Alternating Current

A direct current (DC) — the kind a battery gives — flows steadily in one direction only. An alternating current (AC) changes direction periodically, growing, falling to zero, reversing, and returning, over and over. The wall socket in your home is AC: in Pakistan the supply alternates 50 times every second (50 Hz) at about 220 V.

If you plug both into an oscilloscope the difference is obvious: DC draws a flat horizontal line, while AC draws a smooth sine wave rising above and dipping below zero.

• Instantaneous value (V, I) — the value at a particular instant; it changes every moment.
• Peak value (V₀, I₀) — the maximum the wave reaches in either direction; the amplitude of the sine.
• Cycle — one complete swing: zero → peak → zero → trough → zero.
• Frequency (f) — cycles per second, in hertz (Hz). f = 1/T.

Because AC is always changing, we need a single number that tells us how much it can do. The trick: ask what steady DC current would heat a resistor at the same rate. That equivalent steady value is the root-mean-square (rms) value — the "effective" value of the AC.

1. The average of a full cycle is zero — the positive half cancels the negative half.
2. So we average the square (always positive), then take the root — hence root-mean-square.
3. AC meters and the "220 V" rating are rms values, not peaks.
4. The peak of the 220 V mains is V₀ = √2 × 220 ≈ 311 V.

Connect a pure resistor (a heater element, a lamp) across an AC supply. The current follows Ohm's law at every instant, so it rises and falls in perfect step with the voltage — the two waves peak together and cross zero together. We say voltage and current are in phase (phase difference = 0).

1. Phase difference between V and I is zero — they are in phase.
2. A resistor opposes AC and DC equally; its opposition does not depend on frequency.
3. Real power is dissipated as heat: P = I²_rms R (the bulb glows, the element warms).

A capacitor stores charge on its plates. On DC it charges up and then blocks all flow. On AC, the plates charge and discharge every half-cycle, so a current keeps flowing through the circuit — and the faster the AC alternates, the more easily charge sloshes back and forth. Its opposition to AC is the capacitive reactance X_C.

1. X_C ∝ 1/f: high frequency → small reactance → the capacitor passes it easily.
2. At very low f (and DC, f = 0) X_C → ∞: the capacitor blocks it.
3. In a pure capacitor the current leads the voltage by 90° (a quarter cycle) — current peaks before voltage.
4. Average power in a pure capacitor is zero — it stores and returns energy.

An inductor (a coil) resists any change in the current through it: a changing current induces a back-emf that opposes the change (Lenz's law). The faster the AC alternates, the harder the coil fights it. This frequency-dependent opposition is the inductive reactance X_L.

1. X_L ∝ f: high frequency → large reactance → the inductor blocks it.
2. At DC (f = 0) X_L = 0: a coil is just a wire — it passes DC freely.
3. In a pure inductor the current lags the voltage by 90° — current peaks after voltage.
4. Average power in a pure inductor is zero — energy is stored in the magnetic field and returned.

Put a resistor, an inductor and a capacitor in series across AC. Because the inductor and capacitor push the current opposite ways in phase (one lags, one leads), their reactances partly cancel. The resistor's voltage is in phase with the current, but the reactive voltages are 90° away — so we add them as perpendicular phasors, not as plain numbers.

1. Impedance Z is the total opposition to AC (in ohms) — it replaces "resistance" for AC circuits.
2. φ is the phase angle between supply voltage and current. cos φ is the power factor.
3. If X_L > X_C the circuit is inductive (current lags); if X_C > X_L it is capacitive (current leads).
4. Only the resistor dissipates real power; that is why the power factor cos φ appears.

As you change the frequency, X_L rises and X_C falls. At one special frequency they become equal and cancel exactly. The impedance then drops to its smallest value (just R), and the current shoots up to a maximum. This is resonance, and the frequency where it happens is the resonant frequency f₀.

1. At f₀ the circuit is purely resistive: voltage and current are back in phase (φ = 0, power factor = 1).
2. The current peaks sharply; the sharpness is the quality factor Q.
3. A radio tunes by varying C: when f₀ matches a station's broadcast frequency, that signal resonates loudest while others are rejected.

AC is the backbone of the electricity grid because transformers can raise its voltage for efficient long-distance transmission and lower it again for safe use. Reactance and resonance let us filter and tune signals — the heart of every radio and TV. Rectifiers turn AC back into DC for electronics.

1. AC reverses direction periodically: V = V₀ sin ωt, ω = 2πf.
2. Effective values: I_rms = I₀/√2, V_rms = V₀/√2; full-cycle average = 0.
3. Resistor: V and I in phase; P = I²_rms R.
4. Capacitor: X_C = 1/(2πfC); passes high f; I leads V by 90°.
5. Inductor: X_L = 2πfL; blocks high f; I lags V by 90°.
6. Series R–L–C: Z = √(R² + (X_L − X_C)²); tan φ = (X_L − X_C)/R.
7. Resonance: f₀ = 1/(2π√(LC)); Z minimum, current maximum.