Alternating Current

Alternating Current (AC) extends the study of circuits from the steady Direct Current (DC) you have seen earlier. This chapter deals with voltages and currents that vary sinusoidally with time. You will learn how the three basic circuit elements, resistors, inductors, and capacitors, behave when connected to an AC source, both individually and in combination. The core of the analysis involves understanding the phase relationship between the voltage across an element and the current flowing through it.

A key tool for this analysis is the phasor diagram, which represents oscillating quantities as rotating vectors. Using phasors, you will analyze series R, L, and C circuits, leading to the concept of impedance, which is the AC equivalent of resistance. The chapter then builds up to the series LCR circuit, a central topic. You will study its behaviour at different frequencies, including the special condition of resonance, where the current in the circuit becomes maximum.

The final part of the chapter covers power calculations in AC circuits. You will learn about RMS (root mean square) values, which are the effective values of AC voltage and current, and the concept of the power factor, which determines the actual power dissipated in a circuit. The chapter usually concludes with the principles of LC oscillations and the working of a transformer. The most challenging part for many is mastering phasor diagrams and the frequency-dependent behaviour of LCR circuits. A strong foundation in Electromagnetic Induction is essential.

Formulas for Alternating Current

Instantaneous AC Voltage: $V = V_m \sin(\omega t)$ V_m is the peak voltage and ω is the angular frequency.
RMS (Root Mean Square) Voltage: $V_{rms} = \frac{V_m}{\sqrt{2}}$ The effective value of AC voltage, used for power calculations.
Inductive Reactance: $X_L = \omega L$ The opposition to current flow offered by an inductor.
Capacitive Reactance: $X_C = \frac{1}{\omega C}$ The opposition to current flow offered by a capacitor.
Impedance of Series LCR Circuit: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ The total effective resistance of a series LCR circuit.
Phase Angle in Series LCR Circuit: $\tan \phi = \frac{X_L - X_C}{R}$ ϕ is the phase difference between the total voltage and the current.
Resonant Frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$ The angular frequency at which current is maximum (X_L = X_C).
Average Power Dissipated: $P_{avg} = V_{rms} I_{rms} \cos \phi$ Power is dissipated only in the resistor.
Power Factor: $\cos \phi = \frac{R}{Z}$ Represents the fraction of total power that is used for work.
Quality Factor (Q-factor): $Q = \frac{\omega_0 L}{R}$ A measure of the sharpness of the resonance peak.
Transformer Equation: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$ Relates voltages (V) and number of turns (N) in the primary (p) and secondary (s) coils.