LC Resonant Circuit Explained: Tank Circuits for Ham Radio

The LC resonant circuit is the oldest building block of radio. A coil and a capacitor, connected together, ring at one frequency and respond strongly to signals at that frequency while ignoring others. It sits inside the front end of a receiver, the output network of a transmitter, the band-pass filter of a Field Day station, the traps in a multiband dipole and the tuner that you adjust when you load a long wire. Understanding how it works takes only one formula and a few ideas, and with them you can predict what a filter will do, choose parts for a simple project and see why antenna tuners behave as they do. This guide explains the circuit step by step with numbers for the HF bands.

What Makes a Circuit Resonate

The LC circuit, also called a resonant circuit, tank circuit or tuned circuit, consists of an inductor, with the symbol L, and a capacitor, with the symbol C, connected together. A capacitor stores energy in its electric field and an inductor stores energy in its magnetic field. In the circuit, the charge flows back and forth between the two, so that the energy swings from the capacitor to the inductor and back, until resistance in the circuit makes the oscillation die out. The swing has a natural frequency, set only by the values of L and C, and that is the resonant frequency.

Another way to see it is through reactance. An inductor opposes the flow of alternating current more strongly as the frequency rises, and a capacitor opposes it more strongly as the frequency falls. At some frequency the two reactances are equal in size, and in the circuit they cancel because they have opposite signs. This is resonance. The same Wikipedia article states that resonance occurs when the inductive and capacitive reactances are equal in magnitude, and it gives the angular frequency as one divided by the square root of L times C.

The Formula You Need

In the units we use in the shack, the resonant frequency in hertz is:

f = 1 / (2π √(L × C))

with L in henries and C in farads. For practical work it helps to remember the order of magnitude: radio circuits use microhenries and picofarads, so convert carefully. A handy form is that the product of L and C, when f is in MHz, L in microhenries and C in picofarads, is about 25330 divided by f squared. In other words, L (µH) × C (pF) = 25330 / f² (MHz). Check it with a number: at 7.1 MHz the square is 50.41, so L × C is about 502, and with 1 µH the capacitor must be about 502 pF. At 14.2 MHz, with the same coil, the capacitor is a quarter as large, 125.6 pF, because doubling the frequency requires one quarter of the product.

The table shows another useful fact. For a given frequency, many pairs of L and C resonate. A large L with a small C gives a high reactance at resonance, and a small L with a large C gives a low one. The choice of the pair is a matter of the impedance that the circuit has to work with, and this is why a filter for a 50 ohm system uses values whose reactances are in the order of 50 ohm, while a vacuum tube tank circuit works at much higher impedance.

Series and Parallel Behaviour

The same two parts behave in opposite ways depending on how they are connected, and that is the central point of the subject. A series resonant circuit has a very low impedance at resonance, because the two reactances cancel and only the small resistance of the circuit remains. It passes a signal at its resonant frequency and blocks others. The Wikipedia article puts it simply: a series resonant circuit provides voltage magnification, and a parallel resonant circuit provides current magnification. A parallel circuit has a very high impedance at resonance, so a signal of the resonant frequency finds it hard to pass to ground and it appears across the circuit at high voltage, while other frequencies are shorted by the coil or the capacitor.

• Series circuit in the signal path acts as a band-pass filter: it passes the resonant frequency and blocks the others.
• Parallel circuit across the signal path (to ground) is also a band-pass filter: it lets the resonant frequency go on and shorts the others.
• Parallel circuit in the signal path acts as a trap or notch: it blocks the resonant frequency, which is the principle of the trap in a multiband dipole.
• Series circuit to ground acts as a notch: it removes the resonant frequency by shorting it.

This is why the traps in a trap dipole are parallel circuits. At the trap’s resonant frequency the high impedance isolates the outer part of the wire, and the antenna is shorter on that band. Below that frequency the trap works as an inductor and loads the wire, so the antenna can also work on a lower band. Our guide to the trap dipole shows how this works in a real antenna.

Q Factor and Bandwidth

No real circuit resonates forever. The resistance of the coil’s wire, the losses in the capacitor and the load that you connect all damp the oscillation. The Q factor measures how weak the damping is. A higher Q means that the resonator rings for longer, responds more strongly at its frequency and has a narrower range of frequencies around the peak. For a high-Q circuit, the Wikipedia article gives the useful definition as the ratio of the resonant frequency to the bandwidth, where the bandwidth is the width between the half-power points. So Q = f / Δf.

This gives a quick estimate. A circuit with a Q of 100 at 7.1 MHz has a bandwidth of about 71 kHz, so it would hardly distinguish stations in the same part of the CW band but would reject a signal in the neighbouring band. A Q of 10 gives a bandwidth of 710 kHz, enough for a whole band segment. A high Q is good for selectivity and bad for tuning, since a high-Q circuit must be tuned precisely and small changes in the part values or in the temperature move the peak out of the band. The idea is the same for antennas, where the narrow bandwidth of a small loop or a short antenna is a consequence of its high Q.

