Current in an R L circuit The accompanying diagram repre-...

Current in an $R L$ circuit The accompanying diagram repre- sents an electrical circuit whose total resistance is a constant $R$ ohms and whose self-inductance, shown as a coil, is $L$ henries, also a constant. There is a switch whose terminals at $a$ and $b$ can be closed to connect a constant electrical source of $V$ volts. From Section $9.2,$ we have $$L \frac{d i}{d t}+R i=V,$$ where $i$ is the current in amperes and $t$ is the time in seconds. Use a phase line analysis to sketch the solution curve assuming that the switch in the $R L$ circuit is closed at time $t=0 .$ What happens to the current as $t \rightarrow \infty ?$ This value is called the steady-state solution.

Current in an $R L$ circuit The accompanying diagram repre- sents an electrical circuit whose total resistance is a constant $R$ ohms and whose self-inductance, shown as a coil, is $L$ henries, also a constant. There is a switch whose terminals at $a$ and $b$ can be closed to connect a constant electrical source of $V$ volts. From Section $9.2,$ we have
$$L \frac{d i}{d t}+R i=V,$$
where $i$ is the current in amperes and $t$ is the time in seconds.

Use a phase line analysis to sketch the solution curve assuming that the switch in the $R L$ circuit is closed at time $t=0 .$ What happens to the current as $t \rightarrow \infty ?$ This value is called the steady-state solution.

Key Concepts

RL Circuit

An RL circuit is an electrical circuit that consists of a resistor (R) and an inductor (L) connected in series or parallel. The inductor opposes changes in the current by storing energy in its magnetic field, while the resistor dissipates energy as heat. This configuration is used to study the transient and steady-state behavior of electrical circuits when subjected to a voltage source.

Linear First-Order Differential Equation

The behavior of the current in an RL circuit is governed by a linear first-order differential equation. This type of equation relates the rate of change of the current to the current itself and any external forcing (such as a voltage source), and it typically exhibits exponential behavior. Such equations are fundamental in modeling time-dependent processes in physics and engineering.

Phase Line Analysis

Phase line analysis is a qualitative technique used to examine the dynamics of one-dimensional systems described by differential equations. By analyzing the sign of the derivative at different values of the variable (in this case, the current), one can determine the stability of equilibrium points and understand the overall behavior of the system without needing an explicit solution.

Steady-State Solution

The steady-state solution refers to the long-term behavior of a system after all transient effects have diminished. For an RL circuit, this occurs when the rate of change of current becomes zero, indicating that the system has reached an equilibrium. At this point, the inductor no longer influences the current, which then settles to a constant value determined by Ohm’s law.

Transient Behavior

Transient behavior describes the initial period after a sudden change—in this case, after closing the switch—when the current adjusts from its initial state toward the steady state. During the transient period, the effects of inductance are significant, and the current typically changes exponentially over time before settling to its final steady value.

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