Parallel Networks

Example #1

Resistors A, B and C are connected together in parallel. Together, they are equivalent to the single resistor on the left labeled ‘ABC’. The reciprocal of the resistance of the entire three-resistor network is equal to the sum of the reciprocals of the individual resistances in the network:

where R_ABC, R_A, R_B and R_C are the resistances of resistors ABC, A, B and C respectively. Substituting in the values from the diagram gives:

.

Taking the reciprocal of both sides of the equation gives:

R_ABC = 2Ω.

Example #2

Resistors A, B and C are connected together in parallel. Together, they are equivalent to the single resistor on the left labeled ‘ABC’. The reciprocal of the resistance of the entire two-resistor network is equal to the sum of the reciprocals of the resistances across the individual resistors:

where R_ABC, R_A and R_B are the resistances of resistors ABC, A and B respectively. Substituting in the values from the diagram gives:

.

Solving for 1/R_A reveals:

.

Taking the reciprocal of both sides gives:

R_A = 12Ω.

Note: The equation

gives a value for 1/R_parallel, not R_parallel. Hence, after plugging numbers into this equation, don’t forget to take the reciprocal of the answer to find R_parallel.