3.3 Series and parallel resistor circuits

Resistors in series and parallel are key to understanding circuit behavior. Series connections add resistances, while parallel connections divide current. These concepts are crucial for analyzing complex circuits and solving real-world electrical problems.

Mastering series and parallel connections helps you simplify circuits and calculate equivalent resistances. This knowledge is essential for designing efficient electrical systems and troubleshooting issues in various devices and applications.

Series and Parallel Connections

Connecting Resistors in Series

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• Series connection occurs when resistors are connected end-to-end, forming a single path for current to flow through
• In a series connection, the current remains the same through each resistor as there is only one path for it to follow
• The total voltage across a series connection is equal to the sum of the individual voltage drops across each resistor (Kirchhoff's voltage law)
• To calculate the equivalent resistance of resistors in series, add the individual resistances together:
  • $R_{eq} = R_1 + R_2 + R_3 + ...$
• Example: Three resistors with values of 10Ω, 20Ω, and 30Ω connected in series have an equivalent resistance of 60Ω (10Ω + 20Ω + 30Ω)

Connecting Resistors in Parallel

• Parallel connection occurs when resistors are connected side-by-side, forming multiple paths for current to flow through
• In a parallel connection, the voltage remains the same across each resistor as they share the same two nodes
• The total current in a parallel connection is equal to the sum of the individual currents through each resistor (Kirchhoff's current law)
• To calculate the equivalent resistance of resistors in parallel, use the reciprocal formula:
  • $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...$
• Example: Three resistors with values of 10Ω, 20Ω, and 30Ω connected in parallel have an equivalent resistance of approximately 5.45Ω ($\frac{1}{10} + \frac{1}{20} + \frac{1}{30} = \frac{1}{5.45}$)

Equivalent Resistance in Complex Circuits

• Complex circuits often contain a combination of series and parallel connections
• To simplify a complex circuit, identify series and parallel sections and calculate their equivalent resistances
• Replace each series or parallel section with its equivalent resistance until the circuit is reduced to a single equivalent resistance
• Example: A circuit with two 10Ω resistors in series, connected in parallel with a 20Ω resistor, has an equivalent resistance of 10Ω (($10Ω + 10Ω$) in parallel with $20Ω$)

Circuit Analysis Techniques

Current and Voltage Division

• Current division is used to determine the current through each branch of a parallel circuit
• The current through a specific branch is proportional to the conductance (reciprocal of resistance) of that branch:
  • $I_x = I_{total} \times \frac{G_x}{G_{total}}$, where $G = \frac{1}{R}$
• Voltage division is used to determine the voltage across each component in a series circuit
• The voltage across a specific component is proportional to its resistance:
  • $V_x = V_{total} \times \frac{R_x}{R_{total}}$
• Example: In a series circuit with a 10Ω and a 20Ω resistor, if the total voltage is 15V, the voltage across the 10Ω resistor is 5V ($15V \times \frac{10Ω}{30Ω}$)

Kirchhoff's Laws and Circuit Simplification

• Kirchhoff's current law (KCL) states that the sum of currents entering a node equals the sum of currents leaving the node
• Kirchhoff's voltage law (KVL) states that the sum of voltage drops around any closed loop in a circuit is zero
• Use KCL and KVL to set up equations and solve for unknown currents and voltages in a circuit
• Circuit simplification involves combining series and parallel sections to reduce the complexity of the circuit
• Identify and combine resistors in series and parallel, replacing them with their equivalent resistances
• Repeat the process until the circuit is simplified to a single equivalent resistance or a manageable form
• Example: In a circuit with a 10Ω resistor in series with two 20Ω resistors in parallel, simplify by first combining the parallel resistors (equivalent resistance of 10Ω), then adding the series resistor for a total equivalent resistance of 20Ω