4.3.1 - Series, parallel and multi-source circuits

Kirchhoff Laws as Conservation Rules

Kirchhoff's laws are not extra circuit equations to memorise separately from physics. They are conservation laws written in electrical language.

This is conservation of charge. Charge does not build up at an ordinary junction in a steady d.c. circuit, so every coulomb per second entering must leave.

This is conservation of energy. A source transfers energy to each coulomb of charge; components transfer energy from each coulomb to the surroundings. If you return to the same point in the circuit, the net change in energy per unit charge is zero.

For calculations, choose directions for currents before you start. If the final value of a current is negative, the current actually flows in the opposite direction to your assumption. The negative sign is useful information, not a failed calculation.

Worked example: three wires meet at a junction. A current of $0.82\,\text{A}$ enters the junction. Currents of $0.31\,\text{A}$ and $I$ leave the junction.

Using Kirchhoff's first law:

So $0.51\,\text{A}$ leaves through the third wire.

[DIAGRAM: asset_slug: 4_03_1_series_parallel_and_multi_source_circuits__diagram_01; description: Kirchhoff’s first law conserves charge at a junction and the second conserves energy around a loop.; caption: Kirchhoff’s laws express conservation of charge and energy.]

Series and Parallel Resistor Rules

Components are in series when they are connected in one path, so the same current passes through each component. Components are in parallel when they are connected between the same two nodes, so the same p.d. is across each branch.

In a series chain, each resistor transfers some energy from each coulomb of charge. The p.d.s add, so the resistances add.

Here $R_\text{T}$ is the equivalent resistance in ohms, $\Omega$. This relationship is in the OCR data booklet, but you must recognise when the components really are in series.

In a parallel network, each extra branch gives charge another conducting route. The total current from the supply increases, so the equivalent resistance decreases.

For two resistors in parallel, the equivalent resistance is always less than the smaller resistor. That is a useful check on your calculation.

Worked example: find the total resistance of $18\,\Omega$ and $9.0\,\Omega$ connected in parallel.

The answer is below $9.0\,\Omega$, so the size is physically sensible.

Misconception contrast: current is not used up by a resistor. In a series chain the same current is everywhere, but energy per coulomb is transferred in each component, so the p.d. changes from component to component.

The rules follow directly from the conservation laws. In series, $V=V_1+V_2=IR_1+IR_2=I(R_1+R_2)$, so $R_{\rm T}=R_1+R_2$. In parallel, $I=I_1+I_2=V/R_1+V/R_2$. Dividing by the common p.d. gives $1/R_{\rm T}=1/R_1+1/R_2$. This derivation also explains why an added branch reduces total resistance: at fixed p.d. it adds current.

Reducing Mixed Circuits

A mixed circuit contains some components in series and some in parallel. The safest method is to simplify one clear section at a time, replacing it with an equivalent resistance.

Worked example: a $9.0\,\text{V}$ supply is connected to a $12\,\Omega$ resistor in series with a parallel pair of $18\,\Omega$ and $9.0\,\Omega$ resistors. Find the total current and the current in each parallel branch.

First reduce the parallel section:

Now add the series resistor:

Use $V=IR$ for the whole circuit:

The $12\,\Omega$ resistor is in series with the supply, so it carries $0.50\,\text{A}$. Its p.d. is:

The remaining p.d. across the parallel section is:

Each parallel branch has $3.0\,\text{V}$ across it:

The branch currents add back to $0.500\,\text{A}$, which checks Kirchhoff's first law.

If a circuit is not reducible by obvious series and parallel groups, use Kirchhoff's laws directly. OCR questions at this level usually make the reducible section identifiable, but the reason the reduction works is always Kirchhoff's first and second laws.

[DIAGRAM: asset_slug: 4_03_1_series_parallel_and_multi_source_circuits__diagram_02; description: A 12 Ω resistor in series with an 18 Ω and 9 Ω parallel pair.; caption: A 12 Ω resistor in series with an 18 Ω and 9 Ω parallel pair.]