AC

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Alternating Current (AC) is electric current that reverses direction periodically. It’s the form of electricity used in homes and industries because it can be easily transformed to different voltages. This chapter covers AC fundamentals, AC circuits, and electromagnetic induction (EMI).

Key Definitions:

• Alternating EMF: $\varepsilon = \varepsilon_0 \sin(\omega t)$ or $\varepsilon = \varepsilon_0 \cos(\omega t)$
• RMS Value: $\varepsilon_{rms} = \frac{\varepsilon_0}{\sqrt{2}}$ (for sinusoidal AC)
• Angular Frequency: $\omega = 2\pi f$ (rad/s)
• Phasor: Rotating vector representing sinusoidal quantity

Important Formulas:

Quantity	Formula	Unit
Average value (half cycle)	$\varepsilon_{av} = \frac{2\varepsilon_0}{\pi}$	Volt
Form factor	$\frac{\varepsilon_{rms}}{\varepsilon_{av}}$	Dimensionless
Peak factor	$\frac{\varepsilon_0}{\varepsilon_{rms}}$	Dimensionless

Faraday’s Law of EMI: Induced EMF = negative rate of change of magnetic flux: $$\varepsilon = -\frac{d\Phi}{dt}$$

Lenz’s Law: Induced current opposes the change in magnetic flux that causes it.

⚡ CUET Exam Tips:

• For sinusoidal AC: $I_{rms} = I_0/\sqrt{2}$
• Inductive reactance $X_L = \omega L = 2\pi fL$
• Capacitive reactance $X_C = 1/(\omega C) = 1/(2\pi fC)$
• At resonance in LCR circuit: $X_L = X_C$, current is maximum

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

AC Through Different Elements:

Element	Impedance	Phase
Resistor (R)	Z = R	Voltage and current in phase
Inductor (L)	Z = ωL	Voltage leads current by 90°
Capacitor (C)	Z = 1/ωC	Voltage lags current by 90°

Series LCR Circuit:

$$Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$$

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

• When $\omega L > 1/(\omega C)$: Circuit is inductively dominant (current lags voltage)
• When $\omega L < 1/(\omega C)$: Circuit is capacitively dominant (current leads voltage)
• When $\omega L = 1/(\omega C)$: Resonance occurs

Resonance in LCR Circuit:

At resonance: $$\omega_0 = \frac{1}{\sqrt{LC}}$$

$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$

• At resonance: Z = R (minimum impedance)
• Current is maximum: $I_{max} = \varepsilon_{rms}/R$
• Power factor = 1 (purely resistive)

Quality Factor (Q):

$$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR}$$

• Q factor measures sharpness of resonance
• High Q = narrow bandwidth = selective circuit

AC Power:

Power Type	Formula	Unit
Instantaneous	$P = \varepsilon i$	Watt
True (Real)	$P = \varepsilon_{rms} I_{rms} \cos\phi$	Watt
Reactive	$P_r = \varepsilon_{rms} I_{rms} \sin\phi$	VAR
Apparent	$P_a = \varepsilon_{rms} I_{rms}$	VA

$$\cos\phi = \frac{R}{Z} = \text{Power Factor}$$

Self-Induction:

When current in a coil changes, an induced EMF opposes the change: $$\varepsilon = -L\frac{dI}{dt}$$

$$L = \frac{N\Phi}{I} \quad \text{(Henry, H)}$$

Energy stored in inductor: $$U = \frac{1}{2}LI^2$$

⚠️ CUET Common Mistakes:

1. Confusing RMS with average values
2. Forgetting that capacitive reactance decreases with frequency (opposite of inductor)
3. Not understanding power factor’s role
4. Confusing phasor representation with vectors

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage with detailed derivations, transformer theory, and previous year CUET patterns.

