Alternating Current Notes

Introduction

Alternating Current (AC) is a fundamental concept in the field of electromagnetism and electrical engineering. In contrast to Direct Current (DC), where the flow of electric charge is unidirectional, AC involves the flow of electric charge that periodically reverses direction. This document provides a comprehensive study of AC, tailored for JEE Advanced Physics, covering all relevant concepts, equations, and examples.

Basic Concepts of Alternating Current

Definition

Alternating Current (AC) is an electric current that reverses its direction periodically. The voltage in an AC circuit also alternates in a sinusoidal manner.

Sinusoidal Waveform

The most common form of AC is a sinusoidal waveform, which can be described mathematically as: $$ v(t) = V_m \sin(\omega t + \phi) $$ where:

• $v(t)$ is the instantaneous voltage
• $V_m$ is the peak voltage
• $\omega$ is the angular frequency ($\omega = 2\pi f$)
• $t$ is the time
• $\phi$ is the phase angle

Frequency and Period

• Frequency (f): The number of cycles per second, measured in Hertz (Hz).
• Period (T): The time taken to complete one cycle, given by $T = \frac{1}{f}$.

RMS Value

The Root Mean Square (RMS) value of an AC is a measure of the effective value of the alternating current or voltage. For a sinusoidal AC: $$ V_{rms} = \frac{V_m}{\sqrt{2}} $$ $$ I_{rms} = \frac{I_m}{\sqrt{2}} $$ where $V_m$ and $I_m$ are the peak voltage and current, respectively.

AC Circuit Elements

Resistive Circuit (R)

In a purely resistive circuit, the voltage and current are in phase. Ohm's law applies as: $$ V = IR $$

Inductive Circuit (L)

In a purely inductive circuit, the current lags the voltage by 90 degrees. The inductive reactance is given by: $$ X_L = \omega L = 2\pi f L $$ The voltage-current relationship is: $$ V_L = I X_L = I \omega L $$

Capacitive Circuit (C)

In a purely capacitive circuit, the current leads the voltage by 90 degrees. The capacitive reactance is given by: $$ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} $$ The voltage-current relationship is: $$ V_C = I X_C = \frac{I}{\omega C} $$

AC Circuit Analysis

Series RLC Circuit

In a series RLC circuit, the resistor (R), inductor (L), and capacitor (C) are connected in series. The total impedance (Z) is: $$ Z = \sqrt{R^2 + (X_L - X_C)^2} $$ The phase angle ($\phi$) between the voltage and current is: $$ \tan \phi = \frac{X_L - X_C}{R} $$

Power in AC Circuits

The power in AC circuits is given by: $$ P = V_{rms} I_{rms} \cos \phi $$ where $\cos \phi$ is the power factor.

Resonance in AC Circuits

Series Resonance

Occurs when the inductive reactance equals the capacitive reactance ($X_L = X_C$). The resonance frequency ($f_0$) is: $$ f_0 = \frac{1}{2\pi \sqrt{LC}} $$

Parallel Resonance

Occurs when the total impedance of the circuit is at its maximum and the circuit behaves like a purely resistive circuit.

Conclusion

Understanding alternating current is vital for mastering electromagnetism and AC circuit analysis. This study note has covered the fundamental concepts, including sinusoidal waveforms, RMS values, circuit elements, and resonance. By grasping these principles and practicing problem-solving, students can excel in the AC-related questions in JEE Advanced Physics.