Module 2: AC Circuits (Single-Phase & Three-Phase Systems)

1. 5 Key Ideas: Fundamental Principles of AC & Polyphase Systems

Idea 1: Sinusoidal EMF Generation & Faraday's Principle

Alternating voltage is generated mechanically in AC alternators by rotating a planar coil of $N$ turns and cross-sectional area $A$ with uniform angular velocity $\omega$ (rad/s) inside a stationary magnetic field of flux density $B$.

1. When the coil's plane is aligned with the magnetic axis, the magnetic flux linking the coil is at its maximum: $\Phi_m = B_m A$.
2. In time $t$ seconds, the coil rotates through an angular displacement $\theta = \omega t$.
3. The instantaneous perpendicular component of flux linking the coil becomes: $\Phi(t) = \Phi_m \cos(\omega t)$
4. By Faraday's Law of Electromagnetic Induction, the instantaneous induced EMF across the coil's terminals is: $e(t) = -\frac{d}{dt}\left[N \Phi(t)\right] = -N \frac{d}{dt}\left[\Phi_m \cos(\omega t)\right] = N \Phi_m \omega \sin(\omega t)$
5. The maximum peak EMF occurs at $\omega t = 90^\circ$ ($\sin\omega t = 1$): $E_m = \omega N \Phi_m = 2\pi f N B_m A \quad [\text{Volts}]$ $\implies e(t) = E_m \sin(\omega t), \qquad i(t) = I_m \sin(\omega t)$

Idea 2: AC Waveform Metrics — RMS, Average, Form & Peak Factors

Because AC sinusoidal quantities reverse direction periodically, their mathematical properties are quantified through standardized metrics:

• Cycle: One complete set of positive and negative alternating values ($360^\circ$ or $2\pi\text{ radians}$).
• Time Period ($T$): The duration required to complete one full cycle ($T = 1/f$ seconds).
• Frequency ($f$): Number of complete cycles executed per second ($f = 1/T\text{ Hertz}$).
• Root Mean Square (RMS) / Effective Value ($V_{\text{rms}}, I_{\text{rms}}$): The steady direct current (DC) that produces the exact same amount of thermal heat in an identical resistor over the same duration: $I_{\text{rms}} = \sqrt{\frac{1}{2\pi} \int_0^{2\pi} i^2(\theta) \, d\theta} = \frac{I_m}{\sqrt{2}} \approx 0.7071 \, I_m$
• Average Value ($V_{\text{avg}}, I_{\text{avg}}$): The arithmetic mean of mid-ordinates taken over one half-cycle (the full-cycle average of any symmetrical wave is zero): $I_{\text{avg}} = \frac{1}{\pi} \int_0^\pi I_m \sin\theta \, d\theta = \frac{2 I_m}{\pi} \approx 0.6366 \, I_m$
• Form Factor ($K_f$): The ratio of the RMS value to the half-cycle average value. For an ideal sinusoidal wave: $K_f = \frac{\text{RMS Value}}{\text{Average Value}} = \frac{0.7071 \, I_m}{0.6366 \, I_m} = \frac{\pi}{2\sqrt{2}} \approx 1.11$
• Crest / Peak / Amplitude Factor ($K_p$): The ratio of the maximum peak amplitude to the RMS value: $K_p = \frac{\text{Maximum Value}}{\text{RMS Value}} = \frac{I_m}{I_m / \sqrt{2}} = \sqrt{2} \approx 1.414$

Idea 3: Single-Element AC Circuit Dynamics ($R$, $L$, $C$)

• Pure Resistance ($R$): The voltage and current waves pass through zero and peak at the exact same instant: $v(t) = V_m \sin(\omega t), \quad i(t) = I_m \sin(\omega t) \implies \phi = 0^\circ \;\; (\text{In-Phase})$ The instantaneous power $p(t) = v(t) \cdot i(t) = \frac{V_m I_m}{2} - \frac{V_m I_m}{2}\cos(2\omega t)$ pulsates at twice the supply frequency ($2f$). Its average value is strictly positive: $P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} = I^2 R$ [Watts].
• Pure Inductance ($L$): The induced back-EMF ($e = -L \, di/dt$) opposes changes in current. By integration, the current lags the applied voltage by $90^\circ$ ($\pi/2\text{ rad}$): $i(t) = \frac{V_m}{\omega L} \sin\left(\omega t - \frac{\pi}{2}\right) = I_m \sin\left(\omega t - \frac{\pi}{2}\right), \quad X_L = \omega L = 2\pi f L \; [\Omega]$ The instantaneous power $p(t) = -\frac{V_m I_m}{2}\sin(2\omega t)$ is a pure sine wave oscillating at $2f$. Energy is alternately absorbed into and released from the magnetic field; average real power consumed is strictly zero ($P_{\text{avg}} = 0\text{ W}$).
• Pure Capacitance ($C$): The current is governed by charge accumulation ($i = dq/dt = C \, dv/dt$). The current leads the applied voltage by $90^\circ$ ($\pi/2\text{ rad}$): $i(t) = \omega C V_m \cos(\omega t) = I_m \sin\left(\omega t + \frac{\pi}{2}\right), \quad X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \; [\Omega]$ Energy is alternately stored in and discharged from the electrostatic field; average real power is zero ($P_{\text{avg}} = 0\text{ W}$).

