Series and Parallel Circuits Explained

Ohm's law

For a resistor, the voltage across it is proportional to the current through it: V = I R. A 100 Ω resistor with 0.05 A through it has 5 V across it. Voltage is the energy each coulomb of charge gives up crossing the component; current is how many coulombs pass each second; their product is the power turned into heat or light, P = V I = I² R.

Series

Components in series sit in a single line, so the same current flows through every one. Their voltages add up to the supply voltage, and their resistances simply add:

R = R₁ + R₂ + R₃ + …

The biggest resistor takes the biggest share of the voltage. Break the circuit anywhere, as with a failed bulb in an old string of fairy lights, and the current stops everywhere.

Parallel

Components in parallel each connect across the same two points, so each has the full voltage across it and the currents add up:

1 ÷ R = 1 ÷ R₁ + 1 ÷ R₂ + 1 ÷ R₃ + …

The total is always smaller than the smallest branch, because every branch is another path for current. For two resistors, R = R₁R₂ ÷ (R₁ + R₂). House wiring is parallel: every socket gets mains voltage, and switching one lamp off leaves the rest on.

Kirchhoff's laws

• Current law: the current flowing into any junction equals the current flowing out. Charge is conserved and does not pile up.
• Voltage law: round any closed loop, the voltage rises (batteries) equal the voltage drops (resistors). Energy per charge is conserved.

With Ohm's law these solve any network of resistors and sources, however tangled; circuit simulators write one current-law equation per junction and solve them all together, which is what this simulation does.

A worked example

A 9 V battery feeds R₁ = 100 Ω in series with R₂ = 220 Ω and R₃ = 330 Ω in parallel. Ignore the battery's internal resistance for now.

1. The parallel pair: 220 × 330 ÷ (220 + 330) = 132 Ω.
2. The whole circuit: 100 + 132 = 232 Ω.
3. Battery current: 9 ÷ 232 = 38.8 mA, all of it through R₁, which drops 3.88 V.
4. The pair gets the remaining 9 − 3.88 = 5.12 V: 23.3 mA through R₂ and 15.5 mA through R₃, which add to 38.8 mA, as the current law requires.
5. Power: R₁ 0.151 W, R₂ 0.119 W, R₃ 0.079 W, total 0.349 W = 9 V × 38.8 mA.

Real batteries: internal resistance

A battery behaves like an ideal voltage E (its electromotive force) in series with a small internal resistance r. Drawing current I drops I r inside it, so the voltage at the terminals is V = E − I r. A fresh AA cell has about 0.1 to 0.3 Ω; as it runs down r rises, which is why a weak battery reads nearly full on a meter (almost no current) yet sags under load. With r = 0.5 Ω in the example the current falls slightly to 38.7 mA and the terminal voltage to 8.98 V.

Dividers and bridges

Two resistors in series divide a voltage: V_out = V × R₂ ÷ (R₁ + R₂). Anything connected across R₂ is in parallel with it and pulls V_out down, so a divider only gives its nominal voltage into a load much larger than R₂. A Wheatstone bridge is two dividers side by side with a meter between their midpoints; it reads zero when R₁ ÷ R₂ = R₃ ÷ R₄, which lets an unknown resistance be measured very precisely against known ones, as in strain gauges and old laboratory bridges.

Using the simulation

In the Series and Parallel Circuit Simulator, pick a circuit and set the battery, its internal resistance and each resistance. Every current, voltage and power is listed, the bulbs glow with their power, and the electrons drift at speeds proportional to the current. Things to try:

• In series, make R₃ large: it takes most of the voltage and glows brightest, while the others dim.
• In parallel, open the switch: R₃ goes dark but R₁ and R₂ are unchanged (with r = 0); with a large internal resistance they brighten slightly, since the battery is less loaded.
• In the bridge, set R₁ = 100, R₂ = 220, R₃ = 150 and adjust R₄ to 330: the meter goes green and reads zero.
• Watch the chart: the power in R₁ peaks when R₁ equals the resistance the rest of the circuit presents to it, the maximum power transfer theorem.

What the model assumes

• Ohmic resistors. Each bulb is a fixed resistance. The resistance does not change with current or temperature.
• Ideal wires and switch. Wires have 0.001 Ω, a closed switch the same, an open switch 10⁹ Ω; the meter in the bridge has 50 Ω.
• Direct current, steady state. Everything settles instantly; there is no capacitance or inductance, so no delay or ringing when the switch moves.
• A battery that never runs down, with a constant E and r.
• Electron speeds scaled for display. Real drift speeds are fractions of a millimeter per second.

Edge cases

• Open switch in series: the current is zero everywhere and the full battery voltage appears across the open switch, which is why an open switch in a mains circuit can still give a shock.
• Zero internal resistance with a very small load: the current grows without limit in the ideal model. Real batteries are limited by r, and short-circuiting one can heat it dangerously.
• Equal resistors in parallel: n equal resistors R in parallel give R ÷ n, and the current splits equally.
• Balanced bridge: no current through the meter, so the bridge behaves like two independent dividers and the meter's own resistance does not matter.

Where the model stops being right

• Real bulbs are not ohmic. A tungsten filament's resistance rises about tenfold from cold to white hot, so it draws a large surge when switched on and its brightness is not proportional to V². LEDs and diodes are even less ohmic: almost no current below a threshold voltage.
• Alternating current. Capacitors and inductors make current and voltage get out of step, and impedance replaces resistance; the RC and RL Circuit Simulator shows their charging curves.
• Fast signals. At high frequency the wires themselves have inductance and capacitance and behave as transmission lines; signals take time to travel, and Kirchhoff's laws in this simple form no longer hold.
• Heating and limits. Resistors have power ratings and warm up; wires have resistance that matters for long runs, as the voltage drop in a cable shows; batteries' E and r change with charge and temperature.

Calculators

Work out V, I, R and P with the Ohm's Law Calculator, combine resistors with the Series and Parallel Calculator, and design dividers with the Voltage Divider Calculator; the guides How to Use Ohm's Law and How a Voltage Divider Works go further.