Voltage and Current Dividers

Voltage Division

When we have a voltage source and resistors connected in series, we can express the voltage across a resistor as a ratio of voltages and resistances, without ever knowing the current.

In the circuit above,

${\frac {v_{1}}{v_{s}}}={\frac {R_{1}}{R_{1}+R_{2}}}$

or

$v_{1}={\frac {R_{1}}{R_{1}+R_{2}}}v_{s}$

We can generalize this equation for $N$ number of resistances in series with the equation:

$v_{k}={\frac {R_{k}}{R_{1}+R_{2}+\cdot \cdot \cdot +R_{N}}}v_{s}$

where $v_{s}$ is the voltage across the whole string of resistors.