Series and Parallel

You need a 15 kΩ resistor and your kit does not have one. The "k" stands for kilo-, a thousand, so 15 kΩ is 15 000 Ω. You can build that value from resistors you do have, once you know how resistors combine.

Parts connect in two basic ways: in series and in parallel. Each way has a rule that turns a group of resistors into one equivalent resistance: the single resistor that would take the same current from the same supply. Once a group is one value, Ohm's law works on it directly.

Series: one path

Parts are in series when they connect end to end on one path. All the current that goes through one must go through the next, because it has nowhere else to go. So the same current flows through every part in a series chain.

R_total = R₁ + R₂ + R₃ + …

The supply voltage splits between the resistors, and the bigger resistor gets the bigger share. A 1 kΩ and a 9 kΩ in series across 10 V get 1 V and 9 V. The next lesson builds on this.

The two AA cells in a TV remote are in series: 1.5 V plus 1.5 V gives 3 V, and the same current flows through both. The LED and its resistor in the last lesson were in series too.

Parallel: more than one path

Parts are in parallel when both their ends connect to the same two points. Each separate path is called a branch. Every branch has the same voltage across it, and the current splits between the branches. The branch with the lower resistance takes the bigger share of the current.

1 / R_total = 1 / R₁ + 1 / R₂ + …

For two resistors, this shortcut is easier:

R_total = (R₁ × R₂) / (R₁ + R₂)

A car's headlights are wired in parallel across its 12 V system. Each one gets the full 12 V, and if one fails, the other stays on.

Where the rules come from

Both rules come from two more general ones, Kirchhoff's laws. These use two new words. A node is a point where two or more parts connect. All the wire joining them counts as one node, and some books call it a junction. A loop is any path around a circuit that ends where it started.

• The current law: the current flowing into a node equals the current flowing out. Charge does not pile up or vanish at a node.
• The voltage law: the voltages around any loop add up to zero. The rise across the supply equals the drops across the parts added together.

The same current in a series chain is the current law at work. The same voltage across parallel branches is the voltage law. Kirchhoff's laws get their own lesson later in this module, for circuits that the series and parallel rules cannot reduce.

Build a group of resistors below. Use Arrangement to switch between series and parallel, and watch the equivalent resistance, the current in each branch and the power in each resistor. It starts with 1 kΩ and 4.7 kΩ in series, which make 5.7 kΩ and take 1.58 mA from the 9 V supply.

Two quick checks

• A series total is always larger than the largest resistor. You made the one path harder to get through.
• A parallel total is always smaller than the smallest resistor. You added another path, so more current gets through for the same push.

If your answer breaks either rule, the arithmetic is wrong. In the instrument, switch the starting pair to Parallel: the total drops to 825 Ω, below the smaller 1 kΩ. Two equal resistors in parallel give half the value of one. You will meet this case often.

Where you use these rules

Making a value you do not have. Resistors are sold in standard lists of values. The E12 list has 12 values in each factor of ten: 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68 and 82, then 100, 120 and so on. Kit values usually come from it. Say you need 15 kΩ and your drawer only has 10 kΩ and 33 kΩ. Two 10 kΩ in parallel make 5 kΩ. A third 10 kΩ in series with that pair brings the total to 15 kΩ.

Sharing the heat. Two 1 Ω resistors in parallel make 0.5 Ω, and each one carries half the current. Since P = I² × R, each one turns (I/2)² × 1 Ω = I²/4 into heat. A single 0.5 Ω resistor carrying all the current would turn I² × 0.5 Ω = I²/2 into heat. So each resistor in the pair takes half the heat of the single part it replaces. This is a common fix when one resistor's power rating is too low: two equal 1/4 W resistors in parallel can share 1/2 W between them.

Connecting a meter adds a parallel path

When you connect anything between two nodes, you put it in parallel with whatever is already between those nodes, even by accident.

A multimeter is the handheld meter that measures volts, amps and ohms. When it measures volts, it puts its own resistance between its two probes (the leads you touch to the circuit). This is its input resistance, and a typical multimeter has 10 MΩ. The "M" stands for mega-, a million, so 10 MΩ is ten million ohms.

Across a 1 kΩ resistor, that 10 MΩ in parallel changes the resistance by 0.01 %, too little to notice. Across a 1 MΩ resistor it changes it by 9 %, so the meter itself throws the reading off. The multimeter gets its own lesson later in this module, including what it does to the circuit it measures.

This effect is called loading: connecting something changes the circuit you connect it to. The next lesson shows a circuit where loading matters a lot, the voltage divider.