Flow divider

The current divider is a parallel connection of passive electrical or magnetic two-pole connections , by means of which an electrical current or a magnetic flux is divided into several partial currents / flows.

Current dividers for alternating current can also be implemented with transformers, they are then called current transformers .

General flow divider rule

The flow divider rule can be used to easily calculate the partial flows. This rule only applies if all branches on which the total current is divided are passive. With direct current , these are ohmic resistances . In the case of alternating current , capacitors ( capacitive current divider ) and coils ( inductive current divider ) would also be possible. In magnetic circuits there is only magnetic resistance . As soon as active components such as sources appear, the mesh flow method must be used. The flow divider rule is also used when calculating a network using the superposition method.

The flow divider rule is:

${\displaystyle {\frac {\text{Teilstrom}}{\text{Gesamtstrom}}}={\frac {\text{Gesamtwiderstand}}{\text{Vom Teilstrom durchflossener Widerstand}}}}$

or expressed with guide values:

${\displaystyle {\frac {\text{Teilstrom}}{\text{Gesamtstrom}}}={\frac {\text{Vom Teilstrom durchflossener Leitwert}}{\text{Gesamtleitwert}}}}$

With

${\displaystyle {\text{Leitwert}}={\frac {1}{\text{Widerstand}}}}$

Current divider with ohmic resistors

Generalized to n parallel branches ( i = 1 ... n ) the following results for the current in branch k :

• for ohmic circuits

${\displaystyle {\frac {I_{k}}{I}}={\frac {R}{R_{k}}}={\frac {G_{k}}{G}}}$

with the total resistance and the total conductance ${\displaystyle {\frac {1}{R}}=\sum _{i=1}^{n}{\frac {1}{R_{i}}}}$ ${\displaystyle G=\sum _{i=1}^{n}G_{i}}$

• for complex circuits

${\displaystyle {\frac {I_{k}}{I}}={\frac {Z}{Z_{k}}}={\frac {Y_{k}}{Y}}}$

with the total impedance and the total admittance ${\displaystyle {\frac {1}{Z}}=\sum _{i=1}^{n}{\frac {1}{Z_{i}}}}$ ${\displaystyle Y=\sum _{i=1}^{n}Y_{i}}$

• for magnetic circuits

${\displaystyle {\frac {\Phi _{k}}{\Phi }}={\frac {R_{m}}{R_{m_{k}}}}={\frac {G_{m_{k}}}{G_{m}}}}$

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

The resistances of each branch must first be combined into one resistance per branch in order to correspond to the equations in the form shown above. The total resistance only refers to the considered parallel connection , in which the total current is divided. Any resistances that are in series before or after the parallel connection are not taken into account. In the case of more complex circuits with multiple branches, the formula may have to be applied several times in order to obtain the partial flow sought.

For a rough control of the currents calculated with this rule, two simple phrases are suitable. On the one hand, each partial flow is smaller than the total flow, since this corresponds to the sum of all partial flows. On the other hand, the partial currents in the branches are inversely proportional to their branch resistances. This means that the smaller (larger) the branch resistance, the larger (smaller) the partial flow.

In some sources the rule is expressed somewhat modified. At first this variant seems a little more difficult, but over time it becomes just as easy for experienced users as the first variant. It is as follows:

${\displaystyle {\frac {\text{Teilstrom}}{\text{Gesamtstrom}}}={\frac {\text{Vom Teilstrom nicht durchflossener Teilwiderstand}}{\text{Ringwiderstand der Masche}}}}$

Derivation of the rule for a simple example

Current divider made up of two ohmic resistors connected in parallel

According to Kirchhoff's rules , the total current is divided into the two branches: ${\displaystyle \,I}$

${\displaystyle \,I=I_{1}+I_{2}}$

Since the same voltage drops across the two resistors connected in parallel, Ohm's law applies :

