RL Circuits: Current Growth & Decay — Analyze current growth/decay in RL circuits - Physics 2

SECTION 1

Historical Context & Motivation

The study of RL circuits — circuits containing a resistor (R) and an inductor (L) in series — grew directly from the nineteenth-century quest to understand how changing electric currents interact with magnetic fields. Michael Faraday's discovery that a time-varying magnetic flux induces an electromotive force (EMF) laid the groundwork, but it was Joseph Henry's independent discovery of self-inductance that revealed why certain coils resisted sudden changes in current. Understanding the transient response of RL circuits became essential not only for telegraph and telephone engineering in the late 1800s but also for modern power electronics, motor drives, and signal-processing filters. The exponential behavior of current growth and decay in these circuits represents one of the most fundamental first-order differential equations in electrical physics, bridging Faraday's law with Ohm's law in a single, elegant framework.

1831 • Faraday's Law of Induction

Michael Faraday demonstrates that a changing magnetic flux through a loop induces an EMF, establishing the physical basis for inductance and transformer action.

1832 • Henry's Self-Inductance

Joseph Henry independently discovers electromagnetic induction and identifies self-inductance — the property by which a coil opposes changes in its own current — providing the L in RL circuits.

1845 • Kirchhoff's Circuit Laws

Gustav Kirchhoff formalizes the voltage and current loop laws that allow systematic analysis of circuits containing resistors and inductors, enabling the derivation of transient equations.

1886 • Heaviside's Operational Calculus

Oliver Heaviside develops operational methods for solving differential equations in circuits, streamlining the analysis of RL transients and paving the way for modern Laplace-transform techniques.

1940s • Radar & Power Electronics

World War II accelerates the application of RL circuit theory to pulse-forming networks in radar, switching regulators, and electromagnetic relays — applications that remain central to electrical engineering today.

The central question this lesson addresses is both conceptual and quantitative: when a DC voltage source is suddenly connected to or disconnected from an RL circuit, how does the current evolve in time, and what determines the rate of that evolution? Answering this question requires synthesizing Faraday's law of induction with Kirchhoff's voltage law and solving the resulting first-order ordinary differential equation — a procedure that yields the characteristic exponential curves of current growth and decay.

SECTION 2

Core Principles & Definitions

Before diving into the mathematics, it is important to consolidate the foundational ideas that govern RL circuit behavior. An inductor is a passive two-terminal component — typically a coil of wire — that stores energy in a magnetic field when current flows through it. Its defining property is inductance L, measured in henrys (H), which quantifies the magnitude of the induced back-EMF per unit rate of change of current. Combining an inductor with a resistor and a voltage source produces a first-order system whose transient behavior is entirely characterized by the ratio L/R, known as the time constant τ.

Self-Inductance (L)

The proportionality constant between the magnetic flux linkage through a coil and the current flowing in it: Φ = LI. A larger L means a greater back-EMF for the same rate of current change.

Back-EMF (Lenz's Law)

An inductor generates a voltage v_L = −L(dI/dt) that opposes any change in current. This is the circuit-level manifestation of Lenz's law, ensuring energy conservation.

Time Constant (τ = L/R)

The characteristic time scale over which current rises to ≈ 63.2 % of its final value during growth, or falls to ≈ 36.8 % of its initial value during decay. After 5τ the transient is effectively complete.

Steady-State Current

When dI/dt → 0, the inductor behaves as a short circuit (ideal case), and the current reaches I_max = V/R, determined solely by Ohm's law.

Energy Stored in an Inductor

The magnetic field energy is U = ½LI². During current growth, energy flows from the source into both the resistor (as heat) and the inductor (as stored magnetic energy). During decay, the stored energy is dissipated entirely in the resistor.

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KEY TAKEAWAY

Think of an inductor as the electrical analogue of a massive flywheel on a mechanical shaft. Just as a heavy flywheel resists sudden changes in rotational speed — slowly spinning up when torque is applied and coasting for a long time after the drive is removed — an inductor resists sudden changes in current. The time constant τ = L/R plays the role of the flywheel's moment of inertia divided by friction: a large L (heavy flywheel) or a small R (low friction) means the system takes longer to reach its new equilibrium.

SECTION 3

Visual Explanation — RL Circuit & Current-Time Curves

The circuit above shows a DC voltage source V in series with a switch S, a resistor R (pink), and an inductor L (violet). When the switch closes at t = 0, current I(t) begins to grow from zero toward V/R, opposed at every instant by the inductor's back-EMF.

