# Maximize Power Delivered to Circuits Optimization Problem

The first derivative is used to maximize (optimize) the power delivered to a load in electronic circuits.

## Problem

In the electronic circuit shown below, the voltage E (in Volts) and resistance r (in Ohms) are constant. R is the resistance of a load. In such a circuit, the electric current i is given by
i = $$\dfrac{E}{(r + R)}$$

and the power $$P$$ delivered to the load
R is given by
P = $$R i^2$$

$$r$$ and $$R$$ being positive, determine
R so that the power $$P$$ delivered to $$R$$ is maximum.

Solution to the Problem

We first express power $$P$$ in terms of $$E$$, $$r$$, and the variable $$R$$ by substituting i = $$\dfrac{E}{(r + R)}$$ into P = $$R i^2$$.
$\text{P(R)} = \frac{R E^2}{(r + R)^2}$
We now differentiate $$P$$ with respect to the variable $$R$$ $\frac{dP}{dR} = \frac{E^2 [(r + R)^2 - R 2 (r + R)]}{[(r + R)^4]} = \frac{E^2 [(r + R) - 2 R]}{(r + R)^3} = \frac{E^2 (r - R)}{(r + R)^3}$ To find out whether $$P$$ has a local maximum, we need to find the critical points by setting $$\dfrac{dP}{dR} = 0$$ and solve for $$R$$.
Since $$r$$ and $$R$$ are both positive (resistances), $$\dfrac{dP}{dR}$$ has only one critical point at $$R = r$$. Also for $$R \lt r$$, $$\dfrac{dP}{dR}$$ is positive and $$P$$ increases, and for $$R > r$$, $$\dfrac{dP}{dR}$$ is negative and $$P$$ decreases. Hence $$P$$ has a maximum value at $$R = r$$. The maximum power is found by setting $$R = r$$ in $$P(R)$$
$\text{P(r)} = \frac{r E^2}{(r + r)^2} = \frac{E^2}{4r}$
So in order to have maximum power transfer from the electronic circuit to the load $$R$$, the resistance of $$R$$ has to be equal to $$r$$.
As an example, the plot of $$P(R)$$ for $$E = 5$$ volts and $$r = 100$$ Ohms is shown below and it clearly shows that $$P$$ is maximum when $$R = 100$$ Ohms = $$r$$.

Let us examine $$P(R)$$ again. If $$R$$ approaches zero, $$P(R)$$ also approaches zero. If $$R$$ increases indefinitely, $$P(R)$$ approaches zero since the horizontal asymptote of the graph of $$P(R)$$ is the horizontal axis. So that somewhere for a finite value (found to be $$r$$) $$P(R)$$ has a maximum value.

## References and Links

calculus problems

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