 777.6 Ω
XC = 1 / (2π × 2.476×10⁶ × 100×10⁻¹²) = 642.8 Ω
Xnet = 777.6 − 642.8 = +134.8 Ω
Z = √(10² + 134.8²) = √(100 + 18171) = √18271 = 135.2 Ω
Result: At resonance Z = 10 Ω; 10% below: 149 Ω; 10% above: 135 Ω. Moving just 10% away from resonance increases impedance by a factor of 13–15, demonstrating the sharp selectivity of this circuit.
Q Factor: The Measure of Circuit Quality
The Q factor — Quality factor — is the single most important specification for a resonant circuit. It tells you how selective the circuit is: how sharply it responds to the resonant frequency while rejecting others. A high-Q circuit is very selective; a low-Q circuit is broad and accepts a wide range of frequencies.
For a series resonant circuit, Q is defined as the ratio of the reactive power stored in the circuit to the real power dissipated by the resistance. In practical terms, Q equals the reactance of either component at resonance divided by the series resistance:
Q = XL / R = XC / R = (2πf0L) / R = 1 / (2πf0CR)
where all values are evaluated at the resonant frequency f0
Because XL = XC at resonance, either reactance gives the same Q value. The formula using L is usually more convenient because inductors dominate loss in real circuits.
Q is a dimensionless number — no units. Typical values in practical circuits:
| Circuit Type | Typical Q Range | Application |
|---|---|---|
| Air-core coil, unshielded | 50–200 | High-frequency tuned circuits and filters, oscillator tanks |
| Toroidal inductor (powdered iron) | 100–300 | Low-pass filters, impedance matching |
| Toroidal inductor (ferrite) | 20–80 | Audio and IF transformers |
| Crystal resonator | 10,000–100,000 | Oscillator frequency control, narrow filters |
| Ceramic resonator | 1,000–5,000 | Low-cost oscillators |
The resistance R in the Q formula includes all sources of loss in the circuit — the DC resistance of the wire, eddy-current losses in the core, skin-effect losses at high frequency, and any external resistance deliberately added. This total loss resistance is often called the equivalent series resistance (ESR) or the circuit resistance.
A tuned circuit for a 7.1 MHz oscillator uses an air-core coil with L = 2.5 µH and ESR = 0.5 Ω. Find the capacitance needed for resonance and the Q.
Step 1 — Capacitance for resonance:
C = 1/(4π² × f² × L) = 1/(39.478 × 50.41×10¹² × 2.5×10⁻⁶) = 201.6 pF
Step 2 — Reactance at 7.1 MHz:
XL = 2π × 7.1×10⁶ × 2.5×10⁻⁶ = 111.5 Ω
Step 3 — Q:
Q = XL / R = 111.5 / 0.5 = 223
This is an excellent Q for an oscillator tank circuit. With this Q, the oscillator will be stable and its output will be close to a pure sine wave, because the circuit strongly favors the resonant frequency over its harmonics.
Bandwidth and the 3 dB Points
Q factor and bandwidth are two sides of the same coin. A high-Q circuit is very selective — it responds strongly only to frequencies very close to f0. A low-Q circuit is broad — it passes a wide range of frequencies around f0. The bandwidth is the frequency range over which the circuit passes signals acceptably well.
The standard definition of bandwidth uses the half-power points — the frequencies at which the output power falls to half its maximum value. Since power is proportional to I², the half-power points occur where current falls to 1/√2 ≈ 0.707 of its peak value. A 3 dB drop in power corresponds to a 0.707 drop in current (because 10 log(0.5) = −3 dB). These frequencies are called the upper and lower −3 dB frequencies, or the half-power bandwidth points.
BW = f0 / Q
where:
BW = bandwidth in Hz
f0 = resonant frequency in Hz
Q = quality factor (dimensionless)
The −3 dB frequencies are: f1 = f0 − BW/2 and f2 = f0 + BW/2
This formula reveals a fundamental trade-off: higher Q gives better selectivity (narrower bandwidth) but also means a more sensitive circuit that requires more precise tuning. Lower Q gives a more tolerant circuit that accepts a broader range of frequencies but does a poorer job of rejecting adjacent signals.
A metal detector's search coil forms a series resonant circuit at f0 = 100 kHz with Q = 20. Find the bandwidth and the −3 dB frequencies.