Idea 4: Series $R$-$L$ Circuit & The Power Triangle

When a resistance $R$ and an inductance $L$ are connected in series across an AC supply $v(t) = V_m \sin(\omega t)$:

• The resistive drop $V_R = IR$ is in phase with current $I$.
• The inductive drop $V_L = I X_L$ leads current $I$ by $90^\circ$.
• By phasor vector addition, total applied voltage is: $V = \sqrt{V_R^2 + V_L^2} = I \sqrt{R^2 + X_L^2} = I Z$
• Circuit Impedance ($Z$): $Z = \sqrt{R^2 + X_L^2} \quad [\Omega], \qquad \phi = \arctan\left(\frac{X_L}{R}\right)$
• The current lags the applied voltage by phase angle $\phi$: $i(t) = I_m \sin(\omega t - \phi)$.
• The Three Powers:
  1. Active / Real Power ($P$): Rate of electrical energy permanently converted to heat and mechanical work: $P = V I \cos\phi = I^2 R \quad [\text{Watts (W)}]$
  2. Reactive Power ($Q$): Energy oscillating between supply and magnetic/electric fields without doing real work: $Q = V I \sin\phi = I^2 X_L \quad [\text{Volt-Amperes Reactive (VAR)}]$
  3. Apparent Power ($S$): The total capacity of the electrical supply apparatus (transformers and cables): $S = V I = I^2 Z = \sqrt{P^2 + Q^2} \quad [\text{Volt-Amperes (VA)}]$
  4. Power Factor ($\text{p.f.}$): $\text{p.f.} = \cos\phi = \frac{R}{Z} = \frac{P}{S} \quad (\text{Lagging for inductive circuits})$

Idea 5: Polyphase Systems & The Star ($Y$) vs. Delta ($\Delta$) Topologies

A three-phase alternator generates three identical sinusoidal EMFs mutually displaced by $120^\circ$ ($2\pi/3\text{ rad}$): $e_R = E_m \sin(\omega t), \quad e_Y = E_m \sin(\omega t - 120^\circ), \quad e_B = E_m \sin(\omega t + 120^\circ)$

Seven Advantages of 3-Phase Systems Over Single-Phase:

1. Conductor Economy: Requires less conductor mass to transmit equal power over equal distance at identical loss.
2. Uniform Constant Torque: Produces a smooth rotating magnetic field (single-phase machines produce pulsating torque and vibration).
3. Inherently Self-Starting: 3-phase induction motors start autonomously without auxiliary starting switches or capacitors.
4. Higher Power Density & Lower Cost: 3-phase machines are physically smaller and less costly for the same kVA rating.
5. No Harmonics in Parallel Alternators: Third harmonics cancel out in closed delta windings or balanced star connections.
6. Dual-Voltage Availability: Star networks provide two voltages: Line-to-Line ($400\text{ V}$ for motors) and Line-to-Neutral ($230\text{ V}$ for domestic appliances).
7. Balanced Neutral Cancellation: In a balanced star system, the vector sum of currents in the neutral wire is zero ($I_R + I_Y + I_B = 0$).

2. Formula & Unit Quick-Reference Strip

3. Mathematical Derivations: Sinusoidal Generation, Reactance & 3-Phase Systems

4. Typical Laboratory Hardware Values & Grid Standards

5. Worked Examples: Presidency University Curriculum Solutions

6. Common Exam Traps & Where Marks Are Lost

7. Interactive Circuit Simulator: AC Waveforms & Three-Phase Lab

Experiment with rotating coil generation, pure element phase dynamics, series $R$-$L$ impedance triangles, and live Star vs. Delta three-phase power scaling in the bespoke simulator below:

Interactive AC Rotating Phasor Vector & Power Triangle

Inspect how voltage and current rotating vectors generate active, reactive, and apparent power:

8. Parametrised Practice Problem

9. Unit Mastery Quiz

10. References & Academic Attribution

1. Dr. PL. Somasundaram, Assistant Professor – Senior Scale, Department of Electrical & Electronics Engineering, School of Engineering, Presidency University, Bengaluru — Lecture Slides for EEE1200: Basics of Electrical & Electronics Engineering (Module 2: AC Circuits).
2. B.L. Theraja & A.K. Theraja, A Textbook of Electrical Technology: Volume I (Basic Electrical Engineering), S. Chand & Company Ltd.
3. Vincent Del Toro, Electrical Engineering Fundamentals, 2nd Edition, Prentice-Hall International.
4. Bureau of Indian Standards (BIS), IS 12360: Voltage bands for electrical installations including standard voltages and frequencies.