Voltage divider

The voltage divider is a series circuit made up of passive electrical two-pole terminals that divide an electrical voltage .

In relation to magnetic circuits , the term voltage divider is also used to describe the division of the magnetic voltage (flux) along magnetic resistances .

Simple voltage divider with two ohmic resistors

Voltage divider made up of two ohmic resistors connected in series

The voltage divider is described in the standard case by the series connection of two ohmic resistors .

To calculate the partial voltage U ₂ across R ₂ , the total resistance is first calculated according to the rule for series connections as follows:

${\displaystyle R_{\mathrm {ges} }=R_{1}+R_{2}\!\,}$

The total voltage and the values ​​of the resistors are generally known, which means that the current I can be determined using Ohm's law :

${\displaystyle I={\frac {U}{R_{\mathrm {ges} }}}={\frac {U}{R_{1}+R_{2}}}}$

According to the rules for series connections, the current through all components is identical and the U _{2 you are looking} for is:

${\displaystyle U_{2}=I\cdot R_{2}}$

If the formula for the common current is used here, the output voltage results as a function of the divider resistances and the input voltage in general:

${\displaystyle U_{2}={\frac {U}{R_{\mathrm {ges} }}}\cdot R_{2}}$

Generalized by equivalence conversion, the ratio between input and output voltage is dependent on the divider resistances.

${\displaystyle U_{2}={\frac {U}{R_{1}+R_{2}}}\cdot R_{2}=U\cdot {\frac {R_{2}}{R_{1}+R_{2}}}\quad \Leftrightarrow \quad {\frac {U_{2}}{U}}={\frac {R_{2}}{R_{1}+R_{2}}}}$

This reformed expression shows that the ratio of the voltage drop over to the total voltage is identical to the ratio of resistance to the total resistance from and . ${\displaystyle U_{2}}$ ${\displaystyle R_{2}}$ ${\displaystyle U}$ ${\displaystyle R_{2}}$ ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$

Further equivalence transformations result in the following practical equations:

${\displaystyle {\frac {R_{1}}{U_{1}}}={\frac {R_{2}}{U_{2}}}}$

respectively:

${\displaystyle {\frac {U_{1}}{R_{1}}}={\frac {U_{2}}{R_{2}}}}$

These equations are often used when dimensioning (selecting suitable resistors with regard to the power dissipation, the level of the output voltage and consequently also the level of the load and cross current) of the voltage divider.

Voltage divider rule

With the help of the voltage divider rule, partial voltages can be calculated directly from the partial resistances and the total voltage. The last equation for the previous circuit represents the special case of the voltage divider rule for exactly two partial resistances. The voltage divider rule is only applicable if all the components on which the total voltage is divided are linear and passive. As soon as active components such as sources appear, the node potential method or mesh current method must be used.

Verbally the voltage divider rule is:

${\displaystyle {\frac {\text{Teilspannung}}{\text{Gesamtspannung}}}={\frac {\text{Widerstand unter Teilspannungsabfall}}{\text{Gesamtwiderstand}}}}$

Generalized to n resistors connected in series (i = 1, ..., n), the following equations for the respective applications result for the partial voltage across the resistor k (with n and k integer, n ≥ 1, 1 ≤ k ≤ n ). Resistors connected in parallel must first be combined into one resistor in order to correspond to the equations in the form shown. The total resistance only refers to the resistances over which the total voltage drops. Any resistances that are before, after or in parallel branches to the section under consideration are not taken into account. In the case of circuits with internal parallel branches, the formula may have to be used several times in order to obtain the partial voltage sought.

DC voltage drop

Voltage divider with ohmic resistors

With direct voltage , only real-valued resistance values, so-called ohmic resistances, occur.

${\displaystyle {\frac {U_{k}}{U}}={\frac {R_{k}}{R}}}$with the total resistance ${\displaystyle R=\sum _{i=1}^{n}R_{i}}$

In the case of direct voltage, the individual partial voltages are always smaller than the total voltage. The ratio of partial stress to total stress takes on values ​​between 0 and 1. A typical example of an adjustable voltage divider is a potentiometer in which the division ratio can be variably set via a sliding contact on a continuous resistor body. Partial voltages are proportional to the resistances over which they drop. This means that the smaller (larger) the resistance, the smaller (larger) the partial voltage.

AC voltage drop

In the case of harmonic alternating voltage with a constant angular frequency ω, complex resistances, so-called impedances , in the form of capacitances ( capacitive voltage divider ) and inductances ( inductive voltage divider ) can also occur. The calculation of a voltage divider is then part of the complex AC calculation .

${\displaystyle {\frac {U_{k}}{U}}={\frac {Z_{k}}{Z}}}$with the total impedance ${\displaystyle Z=\sum _{i=1}^{n}Z_{i}}$

In the case of alternating voltage and the impedances occurring, the partial voltages at the capacitances and inductances can become greater than the total voltage due to resonance due to the energy storage in the impedances . When using the voltage divider rule with alternating voltage, it is important that the impedances, in particular inductances, are not coupled to one another via their energy stored in the electric or magnetic field . This fact is synonymous with the requirement of passive two-pole systems that do not have any voltage or current sources.

Magnetic circles

In magnetic circuits , the magnetic voltage is only divided between magnetic resistances .

${\displaystyle {\frac {U_{m_{k}}}{U_{m}}}={\frac {R_{m_{k}}}{R_{m}}}}$with the total resistance ${\displaystyle R_{m}=\sum _{i=1}^{n}R_{m_{i}}}$

Examples

Example with multiple applications

Voltage divider with an inner branch

The voltage is searched for in the adjacent circuit. Due to the nested position of the resistor, multiple use of the voltage divider rule is necessary. To do this, the voltage across the parallel connection is first calculated. The voltage divider rule gives the equation: ${\displaystyle R_{21}}$ ${\displaystyle U_{2}}$

${\displaystyle {\frac {U_{2}}{U}}={\frac {R_{2}}{R_{1}+R_{2}}}}$

With ${\displaystyle \,R_{2}=\left(R_{21}+R_{22}\right)\parallel R_{23}}$

The partial voltage divided the series combination and on. By repeated application of the voltage divider rule, the voltage is dependent on determined: ${\displaystyle U_{2}}$ ${\displaystyle R_{21}}$ ${\displaystyle R_{22}}$ ${\displaystyle R_{21}}$ ${\displaystyle U_{2}}$

${\displaystyle {\frac {U_{21}}{U_{2}}}={\frac {R_{21}}{R_{21}+R_{22}}}}$

If both equations are multiplied together, the result is an overall equation which is directly dependent on U: ${\displaystyle U_{21}}$

${\displaystyle {\frac {U_{2}}{U}}\cdot {\frac {U_{21}}{U_{2}}}={\frac {U_{21}}{U}}={\frac {R_{2}}{R_{1}+R_{2}}}\cdot {\frac {R_{21}}{R_{21}+R_{22}}}}$

Example of magnetic circuit

magnetic voltage divider made of two resistors

The same rule applies in magnetic circles. The equations for the partial voltages via and result: ${\displaystyle R_{m_{\text{Fe}}}}$ ${\displaystyle R_{m_{\text{Air}}}}$

${\displaystyle {\frac {U_{m_{\text{Fe}}}}{\Theta }}={\frac {R_{m_{\text{Fe}}}}{R_{m}}}}$

or for the other branch

${\displaystyle {\frac {U_{m_{\text{Air}}}}{\Theta }}={\frac {R_{m_{\text{Air}}}}{R_{m}}}}$

with the total resistance:

${\displaystyle R_{m}=R_{m_{\text{Fe}}}+R_{m_{\text{Air}}}}$

Loaded voltage divider

Loaded voltage divider with load resistance R _L parallel to R ₂

In the circuit in the section Simple voltage divider with two ohmic resistors, let the output be at the connections of R ₂ . If a consumer is connected there with the resistor parallel to R ₂ , a loaded voltage divider is created for which the voltage calculations must be carried out again. The resistance of the parallel connection of and is smaller than the smallest part of resistance of the parallel connection. It is calculated with: ${\displaystyle R_{L}}$ ${\displaystyle R_{P}}$ ${\displaystyle R_{2}}$ ${\displaystyle R_{L}}$