When the switch closes, the inductor initially acts almost like an open circuit because the rate of change of current, dI/dt, is at its maximum value, producing the largest possible back-EMF. As time progresses, the current rises and dI/dt decreases, reducing the back-EMF. The result is a characteristic exponential approach toward the steady-state current I_max = V/R. Conversely, if the source is suddenly removed (the circuit is opened and the inductor can discharge through R), the current decays exponentially from its initial value toward zero. Both processes are governed by the same time constant τ = L/R, and after approximately 5τ the transient is considered complete — the current has reached within 0.7 % of its final value.

SECTION 4

Mathematical Framework

Current Growth — Energizing the RL Circuit

Applying Kirchhoff's voltage law around the series loop after the switch closes at t = 0, the sum of the voltage drops across R and L must equal the source voltage V. The inductor's voltage is v_L = L(dI/dt), and the resistor's voltage is v_R = IR. This yields the first-order linear ODE whose solution is the current growth equation.

KIRCHHOFF'S VOLTAGE LAW (KVL)

V = IR + L(dI/dt)

V = source EMF (volts), I = instantaneous current (amperes), R = resistance (ohms), L = inductance (henrys), dI/dt = rate of change of current (A/s).

CURRENT GROWTH EQUATION

I(t) = (V/R)(1 − e^(−Rt/L)) = I_max(1 − e^(−t/τ))

I_max = V/R is the steady-state current; τ = L/R is the inductive time constant. At t = τ, the current has reached about 63.2 % of I_max.

Current Decay — De-energizing the RL Circuit

When the voltage source is removed (or the switch reroutes the inductor through the resistor), the stored magnetic energy drives a decaying current through R. The KVL equation becomes 0 = IR + L(dI/dt), a homogeneous first-order ODE whose solution is a simple decaying exponential.

CURRENT DECAY EQUATION

I(t) = I₀ e^(−t/τ) = I₀ e^(−Rt/L)

I₀ is the current at the instant the source is disconnected (often V/R if the circuit was in steady state). The current drops to 36.8 % of I₀ after one time constant and to less than 1 % after 5τ.

VOLTAGE ACROSS THE INDUCTOR

v_L(t) = V e^(−t/τ) (growth) | v_L(t) = −I₀R e^(−t/τ) (decay)

During growth, v_L starts at V and decays to zero. During decay, the inductor drives a voltage of magnitude I₀R that also decays exponentially. The sign reflects Lenz's law: the induced EMF always opposes the change in current.

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Derivation Sketch

To solve V = IR + L(dI/dt), rearrange to dI/(V/R − I) = (R/L)dt. Integrating both sides from 0 to t with I(0) = 0 yields −ln(1 − IR/V) = Rt/L, which exponentiates to give I(t) = (V/R)(1 − e^−Rt/L). The decay case follows analogously with V = 0 and I(0) = I₀.

SECTION 5

Growth vs. Decay — Exponential Curves in Detail

The green curve shows current growth: I(t) = I_max(1 − e^−t/τ). The red curve shows current decay: I(t) = I₀e^−t/τ. At t = τ, the growth curve has reached 63.2 % of I_max while the decay curve has fallen to 36.8 % of I₀. Both curves are essentially flat after 5τ.

Time	Growth I(t)/I_max	Decay I(t)/I₀
0	0 %	100 %
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 %

Percentage of final (growth) or initial (decay) current at integer multiples of τ

The symmetry between growth and decay is noteworthy: if growth reaches fraction f of I_max at time t, then decay has fallen to fraction (1 − f) of I₀ at the same elapsed time. This complementary relationship arises because both curves are solutions to the same first-order linear ODE differing only in boundary conditions. Engineers often adopt the "five time-constant rule": after 5τ, the transient is treated as complete for all practical purposes, since the remaining deviation from steady state is less than 1 %.

SECTION 6

Worked Example — Current Growth in an RL Circuit

A series RL circuit consists of a 12 V DC source, a 4 Ω resistor, and a 20 mH inductor. The switch closes at t = 0. Find (a) the time constant, (b) the steady-state current, (c) the current at t = 5 ms, and (d) the time at which the current reaches 2.5 A.

RL Current Growth

1
Step 1 — Identify Given Values

V = 12 V, R = 4 Ω, L = 20 mH = 0.020 H. The initial current is I(0) = 0 because the inductor prevents an instantaneous jump in current.
2
Step 2 — Calculate the Time Constant τ

τ = L / R = 0.020 H / 4 Ω = 0.005 s = 5 ms. This means the transient will be essentially complete after 5 × 5 ms = 25 ms.

