Result

Combining resistors

(E6.2.1) E6.2 · Resistors in Series and Parallel →

R_{ser} = \sum R_{i}, \frac{1}{R _{par}} = \sum \frac{1}{R _{i}}

Statement

Collapse a group of resistors into one equivalent value.

Symbols and units

$R_{eq}$	single resistor that replaces the group	Ω
$R_{i}$	each resistor in the group	Ω

When it applies

Ohmic resistors in steady state.

Don't use when

Assumes ideal, purely ohmic resistors — a real resistor's value can drift with temperature as current flows, breaking the simple series/parallel combination over time.

Derivation

Series — charge cannot pile up, so the same I passes through both, and the drops add:

Δ V = I R_{1} + I R_{2} = I (R_{1} + R_{2})

Compare with ΔV = IR_eq:

R_{eq} = R_{1} + R_{2}

Parallel — both ends are common, so both see the same ΔV and the currents add:

I = \frac{Δ V}{R _{1}} + \frac{Δ V}{R _{2}}

Compare with I = ΔV/R_eq:

\frac{1}{R _{eq}} = \frac{1}{R _{1}} + \frac{1}{R _{2}}

Worked examples

6 Ω with 3 Ω

R_{ser} = 9 Ω, R_{par} = \frac{( 6 ) ( 3 )}{6 + 3} = 2 Ω

Voltage divider, R₁ = 2.0 kΩ, R₂ = 8.0 kΩ, ΔV = 10 V

Δ V_{1} = Δ V \frac{R _{1}}{R _{1} + R _{2}} = 10 \times \frac{2.0}{10} = 2.0 V

The standard sensor circuit: a thermistor in one leg, read at the midpoint.

Watch out for

A parallel combination is always smaller than the smallest resistor — adding a path can only make it easier for current to flow. Two equal resistors in parallel give R/2.

In practice

An <b>ammeter</b> goes in series and needs very low resistance; a <b>voltmeter</b> goes in parallel and needs very high resistance. Both are built to disturb the circuit as little as possible.

Connects to

Parallel / series Junction and loop rules Terminal voltage