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Connecting Resistors in Parallel

Whereas in simple circuits and series connections the current only finds one path through the circuit, in parallel circuit configurations (shunted connection) there is a branching of the circuit (see the following figure). At the current's branching point (node) A the total current I is divided into partial currents I1, I2 and I3, at the current branching point B the partial currents merge again into the total current I.

Resistor Circuits !

A fundamental attribute of a parallel circuit is immediately obvious from the structure of the circuit itself: All of the resistors connected in parallel are located between the same points (namely A and B) and are thus all connected to the same voltage U:

When resistors are connected in parallel one and the same voltage is applied across all of the resistors.

If you now measure the partial currents flowing in the individual branches and compare them to the total current I, you arrive at a principle which constitutes a simplified form of Kirchhoff's first law:

In a parallel circuit the total current is equal to the sum of its partial currents, i.e. it is true that

Resistor Circuits !

In its basic form the law states that

The sum of the currents flowing to a node is equal to the sum of currents flowing out of the node

and for this reason is sometimes referred to in German as the "node" rule.

Since Ohm's law naturally applies for the parallel circuit and equally for each of the resistors as well as their parallel connection, it is possible to obtain the following equation for the connection of resistors in parallel based on this simplified version of Kirchhoff's first law

Resistor Circuits !

Rges = Rtot

whereby Rtot is the total resistance (equivalent resistance) of the parallel connection. If both sides of the equation are divided by U, you obtain the following relationship for the resistance of the parallel circuit

Resistor Circuits !

If in this expression you substitute the resistances by their reciprocal conductances (i.e. the inverse of the resistances), you obtain the total conductance of the parallel connection

Resistor Circuits !

Hence it holds true that:

In a parallel connection of resistors the reciprocal of the total resistance is equal to the sum of the reciprocal of the individual resistors and the total conductance is equal to the sum of the conductances of the individual resistors.

In particular this means that in a parallel circuit the total resistance is always lower than the lowest resistance of any individual resistor.

Since all of the resistors are connected to the same voltage, due to Ohm's law the partial currents must respond in the completely opposite fashion to the partial resistances, i.e. it is true that 

In a parallel connection of resistors the partial currents respond in inverse üroportion to the partial resistors.

Example: A parallel circuit comprising resistors R1 = 20 W, R2 = 30 W and R3 = 60 W is connected to a voltage of U = 10 V. It is then true for the total resistance that

Resistor Circuits !

Then according to Ohm's law the following is also true for the total current I = 10 V/10 W = 1 A and for the partial currents

Resistor Circuits !

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