Voltage Division Formula and Derivation

Voltage Divider Rule

Voltage divider rule is used to find the voltage across an element connected in a series network.

If there are N resistors R₁, R₂,..….R_N; connected in series with a voltage source of voltage V_S as shown in the below figure. Then the voltage across i th resistor is,

V_Ri = V_SR_i/(R₁ + R₂ +…..R_N).

Series connected resistors

For example the voltage across resistor R₁ is V_R1 = V_SR₁/(R₁ + R₂ +…..R_N).

Similarly the formula can be applied to find the voltage across an i th element in an AC network having impedances as,

V_Zi = V_SZ_i/(Z₁ + Z₂ +…..Z_N).

Hint: In voltage division technique the voltage across an element in series connected network can be found by dividing the product of the applied voltage and the impedance (or resistance in a resistive network) of the element where the voltage should be measured with the total impedance.

Voltage Divider Rule Formula Derivation

Consider the following circuit, in which the voltage V_R2 should be measured.

Circuit for deriving voltage divider formula

By applying Kirchhoff’s voltage law,
V_S = IR₁ + IR₂ + IR₃
⇒ V_S =I (R₁ + R₂ + R₃)
⇒ I = V_S/ (R₁ + R₂ + R₃) -----(1)
The voltage at R₂, V_R2 = IR₂ -----(2)
Sub equation (1) in (2) we get the voltage across the resistor R₂ as,
V_R2 = V_SR₂/ (R₁ + R₂ + R₃)