Resonant Frequency

Q: How does this formula apply to radio tuning?

Turning a variable capacitor changes $C$, which moves $f_0$. The radio then picks out the station whose frequency matches the new resonant frequency.

In one sentence: The resonant frequency is the one frequency at which a coil and a capacitor swap energy back and forth most easily, and a radio tunes to a station by matching it.

$f_0 = \frac{1}{2\pi\sqrt{LC}}$

Read it out loud: “f zero equals one divided by two times pi times the square root of L times C. f zero is the resonant frequency in hertz. The small zero is a subscript that marks it as the resonant frequency. L is the inductance in henries, and C is the capacitance in farads. The square root sign covers the whole product of L and C.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Frequency, hertz, kilohertz and megahertz: The frequency of a wave is how many complete cycles happen each second, measured in hertz (Hz). One hertz is one cycle per second. Radio signals are far faster, so their frequencies are counted in thousands or millions of hertz. The prefix kilo (k) means a thousand, and mega (M) means a million, so 1 MHz, a megahertz, is one million cycles per second.
Swapping energy between a coil and a capacitor: A capacitor stores energy in its electric field, and a coil stores energy in its magnetic field. In an LC circuit, energy can move from one to the other and back again. This swapping is the heart of resonance: the energy passes from the electric field to the magnetic field and back many times each second, much as a pendulum swings back and forth.
Square roots, like $\sqrt{x}$: The square root of $x$ , written $\sqrt{x}$ , is the number that, multiplied by itself, gives $x$ . For example, $\sqrt{9} = 3$ because $3 \times 3 = 9$ . The square root of a product can be found by taking the square root of each part, so $\sqrt{LC}$ is the square root of the whole product $LC$ . The formula puts the square root over the whole product, not over one part of it.
Powers of ten, like $10^{-6}$ and $10^{-12}$: A power of ten is 10 with a small raised exponent that says how many times to multiply by ten. The prefix micro, written $\mu$ , means $10^{-6}$ , a millionth. The prefix pico, written p, means $10^{-12}$ , a trillionth. So 250 microhenries is $250 \times 10^{-6}$ H. When powers of ten are multiplied, their exponents add, and a square root halves an exponent. The sign $\approx$ means approximately equal. It is used when a result has been rounded, so the number after it is close to the exact value but not exact.
Pi, $2\pi$ and the period: Pi, written $\pi$ , is about 3.14159, and $2\pi$ is about 6.28. The period is the time for one complete swing, and the frequency is one divided by the period. In this formula $2\pi$ appears because one full swing of the circuit covers $2\pi$ radians. The formula then turns that time into a count of swings per second. The angular resonant frequency uses the Greek letter omega, written $\omega_0$ and said ‘oh-MAY-guh zero’, and it is also measured in radians per second.

What each symbol means

Symbol	Meaning
$f_0$	The resonant frequency of the LC circuit, measured in hertz (Hz). It is the frequency at which the coil and the capacitor swap energy most easily. The worked example gives about 1 MHz, or 1.01 million hertz.
$\pi$	The number pi, about 3.14159, with no unit. It appears in the formula because one full swing of the circuit covers $2\pi$ radians.
$L$	The inductance of the coil, measured in henries (H). A bigger coil takes longer to change its current, so it lowers the resonant frequency. The worked example uses 250 microhenries.
$C$	The capacitance of the capacitor, measured in farads (F). A bigger capacitor holds more charge and takes longer to fill and empty, so it lowers the resonant frequency. The worked example uses 100 picofarads.

