Series and Parallel Resistor Networks (Revision) Revision Notes for Grade 12 NSC Matric Physical Sciences

Series and Parallel Resistor Networks (Revision)

Fundamental electrical quantities

When analysing electric circuits, we work with three fundamental quantities that are closely related through Ohm's Law.

Electrical current (I) is the rate of flow of electric charge through a circuit. Current is measured in amperes (A) and represents how much charge passes through a point per unit time.

Potential difference or voltage (V) relates to the energy gained or lost per unit charge as it moves between two points in a circuit. When charge moves through a battery, it gains energy. When it moves through a resistor, it loses energy. Voltage is measured in volts (V).

Resistance (R) is an internal property of circuit elements that opposes the flow of electric charge. Work must be done to move charge through a resistor. Resistance is measured in ohms (Ω).

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These three quantities are interconnected — understanding their relationship is crucial for circuit analysis. Current flows due to potential difference, but the amount of current depends on the resistance present in the circuit.

Ohm's law

For resistors at constant temperature, the ratio $V/I$ remains constant. This constant ratio is called the resistance of the conductor.

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Ohm's Law states: The electric current through a metal conductor at constant temperature is proportional to the voltage across the conductor.

Formula: $I = \dfrac{V}{R}$

This relationship means that at constant temperature, the resistance of a conductor remains constant, regardless of the voltage applied or current passing through it.

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Temperature changes can cause deviations from Ohm's Law. If a graph of current versus voltage becomes non-linear, this indicates that temperature was not kept constant during the investigation.

Series resistor networks

Definition: Resistors are connected in series when they are consecutive elements in the sequence of the circuit with no branches between them.

Key properties of series circuits

Equivalent resistance:
$R_s = R_1 + R_2 + R_3 + \dots + R_n$

Voltage distribution:
$V_\text{total} = V_1 + V_2 + V_3 + \dots + V_n$

Current:
$I_\text{total} = I_1 = I_2 = I_3 = \dots = I_n$

Why these properties make sense

The series properties align with fundamental conservation laws:

Conservation of charge: Since charges cannot be created or destroyed, and there are no branches for charge to accumulate, the current must remain the same throughout the series network.
Conservation of energy: The total work done to move charge through the entire network equals the sum of work done through each individual resistor. Since voltage represents energy per unit charge, the voltages must add up.

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Worked Example: Series Circuit Analysis

A series circuit contains three resistors: $R_1 = 2.00 \,\Omega$ , $R_2 = 6.00 \,\Omega$ , and $R_3$ (unknown value). The circuit is connected to a $36.00 \,\text{V}$ battery and carries a current of $2.00 \,\text{A}$ . Calculate the potential differences across each resistor and determine the resistance of $R_3$ .

Solution:

Step 1:
$V_1 = I \times R_1 = 2.00 \times 2.00 = 4.00 \,\text{V}$

Step 2:
$V_2 = I \times R_2 = 2.00 \times 6.00 = 12.00 \,\text{V}$

Step 3:
$V_\text{total} = V_1 + V_2 + V_3$
$36.00 = 4.00 + 12.00 + V_3$
$V_3 = 20.00 \,\text{V}$

Step 4:
$R_3 = \dfrac{V_3}{I} = \dfrac{20.00}{2.00} = 10.00 \,\Omega$

Final answers:

$V_1 = 4.00 \,\text{V}$
$V_2 = 12.00 \,\text{V}$
$V_3 = 20.00 \,\text{V}$
$R_3 = 10.00 \,\Omega$

Parallel resistor networks

Definition: A parallel configuration occurs when current splits into multiple branches, with each branch containing one or more components.

Key properties of parallel circuits

Equivalent resistance:
$\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} + \dots + \dfrac{1}{R_n}$

Voltage:
$V_\text{total} = V_1 = V_2 = V_3 = \dots = V_n$

Current distribution:
$I_\text{total} = I_1 + I_2 + I_3 + \dots + I_n$

Why these properties make sense

The parallel properties also align with conservation laws:

Conservation of charge: The total number of charges entering the parallel network must equal the total number leaving. Since charges cannot accumulate at junction points, the currents must add up.
Conservation of energy: Each parallel branch connects to the same two points in the circuit, so the energy per unit charge (voltage) must be the same for each branch.

lightbulbExample

Worked Example: Parallel Circuit Analysis

Two resistors are connected in parallel: $R_1 = 2.00 \,\Omega$ and $R_2 = 4.00 \,\Omega$ . The circuit is connected to a $12.00 \,\text{V}$ battery. Calculate the total current flowing in the circuit.

Solution:

Step 1: Equivalent resistance
$\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} = \dfrac{1}{2.00} + \dfrac{1}{4.00} = \dfrac{3}{4}$
$R = \dfrac{4}{3} = 1.33 \,\Omega$

Step 2: Total current
$I = \dfrac{V}{R} = \dfrac{12.00}{1.33} = 9.00 \,\text{A}$

Final answer: The total current flowing in the circuit is $9.00 \,\text{A}$ .

Alternative approach for parallel circuits

Current through $R_1$ : $I_1 = \dfrac{V}{R_1} = \dfrac{12.00}{2.00} = 6.00 \,\text{A}$
Current through $R_2$ : $I_2 = \dfrac{V}{R_2} = \dfrac{12.00}{4.00} = 3.00 \,\text{A}$
Total current: $I_\text{total} = I_1 + I_2 = 9.00 \,\text{A}$

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Both methods give the same result, but the equivalent resistance method is often quicker for finding total current.

Problem-solving strategy

When approaching circuit problems, follow these systematic steps:

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Circuit Analysis Strategy:

Identify the circuit type — series, parallel, or combination
List known values — voltages, currents, and resistances
Choose appropriate formulas — series or parallel relationships
Apply Ohm's Law systematically to find unknown quantities
Check your answer — ensure it makes physical sense

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Important exam tips:

Always draw and label circuit diagrams clearly
Show all working steps systematically
Remember that series adds resistances directly, while parallel uses reciprocals
Check that your calculated values are reasonable
Units are crucial — ensure consistent use of volts, amperes, and ohms

Mixed series-parallel networks

Real circuits often combine series and parallel elements. Treat each section independently by applying the appropriate rules, then combine the results systematically.

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For complex networks, work step by step:

Identify which components are in series or parallel
Calculate equivalent resistance for each section
Combine sections until you have the total circuit resistance
Work backwards to find currents and voltages in individual components

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Key Points to Remember:

Ohm's Law: $I = \dfrac{V}{R}$ applies to individual components and complete circuits
Series circuits: Same current throughout, voltages add up, resistances add up
Parallel circuits: Same voltage across branches, currents add up, reciprocal rule for resistance
Conservation laws: Charge and energy conservation explain why circuit rules work
Problem-solving: Always identify circuit type first, then apply appropriate formulas systematically

Series and Parallel Resistor Networks (Revision) (Grade 12 NSC Matric Physical Sciences): Revision Notes