${\displaystyle U=R_{1}\cdot I_{1}=R_{2}\cdot I_{2}}$

If you solve this equation for ${\displaystyle I_{2}}$

${\displaystyle I_{2}={\frac {R_{1}}{R_{2}}}\cdot I_{1}}$

and insert the result in , we get: ${\displaystyle I=I_{1}+I_{2}}$

${\displaystyle I=I_{1}\cdot \left(1+{\frac {R_{1}}{R_{2}}}\right)=I_{1}\cdot {\frac {R_{1}+R_{2}}{R_{2}}}=I_{1}\cdot {\frac {R_{1}+R_{2}}{R_{2}}}\cdot {\frac {(R_{1}\cdot R_{2})}{(R_{1}\cdot R_{2})}}=I_{1}\cdot {\frac {R_{1}}{R_{1}\parallel R_{2}}}}$

If you divide by and form the reciprocal value on both sides, the result is the same as for the flow divider rule: ${\displaystyle I_{1}}$

${\displaystyle {\frac {I_{1}}{I}}={\frac {R}{R_{1}}}}$and for the other branch with the total resistance ${\displaystyle {\frac {I_{2}}{I}}={\frac {R}{R_{2}}}}$ ${\displaystyle R=R_{1}\parallel R_{2}}$

The total current and the values ​​of the resistors are generally known.

Example with multiple use

Flow divider from three branches with an inner branch in the lowest branch

The current is sought through . To do this, the current in the lowest branch is first calculated. The flow divider rule gives the equation: ${\displaystyle R_{32}}$ ${\displaystyle I_{3}}$

${\displaystyle {\frac {I_{3}}{I}}={\frac {R_{1}\parallel R_{2}\parallel R_{3}}{R_{3}}}}$

with and ${\displaystyle \,R_{2}=R_{21}+R_{22}}$ ${\displaystyle R_{3}=R_{31}+\left(R_{32}\parallel R_{33}\right)}$

The partial current flows out of and through the parallel connection . By applying the current divider rule again, the current is determined as a function of : ${\displaystyle I_{3}}$ ${\displaystyle R_{32}}$ ${\displaystyle R_{33}}$ ${\displaystyle R_{32}}$ ${\displaystyle I_{3}}$

${\displaystyle {\frac {I_{32}}{I_{3}}}={\frac {R_{32}\parallel R_{33}}{R_{32}}}}$

If both equations are multiplied with each other, the result is an overall equation in which is directly dependent on I: ${\displaystyle I_{32}}$

${\displaystyle {\frac {I_{3}}{I}}\cdot {\frac {I_{32}}{I_{3}}}={\frac {I_{32}}{I}}={\frac {R_{1}\parallel R_{2}\parallel R_{3}}{R_{3}}}\cdot {\frac {R_{32}\parallel R_{33}}{R_{32}}}}$

Example of magnetic circuit

magnetic flux divider from two branches

The same rule applies to magnetic circuits. The equations for the partial flows through and result: ${\displaystyle R_{m_{2}}}$ ${\displaystyle R_{m_{3}}}$

${\displaystyle {\frac {\Phi _{2}}{\Phi }}={\frac {R_{m_{23}}}{R_{m_{2}}}}}$and for the other branch with the total resistance of the parallel connection ${\displaystyle {\frac {\Phi _{3}}{\Phi }}={\frac {R_{m_{23}}}{R_{m_{3}}}}}$ ${\displaystyle R_{m_{23}}=R_{m_{2}}\parallel R_{m_{3}}}$

application

Current dividers are used in particular to measure high currents; they are then called shunt , with the measuring device forming one of the current paths. Essentially, however, it measures the voltage dropping on the main path, since only a very small partial current flows through it. In multimeters there are switchable current dividers for current measurement in different areas.

See also

• Voltage divider

Web links

• Exercises on the flow divider rule (PDF file; 111 kB)

Individual evidence

1. ^ Rainer Ose: Electrical engineering for engineers: Fundamentals . Carl Hanser, 2013, ISBN 978-3-446-43955-9 , pp. 378 ( limited preview in Google Book search).
2. ↑ Reiner Johannes Schütt: Electrotechnical basics for industrial engineers: Generating, transmitting, converting and using electrical energy and electrical messages . Springer, 2013, ISBN 978-3-658-02763-6 , pp. 35 ( limited preview in Google Book search).