RLC circuits
Study inductance, core materials and the energy an inductor stores, then analyse RL and LC transients and the damping regimes of RLC circuits.
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Before you begin
An inductor opposes changes in current and stores energy in its magnetic field. The lessons start with self-inductance, combinations of inductors and the effect of dielectric and ferromagnetic materials, then find the energy stored in an inductor. They finish by modelling circuits with differential equations: exponential growth and decay in RL circuits, oscillation in LC circuits and damped oscillation in RLC circuits.
Be comfortable with: D.C. circuits (H2), electromagnetic induction (H2) and electric and magnetic fields (H3).
Learning goals
- Define self-inductance as induced e.m.f. per unit rate of change of current, and use V = L dI/dt
- Describe mutual inductance as a circuit's tendency to oppose a change of current in a nearby circuit
- Calculate the combined inductance of inductors in series and in parallel
- Explain qualitatively how a dielectric increases capacitance and why a strong enough field causes dielectric breakdown
- Explain qualitatively how a ferromagnetic core increases inductance, non-linearly near saturation
- Apply U = ½LI² to the energy stored in an inductor.
- Derive the expression U = ½LI² for the potential energy stored in an inductor by considering the work done on charges.
- Solve RL series circuits with a constant e.m.f. using first-order differential equations
- Solve LC series circuits without an e.m.f. source using second-order differential equations
- Solve problems on RLC series circuits without an e.m.f. source by verifying and using given solutions
Syllabus statements covered
- define self-inductance as the ratio of the e.m.f. induced in an electrical circuit / component to the rate of dI change of current causing it and use V = L to solve problems dt
- show an understanding that mutual inductance is the tendency of an electrical circuit / component to oppose a change in the current in a nearby electrical circuit / component
- show a qualitative understanding that dielectric materials enhance capacitance, and that dielectric breakdown can occur when the electric field is sufficiently strong (knowledge of the quantitative modification of electric fields in matter through the permittivity is not required)
- show a qualitative understanding that ferromagnetic materials enhance inductance and that this enhancement is non-linear especially near saturation (knowledge of the quantitative modification of magnetic fields in matter through the permeability is not required)
- derive, by considering work done on charges, the expression for potential energy stored in an inductor, 1 U = LI 2 , and use this to solve problems 2
- solve problems using the formulae for the combined inductance of two or more inductors in series and in parallel
- solve problems involving circuits with resistors, inductors, and sources of constant e.m.f. (includes solving first-order differential equations) [RL series circuits with constant e.m.f. source]
- solve problems involving circuits with inductors and capacitors only (includes solving second-order differential equations) [LC series circuits without e.m.f. source]
- solve problems involving circuits with resistors, inductors and capacitors only (candidates are not expected to solve the general second-order differential equations, though they can be asked to verify and use particular solutions). [RLC series circuits without e.m.f. source]
Lessons
Work through them in order.
- InductanceDefine self-inductance, use consistent voltage signs and explain mutual induction between nearby circuits.
- Combining inductorsDerive series and parallel equivalent inductance from shared current or voltage, then reduce an uncoupled network.
- Dielectrics, capacitance and breakdownExplain dielectric polarisation, distinguish fixed charge from fixed voltage and identify the breakdown limit.
- Ferromagnetic cores and saturationExplain how magnetic domains enhance inductance and why the additional core response diminishes near saturation.
- Energy in an inductorDerive and apply U = ½LI² for the energy stored in an inductor.
- RL circuitsDerive current growth and decay from a first-order differential equation.
- LC circuitsModel ideal electromagnetic oscillations using charge, current, phase, and energy.
- RLC circuit analysisClassify source-free series responses and verify particular solutions.
Practise and check
Or choose
Topic reference
Quick reference
| Idea | Relation | Notes |
|---|---|---|
| inductor voltage | VL = L dI/dt | voltage drop in the positive-current direction |
| combining inductors | series L₁ + L₂; parallel 1/L_eq = 1/L₁ + 1/L₂ | uncoupled coils only |
| energy stored | U = (1/2)LI² | constant L |
| RL circuit | τ = L/R; I = I_∞(1-e^(-t/τ)) or I = I₀e^(-t/τ) | growth from zero or source-free decay |
| LC circuit | ω₀ = 1/square root of LC, T = 2π square root of LC | q = Q cos ω₀t when I(0) = 0 |
| series RLC circuit | Lq double dot + Rq dot + q/C = 0, Rc = 2 square root of (L/C) | γ = R/(2L), ωd = square root of (ω₀²-γ²) |
Core knowledge to remember
- The induced e.m.f. opposes the change in current, not the current. An ideal inductor carrying a steady current has no voltage across it but still stores energy.
- The current through an inductor cannot change instantly while the voltage across it is finite.
- A dielectric increases capacitance. A ferromagnetic core increases inductance, but its effect shrinks as the core saturates.
- Use current I(t) as the variable for an RL circuit. Use capacitor charge q(t) for LC and RLC circuits, then find the current from I = dq/dt.
- In an LC circuit, energy passes between the capacitor and the inductor while the total stays constant. In an RLC circuit, the resistor dissipates energy at the rate RI².
- An RLC circuit is underdamped if R < Rc, critically damped if R = Rc and overdamped if R > Rc.
Common mistakes
- Saying an inductor opposes the current rather than a change in current.
- Mixing up I and q; remember that I = dq/dt.
- Using the growth expression for a decaying current, or the reverse.
- Using ω₀ = 1/square root of LC as the frequency of a damped oscillation without checking the regime.
- Applying the series and parallel inductance rules when the coils are magnetically coupled.