Alternating Currents

• Period, frequency and peak value; x = x₀ sin ωt with ω = 2πf.
• Mean power in a resistor = half the peak power for a sinusoidal a.c.
• I_r.m.s. = I₀/√2 and V_r.m.s. = V₀/√2.
• Half-wave rectification: one diode. Full-wave: four diodes in a bridge.
• A smoothing capacitor reduces ripple; larger C or larger load R (larger RC) gives less ripple.

Peak and r.m.s.

A sinusoidal alternating current or voltage is described by x = x₀ sin ωt, where x₀ is the peak value, the period T is the time for one cycle and the frequency f = 1/T, with ω = 2πf.

The power in a resistor is P = I²R. Its peak is I₀²R, and the average of sin² over a cycle is ½, so the mean power is half the maximum power. The root-mean-square value is the steady direct current that would dissipate the same mean power in a resistor: I_r.m.s.²R = ½I₀²R, so I_r.m.s. = I₀/√2. Mains voltages are quoted as r.m.s. values.

From a.c. to steady d.c.

A diode conducts in one direction only. Half-wave rectification: a single diode in series with the load passes only the positive half-cycles; the output is a series of positive humps with gaps between them. Full-wave rectification: a bridge of four diodes routes the current through the load in the same direction on both half-cycles; two diodes conduct on each half-cycle, and the output is continuous positive humps at twice the supply frequency.

Smoothing: ripple