Difficulty: Intermediate | Prerequisites: DC circuits (Ohm's law, Kirchhoff's rules), basic capacitor and inductor behaviour.
Tags: LRC circuit, RLC circuit, inductor, capacitor, resistor, steady state, transient response, DC steady state, energy storage, LC oscillation, PHYS 212, University Physics, electromagnetism
This material sits at the junction between DC circuit analysis and AC/oscillatory behaviour. You already know how resistors, capacitors, and inductors behave individually. Now you are asked what happens when all three appear in one circuit, particularly after a switch has been closed (or reopened) for a long time. The "long time" language is the exam's way of asking you about steady-state behaviour, which simplifies the analysis considerably. Understanding these steady states is also the foundation for analysing LC and LRC oscillations later.
After a switch has been closed for a long time in a DC circuit, capacitors act as open circuits (no current through them) and inductors act as short circuits (no voltage across them). When conditions change suddenly (a switch opens or closes), the inductor tries to maintain current and the capacitor tries to maintain voltage, which sets the initial conditions for the transient that follows.
Steady state (DC)
The condition reached after all transients have died away. In a DC circuit, this means nothing is changing with time any longer. In simple terms, it is the "final answer" the circuit settles into if you wait long enough.
Capacitor in DC steady state
A fully charged capacitor passes no current. It behaves as an open circuit (a break in the wire). Think of it as a bucket that is already full: no more water flows in.
Inductor in DC steady state
A fully "charged" inductor has a constant current through it and zero voltage across it. It behaves as a short circuit (a plain wire). Think of it as a heavy flywheel that is already spinning at constant speed: it takes no extra push to keep it going.
Time constant (tau, τ)
The characteristic time for an exponential transient to decay. For an RC circuit, τ = RC. For an RL circuit, τ = L/R. After roughly 5τ, the transient is effectively gone. In simple terms, this is how long you have to wait before "a long time" is a fair description.
LC oscillation
When a charged capacitor is connected to an inductor (with no resistance), energy sloshes back and forth between the electric field of the capacitor and the magnetic field of the inductor at a natural frequency ω₀ = 1/√(LC). Think of it as a spring-mass system: the capacitor is the spring and the inductor is the mass.
Q = CV
The fundamental relation between charge on a capacitor, its capacitance, and the voltage across it. In simple terms, the amount of charge stored is proportional to the voltage you apply, with C as the conversion factor.
When the problem says the switch has been closed "for a long time," all transients have died out and the circuit is in DC steady state.
In DC steady state:
The capacitor carries no current → it is an open circuit.
The inductor carries whatever current flows, but has zero voltage across it → it is a short circuit (just a wire).
Because the capacitor is open, no current flows anywhere in a series RLC loop at steady state. That means:
The current through R is zero.
The voltage across R is V_R = IR = 0.
The voltage across L is V_L = L(dI/dt) = 0 (current is constant at zero).
The entire battery voltage appears across the capacitor: V_C = V (the battery EMF).
Once you know V_C, use Q = CV.
Example: V = 10 V, C = 20 pF → Q = 20 pF × 10 V = 200 pC = 0.2 nC.
Just before reopening, the capacitor is charged to V_C = 10 V and the current is zero.
When the switch opens, the battery is disconnected. The capacitor, inductor, and resistor now form a closed loop.
The capacitor begins to discharge through R and L.
The initial current is determined by the capacitor's voltage and the total impedance of the remaining loop. At t = 0⁺ (just after reopening), the inductor still has zero current (inductors resist instantaneous changes in current). So the initial current through R is also zero, because in a series loop the current must be the same everywhere, and the inductor forces it to start at its previous value.
If L, R, and C are in series and the current was zero before the switch opened, the initial current through R is 0 A.
Important caveat: if the circuit topology means the inductor was carrying current before the switch opened (for example, if L and C were in parallel rather than in series), the analysis changes. Always trace the actual path of current before and after the switching event.
Energy in a capacitor: U_C = ½CV²
Energy in an inductor: U_L = ½LI²
In an LC circuit (no resistance), total energy is conserved: U_C + U_L = constant.
