Series and Parallel Circuits

Spec mapping: OCR H556 Module 4.3 — Electrical circuits (resistors in series $R_{\text{series}} = \sum R_i$ and in parallel $1/R_{\text{parallel}} = \sum 1/R_i$ ; current and voltage divisions; ammeter vs voltmeter placement; mixed networks reducible by sequential series-parallel collapse). Refer to the official OCR H556 specification document for exact wording.

Every realistic electronic circuit is a combination of series and parallel resistor segments. To analyse them cleanly you need three things: (i) compact formulas for combining resistors, (ii) intuition for how current and voltage distribute across a configuration, and (iii) a systematic "reduce and re-expand" workflow for mixed networks. This lesson covers all three from Kirchhoff-1 and Ohm's law, gives worked examples of series, parallel and mixed circuits, and connects forward to potential dividers (lesson 12) and Kirchhoff-2 (lesson 11).

By the end you should be able to (a) compute total resistance and main current for any reducible network without hesitation; (b) identify when a network is not reducible (lesson 11 territory); and (c) get current/pd distributions correct first time without algebraic slips.

Components in Series — The Single-Path Rule

Two or more components are in series if they form a single path with no junctions between them — current must flow through each in turn. The defining property: the same current flows through every series component.

flowchart LR
  PLUS["+"] --> R1["R_1"]
  R1 --> R2["R_2"]
  R2 --> R3["R_3"]
  R3 --> MINUS["-"]

Current in Series

By Kirchhoff-1 at any junction within a series chain, the current is the same everywhere:

$I = I_1 = I_2 = I_3 = \ldots$

Once you have the main current $I$ , every component carries that same $I$ and the pd across each follows from $V = IR$ .

Voltage in Series

The pds across the components add up to the source pd (Kirchhoff-2, lesson 11):

$V_{\text{total}} = V_1 + V_2 + V_3 + \ldots$

Resistance in Series

Apply $V = IR$ to each component with the same $I$ : $V_1 = IR_1$ , etc. Sum: $V_{\text{total}} = I(R_1 + R_2 + R_3)$ . But also $V_{\text{total}} = I R_{\text{total}}$ , so:

$\boxed{\; R_{\text{series}} = R_1 + R_2 + R_3 + \ldots \;}$

Series resistances add directly. No reciprocals. A series chain of identical resistors $R$ has total resistance $nR$ .

Worked Example 1 — Series Network

$R_1 = 100$ $\Omega$ , $R_2 = 220$ $\Omega$ , $R_3 = 470$ $\Omega$ in series across $9.0$ V (negligible internal resistance).

$R_{\text{total}} = 100 + 220 + 470 = \mathbf{790}$ $\Omega$ .
$I = 9.0/790 = \mathbf{0.0114}$ A $\approx 11.4$ mA.
$V_1 = IR_1 = \mathbf{1.14}$ V; $V_2 = IR_2 = \mathbf{2.51}$ V; $V_3 = IR_3 = \mathbf{5.36}$ V.

Check: $V_1 + V_2 + V_3 = 9.01 \approx 9.0$ V ✓. The largest resistor drops the largest pd; the pd ratio matches the resistance ratio. This is the heart of the potential divider (lesson 12).

Components in Parallel — The Same-Voltage Rule

Two or more components are in parallel if they share the same pair of nodes. Defining property: the same pd exists across every parallel component.

flowchart LR
  PLUS["+"] --> J1((Top node))
  J1 --> R1["R_1"]
  J1 --> R2["R_2"]
  J1 --> R3["R_3"]
  R1 --> J2((Bottom node))
  R2 --> J2
  R3 --> J2
  J2 --> MINUS["-"]

Voltage in Parallel

$V = V_1 = V_2 = V_3 = \ldots$ — same across every branch.

Current in Parallel

Kirchhoff-1 at the top node: $I_{\text{total}} = I_1 + I_2 + I_3 + \ldots$

Resistance in Parallel

Apply $I = V/R$ to each branch with the same $V$ : $I_{\text{total}} = V(1/R_1 + 1/R_2 + 1/R_3)$ . But $I_{\text{total}} = V/R_{\text{parallel}}$ , so:

$\boxed{\; \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots \;}$

Reciprocals add. Parallel total is always less than the smallest individual resistor — adding a parallel branch always reduces total resistance.

The Two-Resistor Shortcut

$R_{\text{parallel}} = \frac{R_1 R_2}{R_1 + R_2} \quad \text{(two resistors only)}$

For three or more, use the reciprocal formula.

Special Cases

$n$ equal resistors $R$ : $R_{\text{parallel}} = R/n$ .
$R_1 \gg R_2$ : $R_{\text{parallel}} \approx R_2$ (the larger is essentially open-circuit).

Worked Example 2 — Parallel Network

Same three resistors $100$ $\Omega$ , $220$ $\Omega$ , $470$ $\Omega$ in parallel across $9.0$ V.

