Alternating Current (AC)

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your MDCAT exam.

What is AC for MDCAT? AC is current that changes direction periodically. In Pakistan (and globally), mains electricity is 220-240V, 50 Hz AC. You need to understand AC waveform parameters, phasor diagrams, and reactive circuits (LR, CR, LCR).

AC Parameters:

• Instantaneous current: $i = I_0 \sin(\omega t)$ or $i = I_0 \cos(\omega t)$
• Angular frequency: $\omega = 2\pi f$ (rad/s)
• Frequency $f = 50$ Hz (Pakistan), $T = 1/f = 0.02$ s = 20 ms
• RMS values: $I_{rms} = I_0/\sqrt{2}$, $V_{rms} = V_0/\sqrt{2}$

Why RMS? AC constantly changes — we need a value that represents the same power as an equivalent DC. RMS is the effective value of AC for power calculations.

⚡ MDCAT Tip: Mains voltage 220V AC in Pakistan is the RMS value. The peak voltage is $220 \times \sqrt{2} \approx 311$ V. This matters when dealing with capacitors and peak voltage ratings.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Reactance — Opposition to AC:

Inductive Reactance: $X_L = \omega L = 2\pi f L$ (in Ohms)

• Inductor opposes changes in current (like inertia in mechanics)
• Current lags voltage by 90° in a purely inductive circuit

Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$

• Capacitor opposes changes in voltage
• Current leads voltage by 90° in a purely capacitive circuit

Key difference: $X_L$ increases with frequency (inductive reactance rises at high frequency), while $X_C$ decreases with frequency (capacitive reactance falls at high frequency).

Series LR Circuit: $$Z = \sqrt{R^2 + X_L^2} = \sqrt{R^2 + (\omega L)^2}$$ $$\tan \phi = \frac{X_L}{R}$$ Current lags voltage by angle φ.

Series CR Circuit: $$Z = \sqrt{R^2 + X_C^2} = \sqrt{R^2 + (1/\omega C)^2}$$ $$\tan \phi = \frac{X_C}{R}$$ Current leads voltage by angle φ.

Power in AC Circuits:

• Real power (P): $P = V_{rms} I_{rms} \cos\phi$ (Watts) — dissipated in resistive components
• Reactive power (Q): $Q = V_{rms} I_{rms} \sin\phi$ (VAR) — exchanged between source and reactive components
• Apparent power (S): $S = V_{rms} I_{rms}$ (VA)
• Power factor $\cos\phi = R/Z$

⚡ MDCAT Tip: In pure inductance or pure capacitance, $\cos\phi = 0$ — all power is reactive, no real power dissipation. In pure resistance, $\cos\phi = 1$ — all power is real.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Series LCR Circuit — Complete Analysis:

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

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

Resonance condition: When $X_L = X_C$, i.e., $\omega L = 1/\omega C$ $$\omega_r = \frac{1}{\sqrt{LC}}, \quad f_r = \frac{1}{2\pi\sqrt{LC}}$$

At resonance:

• Impedance is minimum ($Z = R$)
• Current is maximum for a given voltage
• Power factor = 1 (current in phase with voltage)
• This is the condition used in tuning circuits (radio, TV)

Quality Factor (Q-factor): $$Q = \frac{\omega_r L}{R} = \frac{1}{\omega_r CR}$$

• High Q = sharp resonance (narrow bandwidth) — used in oscillators
• Low Q = broad resonance (wide bandwidth) — used in power transmission

Transformer — AC Power Transmission: $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

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

Efficiency: $\eta = \frac{P_{out}}{P_{in}} \times 100%$. In an ideal transformer: $P_{in} = P_{out}$ → $V_p I_p = V_s I_s$.

AC Generator (Dynamo) — Working Principle: A coil rotating in a magnetic field produces alternating EMF: $$\varepsilon = \varepsilon_0 \sin(\omega t) \quad \text{where} \quad \varepsilon_0 = NBA\omega$$

• $N$ = number of turns
• $B$ = magnetic field strength
• $A$ = area of coil
• $\omega$ = angular velocity of rotation

Choke Coil (Inductor) in AC Circuits: A coil with high inductance used to limit AC current without power loss (unlike a resistor). Two types:

• Iron choke: Used in low-frequency applications (mains frequency 50 Hz)
• Air choke: Used in high-frequency circuits

MDCAT Patterns — Frequently Asked Questions:

1. Find current and phase angle in a series LCR circuit at a given frequency
2. Calculate resonant frequency and current at resonance
3. Determine RMS current and power dissipated in a circuit
4. Find the turns ratio of a transformer for a given input/output voltage
5. Phase relationship between current and voltage in pure R, L, and C circuits

MDCAT Common Mistakes:

1. Confusing peak and RMS values — always convert to RMS for power calculations
2. Adding reactances without considering their opposite signs ($X_L - X_C$)
3. Forgetting that the sign of $\phi$ determines whether current leads or lags
4. In resonance, confusing $X_L$ and $X_C$ — they are equal but NOT zero individually
5. Mixing up $\omega$ and $f$ in formulas ($X_L = \omega L = 2\pi f L$ — both forms are valid)

Priority Order for MDCAT: AC parameters → RMS values → Reactance → Series circuits → Resonance → Transformer

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