Unit 2 - Notes

ECE131

Unit 2: Fundamentals of A.C. Circuits

1. Introduction to Alternating Current (A.C.) Basics

Alternating Current and Voltage

An alternating quantity (current or voltage) is one whose magnitude changes continuously with time and whose direction reverses periodically.

Generation: Usually generated by an alternator based on Faraday’s Law of Electromagnetic Induction.
Standard Waveform: The sinusoidal wave is the standard waveform for AC power systems because it is the only waveform that remains unchanged in shape (though amplitude and phase may change) when passed through linear circuit elements (R, L, C).

Standard Equation of an AC Voltage:

v(t) = V_m \sin(\omega t + \theta)

Where:

$v(t)$ : Instantaneous voltage at time $t$
$V_m$ : Maximum or Peak amplitude
$\omega$ : Angular frequency (rad/s)
$t$ : Time (seconds)
$\theta$ : Phase angle

Concept of Notations ( $i, v, I, V$ )

In Electrical Engineering, strict notation conventions are used to distinguish between different types of values:

Lower case letters ( $i, v$ ): Represent Instantaneous values. These are time-dependent functions (e.g., $v = 100 \sin(314t)$ ).
Upper case letters ( $I, V$ ): Represent RMS (Root Mean Square) values. Unless specified otherwise, AC ratings (like 230V AC) always refer to RMS values.
Upper case with subscript 'm' ( $I_m, V_m$ ): Represent Maximum or Peak values.

Key Definitions

1. Amplitude (Peak Value)

The maximum positive or negative value attained by an alternating quantity during one cycle.

Denoted by $V_m$ or $I_m$ .

2. Phase

The phase of an alternating quantity represents the fraction of the time period or cycle that has elapsed since the quantity last passed through the zero position of reference. It defines the state of the wave at $t=0$ .

3. Phase Difference ( $\phi$ )

When two alternating quantities of the same frequency do not pass through their zero values at the same instant, they are said to have a phase difference.

In Phase: Both waves reach zero and peak simultaneously. ( $\phi = 0^\circ$ ).
Leading: A wave reaches its maximum value earlier than the reference wave.
Lagging: A wave reaches its maximum value later than the reference wave.

4. Average Value

The arithmetic mean of all instantaneous values over a period.

For a symmetrical sine wave, the average value over one full cycle is zero.
Therefore, the average value is calculated over a half-cycle.

Formula:

V_{avg} = \frac{1}{\pi} \int_{0}^{\pi} V_m \sin(\theta) \, d\theta = \frac{2V_m}{\pi}

V_{avg} \approx 0.637 V_m

5. RMS Value (Root Mean Square)

The "Effective Value" of AC. It is defined as that value of DC current which, when flowing through a given resistance for a given time, produces the same amount of heat as produced by the AC current flowing through the same resistance for the same time.

Formula:

V_{rms} = \sqrt{\frac{1}{2\pi} \int_{0}^{2\pi} [V_m \sin(\theta)]^2 \, d\theta} = \frac{V_m}{\sqrt{2}}

V_{rms} \approx 0.707 V_m

2. Complex Representation of Impedance

To analyze AC circuits efficiently, we use the complex number plane (Phasors), eliminating the need for differential equations.

The Operator ' $j$ '

$j = \sqrt{-1}$ .
Multiplying a phasor by $j$ rotates it $90^\circ$ counter-clockwise.
Multiplying by $-j$ rotates it $90^\circ$ clockwise.

Impedance ( $Z$ )

Impedance is the total opposition offered to the flow of AC current, combining resistance and reactance.

Rectangular Form:

Z = R + jX

Where:

$R$ = Resistance (Real part)
$X$ = Reactance (Imaginary part)

Polar Form:

Z = |Z| \angle \phi

Where

|Z| = \sqrt{R^2 + X^2}

and

\phi = \tan^{-1}(X/R)

Circuit Elements in Complex Domain

Resistor ( $R$ ):
- Impedance: $Z_R = R \angle 0^\circ = R + j0$
- Current and Voltage are in phase.
Inductor ( $L$ ):
- Inductive Reactance: $X_L = \omega L = 2\pi f L$
- Impedance: $Z_L = jX_L = X_L \angle 90^\circ$
- Voltage leads Current by $90^\circ$ .
Capacitor ( $C$ ):
- Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$
- Impedance: $Z_C = -jX_C = \frac{1}{j\omega C} = X_C \angle -90^\circ$
- Voltage lags Current by $90^\circ$ .

3. Steady State Analysis of Series AC Circuits

Series RL Circuit

A pure resistance $R$ is connected in series with a pure inductance $L$ .

Impedance ( $Z$ ): $Z = R + jX_L$
Magnitude: $|Z| = \sqrt{R^2 + X_L^2}$
Phase Angle ( $\phi$ ): $\phi = \tan^{-1}\left(\frac{X_L}{R}\right)$
Nature: Inductive (Current lags Voltage by $\phi$ ).
Voltage Triangle: $V^2 = V_R^2 + V_L^2$

Series RC Circuit

A pure resistance $R$ is connected in series with a pure capacitance $C$ .