$\frac{1}{R_{\text{parallel}}} = \frac{1}{100} + \frac{1}{220} + \frac{1}{470} = 0.01667 \text{ }\Omega^{-1} \Rightarrow R_{\text{parallel}} = \mathbf{60.0 \text{ }\Omega}$

$I_{\text{total}} = 9.0/60.0 = \mathbf{0.150}$ A.
Branch currents: $I_1 = 0.090$ A, $I_2 = 0.041$ A, $I_3 = 0.019$ A. Sum $= 0.150$ A ✓.

Three observations: (i) $R_{\text{parallel}} = 60$ $\Omega < 100$ $\Omega$ . (ii) Smallest $R$ carries largest $I$ . (iii) Parallel draws $13\times$ more current than the same resistors in series.

Current and Voltage Distribution

Configuration	Same quantity	Distributed	Rule
Series	$I$	$V$	$V_i \propto R_i$
Parallel	$V$	$I$	$I_i \propto 1/R_i$

"Series: same I, voltage splits. Parallel: same V, current splits."

Worked Example 3 — Mixed Network (Reduce-Then-Expand)

A $12$ V battery feeds $R_1 = 4.0$ $\Omega$ in series with the parallel pair $R_2 = 6.0$ $\Omega$ , $R_3 = 3.0$ $\Omega$ .

$R_{23} = 6 \times 3/(6+3) = 2.0$ $\Omega$ .
$R_{\text{total}} = 4 + 2 = \mathbf{6.0}$ $\Omega$ .
$I_{\text{main}} = 12/6.0 = \mathbf{2.0}$ A.
$V_{23} = I_{\text{main}} R_{23} = \mathbf{4.0}$ V.
$I_2 = 4.0/6.0 = \mathbf{0.67}$ A; $I_3 = 4.0/3.0 = \mathbf{1.33}$ A. Sum $= 2.0$ A $= I_{\text{main}}$ ✓.

Worked Example 4 — Four-Resistor Network

$R_1 = 2$ $\Omega$ + $R_2 = 3$ $\Omega$ in series; that pair in parallel with $R_3 = 6$ $\Omega$ ; the whole in series with $R_4 = 4$ $\Omega$ .

$R_{12} = 5$ $\Omega$ .
$1/R_{123} = 1/5 + 1/6 = 11/30 \Rightarrow R_{123} = 30/11 \approx 2.73$ $\Omega$ .
$R_{\text{total}} = 2.73 + 4 = \mathbf{6.73}$ $\Omega$ .

Worked Example 5 — Diluted Current with a Shunt

A galvanometer of internal resistance $r_g = 50$ $\Omega$ deflects full-scale at $I_{\text{fs}} = 1.0$ mA. To convert it into an ammeter reading $0$ – $1.0$ A full-scale, a shunt resistor $R_s$ is placed in parallel with the galvanometer to bypass most of the current. Find the shunt resistance.

At full-scale: total current $I_{\text{total}} = 1.0$ A; galvanometer current $I_g = 1.0$ mA. Shunt current: $I_s = I_{\text{total}} - I_g = 0.999$ A.

The pd across the parallel combination is $V = I_g r_g = 0.001 \times 50 = 0.050$ V. This same pd appears across the shunt:

$R_s = V/I_s = 0.050/0.999 = \mathbf{0.0500 \text{ }\Omega}$

The shunt carries $99.9\%$ of the current; the galvanometer carries $0.1\%$ . To extend the range further, decrease $R_s$ . This is exactly how a multimeter's "current range" rotary switch works — different shunt resistors are switched in for different full-scale ranges.

Worked Example 6 — Multiplier Resistor for a Voltmeter

The same galvanometer (full-scale $1.0$ mA, $r_g = 50$ $\Omega$ ) is converted into a voltmeter reading $0$ – $10$ V by placing a multiplier resistor $R_m$ in series with the galvanometer.

At full-scale: $I = 1.0$ mA through the series chain ( $r_g + R_m$ ); applied pd $V = 10$ V. So:

$R_m = V/I - r_g = 10/0.001 - 50 = 10\,000 - 50 = \mathbf{9\,950 \text{ }\Omega}$

A digital multimeter's voltmeter ranges work by switching different precision multiplier resistors into series. The dual roles — shunt for ammeter, multiplier for voltmeter — reflect the fundamental insight: ammeters need low effective resistance (parallel shunt drops the effective $R$ to a tiny value), voltmeters need high effective resistance (series multiplier raises the effective $R$ to a huge value).

Worked Example 7 — Star-Delta Sanity Check

A delta network of three $30$ $\Omega$ resistors (one between each pair of three nodes A, B, C) is sometimes more convenient to analyse if converted to an equivalent Y network (also called a "star"). The star has three resistors meeting at a central fourth node, each connecting to one of A, B, C.

