# Which instruments are connected in parallel and in series

## Series and parallel connections¶

The following section deals with the effects that result from a series or parallel connection of several resistors, current sources or capacitors.

### Series and parallel connection of resistors¶

In (almost) every circuit there are several consumers, i.e. electrical resistors. How these work in their entirety depends on whether they are connected in parallel or in series.

Series connection of resistors

With a series connection of Resistances is the total resistance equal to the sum of the individual resistances :

For example, there are three resistors of the size connected in series, they work together like a resistor of magnitude .

The total voltage applied to a series connection divides according to the rule of meshes into Partial stresses on. The total stress is equal to the sum of the individual partial stresses:

The amperage , which flows through several resistors connected in series, is the same at all points of the series connection. Hence:

This fact is used, among other things, to measure the current intensity by inserting an ammeter at any point in the (sub) circuit to be examined as a series connection.

Is the resistance value of a resistor in a series circuit, the voltage across the resistor can be calculated using the above formulas and Ohm's law or the current strength flowing through the resistor getting closed: Parallel connection of resistors

With a parallel connection of Resistances add the reciprocal values ​​of the individual resistances to the reciprocal of the total resistance on:

For example, there will be three resistors of the size connected in parallel to each other, the result is the reciprocal of the total resistance . The total resistance is thus .

At all The same voltage is applied to resistors in a parallel circuit. This is equal to the total voltage :

This fact is used, among other things, to measure voltage by connecting a voltmeter in parallel to the (sub) circuit or component being examined.

The amperage splits up in a parallel connection according to the knot rule Partial currents on:

Is the resistance value of a resistor of a parallel circuit is known, the voltage applied to the resistor can in turn with the help of the above formulas and Ohm's law or the current strength flowing through the resistor getting closed: ### Series and parallel connection of power sources¶

In order to bring about a higher voltage or a greater usable current strength in a circuit, several power sources (e.g. batteries, accumulators or solar cells) can be connected in series or in parallel with one another.

Series connection of power sources

Become When current sources are connected in series, their individual voltage values ​​add up to a total tension : This equation is formally identical to equation (2). The difference lies in the opposite action of current sources and resistors:

One switches similar current sources in series, the resulting current source has a -fold voltage.

Parallel connection of power sources

Become If current sources are connected in parallel, the partial currents are reduced each individual power source to the total amperage contributed. This equation is formally identical to equation (6). At this point it has two equally valid effects:

With a parallel connection of similar current sources, the maximum possible current strength is around the times increased or the individual (partial) current strengths provided by the current sources by times reduced.

### Series and parallel connection of capacitors¶

Capacitors are also one of the most commonly used electronic components. By connecting several capacitors in series or in parallel, their characteristic size, the capacitance , influence.

Series connection of capacitors

Become Capacitors are connected in series so that when the voltage is applied all with the same amperage on an amount of charge charged.

As is usual with series connections, the partial voltages dropping across the individual capacitors add up that can be found using the general capacitor formula as let express: To find out how several capacitors work in their entirety, i.e. what total capacitance from the series connection of the individual capacitors one has to put both sides of the equation above through the constant charge share. The left side of the equation then corresponds to the total capacity , the right side of the sum of all reciprocal values ​​of the individual capacitors:

If another capacitor is connected in series with an existing capacitor, the total capacitance decreases. Capacitors behave in a series connection in a similar way to resistors in a parallel connection.

Parallel connection of capacitors

Become If capacitors are connected in parallel, they all have the same voltage at. The total current divides into Partial currents on, causing the individual capacitors with different charges getting charged.

Using the general capacitor formula can be the total capacity express directly: The individual quotients correspond to the individual capacities the capacitors connected in parallel to each other. Hence:

A parallel connection of two or more capacitors is like a single capacitor with a correspondingly larger capacitance. In a parallel circuit, capacitors behave similarly to resistors in a series circuit.

### Star-delta conversion¶

In order to determine the individual currents and voltages occurring in a circuit with several resistors, these can be gradually replaced by substitute resistors for series and parallel connections of resistors. Occasionally, however, circuits can also occur in which such a replacement is not immediately possible. Such a circuit is shown as an example in the following figure:

In the example circuit above, it is not possible to say immediately whether the resistance is is now connected in series or in parallel with the other resistors. In such a case, however, it is possible to convert a "triangular" circuit into a "star-shaped" one:

With such a "triangle-star conversion", both the arrangement and the names of the resistors are changed. The assignment takes place as with a geometric rectangle, in which, for example, the triangle side the point opposite.

• The values ​​of the resistances resulting from a triangle-star conversion can be calculated as follows: The resulting resistance values ​​are therefore equal to the product of the two resistances applied, divided by the sum of all three resistances.

• If necessary, the conversion can also be carried out in the opposite direction. For the values ​​of the resistances resulting from a star-delta conversion, the following applies accordingly: Example:

Star-delta conversion for capacitors

When connected in series or in parallel, capacitors behave in exactly the opposite way to resistors: when connected in parallel, their capacitance values ​​add up, while when connected in series, the reciprocal values ​​of their capacities are added.

However, the principle of star-delta conversion can also be applied to capacitors, if one calculates with the reciprocal values ​​of their capacities or with the corresponding reactances of the individual capacitors.  The following conversions are obtained:

• With a delta-star conversion of capacitors, the resulting capacitance values ​​can be calculated as follows: • For the values ​​of the capacities resulting from a star-delta conversion, the following applies accordingly: In the case of capacitors, too, star-delta or delta-star conversions are carried out until the resulting equivalent capacitance values ​​result in a circuit that only consists of series and / or parallel connections of capacitors.

Remarks: