Voltage divider

Figure 1: Voltage divider

In electronics, a voltage divider (also known as a potential divider) is a linear circuit that produces an output voltage (V_out) that is a fraction of its input voltage (V_in). Voltage division refers to the partitioning of a voltage among the components of the divider.

An example of a voltage divider consists of two resistors in series or a potentiometer. It is commonly used to create a reference voltage, or to get a low voltage signal proportional to the voltage to be measured, and may also be used as a signal attenuator at low frequencies. For direct current and relatively low frequencies, a voltage divider may be sufficiently accurate if made only of resistors; where frequency response over a wide range is required, (such as in an oscilloscope probe), the voltage divider may have capacitive elements added to allow compensation for load capacitance. In electric power transmission, a capacitive voltage divider is used for measurement of high voltage.

General case

A voltage divider referenced to ground is created by connecting two electrical impedances in series, as shown in Figure 1. The input voltage is applied across the series impedances Z₁ and Z₂ and the output is the voltage across Z₂. Z₁ and Z₂ may be composed of any combination of elements such as resistors, inductors and capacitors.

Applying Ohm's Law, the relationship between the input voltage, V_in, and the output voltage, V_out, can be found:

$V_\mathrm{out} = \frac{Z_2}{Z_1+Z_2} \cdot V_\mathrm{in}$

Proof:

$V_\mathrm{in} = I\cdot(Z_1+Z_2)$

$V_\mathrm{out} = I\cdot Z_2$

$I = \frac {V_\mathrm{in}}{Z_1+Z_2}$

$V_\mathrm{out} = V_\mathrm{in} \cdot\frac {Z_2}{Z_1+Z_2}$

The transfer function (also known as the divider's voltage ratio) of this circuit is simply:

$H = \frac {V_{out}}{V_{in}} = \frac{Z_2}{Z_1+Z_2}$

In general this transfer function is a complex, rational function of frequency.

Resistive divider

Figure 2: Simple resistive voltage divider

A resistive divider is a special case where both impedances, Z₁ and Z₂, are purely resistive (Figure 2).

Substituting Z₁ = R₁ and Z₂ = R₂ into the previous expression gives:

$V_\mathrm{out} = \frac{R_2}{R_1+R_2} \cdot V_\mathrm{in}$

As in the general case, R₁ and R₂ may be any combination of series/parallel resistors.

also can be define The voltage across the resistive elements will divide as the magnitude of the resistance levels. The more resistance the more voltage will be obtained.

Vx=(Rx/RT)*E

where Vx = voltage drop across resistance, Rx= Resistance where voltage will be measured, RT= Total Resistance, E= Input voltage or supply voltage. This rule generally apply for the series circuit.

Examples

Resistive divider

As a simple example, if R₁ = R₂ then

$V_\mathrm{out} = \frac{1}{2} \cdot V_\mathrm{in}$

As a more specific and/or practical example, if V_out=6V and V_in=9V (both commonly used voltages), then:

$\frac{V_\mathrm{out}}{V_\mathrm{in}} = \frac{R_2}{R_1+R_2} = \frac{6}{9} = \frac{2}{3}$

and by solving using algebra, R₂ must be twice the value of R₁.

To solve for R1:

$R_1 = \frac{R_2 \cdot V_\mathrm{in}}{V_\mathrm{out}} - R_2$

To solve for R2:

$R_2 = \frac{R_1} {({\frac{V_\mathrm{in}}{V_\mathrm{out}}-1})}$

Any ratio between 0 and 1 is possible. That is, using resistors alone it is not possible to either invert the voltage or increase V_out above V_in.

Low-pass RC filter

Figure 3: Resistor/capacitor voltage divider

Consider a divider consisting of a resistor and capacitor as shown in Figure 3.

Comparing with the general case, we see Z₁ = R and Z₂ is the impedance of the capacitor, given by

$Z_2 = -jX_{\mathrm{C}} =\frac {1} {j \omega C}$ $\ ,$

where X_C is the reactance of the capacitor, C is the capacitance of the capacitor, j is the imaginary unit, and ω (omega) is the radian frequency of the input voltage.

This divider will then have the voltage ratio:

${V_\mathrm{out} \over V_\mathrm{in}} = {Z_\mathrm{2} \over Z_\mathrm{1} + Z_\mathrm{2}} = {{1 \over j \omega C} \over {1 \over j \omega C} + R} = {1 \over 1 + j \omega \ R C}$.

The product of τ (tau) = RC is called the time constant of the circuit.

The ratio then depends on frequency, in this case decreasing as frequency increases. This circuit is, in fact, a basic (first-order) lowpass filter. The ratio contains an imaginary number, and actually contains both the amplitude and phase shift information of the filter. To extract just the amplitude ratio, calculate the magnitude of the ratio, that is:

$\left| \frac {V_\mathrm{out}} {V_\mathrm{in}} \right| = \frac {1} {\sqrt { 1 + ( \omega R C )^2 } } \ .$

Inductive divider

Inductive dividers split DC input according to resistive divider rules above.

Inductive dividers split AC input according to inductance:

$V_{out} = V_{in} \cdot \frac {L_2} {L_1 + L_2}$

The above equation is for ideal conditions. In the real world the amount of mutual inductance will alter the results.

Capacitive divider

Capacitive dividers do not pass DC input.

For an AC input a simple capacitive equation is:

$V_{out} = V_{in} \cdot \frac {C_1} {C_1 + C_2}$

Capacitive dividers are limited in current by the capacitance of the elements used.

This effect is opposite to resistive division and inductive division.

Loading effect

The voltage output of a voltage divider is not fixed but varies according to the load. To obtain a reasonably stable output voltage the output current should be a small fraction of the input current. The drawback of this is that most of the input current is wasted as heat in the resistors.

Applications

Voltage dividers are used for adjusting the level of a signal, for bias of active devices in amplifiers, and for measurment of voltages. A Wheatstone bridge and a multimeter both include voltage dividers. A potentiometer is used as a variable voltage divider in the volume control of a radio.

Further reading

• Horowitz, Paul; Hill, Winfield (1989). The Art of Electronics. Cambridge University Press. 

External links

• Resistor voltage divider
• Voltage divider or potentiometer calculations
• Voltage divider tutorial video in HD
• Online calculator to choose the values by series E24, E96
• Online voltage divider calculator: chooses the best pair from a given series and also gives the color code
• Voltage divider theory - RC low-pass filter example and voltage divider using Thévenin's theorem