LC Resonance: Frequency and Analogy Revision Notes | A Level AQA Physics
The resonant frequency of a parallel LC circuit is f₀ = 1/(2π√(LC)), where L is the inductance (H) and C is the capacitance (F). 14 questions answered.
LC Resonance: Frequency and Analogy is part of Analogue signal processing (A-level only) in A Level AQA Physics.
7 min
Reading time
34
Lessons in this topic
A Level AQA
Pathway
Reviewed by Revise Right
Content review
What this lesson covers
The resonant frequency of a parallel LC circuit is f₀ = 1/(2π√(LC)), where L is the inductance (H) and C is the capacitance (F). 14 questions answered.
State the resonant frequency formula for a parallel LC circuit and identify each variable
Calculate the resonant frequency of a parallel LC circuit given values of inductance and capacitance
Explain the analogy between an LC circuit and a mass-spring system, mapping inductance to mass and capacitance to spring constant
Compare the energy exchange in an LC circuit with that in a mechanical oscillator
Key ideas to keep in view
Use this lesson to stay inside Analogue signal processing (A-level only) while connecting the detail back to the wider Electronics (A-level only) topic.
Resonant frequency: The specific frequency at which a parallel LC circuit oscillates with maximum amplitude, determined by the values of inductance and capacitance. Given by $f_0 = \frac{1}{2\pi\sqrt{LC}}$.
Inductance (L): A property of a coil that opposes changes in current, measured in henries (H). In the LC analogy, inductance plays the role of mass — it resists changes in motion (current flow).
Capacitance (C): A property of a capacitor that stores charge and opposes changes in voltage, measured in farads (F). In the LC analogy, capacitance plays the role of a spring — it stores energy and provides a restoring force.
Parallel LC circuit: A circuit in which an inductor and capacitor are connected in parallel, capable of sustaining oscillating currents at a characteristic resonant frequency.
Revision notes
Read the core explanations from LC Resonance: Frequency and Analogy before testing yourself from memory.
1. Why LC Circuits Oscillate
Push a swing and it keeps going back and forth. Connect a charged capacitor to an inductor and something remarkably similar happens — charge sloshes back and forth between them at a precise, predictable frequency. That frequency is what this lesson is about.
This behaviour is called LC resonance, and it sits at the heart of radio tuning, signal filtering, and analogue signal processing. The AQA specification focuses on parallel LC circuits only, so every example here uses that arrangement.
Before diving into the formula, it helps to picture what is physically happening inside the circuit. A capacitor stores energy in an electric field between its plates. An inductor stores energy in a magnetic field around its coil. In a parallel LC circuit, these two components continuously trade energy back and forth — and that trading process is the oscillation.
2. The Resonant Frequency Formula
The AQA specification gives you one key equation for this topic. You do not need to derive it — but you do need to use it confidently and understand what each symbol means.
Notice that $f_0$ decreases when either $L$ or $C$ increases. A larger inductance means the circuit is more reluctant to change its current — it oscillates more slowly. A larger capacitance means more charge is stored for the same voltage — again, the oscillation slows down. Both effects pull the resonant frequency downward.
Conversely, reducing either $L$ or $C$ pushes $f_0$ upward. Radio tuners exploit exactly this: a variable capacitor lets you shift $f_0$ to match the frequency of the station you want to receive.
3. Calculating Resonant Frequency: Worked Example
The formula looks compact, but exam questions often give values in non-SI units — microfarads, millihenries — and the square root step catches students out. Work through this example carefully, paying attention to unit conversions before substitution.
Worked example: Calculating the resonant frequency of a parallel LC circuit
A parallel LC circuit has an inductance of $L = 2.0 \text{ mH}$ and a capacitance of $C = 50 \text{ nF}$. Calculate the resonant frequency $f_0$. [3 marks]
$L = 2.0 \text{ mH} = 2.0 \times 10^{-3} \text{ H}$
$C = 50 \text{ nF} = 50 \times 10^{-9} \text{ F} = 5.0 \times 10^{-8} \text{ F}$ — Always convert to SI base units (H and F) before substituting. This is where most marks are lost. [M1]
$LC = 2.0 \times 10^{-3} \times 5.0 \times 10^{-8}$
$= (2.0 \times 5.0) \times 10^{-3-8}$
$= 10.0 \times 10^{-11}$
$= 1.0 \times 10^{-10} \text{ H F}$ — Multiply the coefficients and add the exponents separately. Keep the full precision here — do not round yet.
Here is where the physics becomes genuinely elegant. A mass on a spring and a parallel LC circuit are mathematically identical systems — they obey the same equations of motion, just with different physical quantities playing each role.
Picture a mass hanging from a spring. Pull it down and release it: the spring pulls it back up, it overshoots, the spring then pushes it back down, and so on. Energy alternates between kinetic energy (mass moving) and elastic potential energy (spring stretched or compressed). The system oscillates at a natural frequency determined by the mass and the spring stiffness.