τ = 5 ms
3
Step 3 — Find the Steady-State Current I_max

When dI/dt → 0, the inductor acts as a short circuit. By Ohm's law, I_max = V / R = 12 V / 4 Ω = 3 A.

I_max = 3 A
4
Step 4 — Current at t = 5 ms

I(5 ms) = I_max(1 − e^−t/τ) = 3(1 − e^−5/5) = 3(1 − e⁻¹) = 3(1 − 0.368) = 3 × 0.632 = 1.896 A. Since t = τ, this confirms the 63.2 % rule.

I(5 ms) ≈ 1.90 A
5
Step 5 — Time to Reach 2.5 A

Set I(t) = 2.5 A and solve: 2.5 = 3(1 − e^−t/τ). Dividing: 1 − e^−t/τ = 5/6, so e^−t/τ = 1/6. Taking the natural log: −t/τ = ln(1/6) = −1.792, hence t = 1.792 × 5 ms = 8.96 ms.

t ≈ 8.96 ms

SECTION 7

RL vs. RC Circuits — A Comparative Perspective

RL circuits are often studied alongside RC circuits because both exhibit first-order exponential transients yet store energy in fundamentally different forms. A capacitor stores energy in an electric field (U = ½CV²) and resists changes in voltage, while an inductor stores energy in a magnetic field (U = ½LI²) and resists changes in current. Recognizing the duality between these two circuit types deepens conceptual understanding and simplifies the analysis of more complex RLC systems.

Property	RL Circuit	RC Circuit
Energy-storing element	Inductor (magnetic field)	Capacitor (electric field)
Resists changes in	Current	Voltage
Time constant	τ = L/R	τ = RC
Growth equation	I(t) = (V/R)(1 − e^−t/τ)	V_C(t) = V(1 − e^−t/τ)
Decay equation	I(t) = I₀e^−t/τ	V_C(t) = V₀e^−t/τ
Stored energy	U = ½LI²	U = ½CV²
DC steady-state behavior	Short circuit (wire)	Open circuit (break)

Side-by-side comparison of first-order RL and RC transient circuits

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KEY TAKEAWAY

RL and RC circuits are electromagnetic duals: swap L ↔ C, current ↔ voltage, series ↔ parallel, and every RL equation morphs into the corresponding RC equation. Mastering one circuit type immediately grants you intuition for the other, and this duality extends into AC analysis, where impedances Z_L = jωL and Z_C = 1/(jωC) are reciprocals in form.

SECTION 8

Connections to Advanced Theory — RLC Circuits & AC Analysis

The first-order RL transient is a stepping stone to richer dynamics that appear when capacitors join the circuit. In a series RLC circuit, the governing equation becomes a second-order ODE, and the transient response can be overdamped, critically damped, or underdamped depending on the discriminant R² − 4L/C. The RL time constant τ = L/R reappears as part of the damping factor α = R/(2L), which controls how quickly oscillations die out. Similarly, in AC steady-state analysis, the inductor's impedance Z_L = jωL introduces a frequency-dependent phase shift between voltage and current, forming the basis of low-pass and high-pass filter design.

Feature	RL (First-Order)	RLC (Second-Order)
Order of ODE	1st	2nd
Transient shape	Purely exponential	Exponential × sinusoid (underdamped)
Key parameter	τ = L/R	α = R/(2L), ω₀ = 1/√(LC)
Possible behaviors	Monotonic approach to steady state	Overdamped, critically damped, underdamped
Resonance	Not applicable	At ω = ω₀, impedance is minimized

From RL to RLC: increasing complexity in transient circuit analysis

Beyond pure circuit theory, the RL time constant appears in electromagnetic compatibility (EMC) analysis — modeling how quickly eddy currents decay in conductive shields — and in the design of relay drivers, solenoid actuators, and switched-mode power supplies. Any system where an inductive element switches between states involves the growth and decay mathematics developed in this lesson, making RL transient analysis a foundational skill for electrical engineering, applied physics, and power systems design.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why the current through an inductor cannot change instantaneously when a switch is closed. What physical quantity would have to be infinite for an instantaneous change to occur?

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An instantaneous change in current (dI/dt → ∞) would require the inductor to develop an infinite back-EMF, since v_L = L(dI/dt). This would demand an infinite voltage across the inductor, which no real source can supply. Physically, the magnetic field stores energy U = ½LI², and changing the stored energy instantaneously would require infinite power. Therefore, the inductor enforces continuity of current — it can change only at a finite rate determined by the applied voltage and the inductance.