Impedance ( $Z$ ): $Z = R - jX_C$
Magnitude: $|Z| = \sqrt{R^2 + X_C^2}$
Phase Angle ( $\phi$ ): $\phi = \tan^{-1}\left(\frac{-X_C}{R}\right)$
Nature: Capacitive (Current leads Voltage by $\phi$ ).
Voltage Triangle: $V^2 = V_R^2 + V_C^2$

Series RLC Circuit

A resistance $R$ , inductance $L$ , and capacitance $C$ are connected in series.

Net Reactance ( $X$ ): $X = X_L - X_C$
Impedance ( $Z$ ): $Z = R + j(X_L - X_C)$
Magnitude: $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$
Phase Angle: $\phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$

Three conditions based on reactance:

$X_L > X_C$ : Circuit acts Inductive (Lagging PF).
$X_C > X_L$ : Circuit acts Capacitive (Leading PF).
$X_L = X_C$ : Circuit acts Resistive (Unity PF). This is Resonance.

4. Power and Power Factor

In AC circuits, power is not simply $V \times I$ due to the phase difference.

1. Instantaneous Power

p(t) = v(t) \times i(t)

2. Active Power (Real Power/True Power) - $P$

The power actually consumed or dissipated by the resistive part of the circuit.

Formula: $P = VI \cos \phi$
Unit: Watts (W) or kiloWatts (kW)

3. Reactive Power - $Q$

The power that oscillates back and forth between the source and the reactive components (inductors/capacitors) to establish magnetic/electric fields.

Formula: $Q = VI \sin \phi$
Unit: Volt-Ampere Reactive (VAR) or kVAR

4. Apparent Power - $S$

The product of RMS voltage and RMS current.

Formula: $S = VI$
Unit: Volt-Ampere (VA) or kVA
Relation: $S^2 = P^2 + Q^2$ or $\vec{S} = P + jQ$ (for inductive load).

5. Power Factor (pf)

Defined as the cosine of the angle between voltage and current. It represents the efficiency of the power system.

Formula: $pf = \cos \phi = \frac{R}{Z} = \frac{\text{Active Power}}{\text{Apparent Power}}$
Range: $0$ to $1$.
Descriptive: Must specify "Lagging" (inductive) or "Leading" (capacitive).

5. Resonance in Series RLC Circuit

Resonance occurs in an RLC series circuit when the inductive reactance equals the capacitive reactance.

Condition for Resonance

X_L = X_C \Rightarrow \omega L = \frac{1}{\omega C}

Resonant Frequency ( $f_r$ )

f_r = \frac{1}{2\pi \sqrt{LC}} \text{ Hz}

Characteristics at Resonance

Impedance is Minimum: $Z = R$ (Purely resistive).
Current is Maximum: $I_{max} = V / R$ .
Power Factor is Unity: $\cos \phi = 1$ .
Voltage Magnification: The voltage across $L$ and $C$ can be much higher than the supply voltage.

Quality Factor (Q-Factor)

It measures the "sharpness" of resonance or the voltage magnification.

Q = \frac{\text{Voltage across L or C}}{\text{Applied Voltage}} = \frac{1}{R}\sqrt{\frac{L}{C}}

6. Three-Phase Circuits

Three-phase systems are preferred over single-phase for power generation and transmission because they provide constant power output, are more efficient for the same conductor size, and utilize self-starting induction motors.

Numbering and Phase Sequence

Three windings are displaced by $120^\circ$ electrically.
Standard Colors/Notation: Red (R), Yellow (Y), Blue (B).
Phase Sequence: The order in which the voltages reach their maximum positive value (usually R-Y-B).

Terminology

Phase Voltage ( $V_{ph}$ ): Voltage induced in one coil (between a phase wire and neutral).
Line Voltage ( $V_L$ ): Voltage between two phase wires (e.g., $V_{RY}$ ).
Phase Current ( $I_{ph}$ ): Current flowing through one coil/winding.
Line Current ( $I_L$ ): Current flowing through the transmission line.

Interconnection of Phases

1. Star Connection (Wye or Y)

Connection: Similar ends (Start or Finish) of the three coils are joined together to form a common Neutral point ( $N$ ).
System: 3-Phase, 4-Wire system (R, Y, B, N).

Star Relationships:

Current: Line Current = Phase Current
$I_L = I_{ph}$
Voltage: Line Voltage = $\sqrt{3} \times$ Phase Voltage
$V_L = \sqrt{3} V_{ph}$
Angle: Line voltage leads the respective phase voltage by $30^\circ$ .

2. Delta Connection (Mesh or $\Delta$ )

Connection: Dissimilar ends are connected (End of R to Start of Y, etc.) to form a closed loop. No Neutral point exists.
System: 3-Phase, 3-Wire system.

Delta Relationships:

Voltage: Line Voltage = Phase Voltage
$V_L = V_{ph}$
Current: Line Current = $\sqrt{3} \times$ Phase Current
$I_L = \sqrt{3} I_{ph}$
Angle: Line current lags the respective phase current by $30^\circ$ .

Power in Three-Phase Circuits (Star or Delta)

Regardless of the connection type (Star or Delta), the total power formulas remain the same:

Active Power ( $P$ ):
$P = 3 V_{ph} I_{ph} \cos \phi = \sqrt{3} V_L I_L \cos \phi \quad \text{(Watts)}$
Reactive Power ( $Q$ ):
$Q = \sqrt{3} V_L I_L \sin \phi \quad \text{(VAR)}$
Apparent Power ( $S$ ):
$S = \sqrt{3} V_L I_L \quad \text{(VA)}$