Circuit Theorems

From HvWiki

1 Voltage Divider Theorem
2 Half-power Theorem
3 Superposition Theorem
4 Thevenin and Norton

Voltage Divider Theorem

This theorem tells you how voltages are shared between resistors and how to get any voltage you want from a supply.

When resistors are in series, the voltage across each is in the ratio of the resistance to the total resistance.

For example, if there is a $1 k Ω$ and $2 k Ω$ resistor on a 12V supply, $12V \times \frac{2k\Omega}{2k\Omega + 1k\Omega} = 8V$ will be across the $2 k Ω$ resistor and 4V across the $1 k Ω$ . This is often generalised in the formula:

$V_R = V_s \times \frac{R}{R_{tot}}$

[edit]

Half-power Theorem

This theorem tells you how to get the maximum power from a supply.

The maximum power from a source is obtained when the load impedance is the complex conjugate of the source impedance.

For DC, this just means, "when the load resistance is the same as the output resistance". For AC, it's when the resistances match but the reactances are opposite - ie. inductive rather than capacitive.

Adjusting the load impedance to get the maximum output is called impedance matching.

For example, a car amp has a nominal output impedance of 4 $Ω$ . That means a 4 $Ω$ car speaker will get the maximum power from it.

Note that the efficiency at this maximum power is not high: it's half. The same amount of power it burnt in the load as in the source. This is why it's called the half-power theorem.

Why doesn't decreasing the load impedance from this value increase the output power? Power is voltage times current. Although decreasing the load impedance increases the current, the voltage divider theorem (see above) shows that it also decreases the voltage across the load.

[edit]