Series & Parallel Combinations

PHYCells & Circuits

This subtopic is the algebra of combining resistors: series resistances add, while parallel resistances combine through reciprocals, and most circuits reduce by alternating these two rules. The core idea is that series elements share the same current and parallel elements share the same voltage.

ISC problems mix a parallel block in series with another resistor across a battery and ask for the main current and the branch currents.

Series combination

R_s = R_1 + R_2 + R_3 + \cdots

R_s

equivalent series resistance (

\Omega

). Same current flows through each; total PD is the sum of individual drops.

R_s

is larger than any single resistor.

Parallel combination

\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} + \cdots

R_p

equivalent parallel resistance (

\Omega

). Same PD across each; total current is the sum of branch currents.

R_p

is smaller than the smallest resistor.

Derivation

Three resistors in parallel share the same potential difference $V$ across each (both ends tied to the same pair of nodes), while the source current $I$ splits into the branch currents:
$I = I_1 + I_2 + I_3$
Ohm's law gives each branch current in terms of the common $V$ :
$I_1=\dfrac{V}{R_1}, \quad I_2=\dfrac{V}{R_2}, \quad I_3=\dfrac{V}{R_3}$
Substitute and factor out $V$ ; the equivalent resistor $R_p$ must carry the same total $I$ for the same $V$ :
$I=\dfrac{V}{R_p} = V\left(\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}\right) \;\Rightarrow\; \dfrac{1}{R_p}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}$

Two resistors in parallel (product over sum)

R_p = \dfrac{R_1 R_2}{R_1 + R_2}

Handy shortcut for exactly two resistors. For

6\,\Omega

and

3\,\Omega

this gives

2\,\Omega

Current divider for branch currents

I_1 = I \, \dfrac{R_2}{R_1 + R_2}, \qquad I_2 = I \, \dfrac{R_1}{R_1 + R_2}

I

current entering a two-resistor parallel node,

I_1, I_2

branch currents. Each branch carries a fraction inversely proportional to its own resistance.

Series resistors carry identical current; parallel resistors share identical voltage — identify which is fixed before computing.
Equivalent series resistance exceeds the largest member; equivalent parallel resistance is below the smallest member — use this to sanity-check.
The product-over-sum shortcut applies only to two resistors at a time; for three or more use full reciprocals.
Find the main current from the battery using the total equivalent resistance, then redistribute into branches with the current divider.
Across a parallel block the voltage is common; branch currents follow $I_i = V/R_i$ .
Include internal resistance $r$ in series with the external network when computing the main current if the battery is non-ideal.
In a current divider, the smaller-resistance branch carries the larger current (current prefers the easier path).

Where the marks go

Adding parallel resistances directly instead of adding their reciprocals (or using product-over-sum).
Forgetting to take the reciprocal of $1/R_p$ at the end, reporting $1/R_p$ as the resistance.
Assuming equal current in parallel branches when their resistances differ — current splits inversely with resistance.
Applying product-over-sum to three resistors at once instead of pairing or using full reciprocals.

How the board asks it

Numericaltotal equivalent resistance then current divider
A battery of emf $6\,V$ and internal resistance $0.5\,\Omega$ is connected to a parallel combination of $3\,\Omega$ and $6\,\Omega$ , which is in series with a $1.5\,\Omega$ resistor. Calculate the main current drawn from the battery and the current in each parallel branch.
Numericalparallel combination of three resistors
Three resistors of $4\,\Omega$ , $6\,\Omega$ and $12\,\Omega$ are connected in parallel across a $12\,V$ source. Calculate the equivalent resistance of the combination and the total current supplied by the source.
Derive / proveparallel combination through reciprocals at common voltage
Three resistors $R_1$ , $R_2$ and $R_3$ are connected in parallel across a cell of potential difference $V$ . Obtain an expression for the equivalent resistance of the combination, stating clearly the quantity that is common to all three resistors.
Give reasonsequivalent resistance bounds as a sanity check
Two resistors are connected in parallel. Give reasons why the equivalent resistance of the combination is always less than the smaller of the two individual resistances.
Applicationcurrent divider; current prefers the easier path
A current of $9\,A$ enters a junction and divides between two parallel resistors of $2\,\Omega$ and $4\,\Omega$ . State which branch carries the larger current, giving a reason, and calculate the current in each branch.

Practise this topic

PreviousEMF, Terminal Voltage & Internal Resistance NextGrouping of Cells

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.