Voltage and Current Dividers

From Mech

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,

$\frac{v_1}{v}=\frac{R_1}{R_1+R_2}$

or

$v_1=\frac{R_1}{R_1+R_2}v$

We can generalize this equation for $N$ resistors in series with the equation:

$v_k=\frac{R_k}{R_1+R_2+\cdot\cdot\cdot+R_N}v$

where $v k$ is the voltage across resistor $k$ and $v$ 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

$\frac{i_1}{i}=\frac{R_2}{R_1+R_2}$

or

$i_1=\frac{R_2}{R_1+R_2}i$

where $i$ 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 $N$ resistors in parallel with the equation:

$i_k=\frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+\cdot\cdot\cdot+\frac{1}{R_N}}i$