In an LRC circuit, the resistor dissipates energy as heat, so oscillations are damped.
Quantity | Formula |
|---|---|
Charge on capacitor | Q = CV |
Voltage across resistor | V_R = IR |
Voltage across inductor | V_L = L (dI/dt) |
Voltage across capacitor | V_C = Q/C |
Energy in capacitor | U_C = ½ CV² |
Energy in inductor | U_L = ½ LI² |
LC natural frequency | ω₀ = 1/√(LC) |
RL time constant | τ = L/R |
RC time constant | τ = RC |
The LRC circuit is the electrical analogue of a mechanical spring-mass-damper system. It is the basis of every radio tuner: by adjusting C (or L), you change the resonant frequency and select a different station. The same principle governs band-pass filters in audio equipment and the timing circuits in digital electronics.
Students often assume that "after a long time" means the capacitor is uncharged. In a DC circuit with a battery, the opposite is true: the capacitor charges fully to the battery voltage.
Students sometimes apply V = IR to the whole circuit at steady state and get a nonzero current. Remember, the capacitor blocks DC, so at steady state the current is zero.
Confusing the behaviour of L and C at steady state is common. Capacitors block DC (open circuit). Inductors pass DC freely (short circuit). These roles reverse at high frequency, but for DC steady-state problems, keep them straight.
When a switch opens, students sometimes forget that the inductor resists changes in current. If the inductor current was zero before the switch opened, it remains zero at the instant after.
⚠️ The phrase "after a long time" is a direct signal to use DC steady-state rules. Do not attempt a full transient analysis when you see it.
⚠️ Always identify which components are in series vs. parallel before and after a switching event. The topology determines which steady-state rules apply.
⚠️ Exam problems love to ask about initial conditions after a switch changes state. Identify V_C (unchanged at the instant of switching) and I_L (unchanged at the instant of switching) immediately.
⚠️ Unit traps: pF × V gives pC; convert to nC or μC as needed. 20 pF × 10 V = 200 pC = 0.2 nC, not 20 pC.
True or false: At DC steady state, the voltage across an ideal inductor is zero.
True or false: At DC steady state, the current through a capacitor is zero.
Fill in the blank: The energy stored in a capacitor is U = ½ ___ V².
True or false: When a switch opens, the current through an inductor changes instantaneously to zero.
Fill in the blank: Q = ___ × V for a capacitor.
Answers: 1. True. 2. True. 3. C. 4. False (inductors resist instantaneous current changes). 5. C.
Q: A series RLC circuit is connected to a 10 V battery. After the switch has been closed for a long time, what is the voltage across the capacitor?
A: 10 V. At DC steady state, no current flows (the capacitor blocks it), so there is no voltage drop across R or L. The full battery voltage appears across C.
Q: In the same circuit, what is the voltage across the resistor at steady state?
A: 0 V. No current flows, so V_R = IR = 0.
Q: If C = 20 pF and the capacitor is charged to 10 V, what is the charge stored?
A: Q = CV = (20 × 10⁻¹²)(10) = 2 × 10⁻¹⁰ C = 0.2 nC.
Q: The switch is now opened. The inductor had zero current through it. What is the initial current through R?
A: 0 A. The inductor maintains its current (zero) at the instant of switching, and in a series loop all elements carry the same current.
Q: What is the natural oscillation frequency of an LC circuit with L = 20 mH and C = 20 pF?
A: ω₀ = 1/√(LC) = 1/√(20 × 10⁻³ × 20 × 10⁻¹²) = 1/√(4 × 10⁻¹³) ≈ 1.58 × 10⁶ rad/s.
This connects directly to AC circuit analysis (impedance, phasors, resonance), which uses the same components but with time-varying sources. The energy storage concepts reappear in electromagnetic waves, where E-field energy and B-field energy oscillate in a travelling wave just as capacitor and inductor energy oscillate in an LC circuit.
Related Terms / Search Tags: RLC circuit, LRC circuit, series RLC, DC steady state, transient response, capacitor voltage, inductor current, LC oscillation, natural frequency, time constant, tau, energy storage, PHYS 212, University Physics, electromagnetism, Exam 3