In Exercises 35-38, consider a series circuit (Figure 4)...

In Exercises $35-38,$ 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 (timet in seconds). The current through the circuit $I(t)$ (in amperes) satisfies the differential equation $$\frac{d I}{d t}+\frac{R}{L} I=\frac{1}{L} V(t)$$ Solve Eq. (9) with initial condition $I(0)=0,$ assuming that $R=100$ ohms $(\Omega), L=5$ henries $(\mathrm{H}),$ and $V(t)$ is constant with $V(t)=10$ volts $(\mathrm{V}) .$

In Exercises $35-38,$ 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 (timet in seconds). The current through the circuit $I(t)$ (in amperes) satisfies the differential equation
$$\frac{d I}{d t}+\frac{R}{L} I=\frac{1}{L} V(t)$$

Solve Eq. (9) with initial condition $I(0)=0,$ assuming that $R=100$ ohms $(\Omega), L=5$ henries $(\mathrm{H}),$ and $V(t)$ is constant with $V(t)=10$ volts $(\mathrm{V}) .$

Key Concepts

Integrating Factor Method

This is a common method used to solve first-order linear differential equations. By multiplying both sides of the differential equation by a strategically chosen function (the integrating factor), the left-hand side becomes the derivative of a product, allowing for direct integration and subsequently leading to the general solution.

First-Order Linear Differential Equations

These are differential equations that can be written in the form dy/dt + P(t)y = Q(t), where P(t) and Q(t) are functions of time. They have well-established solving techniques, including the application of an integrating factor, making them fundamental in describing systems with proportional response dynamics, such as circuits and population models.

RL Circuit Dynamics

An RL circuit, which is composed of a resistor (R) and an inductor (L) in series, exhibits behavior that can be modeled by a first-order linear differential equation. This model describes how the current through the circuit changes over time in response to an applied voltage, capturing both the energy storage in the inductor and the energy dissipation in the resistor.

Transient Response

The transient response refers to the behavior of a circuit as it responds to a change or a new input, such as the application of a voltage source. In the context of RL circuits, the transient phase is the period during which the current builds up from an initial value to its steady-state value, a process that is typically governed by exponential growth or decay.

Time Constant

The time constant, denoted as ? and calculated as L/R for an RL circuit, is a measure of how quickly the transient response occurs. It indicates the time required for the system to reach approximately 63% of its final steady-state value. This concept is crucial in assessing circuit dynamics and response speeds in various engineering applications.

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