Consider a series circuit (Figure 4) consisting of a...

Consider a series circuit (Figure 4$)$ consisting of a resistor of $R$ ohms, an inductor of $L$ henries, and a variable voltage source of $V(t)$ volts (time t in seconds). The current through the circuit $I(t)$ (in amperes) satisfes the differential equation $$ \frac{d I}{d t}+\frac{R}{L} I=\frac{1}{L} V(t) $$ Solve for $I(t),$ assuming that $R=500 \Omega, L=4 \mathrm{H},$ and $V=$ 20 $\cos (80) \mathrm{V}^{2}$

Consider a series circuit (Figure 4$)$ consisting of a resistor of $R$ ohms, an inductor of $L$ henries, and a variable voltage source of $V(t)$ volts (time t in seconds). The current through the circuit $I(t)$ (in amperes) satisfes the differential equation
$$
\frac{d I}{d t}+\frac{R}{L} I=\frac{1}{L} V(t)
$$
Solve for $I(t),$ assuming that $R=500 \Omega, L=4 \mathrm{H},$ and $V=$ 20 $\cos (80) \mathrm{V}^{2}$

Key Concepts

Series RL Circuit

A series RL circuit is composed of a resistor (R) and an inductor (L) connected in series. The behavior of such circuits is governed by Kirchhoff’s voltage law and leads to a first?order differential equation that describes how the current changes over time when driven by a voltage source. This analysis is central in studying transient responses in circuits, especially when energy storage in the inductor causes delays in current changes.

First-Order Linear Differential Equation

A first?order linear differential equation has the standard form dy/dt + P(t)y = Q(t). In the context of circuits, this form naturally arises when combining the resistive (dissipative) and reactive (energy storing) elements. The equation represents the interplay between the natural exponential decay of the circuit response and the external forcing from the voltage source.

Integrating Factor Method

The integrating factor method is a systematic technique used to solve first?order linear differential equations. It involves multiplying the entire equation by an integrating factor, usually exp(?P(t) dt), to transform the left-hand side into the derivative of a product. This method simplifies the equation, making it easier to integrate and to obtain a general solution composed of both transient and particular components.

Transient and Steady-State Response

When a linear system is subjected to a forcing function, the overall response typically includes a transient part, which decays over time, and a steady-state (or particular) part that persists. In an RL circuit, the transient response is represented by an exponentially decaying term related to the initial conditions, while the steady-state response mirrors the driving voltage’s form (often a sinusoidal function) after the effects of the initial conditions have dissipated.

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