# Voltage Dividers !

**Loaded Voltage Divider:**

The voltage divider can be loaded by connecting it to a load (resistor *R*_{L }in the diagram below). This load resistance conducts a load current *I*_{L}, while resistor *R*_{2} conducts a parallel current *I*_{Q}. Resistor *R*_{1} conducts the sum of these two currents. The parallel current *I*_{Q} produces heat loss in resistor *R*_{2}.

For an unloaded voltage divider, the voltage across *R*_{2} is proportional to the ratio between *R*_{2} and the total resistance *R*_{1} + *R*_{2}. By contrast, a loaded voltage divider exhibits a curved characteristic whose deviation from the linear characteristic in the unloaded state is inversely proportional to the ratio between the load resistance and the total resistance *R*_{1} + *R*_{2} in the unloaded state, i.e. directly proportional to the ratio between the load current and the parallel current across the divider resistor being loaded. This is because the loaded voltage divider comprises a series connection between *R*_{1} and the parallel connection of *R*_{2} and *R*_{L}. The equivalent resistance *R*_{2}^{*} of this parallel circuit is calculated as follows:

Accordingly, the voltage divider's load voltage *U*_{L} is

The value for the unloaded state is derived by letting the load resistance *R*_{L} approach infinity. In this case, the resistance *R*_{2} is negligible compared with *R*_{L} in both denominator terms:

*R*_{L} can then be reduced to result in the equation for an unloaded voltage divider as determined in the previous section. A voltage divider's load voltage is thus always smaller in the loaded state than in the unloaded (idle) state.

*Given U_{L}, the currents I_{L} and I_{Q} can be calculated using Ohm's law and the total current I is the sum of these two currents.*

*The interactive animation below shows a voltage divider which can be connected to a load resistance R_{L} via the button with the red cross. Set different values for the various resistances and observe the resulting effects on voltage and current in the loaded and unloaded states. Note especially how sharply the load voltage U_{L} drops in the loaded state compared with the unloaded state.*

**Unloaded Voltage Divider:**

For the purposes of measurement, it is often necessary to tap sub-voltages from a primary voltage. This is done by means of *voltage division*. Illustrated below is a voltage divider consisting of two series-connected resistors *R*_{1}and *R*_{2}.

The supply voltage *U* to the outer terminals and this is then divided into the two sub-voltages *U*_{1} and *U*_{2}. According to the voltage division law,

According to Ohm's law, the current in the voltage divider is

and the voltage drop across the two resistors is

If the expression for current as derived above is substituted into these two equations, the following equations are obtained for the two sub-voltages:

These equations apply only when no current is being tapped from the voltage divider, i.e. in the no-load state.

The interactive animation below shows a voltage divider whose supply voltage and resistance values can be adjusted with the mouse. If you modify these variables, you will be able to observe the corresponding effects on both sub-voltages.

**Voltage Divider with a Potentiometer:**

In actual practice, use is mainly made of a continuously adjustable voltage divider in the form of a so-called *potentiometer*. This device has three connections, one of them is a variable sliding contact permitting the potentiometer's total resistance to be divided into two partial resistances *R*_{1} and *R _{2}*. As a result, the output voltage

*U*

_{2}present across the potentiometer's sliding contact can be divided at any point between the full value

*U*(i.e. the value applied to the potentiometer's outer terminals) and zero. The diagrams below illustrate the principle of a potentiometer (left) and its circuit symbol (right).

Potentiometers are available in a variety of designs such as rotary and sliding formats. These two designs are illustrated below (left: rotary potentiometer; right: sliding potentiometer).

The interactive animation below shows a potentiometer whose voltage supply and total resistance are adjustable. The sliding control on the right is used to vary the tapping point for the output voltage *U*_{2}. Move the tap and observe the resulting effects on the partial resistances and output voltage.

**Bridge Circuits:**

A *bridge circuit* consists of two voltage dividers connected in parallel as illustrated below.

If the upper voltage divider (comprising resistors *R*_{1} and *R*_{2}) divides the supply voltage in the same ratio as the lower voltage divider (comprising resistors *R*_{3} and *R*_{4}), the potential between points C and D is zero (*U*_{D} = 0). In this state, the bridge is said to be *balanced*. The balancing condition is

If resistors *R*_{3} and *R*_{4} are replaced by an adjustable resistor, the bridge circuit can be used to measure resistances; this kind of circuit is termed *Wheatstone bridge *after the English physicist of the same name (see the diagram below).

*R*_{x} is the unknown resistance, *R*_{N} a (usually switchable) standard resistance for comparison. To perform the measurement, the bridge is balanced (*U*_{D} = 0) and *R*_{x} determined using the following relationship: