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}}{Σ 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}}}{Σ \frac{1}{R _{i}}} i (t)

Phasor analysis

For impedances in series:

V_{k} = Z_{k} I = \frac{Z _{k}}{Z _{e q}} V

which is the equivalent of DC voltage division.

For impedances in parallel, we have:

I_{k} = Y_{k} V = \frac{Y _{k}}{Y _{e q}} I = \frac{V}{Z _{k}} = \frac{Z _{e q}}{Z _{k}} I

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Divider circuit

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Phasor analysis

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