Phasor Diagram Analysis:

For series LCR circuit:

• Resistor voltage: $V_R$ in phase with $I$
• Inductor voltage: $V_L$ leads $I$ by 90°
• Capacitor voltage: $V_C$ lags $I$ by 90°
• Net voltage: $V = \sqrt{V_R^2 + (V_L - V_C)^2}$

Phasor sum: $\mathbf{V} = \mathbf{V}_R + \mathbf{V}_L + \mathbf{V}_C$

Transformer Theory:

Principle: Mutual induction — changing current in primary induces EMF in secondary

Ideal Transformer Equations: $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

$$P_{in} = P_{out} \quad \text{(for ideal transformer)}$$

Types:

• Step-up: $N_s > N_p$, $V_s > V_p$
• Step-down: $N_s < N_p$, $V_s < V_p$

Practical Transformer Losses:

1. Copper loss: $I^2R$ in windings
2. Core (iron) loss:
   • Hysteresis loss: Proportional to frequency and hysteresis loop area
   • Eddy current loss: $P \propto t^2 f^2 B^2$ (where t is lamination thickness)
3. Flux leakage: Not all flux links both coils
4. Magnetisation current: Core requires magnetising current

Reduction of Eddy Currents: Use laminated core (thin sheets insulated from each other) to break up circulating currents.

Bandwidth and Half-Power Points:

At half-power points, power = $\frac{1}{2}P_{max}$: $$P = I_{rms}^2 R = \frac{V_{rms}^2 R}{Z^2}$$

$$Z^2 = 2R^2 \quad \Rightarrow \quad |X_L - X_C| = R$$

Bandwidth: $\Delta\omega = \frac{\omega_0}{Q}$

Choke Coil:

An inductor used to limit AC current without significant power loss (unlike a resistor).

• Series inductance reduces current in AC circuit
• Since ideal inductor stores no net energy (equal power absorbed and returned), power loss is minimal

LC Oscillations:

Undam LC circuit: $$q = q_0 \cos(\omega t + \phi)$$

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

Energy oscillates between capacitor (electric) and inductor (magnetic).

AC Generator (Dynamo):

• Converts mechanical energy to electrical energy
• Principle: Electromagnetic induction
• Coil rotating in uniform B-field: $\varepsilon = NBA\omega \sin(\omega t)$
• Slip rings and brushes for DC output (with commutator)

Three-Phase AC:

Three alternating currents with 120° phase difference: $$I_1 = I_0 \sin(\omega t)$$ $$I_2 = I_0 \sin(\omega t + 120°)$$ $$I_3 = I_0 \sin(\omega t + 240°)$$

Advantages: More efficient power transmission, constant power delivery.

Previous Year CUET Patterns:

CUET 2022: The average value of AC over a complete cycle is: a) $\varepsilon_0$ b) $\varepsilon_0/2$ c) Zero d) $2\varepsilon_0/\pi$ Answer: c) Zero — Average of sine wave over complete cycle is zero (positive and negative cancel)

CUET 2022: In a series LCR circuit at resonance, the current is: a) Minimum b) Maximum c) Zero d) Independent of R Answer: b) Maximum — At resonance, impedance is minimum (equal to R), so current is maximum

CUET 2023: The unit of inductance is: a) Weber b) Henry c) Tesla d) Ampere/second Answer: b) Henry (Wb/A)

CUET 2023: The power factor of an AC circuit is: a) $R/Z$ b) $X_L/Z$ c) $X_C/Z$ d) $V/I$ Answer: a) R/Z — Power factor = cos(phase angle) = adjacent/hypotenuse = R/Z

Advanced AC Concepts:

Complex Impedance: $$Z = R + jX \quad \text{(where } X = X_L - X_C\text{)}$$

$$Z^* = R - jX \quad \text{(complex conjugate)}$$

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

Admittance: $$Y = \frac{1}{Z} = G + jB \quad \text{(siemens)}$$

Skin Effect: At high frequencies, AC current flows mainly near conductor surface, increasing effective resistance.

Power Factor Improvement: Adding capacitor in parallel to inductive load improves power factor, reducing reactive power demand.

Waveforms:

Waveform	RMS	Average (half-cycle)	Form Factor	Crest Factor
Sine	$E_0/\sqrt{2}$	$2E_0/\pi$	$\frac{\pi}{2\sqrt{2}} \approx 1.11$	$\sqrt{2} \approx 1.414$
Square	$E_0$	$E_0$	1	1
Triangular	$E_0/\sqrt{3}$	$E_0/2$	$2/\sqrt{3} \approx 1.15$	$\sqrt{3} \approx 1.73$

Practical Applications:

• Household AC: 220V, 50Hz in India
• Transmission: High voltage to reduce $I^2R$ losses
• Rectification: AC to DC conversion
• Inverters: DC to AC conversion

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