Many networks can be reduced to series-parallel combinations, reducing the complexity in calculating the circuit parameters such as resistance, voltage and current. When several resistors are connected between two points with only a single current path, they are said to be in series. In a parallel circuit, though, the current is divided among each resistor, such that more current goes through the path of least resistance. A parallel circuit has properties that allow both the individual resistances and the equivalent resistance to be calculated with a single formula. The voltage drop is the same across each resistor in parallel.

For the special case of two resistors in parallel, the currents are inversely proportional to their resistances. The formula V = I1 * R1 = I2 * R2 can be rearranged to give R1 / R2 = I2 / I1.

Get the current and the voltage. This may be a value that is given to you in a theoretical problem or something that you measure, using a voltmeter, ammeter or multimeter. The voltage only needs to be obtained over one resistor, since it is the same for all. However, the current Ij (j = 1,2, …,n) needs to be found for each resistor, where Ij represents the current flowing through the jth resistor in parallel, and there are n resistors in total.

Calculate the resistance Rj (j = 1,2, …,n) of each element, where Rj represents the resistance of the jth resistor in parallel and there are a total of n resistors. The resistance of each element is given by the formula Rj = V / Ij. For example, if you have three resistors in parallel with a voltage drop of 9 Volts and currents I1 = 3 Amps, I2 = 6 Amps and I3 = 2 Amps, the resistances are R1 = 3 Ohms, R2 = 1.5 Ohms and R3 = 4.5 Ohms.

Calculate the equivalent resistance for the circuit, if it is part of a larger network. A group of resistors in parallel can be replaced by a single equivalent resistance Req, which simplifies calculations when trying to obtain network parameters. Now instead of a group of resistors in parallel there is a single equivalent resistance with the original voltage V across it and a current I total flowing through it, that is the sum of all the currents through each of the resistors in parallel. The equivalent resistance Req for a parallel circuit is given by the sum of the reciprocals of the individual resistances as follows

1 / Req = 1 / R1 + 1 / R2 + ….1 / Rn.

The equivalent resistance is always smaller than any of the individual resistances in a parallel circuit. For the example with the three resistors the equivalent resistance is Req=0.82 Ohms. Which means the circuit can be replaced with a single resistor with a resistance of 0.82 Ohms, voltage of 9 Volts and a current of 11 Amps.

#### Tips

References

Tips

- For the special case of two resistors in parallel, the currents are inversely proportional to their resistances. The formula V = I1 * R1 = I2 * R2 can be rearranged to give R1 / R2 = I2 / I1.

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