LC oscillations | PhysicsRank

An ideal capacitor can become uncharged while its circuit current is largest. That moment looks strange only if the capacitor is the entire energy story. The inductor is carrying the energy then, and its current can charge the capacitor with the opposite sign.

The mechanical oscillator provides a useful bridge, but the correspondence must preserve the measured variable: charge behaves like displacement, and current behaves like velocity.

What you will learn to do

Derive the LC clock and current phase from a signed charge convention.
Use both initial field stores after switching.
Classify free RLC decay and distinguish it from driven response.
Calculate source and component voltages with consistent peak or RMS amplitudes.
Distinguish current, charge and free-decay frequencies.
Infer component values from a measured half-power curve.

Useful before this chapterInduction, moving conductors and magnetic energy →Alternating circuits: phase, RMS and real power →Damping, driving and the meaning of resonance →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

Which two stores exchange energy in an ideal LC loop, and what mechanical pair can help you picture them?

Compare your reasoning

The capacitor’s electric field and the inductor’s magnetic field exchange energy. They correspond to spring potential energy and kinetic energy in a linear mechanical oscillator; the analogy concerns the equations, not literal moving masses in the capacitor.

Revisit the explanation →

01 / From LC energy exchange to RLC resonance

A capacitor and inductor exchange two forms of energy

Charge a capacitor and connect it to an ideal inductor in a closed loop. The capacitor initially has electric-field energy. Its voltage drives a changing current, building magnetic-field energy in the inductor. When the capacitor first reaches zero charge, current is largest rather than zero. The current continues and charges the capacitor with opposite polarity.

The inductor does not create charge or energy. It allows a nonzero current to persist while the capacitor voltage passes through zero, just as inertia allows a mass to pass through its equilibrium position. The total energy is shared between two stores. In the ideal lossless circuit it is constant.

We assume lumped elements, negligible propagation delay around the loop, constant positive L and C, and no radiation or stray coupling. Real coils also have winding resistance and parasitic capacitance. The ideal model is useful only when those effects are small or have been included explicitly.

E=\frac{q^2}{2C}+\frac12Li^2

q is the signed charge on a named capacitor plate. Define positive loop current i as the current charging that plate, so i = dq/dt. A discharging initially positive capacitor therefore has negative i in this convention.

WORKED EXAMPLE

Locate the quarter-cycle state by energy

A capacitor C = 16 μF initially at +12 V is connected to an ideal L = 0.25 H inductor, with zero initial current. What is the maximum current magnitude?

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Focus: A capacitor and inductor exchange two forms of energy
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.
Equations in this section (LaTeX):
E=\frac{q^2}{2C}+\frac12Li^2
Meaning: q is the signed charge on a named capacitor plate. Define positive loop current i as the current charging that plate, so i = dq/dt. A discharging initially positive capacitor therefore has negative i in this convention.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / From LC energy exchange to RLC resonance

Derive the phase relation instead of memorizing two sinusoids

For the isolated loop, Kirchhoff’s voltage law gives L di/dt + q/C = 0 with the chosen passive signs. Replace i by dq/dt to obtain the same differential form as a mass on a spring. The capacitor term acts as the restoring term and inductance multiplies the second derivative.

For an initially charged capacitor with no current, q(t) is a cosine. Differentiate it to obtain current, including the minus sign. The current and charge extrema are one quarter-cycle apart. The electric and magnetic energies depend on squares, so each repeats twice during one signed charge cycle.

Knowing the oscillation frequency does not specify an amplitude. L and C set the timescale; the starting voltage and current set how much energy is available. Confusing these roles leads to the claim that changing the starting charge changes the ideal period, which the linear equation does not support.

\ddot q+\omega_0^2q=0,\quad \omega_0=\frac1{\sqrt{LC}},\quad q=Q_0\cos\omega_0t,\quad i=-\omega_0Q_0\sin\omega_0t

This particular solution requires q(0) = Q₀ and i(0) = 0. For a different initial current, include a sine term in q.

WORKED EXAMPLE

Convert between angular frequency and the actual quarter-period

For L = 0.25 H and C = 16 μF, find ω₀, the period and the first time of maximum current magnitude after release from a charged capacitor.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Focus: Derive the phase relation instead of memorizing two sinusoids
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.
Equations in this section (LaTeX):
\ddot q+\omega_0^2q=0,\quad \omega_0=\frac1{\sqrt{LC}},\quad q=Q_0\cos\omega_0t,\quad i=-\omega_0Q_0\sin\omega_0t
Meaning: This particular solution requires q(0) = Q₀ and i(0) = 0. For a different initial current, include a sine term in q.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / From LC energy exchange to RLC resonance

A nonzero starting current changes the amplitude and phase

The two field stores can both contain energy at the moment an isolated LC circuit begins its free motion. There is no reason to discard either initial term. Solve with q(0) = q₀ and i(0) = i₀, or use their total energy to find the largest possible charge or current.

The motion traces an ellipse in the q–i plane. A starting state is a point on that ellipse, while energy determines which ellipse it is. This gives a geometric way to separate phase from amplitude.

