Series and parallel combinations
Many circuits can be simplified by combining resistors. Series and parallel rules let complicated networks collapse into a single equivalent resistance.
Resistors in series
Resistors in series carry the same current, and their voltages add, so the equivalent resistance is the sum: \(R_{eq} = R_1 + R_2 + \dots + R_n\). The equivalent is always larger than the largest individual resistor. Three 100 \(\Omega\) resistors in series act like 300 \(\Omega\).
Resistors in parallel
Resistors in parallel share the same voltage, and their currents add, so conductances add: \(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots\). The equivalent is always smaller than the smallest individual resistor. For two resistors, \(R_{eq} = \frac{R_1 R_2}{R_1 + R_2}\), product over sum; n equal resistors R in parallel give \(\frac{R}{n}\).
Simplifying networks
To reduce a network, repeatedly identify groups that are purely in series (same current, no branch between) or purely in parallel (same two nodes), replace each with its equivalent, and redraw. Working from the side farthest from the terminals of interest inwards usually keeps the steps clear. Not every network reduces this way; bridges need other methods.
Short and open circuits in networks
A resistor in parallel with a short circuit carries no current, since the short offers a zero-resistance path, so it can be removed. A resistor in series with an open circuit carries no current either. Recognising these situations, for example a closed switch shorting out part of a network, simplifies analysis.
Voltage and current in combinations
After finding the total current from the equivalent resistance, work back outwards: series resistors share the current and divide the voltage; parallel branches share the voltage and divide the current. This back-substitution recovers the voltage and current of every element.
Practical uses
Combining standard resistor values approximates values not available off the shelf: two 1 k\(\Omega\) resistors in parallel give 500 \(\Omega\). Series strings spread power dissipation across several parts, and parallel combinations increase power-handling capacity. Strings of LEDs, heating elements, and loudspeaker arrangements all rely on these rules.
Delta-wye transformations
Some networks, such as bridges, contain resistors connected neither in series nor in parallel. A delta (triangle) of resistors can be converted into an equivalent wye (star) and vice versa. For a balanced network of equal resistors, \(R_Y = \frac{R_\Delta}{3}\). After transforming, ordinary series and parallel reductions usually finish the job.
Worked example
Find the resistance between A and B: a 4 \(\Omega\) resistor in series with a parallel pair of 6 \(\Omega\) and 3 \(\Omega\), all in parallel with a 10 \(\Omega\) resistor. The pair gives \(\frac{6 \times 3}{9} = 2\) \(\Omega\); adding 4 gives 6 \(\Omega\); in parallel with 10: \(\frac{6 \times 10}{16} = 3.75\) \(\Omega\). With 15 V applied, total current is 4 A: 2.5 A through the 6 \(\Omega\) branch and 1.5 A through 10 \(\Omega\).
Key takeaways
Series resistors add: R_eq = R1 + R2 + ..., always larger than the largest. Parallel conductances add: 1/R_eq = sum of 1/R; two resistors give product over sum, and n equal resistors give R/n, always smaller than the smallest. Reduce networks by repeatedly combining series and parallel groups, removing shorted or open branches, then work back to find each element's voltage and current. Combinations create non-standard values and spread power. Delta-wye transformations, with R_Y = R_delta / 3 when balanced, handle bridges.
Flashcard
R1 + R2 + ...
Check your understanding
What is the equivalent resistance of 12 ohms and 6 ohms in parallel?