${\displaystyle R_{P}=R_{2}\parallel R_{L}={\frac {R_{2}\cdot R_{L}}{R_{2}+R_{L}}}}$

As a result of the reduction in resistance, the voltage falls proportionally according to the voltage divider rule. It now results in: ${\displaystyle U_{2}}$

${\displaystyle U_{2}={\frac {R_{P}}{R_{1}+R_{P}}}\cdot U}$

Equivalent voltage source or Thévenin equivalent

To illustrate the influence of the load resistance, the original circuit without R _{L can be converted} into a two-pole equivalent circuit with an equivalent voltage source U _{2, LL} and an internal resistance R _i . According to the rules for calculating equivalent circuits for active two-terminal networks, the following equations result for this example:

${\displaystyle U_{2,LL}=U\cdot {\frac {R_{2}}{R_{1}+R_{2}}}}$

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

The load resistance remains unaffected by the changeover and its effect on the output voltage of the voltage divider is clearly evident. It now creates a simpler voltage divider comprising R _i and R _L .

${\displaystyle U_{2}=U_{2,LL}\cdot {\frac {R_{L}}{R_{i}+R_{L}}}}$

With the help of R _i , a recommendation for a suitable choice of resistors can be expressed. So that the load resistance has little influence on the output voltage, the internal resistance should have a significantly lower value than the load.

${\displaystyle R_{i}\ll R_{L}\quad \Rightarrow \quad R_{i}<{\frac {1}{10}}\cdot R_{L}}$

application

The application examples overlap with the applications of potentiometers (adjustable voltage dividers). Voltage dividers are used:

• for level adjustment
• in attenuators , e.g. B. also for volume control
• for voltage measurement ; Multimeters have a switchable voltage divider for measuring in different areas.
• In measuring tips for oscilloscopes: here you can usually find voltage dividers with division ratios of 10 to 1 or 100 to 1. In addition to the resistance voltage divider, these probes have a frequency compensation that balances the line and input capacitance when measuring AC voltage. The compensation can often be set or adjusted. It represents a parallel capacitive voltage divider.
• for high-voltage measurement (high-voltage measuring tips or probes); Divider ratios of 1000: 1 or greater. Input voltages of up to around 40 kV are common. The upper partial resistance is approx. 1… 100 GOhm, often the input resistance of the measuring device (e.g. 1 or 10 MOhm) is taken into account. High-voltage measuring tips are available uncompensated for DC voltage measurements, but also frequency-compensated for AC voltage measurements.
• Inductive and resistive voltage dividers are used for position and angle determination as well as in accelerometers. The inductive voltage dividers used here work without contacts with a movable soft magnetic core like a double variometer .
• Inductive voltage dividers provide high-precision voltage ratios in measurement technology, which are almost exclusively dependent on the number of turns of the transformer used. Inductive voltage dividers are used both with fixed voltage ratios and as adjustable decades.
• for creating a bridge circuit by combining voltage dividers.
• The Kelvin-Varley voltage divider is a special design with fixed resistors and step switches for switching . It allows the divider values ​​of the voltage ratio to be set repeatedly.

See also

• Flow divider
• Capacitive voltage divider

literature

• Reinhold Pregla: Fundamentals of electrical engineering . 7th edition. Hüthig Verlag, Heidelberg 2004, ISBN 3-7785-2867-X .
• Klaus Lunze: Introduction to electrical engineering . 13th edition. Verlag Technik, Berlin 1991, ISBN 3-341-00980-9 .
• Heinz Meister: Electrotechnical basics . 9th edition. Vogel Fachbuchverlag, Würzburg 1991, ISBN 3-8023-0528-0 .

Web links

• Calculator for loaded and unloaded voltage dividers
• The electronics compendium: voltage dividers
• Voltage divider with offset
• Exercises on the voltage divider rule (PDF file; 104 kB)

Wikibooks: Voltage Divider  - Learning and Teaching Materials