Resonant Frequency Calculator
Calculate the resonant frequency of LC and RLC circuits. Find quality factor, bandwidth, damping ratio, and characteristic impedance from inductance.
Resonant Frequency Calculator
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Formula: f = 1 / (2 * pi * sqrt(L * C))
Worked example — 101.3 pF capacitor needed to resonate at 1 MHz with 250 uH
Formula
f = 1 / (2 * pi * sqrt(L * C))
The resonant frequency is calculated using the inductance (L) in henries and capacitance (C) in farads. Q factor = (1/R)*sqrt(L/C) for series RLC. Bandwidth = f/Q. Characteristic impedance = sqrt(L/C).
Worked Examples
Example 1: AM Radio Tuning Circuit
Problem:An AM radio uses a 250 uH inductor. What capacitance tunes to 1000 kHz?
Solution:f = 1 / (2*pi*sqrt(L*C)) Rearranging: C = 1 / (4*pi^2*f^2*L) C = 1 / (4 * 9.8696 * (1e6)^2 * 250e-6) C = 1.013e-10 F = 101.3 pF
Result:101.3 pF capacitor needed to resonate at 1 MHz with 250 uH
Example 2: Audio Crossover Filter
Problem:Design a resonant circuit at 3 kHz using a 10 mH inductor and find the Q with 50 ohm resistance.
Solution:f = 1/(2*pi*sqrt(L*C)), so C = 1/(4*pi^2*f^2*L) C = 1/(4*9.8696*9e6*0.01) = 2.814e-7 F = 281.4 nF Q = (1/50)*sqrt(0.01/2.814e-7) = 3.77
Result:C = 281.4 nF, Q = 3.77, BW = 796 Hz
Frequently Asked Questions
What is resonant frequency in an LC circuit?
Resonant frequency is the frequency at which an LC (inductor-capacitor) circuit naturally oscillates when energy is exchanged between the magnetic field of the inductor and the electric field of the capacitor. At resonance, the inductive reactance equals the capacitive reactance, and they cancel each other out. The formula is f = 1 / (2 * pi * sqrt(L * C)), where L is inductance in henries and C is capacitance in farads.
What is the quality factor (Q) and why does it matter?
The quality factor Q measures how underdamped a resonant circuit is, representing the ratio of energy stored to energy lost per cycle. A higher Q means sharper resonance with a narrower bandwidth and less energy loss. For a series RLC circuit, Q = (1/R) * sqrt(L/C). A circuit with Q of 100 has a bandwidth that is 1/100th of its resonant frequency. High-Q circuits are essential in radio receivers for selecting narrow frequency bands.
How does resistance affect resonant frequency?
In an ideal LC circuit with no resistance, the resonant frequency depends only on L and C. When resistance is added (RLC circuit), the undamped natural frequency remains the same, but the actual oscillation frequency (damped frequency) decreases slightly. The damping ratio determines whether the circuit oscillates (underdamped), critically damps, or overdamps. Resistance also broadens the bandwidth and reduces the Q factor.
Where are resonant circuits used in practice?
Resonant circuits are fundamental to radio tuning, where they select specific frequencies from the electromagnetic spectrum. They are used in bandpass filters, oscillators, impedance matching networks, wireless power transfer systems, and MRI machines. Crystal oscillators in computers use mechanical resonance of quartz crystals. Tesla coils use resonance to achieve extremely high voltages. Even musical instruments rely on acoustic resonance principles.
How much does the resonant frequency shift if I change the inductance or capacitance?
Because f = 1 / (2 pi sqrt(LC)), the frequency depends on the inverse square root of the product LC. Multiplying either the inductance or the capacitance by four halves the resonant frequency, and dividing either by four doubles it. Since only the product matters, a small inductor with a large capacitor can tune to the same frequency as a large inductor with a small capacitor, though their characteristic impedance differs.
What is the difference between a series and a parallel LC resonant circuit?
At resonance the inductive and capacitive reactances cancel in both, but the consequence is opposite. A series LC presents minimum impedance, ideally just the winding resistance, so it acts as a notch or acceptor and passes a large current at resonance. A parallel LC tank presents maximum impedance, so it rejects the resonant frequency in a signal path or forms the high-impedance load of an oscillator.
What is bandwidth and how is it related to the quality factor?
Bandwidth is the width in hertz between the two half-power points, where the response has fallen 3 dB from its peak. It equals the resonant frequency divided by Q, so a 1 MHz circuit with a Q of 100 has a 10 kHz bandwidth. High Q therefore means sharp selectivity but slow response and tight tolerance demands, while low Q gives a broad, forgiving passband.
What is the characteristic impedance of an LC tank and why does it matter?
Characteristic impedance is the square root of L divided by C, and it equals the magnitude of either reactance at resonance. It sets how much circulating voltage and current the tank develops for a given stored energy, and it fixes Q for a given loss resistance. Choosing a high characteristic impedance favours voltage swing, while a low one favours current, which affects how the tank couples to its source and load.
What is the damping ratio and when does an RLC circuit stop oscillating?
The damping ratio for a series RLC is (R / 2) x sqrt(C / L). Below one the circuit is underdamped and rings at a damped frequency slightly below the undamped resonance. At exactly one it is critically damped and returns to rest in the shortest time without overshoot, and above one it is overdamped and creeps back slowly. Filter and transient design turns on which regime is wanted.
Why does the real resistance of an inductor limit the Q that can be achieved?
Real inductors have winding resistance, skin effect and core loss, all of which appear as an equivalent series resistance that dissipates energy each cycle. Since Q is the ratio of energy stored to energy lost per radian, that loss caps the attainable Q no matter how ideal the capacitor is. This is why high-Q coils use thick or litz wire, air or low-loss cores, and generous spacing.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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