Voltage and Current Dividers

Voltage Division

When we have a voltage across a string of resistors connected in series, the voltage across the entire string will be divided up among the resistors. We can express the voltage across a single resistor as a ratio of voltages and resistances, without ever knowing the current.

In the circuit above,

${\displaystyle {\frac {v_{1}}{v}}={\frac {R_{1}}{R_{1}+R_{2}}}}$

or

${\displaystyle v_{1}={\frac {R_{1}}{R_{1}+R_{2}}}v}$

We can generalize this equation for ${\displaystyle N}$ resistors in series with the equation:

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

where ${\displaystyle v_{k}}$ is the voltage across resistor ${\displaystyle k}$ and ${\displaystyle v}$ is the voltage across the whole string of resistors.

Current Division

Resistors in parallel divide up the current. When we have a current flowing through resistors in parallel, we can express the current flowing through a single resistor as ratio of currents and resistances, without ever knowing the voltage.

In the circuit above

${\displaystyle {\frac {i_{1}}{i}}={\frac {R_{2}}{R_{1}+R_{2}}}}$

or

${\displaystyle i_{1}={\frac {R_{2}}{R_{1}+R_{2}}}i}$

where ${\displaystyle i}$ is the current flowing through all the resistors. Note that the numerator on the right is R2, not R1. Remember that a larger resistance will carry a smaller current.

We can generalize the equation for ${\displaystyle N}$ resistors in parallel with the equation:

${\displaystyle i_{k}={\frac {\frac {1}{R_{k}}}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdot \cdot \cdot +{\frac {1}{R_{N}}}}}i}$

where ${\displaystyle i_{k}}$ is the current flowing through resistor ${\displaystyle k}$ and ${\displaystyle i}$ is the current flowing through all the resistors.

Practice Problems

Problem 1

Use voltage division to find ${\displaystyle v_{x}}$ in the circuit below:

Problem 2

Simplify the circuit and then use current division to find ${\displaystyle i_{x}}$ in the circuit below: