RC and RL Circuits (Charging and Discharging)

First-order transient response · τ = R·C (RC) · τ = L/R (RL)

Input parameters

Circuit type

Phase

Circuit schematic

Charging

Source Voltage

V₀ of the DC source

1100

Resistance

R in series with the reactive element

0.11000

Capacitance

Capacitor C (RC only)

0.11000

Results

Time Constant (τ)

Initial Current (I₀)

Maximum Stored Energy

µJ

Cutoff Frequency (f_c)

159.2

Low-pass (output across the capacitor)

Maximum Power in R (P_max)

0.1

Rise Time 10–90%

2.197

Settling 2%

3.912

Settling 1%: 4.605 ms

Approach to steady state

Time	% of final value
1τ1 ms	63.2%
2τ2 ms	86.5%
3τ3 ms	95.0%
4τ4 ms	98.2%
5τ5 ms	99.3%

Voltage and Current vs. Time

Fundamentals & Explanation

First-Order Circuits and Their Differential Equation

A circuit with a single DC source, one resistor and one energy-storage element —a capacitor (RC) or an inductor (RL)— is described by a first-order linear differential equation. Applying Kirchhoff's voltage law (KVL) around the loop gives, respectively:

RC (KVL)

RC\,\frac{dV_C}{dt} + V_C = V_0

RL (KVL)

\frac{L}{R}\,\frac{di}{dt} + i = \frac{V_0}{R}

The solution of any such equation is an exponential that moves from an initial value x₀ to a steady-state value x∞ with a single time scale τ. This universal form generates, by itself, the four charge/discharge configurations:

x(t) = x_\infty + \left(x_0 - x_\infty\right)e^{-t/\tau}

The capacitor opposes sudden changes in voltage (V_C is continuous); the inductor opposes sudden changes in current (i_L is continuous). Hence x₀ is evaluated keeping that state variable unchanged at t = 0⁺.

Time Constant τ

\tau = R\,C

\tau = \frac{L}{R}

τ is the time for the exponential to cover a fraction 1 − e⁻¹ of its full travel. The rising quantity reaches 63.2 % of its final value at t = τ; the decaying one drops to 36.8 %. After 5τ the circuit is at 99.3 % of steady state, conventionally "settled".

Charge and Discharge: the Four Configurations

Each configuration follows from the universal formula evaluated with its pair (x₀, x∞). The minus sign in discharge indicates a reversed direction relative to charging.

Mode	Variable	x₀	x∞	x(t)
RC charge	V_C	0	V₀	V₀(1 − e^−t/τ)
RC charge	i	V₀/R	0	(V₀/R) e^−t/τ
RC discharge	V_C	V₀	0	V₀ e^−t/τ
RC discharge	i	−V₀/R	0	−(V₀/R) e^−t/τ
RL charge	i_L	0	V₀/R	(V₀/R)(1 − e^−t/τ)
RL charge	V_L	V₀	0	V₀ e^−t/τ
RL discharge	i_L	V₀/R	0	(V₀/R) e^−t/τ
RL discharge	V_L	−V₀	0	−V₀ e^−t/τ

Characteristic-current nuance. The value V₀/R is the initial current I₀ (peak at t = 0) for RC and for RL discharge, but the steady-state current I∞ (reached as t → ∞) for RL charging: the same number with a different physical meaning depending on the case.

Stored Energy and Dissipated Power

Capacitor

U_C = \tfrac{1}{2}\,C V_0^2

Inductor

U_L = \tfrac{1}{2}\,L I_\infty^2

Power in R

P_R(t) = i^2(t)\,R

The capacitor stores energy in its electric field and the inductor in its magnetic field. The power dissipated in R peaks at P_max = V₀²/R. A classic result: when charging a capacitor, the energy dissipated in R during the transient is exactly ½CV₀² —equal to the stored energy— independent of R (the source delivers CV₀²; half is stored, half dissipated).

Characteristic Times

All are fixed multiples of τ. The 10–90 % rise/fall time and the settling times are the standard speed metrics in electronics:

\begin{gathered} t_{r}^{10\text{–}90} = \tau\ln 9 \approx 2.20\,\tau \\[2pt] t_{2\%} = \tau\ln 50 \approx 3.91\,\tau \qquad t_{1\%} = \tau\ln 100 \approx 4.61\,\tau \end{gathered}

t	Charge (1 − e⁻ⁿ)	Discharge (e⁻ⁿ)
1τ	63.2 %	36.8 %
2τ	86.5 %	13.5 %
3τ	95.0 %	5.0 %
4τ	98.2 %	1.8 %
5τ	99.3 %	0.7 %

Bridge to the Frequency Domain

The same τ sets the behavior under sinusoidal signals. The cutoff frequency (−3 dB) is where the output drops to 1/√2 ≈ 70.7 % of the input:

f_c = \frac{1}{2\pi\tau} = \frac{1}{2\pi R C} = \frac{R}{2\pi L}

Depending on where the output is taken, the same circuit is a different filter: output across the capacitor → low-pass; across the resistor → high-pass. For RL the relationship reverses (output across the inductor → high-pass). The full frequency response —magnitude and phase over all ω, the Bode plot— follows from evaluating the transfer function H(jω); f_c is the point where |H| drops by 3 dB.

Model Limits and Assumptions

Ideal (LTI) components

R, L and C are assumed linear, constant and time-invariant. Parasitics are not modeled: equivalent series resistance/inductance (ESR/ESL), leakage current, dielectric absorption, or inductor-core saturation.

First-order only

There is a single storage element. Combining L and C yields a second-order RLC circuit with resonance and possible overshoot/damped oscillation, beyond the scope of this first-order model.

Lumped elements (quasi-static)

Ideal step source and switch, no source impedance. The lumped-element approximation holds when circuit dimensions are ≪ λ = c/f. When τ enters the sub-nanosecond range (f_c in the GHz range) the wavelength becomes comparable to the circuit size and transmission-line effects —which this model does not capture— appear. RL discharge assumes the inductor was previously energized to I∞ = V₀/R.

Quantity	Symbol	SI unit
Source voltage	V₀	volt (V)
Resistance	R	ohm (Ω)
Capacitance	C	farad (F)
Inductance	L	henry (H)
Time constant	τ	second (s)
Cutoff frequency	f_c	hertz (Hz)
Energy / Power	U / P	joule (J) / watt (W)

References and Resources

[1] Sadiku, M. & Alexander, C. Fundamentals of Electric Circuits (6th ed.), Ch. 7: "First-Order Circuits". McGraw-Hill.
[2] Nilsson, J. & Riedel, S. Electric Circuits (11th ed.), Ch. 7: "Response of First-Order RL and RC Circuits". Pearson.
[3] Horowitz, P. & Hill, W. The Art of Electronics (3rd ed.), §1.4 (RC time constant, filters). Cambridge University Press.

RC and RL Circuits (Charging and Discharging)

Input parameters

Results

Voltage and Current vs. Time

Fundamentals & Explanation

First-Order Circuits and Their Differential Equation

Time Constant τ

Charge and Discharge: the Four Configurations

Stored Energy and Dissipated Power

Characteristic Times

Bridge to the Frequency Domain

Model Limits and Assumptions

Ideal (LTI) components

First-order only

Lumped elements (quasi-static)

References and Resources

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First-Order Circuits and Their Differential Equation

Time Constant τ

Charge and Discharge: the Four Configurations

Stored Energy and Dissipated Power

Characteristic Times

Bridge to the Frequency Domain

Model Limits and Assumptions

Ideal (LTI) components

First-order only

Lumped elements (quasi-static)

References and Resources