Bandwidth:
BW = f0 / Q = 100,000 / 20 = 5,000 Hz = 5 kHz
Lower −3 dB frequency:
f1 = 100,000 − 5,000/2 = 97,500 Hz = 97.5 kHz
Upper −3 dB frequency:
f2 = 100,000 + 2,500 = 102,500 Hz = 102.5 kHz
Interpretation: The circuit responds strongly only within about 2.5 kHz of its resonant frequency. A metal object near the coil changes its inductance slightly, shifting the resonant frequency; the detector senses the resulting change in the coil's response. A higher Q would make the detector more sensitive to small shifts, but also more sensitive to temperature drift.
Voltage Magnification at Resonance
One of the most striking properties of a series resonant circuit is that the voltage across the inductor or capacitor at resonance can be far larger than the supply voltage. This is not a measurement error — it is a real physical phenomenon called voltage magnification, and it is the reason that high-voltage capacitors are required in resonant circuits even when operating at relatively modest supply voltages.
To understand why this happens, recall that at resonance the current is at its maximum: I = VS/R, where VS is the supply voltage. The voltage across the inductor is:
VL = I × XL = (VS/R) × XL = VS × (XL/R) = VS × Q
Similarly: VC = VS × Q
The voltage across both reactive components at resonance equals the supply voltage multiplied by Q. A circuit with Q = 100 and a supply voltage of 100 V will develop 10,000 V across both the inductor and the capacitor at resonance. This is why capacitors in resonant circuits must be rated for voltages many times higher than the supply rail.
Note that VL and VC are equal in magnitude but exactly opposite in phase — they cancel in the KVL sum around the loop, so the net reactive voltage presented to the source is zero. Only VR = I × R = VS appears from the source's perspective.
A wireless phone charger runs from a 5 V USB supply and drives its 8 µH transmit coil at 140 kHz through a series capacitor. The coil and capacitor have a total loss resistance of 0.1 Ω, and the peak coil current is 4 A. Find the Q and the peak voltage across the capacitor.
Step 1 — XL at 140 kHz:
XL = 2π × 140×10³ × 8×10⁻⁶ = 7.04 Ω
Step 2 — Q:
Q = XL / R = 7.04 / 0.1 = 70
Step 3 — Peak voltage across capacitor:
VC(peak) = Ipeak × XC = 4 × 7.04 = 28 V peak
This means the resonant capacitor must be rated for at least 50 V — even though the charger runs from a 5 V supply. Resonant circuits routinely develop voltages well above their supply, and capacitors rated only for the supply voltage fail.
Voltage magnification in a series resonant circuit. At resonance, the voltage across L and C each equal Q × VS — far exceeding the supply voltage. The voltage across R equals VS at resonance since all supply voltage appears across the resistance.
View LargerSeries Resonant Circuit Calculator
Series Resonant Circuit Calculator
Enter the circuit values to calculate resonant frequency, Q factor, bandwidth, impedance at resonance, and component voltages at resonance.
The Frequency Response Curve
The frequency response of a series resonant circuit — a plot of current or output voltage versus frequency — forms a bell-shaped curve centered on f0. The width of this bell is determined by Q: high Q produces a narrow, tall peak; low Q produces a wide, shallow peak.
The shape of the response curve follows this equation for normalized current (current as a fraction of peak current):
I/Imax = 1 / √(1 + Q²((f/f0) − (f0/f))²)
This shows that Q² multiplies the frequency deviation term — doubling Q makes the circuit four times more selective at the same frequency offset from resonance. This is why high-Q circuits are so desirable in selective filters: they suppress nearby unwanted frequencies by a factor proportional to Q².
The curve is not perfectly symmetrical on a linear frequency scale — it is slightly asymmetric, with the lower half wider than the upper half. However, on a logarithmic frequency scale (which is more natural for decade-spanning measurements) the curve is symmetric. For narrowband circuits (BW much less than f0), the asymmetry is negligible.
| Frequency (relative to f0) | Net Reactance | Relative Current (Q = 50) | Circuit Behavior |
|---|---|---|---|
| 0.90 × f0 | Strongly capacitive | ~10% | Highly attenuated |
| 0.97 × f0 | Moderately capacitive | ~50% | −6 dB below peak |
| 0.99 × f0 | Slightly capacitive | ~71% (−3 dB point) | Half-power bandwidth |
| f0 | Zero (cancelled) | 100% (maximum) | Full resonance |
| 1.01 × f0 | Slightly inductive | ~71% (−3 dB point) | Half-power bandwidth |
| 1.03 × f0 | Moderately inductive | ~50% | −6 dB below peak |
| 1.10 × f0 | Strongly inductive | ~10% | Highly attenuated |
Series Resonance in Practice
Series resonant circuits appear in equipment you use every day:
Wireless Charging
A wireless charger drives a coil through a series capacitor chosen to resonate with it. At resonance the coil current is limited only by resistance, so a large current — and a strong magnetic field — is produced from a low supply voltage. The phone's receiving coil is tuned to the same frequency so it picks up the energy efficiently.