For three identical delta resistors $R_\Delta = 30$ $\Omega$ , the equivalent star resistors are each:

$R_Y = \frac{R_\Delta}{3} = \mathbf{10 \text{ }\Omega}$

The transformation preserves the impedance seen between any pair of external nodes. Some otherwise-irreducible networks (delta-Y bridges, certain three-phase circuits) become reducible after a star-delta transformation. Beyond A-Level in detail, but the existence of this trick is useful general knowledge.

Power Dissipation — Cross-Check

For any network: $\sum P_i = V_{\text{supply}} I_{\text{total}}$ .

Example 3 powers: $P_1 = I^2 R_1 = 16$ W; $P_2 = I_2^2 R_2 = 2.67$ W; $P_3 = I_3^2 R_3 = 5.33$ W. Sum $= 24$ W $= VI$ ✓.

If powers do not sum, you have an arithmetic error somewhere in the analysis. Performing this cross-check on every multi-step problem catches algebraic slips that would otherwise be missed — and OCR mark schemes generally accept the conservation check as evidence of physical reasoning even if numerical work is rounded.

Series vs Parallel — Where Does Most Power Go?

A useful intuition: in a series chain the current is fixed, so $P = I^2 R$ grows linearly with $R$ . The largest resistor dissipates the most power. In a parallel combination the voltage is fixed, so $P = V^2/R$ falls inversely with $R$ . The smallest resistor dissipates the most power.

This explains many practical observations: in a series chain of bulbs (Christmas-tree lights, traditional design) the dimmer bulbs (higher $R$ , lower-wattage) actually shine brighter — counter-intuitive but follows directly from $P = I^2R$ . In a parallel arrangement (modern domestic wiring), each bulb sees the full mains voltage and dissipates its rated power independently of the others.

Ammeter vs Voltmeter

Ammeter: in series with the component. Ideal $R = 0$ , so it does not perturb the current.
Voltmeter: in parallel with the component. Ideal $R = \infty$ , so it draws no current.

flowchart LR
  PS["Battery"] --> SW["Switch"]
  SW --> AM["Ammeter (series, R → 0)"]
  AM --> COMP["Component"]
  COMP --> PS
  COMP -.-> VM["Voltmeter (parallel, R → ∞)"]

Real meters: ammeter $0.01$ – $0.1$ $\Omega$ ; digital voltmeter $10^7$ – $10^9$ $\Omega$ . Mixing the two roles up is a wiring disaster.

Loading Errors

When a non-ideal meter is connected, it perturbs the circuit it is measuring — a "loading error". For a high-resistance circuit, the voltmeter's finite input resistance draws a small but non-zero current, reducing the measured pd below the true open-circuit value (lesson 12 covers this for potential dividers). For a low-resistance circuit, the ammeter's finite resistance adds to the series impedance, reducing the measured current below the true value (lesson 9 mentions this in the context of measuring internal resistance).

Rule of thumb: the meter perturbs the measurement by less than $\sim 1\%$ if its input impedance (voltmeter) or output impedance (ammeter) differs from the local circuit impedance by at least a factor of $100$ . Modern digital multimeters easily satisfy this for everyday A-Level work but precision laboratory measurements (microvolt-level signals, nanoampere currents) require careful meter selection.

When Components Are Neither Series Nor Parallel

A Wheatstone bridge (four resistors in a diamond with a galvanometer across the diagonal) cannot be reduced by series-parallel rules. Use Kirchhoff-2 (lesson 11). Tell-tale sign: cannot label every resistor as "in series with X" or "in parallel with Y" without ambiguity.

Five-Resistor Diamond — A Worked Non-Reducible Example

Consider a square network with one resistor on each side (four sides — $R_1$ , $R_2$ , $R_3$ , $R_4$ — plus a fifth resistor $R_5$ connecting two diagonally opposite corners). Trying to label by series/parallel rules: $R_5$ is in neither series nor parallel with any single side; it sits across a pair of nodes whose voltage relationship to the source nodes depends on the combined network. No purely local series-parallel reduction works.

Series and Parallel Circuits

Series and Parallel Circuits

Components in Series — The Single-Path Rule

Current in Series

Voltage in Series

Resistance in Series

Worked Example 1 — Series Network

Components in Parallel — The Same-Voltage Rule

Voltage in Parallel

Current in Parallel

Resistance in Parallel

The Two-Resistor Shortcut

Special Cases

Worked Example 2 — Parallel Network

Current and Voltage Distribution

Worked Example 3 — Mixed Network (Reduce-Then-Expand)

Worked Example 4 — Four-Resistor Network

Worked Example 5 — Diluted Current with a Shunt

Worked Example 6 — Multiplier Resistor for a Voltmeter

Worked Example 7 — Star-Delta Sanity Check

Power Dissipation — Cross-Check

Series vs Parallel — Where Does Most Power Go?

Ammeter vs Voltmeter

Loading Errors

When Components Are Neither Series Nor Parallel

Five-Resistor Diamond — A Worked Non-Reducible Example

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