Now picture the LC circuit. Give the capacitor an initial charge and connect it to the inductor. The capacitor discharges, driving current through the inductor. The inductor's magnetic field builds up, storing energy. When the capacitor is fully discharged, the inductor's collapsing field drives current back the other way, recharging the capacitor with opposite polarity. The cycle repeats — energy alternates between the electric field of the capacitor and the magnetic field of the inductor.
Mass-Spring System
Parallel LC Circuit
Mass $m$ (kg)
Inductance $L$ (H)
Spring constant $k$ (N/m)
Reciprocal of capacitance $1/C$ (F$^{-1}$)
Displacement $x$ (m)
Charge $Q$ (C)
Velocity $v$ (m/s)
Current $I$ (A)
Kinetic energy $\frac{1}{2}mv^2$
Magnetic energy $\frac{1}{2}LI^2$
Elastic potential energy $\frac{1}{2}kx^2$
Electric potential energy $\frac{Q^2}{2C}$
Natural frequency $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
Resonant frequency $f_0 = \frac{1}{2\pi\sqrt{LC}}$
The structural similarity is striking. In both systems, the oscillation frequency is set by the ratio of a 'restoring' quantity to an 'inertial' quantity. In the spring system, a stiffer spring (larger $k$) raises the frequency; in the LC circuit, a smaller capacitance (larger $1/C$) raises the frequency. In the spring system, a heavier mass (larger $m$) lowers the frequency; in the LC circuit, a larger inductance (larger $L$) lowers the frequency.
4a. Inductance as the Analogue of Mass
When current is flowing through an inductor, the magnetic field stores energy. If you try to change that current rapidly, the inductor generates a back-EMF that opposes the change — exactly as a massive object resists acceleration. The larger $L$ is, the more the circuit 'fights back' against rapid changes in current, slowing the oscillation.
4b. Capacitance as the Analogue of a Spring
A stiffer spring (larger $k$) produces a higher oscillation frequency. The electrical equivalent is a smaller capacitance (larger $1/C$), which also raises $f_0$. A softer spring (smaller $k$) slows the oscillation — as does a larger capacitance. The parallel is exact.
5. Energy Exchange in the LC Circuit
Understanding the energy exchange cycle deepens your ability to explain LC resonance in exam questions that use the command word 'describe' or 'explain'. Walk through one complete oscillation cycle.
Quarter-cycle 1: Capacitor fully charged, no current flowing. All energy is stored as electric potential energy in the capacitor's field.
Quarter-cycle 2: Capacitor discharges through the inductor. Current rises, magnetic field builds. Energy transfers from capacitor to inductor.
Quarter-cycle 3: Capacitor fully discharged, current at maximum. All energy is stored as magnetic energy in the inductor's field.
Quarter-cycle 4: Inductor's collapsing field drives current onward, recharging the capacitor with reversed polarity. Energy transfers back to the capacitor.
After one full cycle: Capacitor is fully charged again (opposite polarity). The process repeats at frequency $f_0$.
In an ideal (lossless) LC circuit, this exchange continues indefinitely. In practice, resistance in the wires and inductor dissipates energy as heat, and the oscillations decay — analogous to a damped mass-spring system where friction gradually reduces the amplitude.
6. Applying the Analogy: Comparing the Two Systems
AQA exam questions on this topic often ask you to 'compare' or 'explain' using the analogy. The key skill is mapping each physical quantity precisely — not vaguely saying 'they are similar'. Use the table from Section 4 as your reference and practise writing explicit comparisons.
For example, if an examiner asks why increasing the inductance lowers the resonant frequency, a full-mark answer connects the analogy explicitly: increasing $L$ is analogous to increasing the mass of the oscillating object. A heavier mass oscillates more slowly for the same spring stiffness, so a larger inductance produces a lower resonant frequency for the same capacitance.
It is worth being precise here: the analogy maps $C$ to $1/k$ (compliance), not to $k$ (stiffness). Some students instinctively say 'capacitance is like the spring stiffness' — but the direction is reversed. A larger $C$ lowers $f_0$, just as a softer spring (smaller $k$) lowers the natural frequency. Always check the direction of the effect when applying the analogy.
7. Exam Technique and Common Mistakes
This topic is compact but precise. The marks available split roughly equally between calculation and conceptual explanation. The following guidance targets the most frequent sources of mark loss in AQA mark schemes.
For extended-response questions that ask you to evaluate or compare, structure your answer as: opening statement of the analogy, mapping of each quantity with justification, statement of the shared mathematical form, and a brief note on where the analogy breaks down (e.g. real circuits have resistance; real springs have damping). That structure reliably captures the higher-order marks.
