Divider circuit

We encounter divider circuits frequently in circuit analysis.

Voltage divider

For the above circuit (a voltage divider), we have:

$$v_{R_1}=\frac{R_1}{R_1+R_2}v(t)$$ $$v_{R_2}=\frac{R_2}{R_1+R_2}v(t)$$ $$v_{R_x}=\frac{R_x}{\Sigma R_i}v(t)$$

We may occasionally see voltage dividers in circuits with op amps: As always, our constraints are that $v_p=v_n$, that $i_p=i_n=0$, and that $R_{in}=\infty$. It follows from Kirchhoff’s laws that:

$$v_n-0=R_2(i_o)$$ $$v_o-v_n=R_1(i_o)$$

equating for $i_o$:

$$\frac{v_n}{R_2}=\frac{v_o-v_n}{R_1}$$ $$\frac{v_n{R}_1}{R_2}+\frac{v_n{R}_2}{R_2}=v_o$$ $$v_o=\frac{R_1+R_2}{R_2}{v}_n$$

and this holds for other funky looking resistor combinations.

Current divider

For the above circuit (a current divider), we have:

$$i_1(t)=\frac{R_2}{R_1+R_2}i(t)=\frac{G_1}{G_1+G_2}i(t)$$ $$i_2(t)=\frac{R_1}{R_1+R_2}i(t)=\frac{G_2}{G_1+G_2}i(t)$$ $$i_{R_x}=\frac{\frac{1}{R_x}}{\Sigma \frac{1}{R_i}}i(t)$$

Phasor analysis

For impedances in series:

$${\text{V}}_k=Z_k\text{I}=\frac{Z_k}{Z_{eq}}\text{V}$$

which is the equivalent of DC voltage division.

For impedances in parallel, we have:

$${\text{I}}_k=Y_k\text{V}=\frac{Y_k}{Y_{eq}}\text{I}=\frac{\text{V}}{Z_k}=\frac{Z_{eq}}{Z_k}\text{I}$$