Chapter 23: RL and LC Circuits

Table of Contents

23.1 Introduction

In this chapter, we will explore RL (resistor-inductor) and LC (inductor-capacitor) circuits. These circuits are essential to understanding the transient behavior and oscillations that occur in various electrical and electronic applications.

23.2 RL Circuits

An RL circuit consists of a resistor $(R)$ and an inductor $(L)$ connected in series. When a voltage is applied across the RL circuit, the current flowing through the circuit changes over time due to the presence of the inductor.

RL circuits have one resistor and one inductor in series with one another.

23.2.1 Time Constant in RL Circuits

The time constant $(\tau )$ of an RL circuit is given by the ratio of the inductance $(L)$ to the resistance $(R)$ :

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

The time constant is a measure of how quickly the current in the RL circuit reaches its steady-state value after a voltage is applied or removed.

23.2.2 Transient Response in RL Circuits

When a voltage is applied to an RL circuit, the current through the inductor increases, inducing a back-EMF that opposes the applied voltage. The current eventually reaches a steady-state value, determined by Ohm’s law:

$I = \dfrac{V}{R}$

The transient response of the current in an RL circuit can be described by the following equation:

$i(t) = \dfrac{V}{R} \left(1 - e^{-t / \tau}\right)$

where $i(t)$ is the current at time $t$ , $V$ is the applied voltage, and $\tau$ is the time constant.

23.3 LC Circuits

An LC circuit consists of an inductor $(L)$ and a capacitor $(C)$ connected in series. When the LC circuit is energized, the energy oscillates between the inductor’s magnetic field and the capacitor’s electric field.

LC circuits have one inductor and one capacitor in series with one another.

23.3.1 Natural Frequency of LC Circuits

The natural frequency $(\omega_0)$ of an LC circuit is given by the following equation:

$\omega_0 = \dfrac{1}{\sqrt{LC}}$

The natural frequency is a measure of how quickly energy oscillates between the inductor and the capacitor in the LC circuit.

23.3.2 Oscillations in LC Circuits

In an ideal LC circuit, the energy oscillates between the inductor and capacitor indefinitely. However, in real-world LC circuits, the oscillations eventually dampen due to energy losses from various sources, such as resistance in the components and radiation.

The oscillations in an LC circuit can be described by the following equation:

$q(t) = Q \cos(\omega_0 t)$

where $q(t)$ is the charge on the capacitor at time $t$ , $Q$ is the maximum charge, and $\omega_0$ is the natural frequency.

Chapter Summary

In this chapter, we explored RL and LC circuits, which are crucial for understanding the transient behavior and oscillations in various electrical applications. We discussed the time constant in RL circuits, which determines the rate at which current reaches its steady-state value. We also examined the natural frequency of LC circuits, which dictates the rate at which energy oscillates between the inductor and capacitor. Understanding RL and LC circuits is essential for designing and analyzing various electrical and electronic systems, such as filters, oscillators, and transient suppression circuits.

Continue to Chapter 24: The LRC Series Circuit

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Chapter 23: RL and LC Circuits

23.1 Introduction