LC resonance ties together inductance, capacitance, and oscillation into a single elegant formula — and the mass-spring analogy gives you a physical intuition that makes the mathematics meaningful rather than arbitrary. Carry both the formula and the analogy into your exam, and you have everything AQA requires on this topic.
For an LC circuit, the resonant frequency is f0 = 1/(2π√(LC)). This equation shows that increasing either the inductance or the capacitance lowers the resonant frequency, because the oscillation takes longer to exchange energy between the magnetic and electric stores.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course.
What is the resonant frequency formula for an LC circuit?
The resonant frequency of a parallel LC circuit is f₀ = 1/(2π√(LC)), where L is the inductance (H) and C is the capacitance (F). Increasing either L or C lowers the resonant frequency, which is why a radio tuner uses a variable capacitor to shift f₀.
How do you calculate the resonant frequency of an LC circuit?
The resonant frequency of an LC circuit is calculated by converting L and C to SI units, taking √(LC), then using f₀ = 1/(2π√(LC)). For L = 2.0 mH and C = 50 nF, LC = 1.0 × 10⁻¹⁰, √(LC) = 1.0 × 10⁻⁵, so f₀ = 1 ÷ (6.2832 × 10⁻⁵) ≈ 1.6 × 10⁴ Hz.
What is the analogy between an LC circuit and a mass-spring system?
In the mass-spring analogy, inductance L plays the role of mass, charge Q matches displacement, current matches velocity, and 1/C matches the spring constant k. Magnetic energy ½LI² matches kinetic energy, and the capacitor's electric energy Q²/2C matches elastic potential energy.
Why is inductance like mass in an LC circuit?
Inductance is like mass because both represent inertia. A large mass resists changes in velocity, and a large inductance resists changes in current by generating a back-EMF that opposes the change, so a larger L slows the oscillation and lowers the resonant frequency.
How does energy move in an LC circuit?
In an LC circuit energy passes back and forth between the capacitor's electric field and the inductor's magnetic field. The charged capacitor discharges through the inductor, building its magnetic field; at maximum current all energy is magnetic; the collapsing field then recharges the capacitor with reversed polarity, and the cycle repeats at f₀.
Why is capacitance like a spring in an LC circuit?
Capacitance is like the spring in an LC circuit because a capacitor stores charge and, when displaced from equilibrium, provides a restoring voltage that drives current back, as a spring's restoring force drives a mass back. More precisely, 1/C maps to the spring constant k, so a smaller capacitance acts like a stiffer spring.
How does a radio tuner use an LC circuit?
A radio tuner uses a variable capacitor in an LC circuit to shift the resonant frequency, f₀ = 1/(2π√(LC)), to match the frequency of the station to be received. Reducing the capacitance raises the resonant frequency and increasing it lowers the frequency.
Why do the oscillations in a real LC circuit die away?
Oscillations in a real LC circuit die away because resistance in the wires and inductor dissipates energy as heat each cycle. In an ideal, lossless LC circuit the energy exchange would continue indefinitely; the decay is analogous to a damped mass-spring system where friction gradually reduces the amplitude.
What is LC resonance?
LC resonance is the oscillation of charge back and forth between a capacitor and an inductor at a precise, predictable frequency, f₀ = 1/(2π√(LC)). The capacitor stores energy in an electric field and the inductor in a magnetic field, and the two trade energy continuously; it is at the heart of radio tuning and signal filtering.
How does capacitance affect resonant frequency?
Increasing the capacitance lowers the resonant frequency of an LC circuit, because more charge is stored for the same voltage and the oscillation slows down. This follows from f₀ = 1/(2π√(LC)); reducing the capacitance raises the resonant frequency, like using a stiffer spring.
Is capacitance analogous to spring stiffness?
Capacitance is not analogous to spring stiffness: the analogy maps C to 1/k, the compliance, so the direction is reversed. A larger capacitance lowers the resonant frequency, just as a softer spring with smaller k lowers the natural frequency of a mass-spring system.
What are the limitations of the LC mass-spring analogy?
The LC mass-spring analogy breaks down because real circuits have resistance and real springs have damping, so neither oscillates indefinitely. Apart from that, the two systems share the same mathematical form, with inductance mapped to mass, 1/C to spring constant, charge to displacement and current to velocity.
What corresponds to kinetic energy in an LC circuit?
In an LC circuit, magnetic energy in the inductor, ½LI², is the analogue of the kinetic energy of a mass on a spring, ½mv². Electric potential energy in the capacitor, Q²/2C, is the analogue of the elastic potential energy of the spring, ½kx².
Why does inductance resist changes in current?
Inductance resists changes in current because the inductor's magnetic field stores energy, and any rapid change in current makes the inductor generate a back-EMF that opposes the change. This is like inertia in a massive object, so a larger inductance slows the oscillation of an LC circuit.