RC and RL transients
When circuits containing capacitors or inductors are switched, voltages and currents settle exponentially. The time constant tells us how fast.
First-order circuits
A circuit with resistors and a single capacitor or single inductor is first-order: its behaviour is described by a first-order differential equation. After a sudden change, such as closing a switch, voltages and currents move exponentially from their initial values towards new final values.
Charging a capacitor
When a capacitor C charges through resistor R from a DC source V, its voltage is \(v(t) = V(1 - e^{-t/\tau})\), where the time constant is \(\tau = RC\). After one time constant the voltage reaches about 63% of V; after 5\(\tau\), over 99%, effectively complete. The charging current starts at \(\frac{V}{R}\) and decays as \(\frac{V}{R}e^{-t/\tau}\).
Discharging
A charged capacitor discharging through a resistor follows \(v(t) = V_0 e^{-t/\tau}\): after one time constant, 37% of the initial voltage remains. This governs how quickly power-supply capacitors bleed down after switch-off and sets the behaviour of simple timers and filters.
RL circuits
For an inductor L in series with resistor R, the time constant is \(\tau = \frac{L}{R}\). When a DC voltage is applied, the current rises as \(i(t) = \frac{V}{R}(1 - e^{-t/\tau})\), and the inductor's voltage decays from V to zero. When the source is removed and a path remains, current decays exponentially with the same time constant.
The general method
Any first-order response can be written as \(x(t) = x(\infty) + [x(0) - x(\infty)]e^{-t/\tau}\). Find the initial value just after switching, using the fact that capacitor voltages and inductor currents cannot jump; the final value, treating capacitors as open and inductors as short circuits at DC; and the time constant from the Thevenin resistance seen by the capacitor or inductor.
Second-order circuits
Circuits with both inductance and capacitance are second-order. A series RLC circuit can respond overdamped (slow, no oscillation), critically damped (fastest without overshoot), or underdamped (oscillating while decaying), depending on R relative to the square root of L/C. Underdamped ringing appears in poorly terminated digital signals and in LC oscillators.
Applications
RC time constants set delays such as power-on reset timers, debounce circuits for mechanical switches, and the classic 555 timer, introduced in 1972. Cameras' flash units and defibrillators charge capacitors over seconds and discharge them in milliseconds. In digital circuits, the RC delay of wires and transistor capacitances limits how fast chips can switch.
Worked example
A 10 k\(\Omega\) resistor charges a 100 \(\mu\)F capacitor from a 9 V battery. The time constant is \(\tau = 10^4 \times 10^{-4} = 1\) s. After 1 s the voltage is \(9(1 - e^{-1}) \approx 5.7\) V; after 2 s, \(9(1 - e^{-2}) \approx 7.8\) V; after 5 s, about 8.94 V. The initial current is \(\frac{9}{10{,}000} = 0.9\) mA, falling to about 0.33 mA after one second.
Key takeaways
First-order RC and RL circuits respond exponentially after switching. Charging follows v = V(1 - e^{-t/tau}) with tau = RC: 63% after one time constant and over 99% after five. Discharge leaves 37% after one time constant. RL circuits have tau = L/R. The general solution is x(t) = x(infinity) + [x(0) - x(infinity)] e^{-t/tau}, using continuity of capacitor voltage and inductor current, DC steady-state rules, and the Thevenin resistance. RLC circuits are overdamped, critically damped, or underdamped. Time constants set timers, debouncing, flashes, and chip speeds.
Flashcard
One capacitor or inductor with resistors
Check your understanding
A 2 kilohm resistor charges a 50 microfarad capacitor. What is the time constant, and roughly when is charging complete?