The formula piece by piece

$L$: The inductance in henries. It is the coil’s share of the swapping: a bigger inductance makes the current change more slowly, which slows the exchange of energy.
$C$: The capacitance in farads. It is the capacitor’s share of the swapping: a bigger capacitance stores more charge, which takes longer to fill and empty, so the exchange is slower.
$LC$: The product of the two. Multiplying them means a bigger value in either part slows the swing, and the two effects combine together. The product has units of henries times farads, which work out to seconds squared.
$\sqrt{LC}$: The square root of the product, which has units of seconds and sets the time scale of the swap. The square root appears because the swing behaves like a mass on a spring, where the rate of swinging depends on a square root of the stiffness-to-mass ratio.
$2\pi\sqrt{LC}$: The period, which is the time for one full swing, measured in seconds. The 2π comes from the radians in one full swing, and the square root gives the time scale.
$f_0 = \frac{1}{2\pi\sqrt{LC}}$: The whole formula in one sentence: the resonant frequency in hertz equals one divided by two pi times the square root of the inductance times the capacitance, which is the number of complete swings the circuit makes each second.
$\omega_0 = \frac{1}{\sqrt{LC}}$: A standard form that uses the angular resonant frequency, in radians per second, instead of hertz. It is one divided by the square root of $LC$ , and it equals $2\pi f_0$ .

What the formula is doing, and why

A coil and a capacitor connected together form an LC circuit. The capacitor stores energy in its electric field, and the coil stores energy in its magnetic field. Energy can slosh back and forth between them, like a pendulum that trades height for speed, and it does so at one natural rhythm. The inputs are $L$ and $C$ , and the output is the resonant frequency $f_0$ in hertz.

Why the formula looks like this? Each part sets how quickly the energy exchange happens. A bigger $L$ or a bigger $C$ slows the exchange, so the frequency is lower, and both appear in the denominator. Because the two effects multiply inside the root, the frequency depends on $\sqrt{LC}$ . Doubling $L$ or $C$ lowers the frequency by a factor of $\sqrt{2}$ , about 0.707, not by a factor of 2.

Why one divided by $2\pi\sqrt{LC}$ ? One complete swing takes $2\pi\sqrt{LC}$ seconds, so the number of swings each second is one divided by that time. The frequency in hertz is the reciprocal of the period, and that is where the one divided by comes from. A tuning circuit with a larger capacitor therefore swings more slowly and tunes to a lower frequency.

Resonance and the peak. In a series RLC circuit, at $f_0$ the coil’s and the capacitor’s reactances are equal and cancel, which leaves only the resistance. So the current is at its largest at this frequency. A circuit with low resistance has a sharp peak at its resonant frequency, so it picks out a narrow band of frequencies, and a radio needs that sharpness to separate one station from the next.

When it applies. The formula gives the natural frequency of an ideal LC circuit with no resistance. Real circuits have some resistance, which lowers and flattens the peak a little. The strongest response then appears close to this value, but not always exactly at it.

An everyday picture

Think of pushing a child on a swing. Push at the swing’s natural rhythm and each push adds height quickly. Push at a very different rhythm and the pushes work against each other. An LC circuit has its own natural rhythm, and a signal at that rhythm builds up the most. A longer swing moves more slowly, and in the same way a bigger coil or capacitor lowers the resonant frequency.

Worked example: A 250 microhenry coil and a 100 picofarad capacitor

Imagine a radio tuning circuit built from a made-up coil with $L = 250\ \mu\text{H}$ and a made-up capacitor with $C = 100$ pF. Find the resonant frequency. A microhenry is $10^{-6}$ H, and a picofarad is $10^{-12}$ F.

