Series & Parallel Circuits

Electric circuits can be arranged in different ways. Two common arrangements are series circuits and parallel circuits. Understanding how current, voltage, and resistance behave in these circuits is essential for analysing electrical systems.

Series Circuits

In a series circuit, all components are connected in a single loop, so the current flows through each component one after the other.

• The current is the same through all components because there is only one path for the charge to flow.
• The total resistance of the circuit is the sum of the individual resistances:

$$R_{\text{total}} = R_1 + R_2 + R_3 + \dots$$

This means adding more resistors in series increases the total resistance.

• The potential difference (voltage) from the power supply divides across the components. The sum of the voltages across each component equals the total voltage supplied:

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

Components with higher resistance have a larger share of the voltage.

For instance, if a 12 V battery powers two resistors in series, one 2 $\Omega$ and one 4 $\Omega$, the total resistance is $2 + 4 = 6\,\Omega$. The current is:

$$I = \frac{V}{R} = \frac{12}{6} = 2\, \text{A}$$

The voltage across each resistor is:

$$V_1 = IR_1 = 2 \times 2 = 4\, \text{V}$$

$$V_2 = IR_2 = 2 \times 4 = 8\, \text{V}$$

Notice the voltages add up to 12 V.

Parallel Circuits

In a parallel circuit, components are connected across separate branches, each branch connected directly to the power supply.

• The potential difference (voltage) across each branch is the same as the voltage of the power supply.
• The total current from the power supply divides among the branches, with more current flowing through branches with lower resistance.
• The total resistance of the circuit is less than the resistance of the smallest branch because the current has multiple paths:

$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$$

Adding more branches in parallel decreases the overall resistance.

For example, if two resistors of 6 $\Omega$ and 3 $\Omega$ are connected in parallel across a 12 V battery, the total resistance is:

$$\frac{1}{R_{\text{total}}} = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$$

$$R_{\text{total}} = 2\, \Omega$$

The total current from the battery is:

$$I_{\text{total}} = \frac{V}{R_{\text{total}}} = \frac{12}{2} = 6\, \text{A}$$

The current splits between the two branches:

$$I_1 = \frac{V}{R_1} = \frac{12}{6} = 2\, \text{A}, \quad I_2 = \frac{12}{3} = 4\, \text{A}$$

If one branch fails (e.g., a component breaks), the current no longer flows through that branch, but other branches continue to operate normally. The total current from the supply will decrease accordingly, and current redistributes among the remaining branches.

Comparing Series & Parallel Circuits

• Current: In series, current is the same everywhere; in parallel, current splits at branches.
• Voltage: In series, voltage divides among components; in parallel, voltage is the same across each branch.
• Resistance: Total resistance increases with more components in series; total resistance decreases with more branches in parallel.
• Component failure: In series, if one component breaks, the whole circuit stops working; in parallel, other branches still work if one fails.
• Typical uses: Series circuits are used in simple devices like old Christmas lights; parallel circuits are used in household wiring so appliances work independently.

Resistors in Series & Parallel

Calculating total resistance in series: Add resistances directly:

$$R_{\text{total}} = R_1 + R_2 + \dots$$

This increases total resistance, reducing current for a given voltage.

Calculating total resistance in parallel: Use the reciprocal formula:

$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$$

This decreases total resistance, increasing current for a given voltage.

For example, two resistors of 4 $\Omega$ and 12 $\Omega$ in parallel:

$$\frac{1}{R_{\text{total}}} = \frac{1}{4} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$$

$$R_{\text{total}} = 3\, \Omega$$

This lower resistance means more current can flow compared to either resistor alone.

AC & DC

Direct Current (DC) is a flow of electric charge in one direction only. Batteries and cells provide DC.

Alternating Current (AC) is a flow of electric charge that reverses direction periodically. The mains electricity supply in UK homes is AC at 230 V and 50 Hz.

AC is used for power distribution because it is easier to transform voltages for efficient transmission.

In circuits, DC provides a constant voltage and current direction, while AC voltage and current constantly change direction and magnitude.