Resonant Frequency
Frequency at which an LC circuit resonates.
Formula
Variables
How it works
In an ideal LC circuit, energy continually swaps between the electric field of the capacitor and the magnetic field of the inductor. The resonant frequency is the natural oscillation rate of that exchange, and it comes from the differential equation for the circuit, which gives an angular frequency \omega_0 = 1/\sqrt{LC}; converting angular frequency to ordinary frequency gives f_0 = 1/(2\pi\sqrt{LC}). At this frequency, the reactive effects of L and C balance in a way that makes the circuit respond most strongly, although real circuits also include resistance that slightly broadens and lowers the peak response.
When to use it
Use this when analyzing or designing tuned circuits such as radio receivers, filters, oscillators, and impedance-matching networks. It is also the right formula for estimating the natural ringing frequency of a practical LC tank circuit before losses are included.
Worked example, step by step
Given: L = 25 H = 25 10^-6 H, C = 100 nF = 100 10^-9 F Step 1: Multiply L and C LC = (25 10^-6)(100 10^-9) = 2500 10^-15 = 2.5 10^-12 Step 2: Take the square root \sqrt{LC} = \sqrt{2.5 10^-12} = 1.5811 10^-6 Step 3: Multiply by 2\pi 2\pi\sqrt{LC} = 2\pi(1.5811 10^-6) 9.934 10^-6 Step 4: Invert to get frequency f_0 = 1 / (9.934 10^-6) 1.01 10^5 Hz Final answer: f_0 1.01 10^5 Hz 101 kHz
Common mistakes to avoid
- Using L in millihenries or C in microfarads without converting to henries and farads first
- Forgetting the 2\pi factor and using the angular frequency formula as if it were ordinary frequency
- Expecting the formula to be exact in a lossy real circuit with resistance, where the true peak response is shifted slightly
Did You Know?
Tuning a radio means adjusting a circuit's resonant frequency to match one station out of the airwaves.
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