Resonance
With a 10 V source, a thousand volts can appear across the capacitor of a resonant circuit. The same phenomenon tunes a radio, sets the time of a quartz clock and powers a SUBE card with no battery.
01Series resonance
In a series RLC circuit the inductive reactance \( \omega L \) grows with frequency and the capacitive reactance \( 1/\omega C \) falls. There is a single frequency at which the two are equal and cancel: the resonant frequency. There the impedance is reduced to \( R \), its minimum, and the current reaches its maximum.
That exchange is the key to the whole phenomenon. The energy goes back and forth between the magnetic field of the inductor and the electric field of the capacitor, and the source only has to replenish what the resistor loses in each cycle. That is why, at resonance, the voltage across \( L \) and across \( C \) can be much greater than that of the source.
02The quality factor Q and the bandwidth
The quality factor measures how much energy the circuit stores compared with what it loses per cycle, and with that it decides two things at once: how sharp the resonance peak is and how much the voltage across the reactive elements is magnified.
| Resonator | Typical Q |
|---|---|
| LC circuit of a crystal AM radio | 50 to 200 |
| Coil of a Qi wireless charger | 50 to 100 |
| Copper microwave cavity | 10,000 |
| 32.768 kHz quartz crystal of a clock | 50,000 to 100,000 |
With \( Q = 100 \) and a 10 V source, 1000 V appears across the capacitor at resonance. It is not a calculation artifact: it is the reason that the capacitors of a resonant circuit are chosen with a wide margin on rated voltage, and that touching the tuning capacitor of a transmitter is dangerous even when the source is modest.
03Tuning in a station
The oldest use of resonance is choosing one station among all those that reach an antenna. A ferrite-rod coil of about 240 µH and a variable capacitor of 30 to 365 pF sweep from about 540 kHz to 1.9 MHz: the whole AM band. In Argentina AM stations are spaced 10 kHz apart, and the design question is how much the circuit attenuates the neighboring station.
240 µH coil. You turn the variable capacitor to tune and choose the loss resistance. The plot shows the resonance curve around the tuned station, with the neighboring stations at ±10 and ±20 kHz.
The result is surprising: with a realistic Q, a single tuned circuit attenuates the neighboring station by barely a few decibels. That limitation is what led, in the 1920s, to the superheterodyne receiver, which shifts every station to a fixed intermediate frequency of 455 kHz and filters it there with several cascaded resonant circuits, with no need to retune them any more.
04Parallel resonance
With \( L \) and \( C \) in parallel the roles are reversed: at resonance the susceptances cancel, the admittance is minimum and the impedance is maximum. The circuit, called a tank, draws almost no current from the source, while inside a current \( Q \) times larger circulates between the inductor and the capacitor.
| Series RLC | Parallel RLC | |
|---|---|---|
| At \( f_0 \), the impedance is | Minimum, equal to \( R \) | Maximum |
| What is magnified | The voltage across \( L \) and \( C \) | The current circulating between \( L \) and \( C \) |
| Typical use | Filter that passes a band, in series with the signal | Load of an oscillator or of a tuned amplifier |
05Where it shows up in electronics
| Application | Frequency | What resonance does |
|---|---|---|
| Quartz clock | 32.768 kHz | The crystal is a mechanical resonator with an enormous Q: it fixes the frequency with parts-per-million precision |
| Qi wireless charging | 100 to 205 kHz | Two resonant circuits at the same frequency transfer energy by induction with good efficiency |
| RFID / SUBE card | 13.56 MHz | The card is powered by the energy its resonant antenna picks up from the reader |
| Electronic ballast of a fluorescent tube | Tens of kHz | The series-resonance overvoltage strikes the tube |
06In the lab
Build a series RLC with 1 mH, 10 nF and 10 Ω, sweep the frequency with a generator and measure the voltage across the resistor. Locate \( f_0 \) and the two −3 dB points, and calculate \( Q \) as \( f_0/B \). Compare with \( (1/R)\sqrt{L/C} \).
In the same circuit, use the oscilloscope probe to measure the voltage across the capacitor at resonance and verify that it is approximately \( Q \) times that of the generator.
Using this page’s lab, determine what \( Q \) would be needed for the neighboring station at 10 kHz to be attenuated 20 dB at 1 MHz, and discuss whether a real coil can achieve it.
07Common mistakes
- Believing that at resonance the voltage across \( L \) and \( C \) is zero. What vanishes is their sum; each one separately is \( Q \) times that of the source.
- Confusing series resonance with parallel resonance. One is a minimum of impedance and the other, a maximum.
- Calculating \( Q \) at the wrong frequency. \( Q = \omega_0 L/R \) is evaluated at resonance, not at any frequency.
- Forgetting the resistance of the coil. In a real tuned circuit, almost all of the \( R \) is that of the coil’s wire and the core losses.
- Expecting a single LC circuit to separate nearby stations. With a realistic Q it is not enough; that is why the superheterodyne exists.
08Self-assessment
What is the impedance of a series RLC at resonance?
\( R \): the inductive and capacitive reactances cancel.
What is the relationship between \( Q \), \( f_0 \) and the bandwidth?
\( B = f_0/Q \): the higher the Q, the narrower the peak.
What voltage appears across the capacitor at resonance?
\( Q \) times the source voltage.
What does a variable capacitor of 30 to 365 pF sweep with a 240 µH coil?
From about 540 kHz to 1.9 MHz: the complete AM band.
What happens to the impedance of a parallel LC tank at resonance?
It is maximum: the circuit draws almost no current from the source.
Why is a quartz crystal such a good frequency standard?
Because it is a resonator with a Q of tens of thousands: its peak is so narrow that the oscillation frequency can hardly move away from it.
09Further reading
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 7th ed., McGraw-Hill, 2021. The frequency response chapter develops series and parallel resonance, Q and bandwidth.
- Frederick E. Terman. Radio Engineers' Handbook. McGraw-Hill, 1943. The classic handbook of vacuum-tube radio: tuned circuits treated from the viewpoint of someone who designed them for production.
- Horowitz and Hill. The Art of Electronics. 3rd ed., Cambridge University Press, 2015. Resonant circuits and crystal oscillators from the practical design point of view.