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# Electrical Parallel Circuits

In a parallel circuit there is more than one path the electrons can flow.

The Parallel Circuit Rules:

1. The total current in parallel circuit is equal to the sum of the currents in all the branches of the circuit.
2. The total voltage across a parallel circuit is equal to the voltage across any of the branches in the circuit.
3. The total resistance of a parallel circuit is calculated by using Ohm's law on the voltage and current of the whole circuit.

The sum of currents in a parallel network can be calculated as

I = I 1 + I 2 + ... + I n (1)

where

I n = parallel current (amps)

I = sum of currents (amps)

The total resistance in a parallel circuit can be calculated as

1 / R = 1 / R 1 + 1 / R 2 + ... + 1 / R n (2)

where

R n = parallel resistance (ohms, Ω)

R = total resistance in the parallel network (ohms, Ω)

### Example - Parallel Network

The total resistance in a network with three resistors R 1 = 10 Ω, R 2 = 20 Ω  and R 3 = 30 Ω can be calculated as

1 / R = 1 / (10 Ω) + 1 / (20 Ω) + 1 / (30 Ω)

= 0.183 (1/Ω)

R = 1 / 0.183

= 5.46 (Ω)

Connected to a 12 V battery the sum of currents can be calculated

I = U / R

=  (12 V) / (5.46 Ω)

= 2.2 Amps)

The current through each resistor can be calculated as

I 1 = U / R 1

=  (12 V) / (10 Ω)

= 1.2 (Amps)

I 2 = U / R 2

=  (12 V) / (20 Ω)

= 0.6 (Amps)

I 3 = U / R 3

=  (12 V) / (30 Ω)

= 0.4 (Amps)

### Resistors in Parallel and resulting Resistance

Two resistors in parallel - resistance ranging 1 - 100 ohm

 R2 ( Ω ) Resulting Resistance (Ω) R1 ( Ω ) 1 1.5 2.2 3.3 4.7 6.8 10 15 22 33 47 68 1 0.5 0.6 0.69 0.77 0.83 0.87 0.91 0.93 0.95 0.97 0.98 0.99 1.5 0.6 0.75 0.89 1.03 1.14 1.22 1.30 1.36 1.40 1.43 1.45 1.46 2.2 0.69 0.89 1.1 1.32 1.50 1.66 1.82 1.92 2.0 2.06 2.10 2.13 3.3 0.77 1.03 1.32 1.65 1.94 2.22 2.48 2.70 2.87 3.00 3.08 3.14 4.7 0.83 1.14 1.50 1.94 2.35 2.78 3.20 3.58 3.87 4.12 4.27 4.39 6.8 0.87 1.22 1.66 2.22 2.78 3.40 4.05 4.68 5.19 5.64 5.94 6.18 10 0.91 1.30 1.82 2.48 3.20 4.05 5.0 6.0 6.9 7.7 8.3 8.7 15 0.93 1.36 1.92 2.70 3.58 4.68 6.0 7.50 8.9 10.3 11.4 12.2 22 0.95 1.40 2.00 2.87 3.87 5.19 6.9 8.9 11.0 13.2 15.0 16.6 33 0.97 1.43 2.06 3.0 4.12 5.64 7.7 10.3 13.2 16.5 19.4 22.2 47 0.98 1.45 2.1 3.08 4.27 5.94 8.3 11.4 15.0 19.4 23.5 27.8 68 0.99 1.46 2.13 3.14 4.39 6.18 8.7 12.2 16.6 22.2 27.8 34.0

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