21. Alternating currents

An alternating voltage or current varies periodically, with period T, frequency f=1/T, angular frequency ω=2πf and peak magnitude I₀ or V₀.

Read peak from the centre line to an extreme, not peak-to-peak, and distinguish the waveform’s cycle time from its angular frequency.

A 50 Hz mains waveform has T=20 ms and ω=100π rad s⁻¹.

The mean of a symmetric AC waveform can be zero while its heating effect is not zero.

A sinusoidal alternating quantity is modelled by x=x₀sin(ωt+φ), where x₀ is peak value and φ sets the phase.

Use the initial value and slope to choose phase; differentiate or inspect the graph to identify when the signal is increasing or decreasing.

A signal starting at zero and rising has φ=0 in x=x₀sinωt.

The sine expression gives instantaneous value, not rms value or average magnitude.

With i=I₀sinωt and fixed resistance R, instantaneous power i²R varies from zero to I₀²R; its cycle average is ½I₀²R.

Use average over a complete cycle and distinguish peak power from mean heating power.

If peak current doubles, mean resistive power quadruples because it depends on I₀².

A zero mean current does not imply zero mean power, since power depends on current squared.

For a sinusoidal AC signal, rms current or voltage produces the same mean power in a resistor as a DC value: I_rms=I₀/√2 and V_rms=V₀/√2.

Use rms values in P=VI or P=I²R for resistive loads, and specify whether a quoted AC value is rms or peak.

A 10 A peak sinusoidal current has I_rms≈7.07 A.

RMS is not the arithmetic average of a symmetric waveform, which is zero.

Half-wave rectification passes one polarity half-cycle; full-wave rectification flips both half-cycles to the same load polarity.

Read the output graph: half-wave has gaps each alternate half-cycle, while full-wave pulses occur twice per input cycle and have higher ripple frequency.

A bridge rectifier uses four diodes to produce full-wave output without a centre-tapped transformer.

Rectification does not by itself make perfectly steady DC; smoothing capacitors reduce ripple after the diode stage.

A diode conducts mainly in forward bias and blocks reverse bias, so one diode passes only one half-cycle of an AC input.

Read diode orientation and output polarity, then identify the gaps in the load voltage waveform.

A positive half-wave output has pulses separated by zero-voltage intervals during the negative input half-cycle.

A diode does not convert AC directly into smooth DC; the output is pulsating and needs smoothing.

In a bridge rectifier, two diodes conduct on each half-cycle so current through the load keeps the same direction.

Trace the conducting pair for positive and negative input halves, then identify the doubled ripple frequency.

The bridge produces full-wave pulsating output without a centre-tapped transformer, though two diode drops appear in each conducting path.

All four diodes do not conduct simultaneously in the ideal bridge, and rectification alone does not remove ripple.

A capacitor across a rectifier load charges when input exceeds its voltage and discharges through the load between peaks, reducing ripple.

Larger C or lighter load slows discharge; check the polarity and allow for diode conduction only near peaks.

A full-wave rectifier with a reservoir capacitor has smaller ripple intervals than a half-wave circuit at the same input frequency.

Smoothing does not make voltage perfectly constant, and an infinitely large capacitor would create unrealistic charging currents.