A switching operation can change which elements form the loop, but finite voltage across an inductor keeps its current continuous, and finite current into a capacitor keeps its voltage continuous. Do not set those quantities to zero merely because the external source has been disconnected. An impulsive idealization would need to be stated and analysed separately.

q(t)=q_0\cos\omega_0t+\frac{i_0}{\omega_0}\sin\omega_0t,\qquad Q_{\max}=\sqrt{q_0^2+(i_0/\omega_0)^2}

The derivative at zero is i₀. The amplitude follows from combining the cosine and sine coefficients, or independently from the conserved energy.

WORKED EXAMPLE

Use both initial stores after switching

An isolated ideal loop has L = 0.25 H, C = 16 μF, initial capacitor voltage +6 V and initial charging current +0.064 A. Find the largest subsequent capacitor voltage magnitude.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

STOP & THINK

For L = 0.25 H and C = 16 μF, an isolated state has VC = 6 V and |i| = 0.064 A. What is total stored energy in mJ?

Answer in mJ.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Focus: A nonzero starting current changes the amplitude and phase
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.
Equations in this section (LaTeX):
q(t)=q_0\cos\omega_0t+\frac{i_0}{\omega_0}\sin\omega_0t,\qquad Q_{\max}=\sqrt{q_0^2+(i_0/\omega_0)^2}
Meaning: The derivative at zero is i₀. The amplitude follows from combining the cosine and sine coefficients, or independently from the conserved energy.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

04 / From LC energy exchange to RLC resonance

Resistance changes energy exchange into a decaying response

Add a constant series resistance R. The loop equation becomes Lq̈ + Rq̇ + q/C = 0. The correspondence with mechanics is m ↔ L, b ↔ R and k ↔ 1/C. Every inference from the previous chapter must preserve that mapping; replacing k with C would give the wrong units and the wrong frequency.

The energy derivative is exactly −Ri². For R < 2√(L/C), charge and current oscillate with damped angular frequency ωd while their amplitude envelopes decay at rate R/(2L). The R/L current-decay rate of a lone RL circuit is not the amplitude-decay rate of an underdamped RLC circuit.

If R reaches or exceeds the critical value, the free response is nonoscillatory. This does not imply that the same circuit cannot pass a sinusoidal current when an external source drives it. Free response and forced response answer different questions.

\gamma=\frac{R}{2L},\quad \omega_d=\sqrt{\frac1{LC}-\frac{R^2}{4L^2}},\quad R_c=2\sqrt{L/C},\quad\frac{dE}{dt}=-Ri^2

The damping classification assumes positive, constant element values. Coil winding resistance and source resistance belong in the total series R if they remain in the loop.

WORKED EXAMPLE

Classify the ring-down without using a driven peak

For L = 0.25 H and C = 16 μF compare free responses with total series R = 20 Ω and R = 300 Ω.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Focus: Resistance changes energy exchange into a decaying response
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.
Equations in this section (LaTeX):
\gamma=\frac{R}{2L},\quad \omega_d=\sqrt{\frac1{LC}-\frac{R^2}{4L^2}},\quad R_c=2\sqrt{L/C},\quad\frac{dE}{dt}=-Ri^2
Meaning: The damping classification assumes positive, constant element values. Coil winding resistance and source resistance belong in the total series R if they remain in the loop.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / From LC energy exchange to RLC resonance

Current resonance comes from cancelling reactive voltage

Drive a series RLC circuit with a sinusoidal source after transients have decayed. The same current flows through all three elements. The resistor voltage is in phase with current; the inductor voltage leads by 90° and the capacitor voltage lags by 90°. Their reactive contributions therefore subtract before combining with the resistor voltage.

For a fixed source voltage amplitude and frequency-independent R, the current is largest when ωL = 1/(ωC). The source then sees a purely resistive impedance even though energy continues moving between the capacitor and inductor. Resonance does not mean those two components have individually zero voltage.

Choose either peak or RMS amplitudes and keep that choice throughout a calculation. The impedance formula works for both. Average resistor power is Irms²R, or Ipeak²R/2. Mixing peak voltage with RMS current manufactures a factor of √2.

Z=R+j\left(\omega L-\frac1{\omega C}\right),\qquad I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{|Z|},\qquad \tan\phi_I=\frac{\omega L-1/(\omega C)}{R}

φI is the phase by which source voltage leads current. Below resonance the current leads the source; above resonance it lags. This current phase differs from the charge/displacement lag in the mechanical analogy.

WORKED EXAMPLE

Resolve large component voltages without violating the source law

A series circuit has L = 0.25 H, C = 16 μF and total R = 20 Ω. A 4.0 V RMS source drives it at ω₀. Find current, inductor voltage, capacitor voltage and average power.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Focus: Current resonance comes from cancelling reactive voltage
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.
Equations in this section (LaTeX):
Z=R+j\left(\omega L-\frac1{\omega C}\right),\qquad I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{|Z|},\qquad \tan\phi_I=\frac{\omega L-1/(\omega C)}{R}
Meaning: φI is the phase by which source voltage leads current. Below resonance the current leads the source; above resonance it lags. This current phase differs from the charge/displacement lag in the mechanical analogy.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / From LC energy exchange to RLC resonance

Charge, capacitor voltage and current have different response peaks

The mechanical displacement corresponds to capacitor charge q, not to current. Because i = dq/dt, their sinusoidal amplitudes satisfy I = ωQ. A current maximum cannot be assigned to Q without dividing by ω first. Capacitor voltage Q/C follows the charge peak when C is constant.

