Voltage-controlled current source or VCCS

Voltage-controlled current source is depicted on the Figure 1.

Figure 1.

Here current $i_O$ is a function of the other circuit values. $i_o=f\left(v_{in}\right)=–g{v}_{in}$, then  $v_o=R_{out}{i}_o=–R_{out}g{v}_{in}$ and $\frac{v_o}{v_{in}}=–R_{out}g$, $g$ is a gain of our circuit.

Let’s represent $v_{in}$ and $v_{out}$ as resistance multiplied with current: $R_{out}{I}_{out}=–R_{in}{I}_{in}g{R}_{out, }$then $\frac{I_{out}}{I_{in}}=–g{R}_{in}$.

$$\begin{gathered} P_{out}=V_{out}{I}_{out}=R_{out}{g}^2{V}_{in }{R}_{in}{I}_{in}, \\ P_{in}=V_{in}{I}_{in} \end{gathered}$$,

then $\frac{P_{out}}{P_{in}}=g^2{R}_{in}{R}_{out}$.