Most circuits are built from resistors wired in series, in parallel, or in some mix of the two. Being able to collapse a network into one equivalent resistance is the first step in analysing almost any DC circuit.
Resistors in series
Series means end-to-end, forming a single path. The same current flows through every resistor, and the resistances simply add:
Rtotal = R1 + R2 + R3 + …
Three resistors of 1 kΩ, 2 kΩ and 3 kΩ in series total 6 kΩ. Across a 12 V supply that gives a loop current of I = 12 V / 6 kΩ = 2 mA. Each resistor drops V = I × R, so the drops are 2 V, 4 V and 6 V — and, as they must, they sum back to the 12 V supply. That is Kirchhoff's Voltage Law in action.
Resistors in parallel
Parallel means both ends of each resistor share the same two nodes, offering multiple paths. Every resistor sees the same voltage, and their conductances (1/R) add:
1 / Rtotal = 1/R1 + 1/R2 + 1/R3 + …
For the common case of exactly two resistors, this simplifies to the "product over sum" form:
Rtotal = (R1 × R2) / (R1 + R2)
Two 1 kΩ resistors in parallel come out to (1k × 1k)/(1k + 1k) = 500 Ω. The equivalent is always smaller than the smallest resistor in the group, because you are adding more paths for current to flow.
- N equal resistors of value R in parallel give R / N.
- Any resistor in parallel with a much smaller one is dominated by the smaller value.
- A short (0 Ω) in parallel with anything gives 0 Ω; an open (∞) in parallel changes nothing.
Reducing a network
For a mixed network, work from the inside out: find the innermost pure series or pure parallel groups, replace each with its single equivalent, and repeat. Eventually the whole thing collapses to one resistor across the source, and Ohm's Law gives you the total current.