# Electronics/Voltage Dividers

## Contents

### Ideal case[edit]

Consider the illustration below. Assume initially that no current is flowing in or out of the terminal marked Vout. In this case, the only path for current is from V_{in} through R_{1} and R_{2} to GND. The equivalent resistance of this configuration is R_{1}+R_{2} since these are resistors in series. From Ohm's law the current flowing through both resistors is

.

Also from Ohm's law, we know that the voltage drops across the resistors are I*R_{1} and I*R_{2} respectively. A quick check shows that the sum of the voltage drops across the resistors adds up to V_{in}. Now we can calculate the voltage at V_{out} (still assuming that no current flows through the terminal V_{out}). In this case the voltage is just

where the 0V is the voltage at GND. If we substitute what we calculated for I, we obtain

.

This is the *voltage divider equation* for the ideal case where no current flows through the output. Another way of saying that no current flows through the output is that the output has infinite resistance. A quick mental check using shows that the voltage calculated divided by infinity equals zero

### Non-ideal case - finite resistance outputs[edit]

In the non-ideal case, we need to consider the resistance of the output. If we assume that the resistance of the output is R_{3} (and it is connected only to GND), we need to modify our analysis as follows. Now we have two resistors in parallel from the V_{out} junction to GND. The equivalent resistance of the parallel reistors then is

and the equivalent reistance of the entire circuit is

.

This yields a current of

.

Now we multiply this by calculated above to obtain the output voltage:

Normally R_{2} will be much smaller than R_{3} so R_{eq} will be approximately equal to R_{2}. Keep in mind that a resistance R_{3} that is 100 times as large as R_{2} results in a voltage sag of about 1% and that a reistance R_{3} that is 10 times as large as R_{2} results in an almost 10% voltage sag.

### Non-ideal case - complex impedance[edit]

Both voltage divider equations hold for complex impedances. Just substitute Z's for R's and do the complex arithmetic. The resulting equations are just the following:

- ideal case - non-ideal case