Question 2: BASIC CALCULATION

A series RL circuit has R = 10 Ω and L = 50 mH connected to a 20 V source. Calculate (a) the time constant τ, (b) the maximum steady-state current, and (c) the current at t = 10 ms.

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(a) τ = L/R = 0.050/10 = 5 ms. (b) I_max = V/R = 20/10 = 2 A. (c) At t = 10 ms = 2τ, I = 2(1 − e⁻²) = 2(1 − 0.1353) = 2 × 0.8647 ≈ 1.73 A. This is consistent with the table value of 86.5 % of I_max at t = 2τ.

Question 3: INTERMEDIATE

An RL circuit carrying a steady-state current of 4 A is suddenly disconnected from its source at t = 0, allowing the current to decay through the resistor. If L = 0.2 H and R = 8 Ω, (a) how long does it take for the current to drop to 1 A, and (b) how much energy has been dissipated in the resistor by that time?

Show answer

(a) τ = L/R = 0.2/8 = 25 ms. Setting 1 = 4e^−t/τ gives e^−t/τ = 0.25, so t = −τ ln(0.25) = 25 × 1.386 ≈ 34.7 ms. (b) The initial stored energy is U₀ = ½LI₀² = ½(0.2)(16) = 1.6 J. The remaining energy when I = 1 A is U = ½(0.2)(1) = 0.1 J. Energy dissipated = 1.6 − 0.1 = 1.5 J. This confirms that the inductor's magnetic field energy is converted to thermal energy in the resistor during decay.

Question 4: APPLIED

A relay coil (modeled as L = 80 mH, R = 16 Ω) must reach at least 90 % of its steady-state current to reliably close. If connected to a 48 V supply, what is the minimum time after switch closure before the relay is guaranteed to operate? What steady-state current does the coil carry?

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τ = L/R = 0.080/16 = 5 ms. The steady-state current is I_max = 48/16 = 3 A. For 90 % of I_max: 0.90 = 1 − e^−t/τ, so e^−t/τ = 0.10, t = −5 ln(0.10) = 5 × 2.303 ≈ 11.5 ms. The relay requires roughly 11.5 ms (about 2.3τ) after switch closure. In practice, designers add a safety margin and may use a higher supply voltage to speed up energization.

Question 5: CRITICAL THINKING

Derive an expression for the total energy dissipated in the resistor during the complete growth transient (from t = 0 to t → ∞) by integrating I²R dt. Compare this to the energy stored in the inductor at steady state and explain where the source energy goes.

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During growth, I(t) = (V/R)(1 − e^−t/τ). The total energy from the source is W_source = ∫₀^∞ V·I dt = (V²/R)∫₀^∞ (1 − e^−t/τ)dt. Evaluating this integral yields W_source → ∞ in the continuous DC sense, so we instead consider a finite observation window. The energy dissipated in R is W_R = ∫₀^∞ I²R dt = (V²/R)∫₀^∞ (1 − e^−t/τ)² dt. Expanding and integrating term by term gives W_R,transient = ½LI_max² = ½L(V/R)². Remarkably, the energy dissipated in the resistor during the transient exactly equals the energy stored in the inductor, ½LI_max². The source thus delivers twice the stored energy during the transient: half goes to R as heat, and half is stored in L's magnetic field. This 50-50 energy split is a universal property of first-order RL growth transients, independent of the values of L and R.

SUMMARY

Lesson Summary

Summary

An RL circuit contains a resistor R and an inductor L in series. When a DC source is connected, the current grows exponentially according to I(t) = (V/R)(1 − e^−t/τ), approaching the steady-state value I_max = V/R. When the source is removed, the current decays exponentially as I(t) = I₀e^−t/τ. Both processes are governed by the time constant τ = L/R, which sets the characteristic timescale: the current reaches 63.2 % of its final value after one τ and is within 1 % after five τ.

The inductor's back-EMF (Lenz's law) is the physical mechanism behind these transients: it opposes changes in current, storing energy in a magnetic field (U = ½LI²) during growth and releasing it through the resistor during decay. RL circuits are electromagnetic duals of RC circuits: swapping inductance for capacitance and current for voltage transforms one set of equations into the other. Mastery of the RL transient provides the foundation for analyzing second-order RLC circuits, AC impedance, and practical applications from relay drivers to power converters.