Where the Losses Come From

The Q of a real tuned circuit is limited by its weakest part. In most amateur circuits the coil is the main source of loss. The resistance of the wire at radio frequency is higher than its direct current resistance because of the skin effect, which pushes the current to the surface of the conductor. Core material in a ferrite or powdered-iron coil adds loss, and so do the supports and the nearby metal, such as a shield can that is too close. A good air-wound coil of thick wire with a spacing between turns has a high Q, while a small coil wound on a lossy core may have a Q of only a few tens. A capacitor of good quality, such as a silver mica or an air variable, has much lower losses than the coil. When a filter does not give the response you expected, measure the coil first.

Using the Circuit: Three Examples

First, a simple band-pass filter in front of a receiver. A parallel LC circuit tuned to the band, connected through small coupling capacitors or taps, rejects out-of-band signals such as strong broadcast stations. The amount of rejection depends on the Q and on the number of circuits that you use. Our guide to band-pass filters for Field Day uses the same principle in a multi-station setting.

Second, an L network and a tuner. The circuit that an antenna tuner uses to match an antenna is built from a coil and capacitors, and its adjustment finds a combination that makes the load look like 50 ohm. The theory in the article on L network matching shows how the same ideas of reactance and resonance lead to the values.

Third, a transmitter low-pass filter, where a chain of coils and capacitors passes the wanted frequency and rejects the harmonics. The filter is not a resonator in the narrow sense, but it is built from the same elements, and its design follows from the same relationships between L, C and frequency. Our low-pass filter guide covers the practical side.

A Worked Calculation, Step by Step

Suppose you want a parallel tuned circuit for the 40 metre band, centred on 7.1 MHz, using a 2.2 µH coil that you have in the drawer. First, square the frequency: 7.1 times 7.1 is 50.41. Divide 25330 by that figure and the result is about 502.5, which is the required product of L and C. Divide 502.5 by the coil value of 2.2 and you get a capacitance of about 228 pF. A fixed capacitor of 220 pF in parallel with a 10 pF trimmer gives you room to adjust the circuit onto the frequency, and the trimmer also covers the stray capacitance of the wiring. Now check the reactance: 2π times 7.1 million times 2.2 microhenries is about 98 ohm, so at resonance the coil and the capacitor each have a reactance of about 98 ohm, which is a sensible value for a receiver input. If you needed a lower impedance, you would choose a smaller coil and a larger capacitor. This two-minute calculation tells you what parts to buy before you pick up the soldering iron.

Practical Tips for Building and Measuring

• Check the parts with a meter. A cheap LC meter or the NanoVNA can measure the inductance and capacitance of the parts. Coils wound at home often differ from the calculated value by 10 per cent or more.
• Measure the resonance with a dip. A simple way to find the resonant frequency of a coil and capacitor is to couple a signal generator or a NanoVNA loosely to the circuit and look for the peak or the dip in the response.
• Keep leads short. A few centimetres of wire add some tens of nanohenries, which matters at 28 MHz and above.
• Mind the stray capacitance. The wiring and the coil have a small capacitance of their own, often several picofarads, which adds to the capacitor and lowers the frequency.
• Use a good capacitor type. NP0 (C0G) ceramic, silver mica and air variable capacitors are stable and low-loss. Capacitors with a high dielectric constant change their value with voltage and temperature.
• Allow for power. The voltage and the current inside a tank can be many times larger than at the input, so choose parts with a sufficient rating for the power in a transmitter.

The Bottom Line

An LC resonant circuit is a coil and a capacitor that resonate where their reactances are equal. The formula f = 1 / (2π √LC) gives the frequency, and the shortcut L (µH) × C (pF) = 25330 / f² (MHz) makes the arithmetic easy. A series circuit has a low impedance at resonance and a parallel circuit has a high one, which decides whether it passes or blocks a signal depending on where you place it. The Q factor, the ratio of the frequency to the bandwidth, describes the selectivity and is limited by the losses in the coil. With these ideas you can understand filters, traps and tuners and calculate the parts for a simple project.

Frequently Asked Questions

What is an LC resonant circuit?

It is a circuit made of an inductor and a capacitor that resonates at a particular frequency, where the reactances of the two parts are equal in size. It is also called a tank circuit or tuned circuit.

What is the formula for the resonant frequency?

f = 1 / (2π √(LC)), with L in henries and C in farads. In convenient units, L in µH times C in pF equals 25330 divided by the frequency in MHz squared.

What is the difference between series and parallel resonance?

A series circuit has a minimum impedance at resonance and passes the resonant frequency. A parallel circuit has a maximum impedance at resonance, so it blocks the signal if placed in the path or shorts other frequencies if placed across it.

What does Q factor mean?

Q describes how weakly the circuit is damped. For a high-Q circuit it is the resonant frequency divided by the bandwidth between the half-power points, so a higher Q means a narrower response.

Why does my homemade filter peak at the wrong frequency?

The coil value may differ from the calculation, and stray capacitance and lead inductance shift the resonance. Measure the parts, shorten the leads and adjust the values.

How are LC circuits used in ham radio?

They are used in band-pass and low-pass filters, antenna tuners, traps in multiband antennas, oscillators and the output networks of transmitters.