Complex combinations of resistors
Complex circuits may have some resistors connected in series, and some connected in parallel. For example, the following combination of resistors shows two resistors connected in series, and three resistors connected in parallel.
In such complex circuits, each relationship for calculating resistors in series and in parallel must be used separately.
In the above example, the total series resistance of \({R_1}\) and \({R_2}\) should be calculated, and the total parallel resistance of \({R_1}\) \({R_3},{R_4},{R_5}\) should be calculated, and the results combined by adding each calculated total together.
Question
Calculate the total resistance between points X and Y.
Parallel: \(\frac{1}{R_{T}} = \frac{1}{R_{1}}\; + \frac{1}{R_{2}} \)
\(\frac{1}{R_{T}} = \frac{1}{16}\; + \frac{1}{4} = 0.3125\)
\(R_{T} = \frac{1}{0.3125}\; = 3.2 \Omega\)
Series: \(R_{T} = R_{1} + R_{2}\)
\(R_{T} = 8 + 5\)
\(R_{T} = 13 \Omega\)
add the total resistances for each parallel and series parts:
\(R_{combined} = R_{P} + R_{S} = 3.2 + 13 = 16.2 \Omega\)
Question
Calculate the resistance between points X and Y
First, work out the total series resistance:
\(R_{T} = R_{1} + R_{2}\)
\(R_{T} = 14 + 6\)
\(R_{T} = 20 \Omega\)
Then work out the parallel combination of this total resistance with the \(5\Omega\) resistor in parallel
Parallel: \(\frac {1}{R_{T}} =\frac {1}{R_{1}} +\frac {1}{R_{2}}\)
\(\frac {1}{R_{T}} = \frac {1}{20} + \frac {1}{5} = 0.25\)
\(R_{T} = \frac {1}{0.25} = 4 \Omega\)
The combined resistance \(R_{combined} = 4 \Omega\)