Write the formula and list the inputs. The formula is $f_0 = \frac{1}{2\pi\sqrt{LC}}$ . We have $L = 250\ \mu\text{H}$ and $C = 100$ pF. Both values have prefixes, so we must convert them to henries and farads before we multiply, or the answer will be wrong by a large factor.
Convert the inductance to henries. Micro means $10^{-6}$ , so $250\ \mu\text{H} = 250 \times 10^{-6}$ H. Writing 250 as $2.5 \times 10^{2}$ and adding the exponents gives $2.5 \times 10^{2} \times 10^{-6} = 2.5 \times 10^{-4}$ H. That is 0.00025 henries, one quarter of a thousandth of a henry.
Convert the capacitance to farads. Pico means $10^{-12}$ , so $100$ pF $= 100 \times 10^{-12}$ F. Writing 100 as $10^{2}$ and adding the exponents gives $10^{2} \times 10^{-12} = 10^{-10}$ F, which is 0.0000000001 farads.
Multiply the two values. $LC = 2.5 \times 10^{-4}\ \text{H} \times 1.0 \times 10^{-10}\ \text{F} = 2.5 \times 10^{-14}\ \text{s}^2$ . The front numbers multiply to 2.5, and the exponents add: $-4 + (-10) = -14$ . The units, henries times farads, come out as seconds squared.
Take the square root. $\sqrt{LC} = \sqrt{2.5 \times 10^{-14}} \approx 1.581 \times 10^{-7}$ s. The square root of 2.5 is about 1.581, and the square root of $10^{-14}$ is $10^{-7}$ , because the exponent is halved: $(10^{-7})^2 = 10^{-14}$ . The result is a time, about 0.0000001581 seconds.
Find the time of one full swing. Multiply by $2\pi$ : $2\pi \times 1.581 \times 10^{-7}$ s $\approx 9.935 \times 10^{-7}$ s. This is the period, the time for one full swing of energy between the coil and the capacitor. It is about 0.9935 microseconds.
Invert the period to get the frequency. The frequency is one over the period: $f_0 = \frac{1}{9.935 \times 10^{-7}\ \text{s}} \approx 1.007 \times 10^{6}$ Hz. Dividing by $9.935 \times 10^{-7}$ is the same as multiplying by $10^{7}$ and dividing by 9.935, which gives 1.007 times $10^{6}$ .
Name the answer. One million hertz is one megahertz (MHz), because mega means $10^{6}$ . So $1.007 \times 10^{6}$ Hz is about 1.01 MHz, which falls inside the AM radio band. Turning a variable capacitor or changing the coil moves this frequency, and that is how a radio tunes to a station.

Answer: The circuit resonates at about 1.01 MHz, which falls inside the AM radio band. Turning a variable capacitor or changing the coil changes this frequency, and that is how a radio tunes to a station.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Mixing up the units of $L$ and $C$ . Microhenries and picofarads must be converted to henries and farads first, or the frequency is off by a large factor.
Putting the square root over the wrong part. The formula is $1/(2\pi\sqrt{LC})$ , with the square root over the whole product $LC$ , not over $L$ alone.
Treating $f_0$ as angular frequency. The formula gives hertz, which is cycles per second. The angular frequency is $\omega = 2\pi f$ , measured in radians per second, and it is a different number.
Expecting a circuit to resonate at any frequency you pick. A tuned circuit has one natural resonant frequency, set by its $L$ and $C$ , and the strongest response occurs there.

Where you will meet it

Radio receivers, where a tuning circuit selects one station from many.
Clock and signal oscillators, where an LC circuit helps set the frequency of the signal.
Wireless power transfer systems, which tune a coil and a capacitor so that the transmitter and receiver share one frequency.
Induction cooktops and heaters, which use resonant circuits to drive their coils at a high frequency.

Questions people ask

What does resonance mean in a circuit?

It means the energy exchange between the coil and the capacitor matches the natural rhythm of the circuit, so a signal at that frequency builds up the most.

Why is the frequency lower when the capacitor is bigger?

A bigger capacitor holds more charge, so it takes longer to fill and empty. A bigger coil also takes longer to change its current, so the same rule applies to $L$ . Both slow the exchange, and a slower exchange means fewer swings each second.

Is the resonant frequency the same as the frequency of the current?

Only when you drive the circuit at that frequency. The resonant frequency depends on the $L$ and $C$ values alone, and the circuit responds most strongly when the drive matches it.

How does this formula apply to radio tuning?

Turning a variable capacitor changes $C$ , which moves $f_0$ . The radio then picks out the station whose frequency matches the new resonant frequency.

Series RLC Impedance: Impedance is the total opposition that a series circuit of a resistor, a coil and a capacitor gives to alternating current, combined into one number in ohms.
Inductive Reactance: Inductive reactance is how strongly a coil pushes back against alternating current, and that push gets stronger as the frequency goes up.
Capacitive Reactance: Capacitive reactance is how strongly a capacitor pushes back against alternating current, and that push gets weaker as the frequency of the current goes up.