Series and Parallel

A string of old holiday lights goes dark from one bad bulb. Fifty bulbs, one burned filament, and the whole strand dies, because the current lighting every bulb travels through each of them in a single line. A house, by contrast, keeps working when the lamp in one room is switched off; the fridge does not notice. The difference between those two behaviors is the difference between the two ways any component can connect, and it is the first question worth asking about any circuit more complicated than the one-resistor loop that your-first-circuit built.

That lesson used exactly two parts: a resistor and an LED in one loop. Real circuits have more parts, and every pair of them relates in one of two ways. Get the two ways right and a page full of symbols collapses into a handful of numbers that work out on paper. Get them wrong and every prediction after that is wrong too.

One path or many

Parts in series sit end to end on a single path, so one current flows through all of them in turn. There is nowhere else for the current to go. The LED loop made this concrete: the 20 milliamps leaving the resistor is exactly the 20 milliamps entering the LED. What the parts share is current. What they divide among themselves is voltage, which is why the 5 volt supply split into 3 volts across the resistor and 2 volts across the LED.

Parts in parallel sit side by side, both ends tied to the same two nodes. A node, from that lesson, is one electrical spot: every point joined by plain wire is the same node, at the same voltage. Because both ends of each part land on the same two nodes, parallel parts share one voltage. What they divide is the current. Two 100 ohm resistors in parallel across a 9 volt battery each see the full 9 volts, each draws its own 90 milliamps, and the battery delivers the sum, 180 milliamps.

The test that sorts any two parts is one question: what do they share? If the same current has no choice but to pass through both, they are in series. If both sit across the same two nodes, they are in parallel. The drawings can lie, since two parts stacked vertically on a schematic may still be in series; the sharing is what counts, not the shape.

Collapsing the network

A group of resistors can often be replaced by one resistor that behaves identically as far as the rest of the circuit can tell. That single value is the equivalent resistance, and finding it is the whole algebra of this section. Series is the easy case: resistances in series simply add, because the current fights each one after another. A 1,000 ohm resistor in series with a 2,200 ohm resistor is 3,200 ohms, always more than the largest part. Across a 9 volt battery that total draws I = 9 / 3200, about 2.8 milliamps.

Parallel is the case people fumble, so it is worth earning properly. Adding a parallel path gives current another route, which lowers the total opposition, so parallel resistance always comes out smaller than the smallest branch, branch being the name for one parallel path. The rule is that the reciprocals add, a reciprocal being one divided by the value:

1/R_total = 1/R1 + 1/R2 + ...      (parallel)
R_total = R1 + R2 + ...            (series)

For two resistors the parallel rule shortens to the product over the sum. A 1,000 ohm and a 2,200 ohm pair give 1000 times 2200 divided by 3200, which is 687.5 ohms. Two identical resistors in parallel are always half the value of one, and three are a third. Check the worked pair against Ohm's law and the sharing makes sense: 9 volts across 687.5 ohms draws about 13.1 milliamps total, of which 9 milliamps goes through the 1,000 ohm branch and about 4.1 through the 2,200 ohm branch. The sums agree because they must.

Series addition also deserves its physical cross-check. The same 2.81 milliamps that 9 volts forces through 3,200 ohms drops 2.81 volts across the 1,000 ohm resistor and 6.19 volts across the 2,200 ohm one, and the two drops sum to the full 9 volts. That additivity is not luck: one current passing two oppositions loses energy at each, and the losses stack. It is also the seed of the divider coming up next.

A short function does the arithmetic for any length of list:

def equivalent(resistors, mode):
    if mode == "series":
        return sum(resistors)
    inverse_sum = sum(1 / r for r in resistors)
    return 1 / inverse_sum

print(equivalent([1000, 2200, 470], "series"))    # 3670
print(equivalent([1000, 2200, 470], "parallel"))  # 279.1576...

Three resistors of 1,000, 2,200, and 470 ohms come to 3,670 ohms in series and about 279 ohms in parallel, smaller than the smallest single resistor, exactly as the rule demands. The parallel branch of the function divides one by the sum of reciprocals, which is the general form of the product-over-sum shortcut. Same physics, no special cases.

The voltage divider

Two resistors in series across a source, tapped at the point where they join, is the voltage divider, and it is the most reused pattern in electronics. Because one current flows through both resistors, the voltage at the tap is just the bottom resistor's share of the total drop. With R1 on top (from the supply to the tap) and R2 on the bottom (from the tap to the negative side of the supply), the tap voltage is:

Vout = Vin * R2 / (R1 + R2)

Worked numbers: a 9 volt battery feeds a 10,000 ohm resistor on top and a 4,700 ohm resistor on the bottom. The same current, 9 / 14700 or about 0.61 milliamps, flows through both, so the tap sits at 0.61 milliamps times 4,700 ohms, which is 2.88 volts. The formula gives the same answer in one step: 9 times 4700 / 14700.

Dividers show up everywhere a circuit needs a voltage smaller than its supply: scaling a sensor's output into a converter's range, setting a reference voltage, or turning a knob's position into a number, which is what a potentiometer is, a divider whose split moves with the knob. The design runs the formula backward. To turn a 5 volt supply into the 3.3 volts many sensors want, pick a ratio of 0.66, and any pair with that fraction works: 10,000 ohms on top and 20,000 ohms on the bottom give 5 times 20000 / 30000, which is 3.33 volts. Smaller resistor pairs waste more current through the divider itself; larger pairs waste less but present a higher resistance to anything connected at the tap. That is the catch, and it gets its full treatment two lessons from now: the tap only holds its predicted voltage while whatever is connected to it draws almost no current. A load that draws real current pulls the tap down, sometimes a long way.

Splitting current the other way

Parallel branches split an arriving current, and the split is worth one formula. The branch with less resistance takes more current, and the share of each branch is proportional to the OTHER branch's resistance: a branch of R1 carries I_total times R2 / (R1 + R2). Say 27 milliamps arrives at a node, one electrical spot where paths meet, and splits into a 100 ohm and a 220 ohm branch. The 100 ohm branch takes 27 times 220 / 320, about 18.6 milliamps, and the 220 ohm branch takes about 8.4 milliamps. The two add back to 27, exactly as charge conservation demands, and the next lesson names that demand the junction rule. The classic slip is putting the branch's own resistance in the numerator, which predicts more current through more resistance and is backwards.

When one part dies

Series punishes failure. In the LED loop, pull the LED out and the resistor's current drops to zero too: one open break stops everything, because the single path is gone. That is the holiday light effect, and it is why cheap series-wired strings came with the tedious game of hunting one dead bulb among fifty. In a series string the voltage also redistributes when a part changes; a bulb that fails short, conducting but not lighting, simply gives its share of the voltage to the rest.

Parallel shrugs it off. Each branch connects directly across the supply, so one branch failing or being switched out changes nothing in the others: same voltage, same current, full brightness. This is why house outlets are wired in parallel. Every outlet sees the same line voltage, 120 volts in North America and 230 volts in much of the rest of the world, appliances switch on and off independently, and each draws exactly the current its own resistance asks for. The tradeoff is that total current adds up branch by branch, which is why a circuit breaker trips on the sum: too many parallel appliances, one hot wire.

Equivalent resistance, the divider, and the current split cover a huge fraction of the circuits that get built, but only the tidy ones: circuits that are purely series, purely parallel, or cleanly built from blocks of each. A circuit with two sources and a shared branch is none of those, and no amount of collapsing cracks it. The next lesson, Kirchhoff's Laws, adds the two conservation rules that solve any circuit, including the ones that refuse to collapse.