Resonant Power Converters
Many modern power supplies, including high-efficiency laptop chargers and LED drivers, use series resonant (LLC) converters. Switching the transistors at or near the resonant frequency lets them turn on and off when the current or voltage is close to zero, greatly reducing switching losses and electrical noise.
Crystal and Ceramic Resonators
Quartz crystals behave as series resonant circuits with an extremely high Q (often 10,000–100,000). They set the frequency of clocks and microcontrollers, and arrays of crystals or ceramic resonators form very narrow filters in wireless chips. The Q formula Q = XL/R still applies — the "L" is the mechanical (motional) inductance of the crystal and the "R" is its mechanical loss.
Notch Filters and Traps
A series LC circuit connected from a signal line to ground short-circuits its resonant frequency while leaving others almost untouched. This trap removes a single unwanted frequency, such as the switching frequency of a power supply leaking into an audio circuit. Loudspeaker crossovers use series LC networks in the same way to tame a driver's resonant peak.
Experiment: Observing Series Resonance with an Audio Oscillator
You need: Audio frequency oscillator or signal generator (20 Hz–20 kHz), 100 mH inductor (available cheaply as an audio choke), 1 µF non-polarized capacitor, 100 Ω resistor, multimeter set to AC volts, connecting wires.
Theory: f0 = 1/(2π√(LC)) = 1/(2π√(0.1 × 10⁻⁶)) = 503 Hz. Q = XL/R = (2π × 503 × 0.1)/100 = 3.16 (low Q due to the resistor, but good for demonstration).
Procedure:
- Wire R, L, and C in series across the oscillator output.
- Connect the multimeter across R to measure current indirectly (VR = I × R).
- Sweep the oscillator from 100 Hz to 1500 Hz while watching VR.
- Find the frequency where VR is maximum — this is f0. Note the frequency; it should be close to 503 Hz.
- At resonance, move the multimeter across the capacitor and measure VC. Compare to the input voltage.
- Note how VC exceeds the input — this is voltage magnification.
- Record VR at f0/2 and 2f0 to see how much attenuation occurs away from resonance.
Expected result: VR peaks sharply at ~503 Hz. VC at resonance ≈ Q × Vin ≈ 3.16 × Vin. At half and double f0, VR is dramatically lower, demonstrating the frequency-selective behavior of the series resonant circuit.
Frequently Asked Questions
If VL and VC both equal Q × VS at resonance, does KVL still hold?
Yes, absolutely — KVL must always hold. VL and VC are equal in magnitude but exactly 180° out of phase with each other at resonance. They cancel when added as phasors, so the sum around the loop is VR + VL − VC = VS + 0 = VS. The large voltages across L and C are real and measurable, but they cancel in the series loop because they are opposite in sign. This is why measuring VC and VL separately with a voltmeter can give readings much higher than VS — the meter measures magnitudes, not the phasor sum.
Why does Q decrease as frequency increases even with the same coil?
Several effects increase coil losses at higher frequencies: skin effect concentrates current in a thin layer on the conductor surface, increasing effective resistance; eddy current losses in any core material increase; and distributed capacitance between coil turns becomes significant, changing the resonant behavior. These effects mean that the same coil has a lower effective Q at 14 MHz than at 7 MHz, which must be considered when designing RF circuits. Manufacturers typically specify Q at a specific test frequency for this reason.
Can series resonance be harmful to equipment?
Yes — the voltage magnification at resonance can destroy components not rated for the resulting voltages. Capacitors in resonant circuits must be rated well above the supply voltage for exactly this reason. In power distribution systems, accidental series resonance between line inductance and power-factor correction capacitors can cause dangerously high voltages. In sensitive amplifiers, a series resonant condition in the wrong place can produce unexpectedly large signal voltages that overdrive the input stage. Understanding series resonance is essential for safe circuit design.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.