Circuit Theory/Maximum Power Transfer

Maximum Power Transfer

Often we would like to transfer the most power from a source to a load placed across the terminals as possible. How can we determine the optimum resistance of the load for this to occur?

Let us consider a source modelled by a Thévenin equivalent (a Norton eqivalent will lead to the same result, as the two are directly equivalent), with a load resistance, R_L. The source resistance is R_s and the open circuit voltage of the source is v_s:

The current in this circuit is found using Ohm's Law:

$i=\frac{v_s}{R_s+R_L}$

The voltage across the load resistor, v_L, is found using the voltage divider rule:

$v_L=v_s\frac{R_L}{R_s+R_L}$

We can now find the power dissipated in the load, P_L as follows:

$P_L=v_{L}i=\frac{R_L{v}_s^2}{{(R_s+R_L)}^2}$

We can now rewrite this to get rid of the R_L on the top:

$P_L=\frac{v_s^2}{{(\frac{R_s}{\sqrt{R_L}}+\sqrt{R_L})}^2}=\frac{v_s^2}{R_s{(\frac{\sqrt{R_s}}{\sqrt{R_L}}+\frac{\sqrt{R_L}}{\sqrt{R_s}})}^2}$

Assuming the source resitance is not changeable, then we optain maximum power by minimising the bracketed part above. it is an elementary mathematical result that $x+x^{-1}$ is at a minimum when x=1. in this case, it is equal to 2. Therefore, the above expresion is mimnimum uder the following condition:

$\frac{\sqrt{R_s}}{\sqrt{R_L}}=1$

This leads to the condition that:

$R_L=R_s$

Efficiency

The efficiency, η of the circuit is the proportion of all the energy dissipated in the cirucit that is dissipated in the load. We can immediately see that at maximum power transfer to the load, the efficiency is 0.5, as the source resistor has half the voltage across it. We can also see that efficiency will increase as the load resistance increases, even though the power transferred will fall.

The efficiency can be calculated using the following equation:

$\eta =\frac{P_L}{P_L+P_s}$

where P_s is the power in the source resistor. This can be found using a simple modification to the equation for P_L:

$P_s=\frac{v_s^2}{R_L{(\frac{\sqrt{R_s}}{\sqrt{R_L}}+\frac{\sqrt{R_L}}{\sqrt{R_s}})}^2}$

The graph below shows the power in the load (as a proprtion of the maximum power, P_max) and the efficency for values of R_L between 0 and 5 times R_s.