Basic Circuit Analysis
Parallel Circuits

Section 2-2 Voltage in Parallel Circuits

 Recall a Definition A parallel circuit has more than one path for current flow.

Equation

Total Resistance of a Parallel Circuit

 RT = 1 1 + 1 + 1 +  ... + 1 R1 R2 R3 Rn

Where

RT = Total resistance of the parallel circuit
R1, R2, R3, Rn = Resistance of the individual resistors

This formal equation for the total resistance of a parallel circuit looks horribly complicated ... at least at first glance. In the days of pencil-and-paper calculation, this equation was generally considered impossible to use. With today's calculator technology, the 1/x function key renders the process of solving the parallel resistance equation fairly straightforward.

Example

Suppose there is a parallel resistor circuit with values of 10W, 15W, and 25W.

1. Set up the equation:

 RT = 1 1 + 1 + 1 R1 R2 R3

2.Subsititue the given values

 RT = 1 1 + 1 + 1 10 15 25

3. Solve with keystrokes:

10 1/x + 15 1/x + 25 1/x = 1/x

4.838709677

4. Present the solution:

RT = 4.8W

More Examples

Endless Examples & Exercises

 Given the value of all the resistors in a parallel circuit, calculate the total resistance.  Do these exercises until you are confident you can handle them.

In practical electronic design and troubleshooting, it is sometimes necessary to adjust the total resistance of a parallel circuit. Suppose, for instance, a circuit is a simple 180W  resistor. For some good reason, you fine it  necessary to lower the resistance to 120W by adding another resistor in parallel. The problem, of  course, is to determine the value of that resistor.

Using the product-over-sum equation, you are given the desired total resistance (RT) and the value of one of the two resistors (R1). Solve the equation for R2.

Procedure

1. Begin with the product-over-sum equation:

 RT = R1R2 R1 + R2

2. Solve it for R2:

 R2 = RTR1 R1 – RT

3. Substitute the known values and do the math:

 R2 = (120)(180) =360 180 – 120

4. Present the solution:

R2 = 360W

1. The total resistance of a parallel circuit is always less than the value of the smallest resistor.

1. When all resistors in a parallel circuit have the same value, the total resistance is equal to the common value divided by the number of resistors. Example: Three 330W resistors in parallel provide a total resistance of 110W.

1. When a resistor is removed  from a parallel circuit, (1) current continues to flow through the remaining resistors, and (2) the total resistance increases

1. When a resistor is shorted in a parallel circuit, the total resistance falls to zero.

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