For the same constant-amplitude source, the current and resistor-power peaks occur at ω₀. The capacitor-voltage peak occurs at √(ω₀² − R²/(2L²)) if that value is real and positive; otherwise its largest amplitude is approached quasistatically. The damped free angular frequency contains R²/(4L²) instead, giving a third distinct quantity.

These differences are not contradictory formulas. They describe different observations: free ring-down, forced capacitor voltage and forced current. Before choosing a formula, name the state of the source, the measured element and whether an amplitude or power is plotted.

INTERACTIVE LAB

Compare the mechanical and electrical response

Select the circuit interpretation. Charge corresponds to displacement and current corresponds to velocity. Keep the source amplitude fixed while sweeping frequency; compare the charge and current peaks.

MEMBERSThe interactive lab is part of membership.Unlock →

WORKED EXAMPLE

Name three frequencies for one physical circuit

For L = 0.25 H, C = 16 μF and R = 20 Ω, compare the current-peak frequency, free ring-down frequency and capacitor-voltage-peak frequency.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Focus: Charge, capacitor voltage and current have different response peaks
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / From LC energy exchange to RLC resonance

Infer circuit values from a response curve

A resonance curve can reveal more than one component if enough independent observations are available. For a series circuit with constant R, half maximum power means |ωL − 1/(ωC)| = R. At those frequencies current amplitude is 1/√2 of its resonant value.

The two positive half-power angular frequencies have difference R/L and product 1/(LC). These identities are exact for the stated series model, without assuming an extremely narrow peak. If you use hertz, convert the angular width with Δω = 2πΔf.

Experimental source resistance, coil resistance and measurement loading can all broaden the peak. The inferred R is their total series effect under the model. A curve with several peaks or strongly frequency-dependent losses should not be forced into a single ideal RLC interpretation.

\omega_+-\omega_-=\frac RL,\qquad \omega_+\omega_-=\omega_0^2,\qquad Q=\frac{\omega_0L}{R}

These are current/power half-width statements. They do not describe the half-height of an arbitrary voltage curve.

WORKED EXAMPLE

Reconstruct a circuit from three measured features

A constant 2.0 V peak source drives a series RLC circuit. The maximum current amplitude is 0.10 A; half-power frequencies are 400 and 625 rad/s. Infer R, L and C.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From LC energy exchange to RLC resonance
Focus: Infer circuit values from a response curve
Learning goals:
- Derive the LC clock and current phase from a signed charge convention.
- Use both initial field stores after switching.
- Classify free RLC decay and distinguish it from driven response.
- Calculate source and component voltages with consistent peak or RMS amplitudes.
- Distinguish current, charge and free-decay frequencies.
- Infer component values from a measured half-power curve.
Equations in this section (LaTeX):
\omega_+-\omega_-=\frac RL,\qquad \omega_+\omega_-=\omega_0^2,\qquad Q=\frac{\omega_0L}{R}
Meaning: These are current/power half-width statements. They do not describe the half-height of an arbitrary voltage curve.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

A capacitor voltage crossing zero does not stop the current.
L and C set the ideal clock; the initial state sets amplitude and phase.
Resonance claims must name the measured quantity and the source condition.
Large internal voltages can cancel at the source while still stressing components.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

For L = 0.040 H and C = 10 μF, what critical series resistance separates oscillatory and nonoscillatory free response? Give at least three significant figures, or an exact fraction.

Answer in Ω.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 2

A series circuit has R = 12 Ω and L = 0.060 H. What is its current-response half-power angular bandwidth?

Answer in rad/s.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

At the first zero of capacitor charge in a lossless release from rest, what is true?

Current is zero
Current magnitude is maximum
All circuit energy has disappeared

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SELF CHECK 4

A free RLC circuit is overdamped. Does that forbid a current maximum versus driving frequency in a driven series experiment?

Yes, because every use of resonance requires free oscillations
No; free response and forced current response have different criteria
Yes, unless the capacitor is removed

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

The capacitor starts with only part of the energyAdvanced · Hints and guided reasoning Reconstruct a capacitor from the response curveAdvanced · Hints and guided reasoning Find the resonant angular frequencyIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

Why must an LC problem specify initial current as well as initial capacitor charge?

Compare your reasoning

Both stores may contain energy, and both initial conditions set the phase. Charge alone does not determine the full motion or its maximum subsequent voltage.

Revisit the explanation →

A few days later: change the situation

Why should you not use the capacitor-voltage peak frequency automatically for the current peak in a driven series RLC circuit?

Compare your reasoning

Current is the time derivative of charge, so its amplitude contains an extra factor of drive frequency. Different measured amplitudes can peak at different frequencies even in the same circuit.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