AC (Alternating Current)

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

AC Circuits — Key Facts

An alternating current (or voltage) periodically reverses direction. For sinusoidal AC: $$I = I_0 \sin(\omega t + \phi)$$

where I₀ = peak current, ω = angular frequency = 2πf, φ = phase angle

RMS Values:

RMS (root mean square) values are used for AC calculations because average value over full cycle is zero.

$$I_{rms} = \frac{I_0}{\sqrt{2}}, \quad V_{rms} = \frac{V_0}{\sqrt{2}}$$

For power calculations, always use RMS values.

⚡ JEE Exam Tip: The peak voltage in Indian mains (230V AC) is $230 \times \sqrt{2} \approx 325$ V. This is why the peak voltage rating of components matters!

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

AC through Different Elements:

Element	Voltage-Current Relation	Phase
Resistor (R)	$V = IR$	In phase
Inductor (L)	$V = IX_L$, $X_L = \omega L$	V leads I by 90°
Capacitor (C)	$V = IX_C$, $X_C = 1/\omega C$	V lags I by 90°

Impedance (Z):

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

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

• If φ > 0: circuit is inductive (V leads I)
• If φ < 0: circuit is capacitive (V lags I)
• If φ = 0: circuit is in resonance

Resonance:

At resonance: $\omega L = \frac{1}{\omega C} \Rightarrow \omega_0 = \frac{1}{\sqrt{LC}}$

At resonance:

• Z = R (minimum impedance)
• I = I_max (maximum current)
• Power factor = 1 (voltage and current in phase)

⚡ JEE Exam Tip: In series resonance, voltage across L and C can individually be much larger than the source voltage. Q-factor measures this magnification: $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR}$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Power in AC Circuits:

Instantaneous power: $p = vi = V_0 I_0 \sin\omega t \sin(\omega t + \phi)$

Average power (real power): $$P_{avg} = V_{rms} I_{rms} \cos\phi$$

where cosφ = power factor

Reactive power: $P_r = V_{rms} I_{rms} \sin\phi$ (VAR) Apparent power: $P_a = V_{rms} I_{rms}$ (VA)

Types of Loads:

• Pure resistive: cosφ = 1, all power consumed
• Pure inductive/capacitive: cosφ = 0, no real power consumed
• Mixed loads: 0 < cosφ < 1

Series RLC at Resonance:

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

$$I_{max} = \frac{V}{R}$$

$$V_L = I X_L = \frac{V}{R} \times \omega_0 L = V \times Q$$

$$V_C = I X_C = \frac{V}{R} \times \frac{1}{\omega_0 C} = V \times Q$$

where $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR}$ is the quality factor.

Bandwidth and Selectivity:

Half-power points: where power = ½ P_max

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

Sharpness of resonance: higher Q → narrower bandwidth → more selective circuit.

Transformers:

Ideal transformer (no losses): $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

Efficiency: $\eta = \frac{P_{out}}{P_{in}}$

Practical transformers have losses due to:

• Copper loss (I²R in windings)
• Iron loss (hysteresis + eddy current in core)
• Flux leakage

AC through Parallel Circuits:

For parallel RLC: $$I = \sqrt{I_R^2 + (I_L - I_C)^2}$$

Resonance in parallel circuits (anti-resonance):

• At $\omega = \omega_0$, the currents through L and C cancel, leaving only current through R
• Impedance is maximum at resonance
• Used in selectivity circuits

Choke Coil:

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

Power loss in pure inductor = 0 (because voltage and current are 90° out of phase).

⚡ JEE Advanced 2023 Analysis: Questions on power factor improvement, resonance in series and parallel circuits, and transformer calculations appeared in recent papers. For power factor correction, adding a capacitor in parallel reduces the reactive component of current, improving overall power factor.

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