AP Physics C E&M: LR Circuit Time Constant

Understanding the transient behavior of an LR circuit is essential for analyzing everything from power supply stabilization to the suppression of voltage spikes in electric motors. The key to this behavior is the time constant, denoted by $\tau$, which dictates how quickly current changes when an inductor is involved. Mastering this concept allows you to predict current over time, calculate stored energy, and understand the fundamental trade-off between inductance and resistance in circuit dynamics.

The Physical Basis: Inductance and Back EMF

At the heart of an LR circuit is the inductor, a coil of wire that opposes changes in current. When the current through an inductor changes, it induces a voltage across its own terminals, a phenomenon known as self-inductance. This induced voltage, or back electromotive force (back EMF), is given by Faraday's Law: ${\epsilon }_L=-L(dI/dt)$, where $L$ is the inductance in henries (H). The negative sign indicates that the induced voltage opposes the change in current (Lenz's Law). In a circuit with both an inductor and a resistor, this opposition to change fights against the driving voltage of the source (like a battery), leading to the characteristic exponential rise and fall of current, rather than an instantaneous jump.

Deriving the Circuit Equation and the Time Constant τ = L/R

Consider a simple series LR circuit connected to a battery of emf $\epsilon$ at time $t=0$. Kirchhoff's loop rule states that the sum of the voltages around the loop is zero. Traversing the loop, we have the battery's emf $\epsilon$, the voltage drop across the resistor $IR$, and the back EMF across the inductor $-L(dI/dt)$. The loop rule gives:

$$\epsilon -IR-L\frac{dI}{dt}=0$$

This is a first-order linear differential equation for current $I(t)$. Rearranging it highlights its structure:

$$L\frac{dI}{dt}+IR=\epsilon$$

We can solve this via separation of variables. Rearranging the original equation:

$$L\frac{dI}{dt}=\epsilon -IR$$ $$\frac{dI}{\epsilon -IR}=\frac{1}{L}dt$$

Integrating both sides from the initial condition $I=0$ at $t=0$ to a general current $I$ at time $t$:

$${\int }_0^I\frac{dI}{\epsilon -IR}={\int }_0^t\frac{1}{L}dt$$

Performing the integration yields:

$$-\frac{1}{R}\ln \left(\frac{\epsilon -IR}{\epsilon }\right)=\frac{t}{L}$$

Solving for $I(t)$:

$$I(t)=\frac{\epsilon }{R}\left(1-e^{-(R/L)t}\right)$$

The quantity in the exponent, $(R/L)t$, must be dimensionless. Therefore, the grouping $L/R$ has units of time (seconds). We define this as the time constant, $\tau$:

$$\tau =\frac{L}{R}$$

The time constant is the hallmark of the circuit's speed. Substituting it into the solution gives the standard form for the energizing (or "charging") inductor:

$$I(t)=\frac{\epsilon }{R}\left(1-e^{-t/\tau }\right)$$

Current as a Function of Time: Energizing and De-Energizing

The derived equation describes the energizing process. Notice that as $t\to \infty$, the exponential term goes to zero, and the current approaches its steady-state value $I_{max}=\epsilon /R$, as if the inductor were just a wire. After one time constant ($t=\tau$), the current reaches $I(\tau )=I_{max}(1-e^{-1})\approx 0.632I_{max}$, or about 63.2% of its final value.

The de-energizing (or "decay") scenario occurs when a source is suddenly removed from a circuit that already has a current $I_0$ flowing through the inductor. Imagine the LR loop with the battery shorted out at $t=0$. Applying Kirchhoff's rule now gives $0-IR-L(dI/dt)=0$, or $L(dI/dt)=-IR$. Solving this with the initial condition $I(0)=I_0$ yields:

$$I(t)=I_0{e}^{-t/\tau }$$

Here, the current decays exponentially from its initial value. After one time constant, it falls to $I_0{e}^{-1}\approx 0.368I_0$, or 36.8% of its initial value. In both processes, the time constant $\tau$ sets the scale: a large $L$ or a small $R$ means a large $\tau$ and a slow change; a small $L$ or a large $R$ means a small $\tau$ and a rapid change.

Energy Stored and Dissipated During Transient Processes

Inductors store energy in their magnetic fields. The energy $U_L$ stored in an inductor carrying current $I$ is given by:

$$U_L=\frac{1}{2}L{I}^2$$

During the energizing process, the battery does work to establish the current. This energy is partitioned between the energy stored in the inductor's magnetic field and the energy dissipated as heat in the resistor. The total energy supplied by the battery from $t=0$ to $t=\infty$ is $\epsilon \cdot Q_{total}$, where $Q_{total}$ is the total charge that eventually flows. It can be shown that exactly half of the battery's total energy output ends up stored in the inductor ($\frac{1}{2}L{I}_{max}^2$), while the other half is dissipated in the resistor during the transient period, regardless of the value of $R$ or $L$.

During the de-energizing process, the inductor acts as the energy source. The magnetic energy stored, $\frac{1}{2}L{I}_0^2$, is completely dissipated as heat in the resistor as the current decays to zero. The power dissipated in the resistor at any moment is $P_R=I^{2}R=I_0^{2}R{e}^{-2t/\tau }$, and integrating this from $t=0$ to $\infty$ confirms that all the initial inductive energy is converted to thermal energy in the resistor.

Common Pitfalls

1. Confusing the time constant for "total time": A common misconception is that the process is "complete" after $t=\tau$. In reality, the exponential asymptotically approaches its final value. After $5\tau$, the current is within 0.7% of its maximum (or minimum), which is often considered functionally complete for engineering purposes.
2. Misapplying the current equations: Students often mix up the energizing and de-energizing formulas. Remember: the energizing equation has a steady-state term and a decaying exponential term $(1-e^{-t/\tau })$. The de-energizing equation is a pure exponential decay $e^{-t/\tau }$. Always identify the initial and final conditions for the current.
3. Forgetting the inductor's behavior at time boundaries: At the instant a switch is thrown ($t=0$), an inductor acts to maintain the current that was flowing through it at that instant. It does this by generating whatever back EMF is necessary. Therefore, at $t=0^{+}$, the current through an inductor cannot change discontinuously. In an energizing circuit, the current starts at zero. In a de-energizing circuit, it starts at its pre-switch value $I_0$.
4. Incorrect energy accounting: It's easy to think the battery's energy goes solely into the inductor. The analysis shows that an equal amount is always dissipated in the resistor during the build-up phase. This dissipation is the unavoidable cost of creating the magnetic field.

Summary

• The time constant $\tau =L/R$ determines the rate of exponential current change in an LR circuit. A larger $L$ or smaller $R$ results in a slower response.
• For an energizing inductor (circuit closed with a source), current rises as $I(t)=I_{max}(1-e^{-t/\tau })$, where $I_{max}=\epsilon /R$.
• For a de-energizing inductor (source removed), current decays as $I(t)=I_0{e}^{-t/\tau }$.
• The energy stored in an inductor's magnetic field is $U_L=\frac{1}{2}L{I}^2$. During energizing, half the battery's total energy output is stored this way, while the other half is dissipated in the resistor.
• During de-energizing, all the energy initially stored in the inductor's magnetic field is dissipated as heat in the resistor.