Alternating Current Notes

Introduction

Alternating Current (AC) is a type of electrical current in which the flow of electric charge periodically reverses direction. This is in contrast to Direct Current (DC), where the electric charge flows in one direction. AC is the form of electrical power commonly delivered to businesses and residences. In this study note, we will delve into the fundamental concepts, mathematical representations, and applications of AC, as well as its significance in the NEET Physics syllabus.

Fundamental Concepts of Alternating Current

Definition and Characteristics

• Alternating Current (AC): An electric current that periodically reverses its direction.
• Frequency (f): The number of cycles per second, measured in Hertz (Hz).
• Period (T): The time taken for one complete cycle, given by $T = \frac{1}{f}$.
• Amplitude (I_0 or V_0): The maximum value of current or voltage.

Mathematical Representation

The instantaneous value of an AC voltage or current can be represented as a sine or cosine function:

$$ V(t) = V_0 \sin(\omega t + \phi) $$

$$ I(t) = I_0 \sin(\omega t + \phi) $$

where:

• $V_0$ and $I_0$ are the peak voltage and peak current, respectively.
• $\omega$ is the angular frequency, given by $\omega = 2\pi f$.
• $\phi$ is the phase angle.

Root Mean Square (RMS) Values

The RMS value of AC is the effective value that represents the DC equivalent for power calculations. It is given by:

$$ V_{\text{rms}} = \frac{V_0}{\sqrt{2}} $$

$$ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} $$

AC Circuit Components

Resistor in an AC Circuit

For a purely resistive circuit, the voltage and current are in phase:

$$ V(t) = V_0 \sin(\omega t) $$

$$ I(t) = I_0 \sin(\omega t) $$

The impedance $Z$ of a resistor $R$ is simply $R$, and the power dissipated is:

$$ P = V_{\text{rms}} I_{\text{rms}} = \frac{V_0 I_0}{2} $$

Inductor in an AC Circuit

For a purely inductive circuit, the current lags the voltage by $90^\circ$ (or $\frac{\pi}{2}$ radians):

$$ V(t) = V_0 \sin(\omega t) $$

$$ I(t) = I_0 \sin\left(\omega t - \frac{\pi}{2}\right) $$

The inductive reactance $X_L$ is given by:

$$ X_L = \omega L $$

Capacitor in an AC Circuit

For a purely capacitive circuit, the current leads the voltage by $90^\circ$ (or $\frac{\pi}{2}$ radians):

$$ V(t) = V_0 \sin(\omega t) $$

$$ I(t) = I_0 \sin\left(\omega t + \frac{\pi}{2}\right) $$

The capacitive reactance $X_C$ is given by:

$$ X_C = \frac{1}{\omega C} $$

Impedance and Phase in AC Circuits

Impedance

The total opposition to current in an AC circuit is called impedance $Z$, which combines resistance $R$, inductive reactance $X_L$, and capacitive reactance $X_C$:

$$ Z = \sqrt{R^2 + (X_L - X_C)^2} $$

Phase Angle

The phase angle $\phi$ between the total voltage and current is given by:

$$ \tan \phi = \frac{X_L - X_C}{R} $$

Resonance in AC Circuits

Series Resonance

In a series RLC circuit, resonance occurs when the inductive reactance equals the capacitive reactance ($X_L = X_C$):

$$ \omega_0 L = \frac{1}{\omega_0 C} $$

The resonant frequency $\omega_0$ is:

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

Parallel Resonance

In a parallel RLC circuit, resonance occurs when the impedance is maximum, and the current through the circuit is minimum.

Power in AC Circuits

Instantaneous Power

The instantaneous power $P(t)$ in an AC circuit is given by:

$$ P(t) = V(t) I(t) $$

Average Power

The average power over a complete cycle is:

$$ P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos \phi $$

where $\cos \phi$ is the power factor.

Conclusion

Understanding alternating current is crucial for mastering the NEET Physics syllabus. AC is not only foundational to modern electrical systems but also rich with concepts such as impedance, phase angle, and resonance. By breaking down these complex ideas, we hope this study note provides a clear and comprehensive understanding of AC.