ElectricalAC CircuitsmediumFE ExamCoursework

Resonant Frequency of an LC Circuit

A series resonant circuit has $L = 10\ \text{mH}$ and $C = 100\ \text{nF}$ .

Find the resonant frequency in hertz.

Given

L	10 mH	Inductance
C	100 nF	Capacitance

Hint 1

At resonance the two reactances cancel — set X_L = X_C and solve.

Hint 2

f₀ = 1/(2π√(LC)).

Hint 3

Convert carefully: 10 mH = 10⁻² H and 100 nF = 10⁻⁷ F, so LC = 10⁻⁹.

Worked solution — try the problem first

At resonance the inductive and capacitive reactances cancel, $X_L = X_C$ , giving

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

Convert to base units — this is where the problem is actually won or lost:

$L = 10\ \text{mH} = 10\times10^{-3}\ \text{H}, \qquad C = 100\ \text{nF} = 100\times10^{-9}\ \text{F}$

$LC = (10\times10^{-3})(100\times10^{-9}) = 1.0\times10^{-9}$

$\sqrt{LC} = \sqrt{1.0\times10^{-9}} = 3.162\times10^{-5}$

$f_0 = \frac{1}{2\pi(3.162\times10^{-5})} = \frac{1}{1.987\times10^{-4}} = 5033\ \text{Hz}$

$\boxed{f_0 \approx 5.03\ \text{kHz}}$

Watch the units of the answer. Omitting the $2\pi$ gives $\omega_0 = 31{,}623$ rad/s, which is correct as an angular frequency but is not the hertz value requested.

Concepts:ResonanceLC circuitsReactanceUnit prefixes

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