maximum real power problem

Question

The task says:

Determine the value of resistance \$ R\$ so that true power of loads is maximum (biggest possible).

Picture of a circuit that came with task:

I solved it this way:

I know that \$ P = I^2 * R_{real} \$ and as \$ I = \frac{U}{\sqrt{(R + 3)^2 + 4^2}} \$ so \$ P = \frac{U^2*(R + 3)}{(R + 3)^2 + 4^2}\$, when I derived that, I get \$ (3+R)^2 + 4 - 2(3+R)^2 = 0 => (3 + R)^2 - 4 = 0 => 9 + 6R + R^2 - 4 = 0 \$ which gives me \$ R_1 = -5, R_2 = -1\$. The answer that I should've got is R = 5, I don't know what I've did wrong.

Answer 1

If the objective is to maximize the power dissipated on the load \$R\$. Use,

\$P_R = \frac{U^2 \cdot R}{(R + 3)^2 + 4^2} = \frac{U^2 \cdot R}{ R^2+6R+9+16} = \frac{U^2}{ R+6+25/R}\$

Then, differentiate the expression to find its maximum, that is, when \$R+6+25/R\$ reaches its minimum.

If you are actually looking for maximizing the power dissipated on both resistors, \$R\$ and \$ 3 \Omega \$, use,

\$P_{total} = \frac{U^2 \cdot (R+3)}{(R + 3)^2 + 4^2} = \frac{U^2}{\frac{(R + 3)^2 + 16}{R+3}} = \frac{U^2}{R + 3 + \frac{16}{R+3}}\$

Then, find when \$R + 3 + \frac{16}{R+3}\$ is at its smallest value.

Answer 2

Well, first of all we can write the input impedance as follows:

$$\underline{\text{Z}}_{\space\text{in}}=3+\text{R}+4\text{j}\tag1$$

The input current is given by:

$$\underline{\text{I}}_{\space\text{in}}=\frac{\hat{\text{V}}\cdot\exp\left(\phi\cdot\text{j}\right)}{3+\text{R}+4\text{j}}\tag2$$

The real power of the load is given by:

$$\text{P}_\text{load}=\text{P}_\text{R}=\text{V}_{\text{R/rms}}\cdot\text{I}_{\text{R/rms}}\cdot\cos\left(\varphi_\text{R}\right)=\text{R}\cdot\text{I}_{\text{R/rms}}^2\cdot\cos\left(\varphi_\text{R}\right)=$$ $$\text{R}\cdot\left(\frac{1}{\sqrt{2}}\cdot\left|\frac{\hat{\text{V}}\cdot\exp\left(\phi\cdot\text{j}\right)}{3+\text{R}+4\text{j}}\right|\right)^2\cdot1=\frac{\text{R}}{2}\cdot\left(\frac{\hat{\text{V}}}{\sqrt{\left(3+\text{R}\right)^2+4^2}}\right)^2=$$ $$\frac{\text{R}}{2}\cdot\frac{\hat{\text{V}}^2}{\left(3+\text{R}\right)^2+4^2}=\frac{\hat{\text{V}}^2}{2}\cdot\frac{\text{R}}{\left(3+\text{R}\right)^2+16}\tag3$$

Now, solve:

$$\frac{\partial\text{P}_\text{load}}{\partial\text{R}}=0\space\Longrightarrow\space\text{R}=5\tag4$$

And the real power of the load is there equal to:

$$\hat{\text{P}}_\text{load}=\frac{\hat{\text{V}}^2}{2}\cdot\frac{5}{\left(3+5\right)^2+16}=\frac{\hat{\text{V}}^2}{32}\tag5$$