Alternating currents

21.1 Characteristics of alternating currents

An alternating current or voltage repeatedly reverses direction. A sinusoidal one has the form:

$x = x_0\sin\omega t$

where $x$ is the instantaneous current or voltage, $x_0$ is the peak value and $\omega = 2\pi f$. The period is the time for one full cycle, and frequency is the number of cycles per unit time.

Root-mean-square values

Current changes all the time, so what value should we use for the heating effect? The root-mean-square (r.m.s.) value of an alternating current is the value of the direct current that would produce the same heating effect (power) in a resistive load.

$I_{\text{r.m.s.}} = \frac{I_0}{\sqrt{2}} \qquad V_{\text{r.m.s.}} = \frac{V_0}{\sqrt{2}}$

The name describes how it is calculated: square the current, take the mean over a cycle, then take the root.

Mean power

For a sinusoidal alternating current in a resistor, the instantaneous power is $I^2R$, which varies between 0 and $I_0^2R$. The mean of $\sin^2$ over a cycle is $\tfrac{1}{2}$, so:

$\langle P \rangle = \tfrac{1}{2}I_0^2R = I_{\text{r.m.s.}}^2R$

21.2 Rectification and smoothing

Rectification converts alternating current to direct current, using diodes, which conduct in one direction only.

Half-wave rectification

A single diode in series with the load passes only one half of each cycle. The other half is blocked, so the output has gaps, and the mean power is low.

Full-wave rectification

A bridge rectifier uses four diodes, arranged so that both halves of the cycle drive current through the load in the same direction. In each half-cycle, two diodes conduct and the other two block.

Two diodes are always in series with the load, so the peak output is about 1.4 V lower than the input peak (0.7 V per silicon diode), which is usually neglected.

Smoothing

A capacitor placed in parallel with the load smooths the output:

1. While the input voltage rises, the capacitor charges to the peak.
2. When the input falls below the capacitor's voltage, the diode(s) stop conducting, and the capacitor discharges through the load, keeping the current flowing.
3. The next rising peak recharges it.

The small remaining variation is called ripple. The greater the time constant $RC$ compared with the period, the smaller the ripple.

So a larger capacitance, or a larger load resistance, gives smoother output. A larger load resistance also means a smaller load current.