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Equivalent Circuits - Example 5- Composite Resistor

  • In some circuits, resistors are connected in different combinations.
  • When resistors are connected in series or parallel, they can be replaced by an equivalent resistor.
  • The equivalent resistor has the same resistance as the combination of resistors.
  • Let’s see an example of a composite resistor.

Example 5

  • Consider three resistors connected in parallel: R1, R2, and R3.
  • The resistor values are: R1 = 4 Ω, R2 = 6 Ω, and R3 = 8 Ω.
  • We need to find the equivalent resistance of this combination.

Solution

  • When resistors are connected in parallel, their equivalent resistance (Rp) is given by: Rp = (1/R1) + (1/R2) + (1/R3)
  • Substituting the given values, we have: Rp = (1/4) + (1/6) + (1/8)

Solution (contd.)

  • Evaluating the equation: Rp = (1/4) + (1/6) + (1/8) = 0.25 + 0.1667 + 0.125
  • Adding the values gives: Rp ≈ 0.5417

Solution (contd.)

  • Therefore, the equivalent resistance (Rp) of the combination is approximately 0.5417 Ω.
  • This means that this combination can be replaced by a single resistor of 0.5417 Ω.

Equivalent Circuit

  • The equivalent circuit for the given combination is shown below: Equivalent Circuit

Equivalent Circuits - Composite Resistor

  • The equivalent resistor represents the combination of resistors in a circuit.
  • It simplifies the circuit analysis by reducing complex configurations.
  • Equivalent resistors are determined based on the type of connection (series or parallel) and their respective formulas.
  • Composite resistors can be replaced by their equivalent resistance for easier calculations.

Equivalent Circuits - Example 6- Series Resistor

  • Similar to parallel combinations, resistors can also be connected in series.
  • The equivalent resistance and circuit simplification for series resistors will now be discussed.

Example 6

  • Consider three resistors connected in series: R4, R5, and R6.
  • The resistor values are: R4 = 2 Ω, R5 = 3 Ω, and R6 = 5 Ω.
  • We need to find the equivalent resistance of this series combination.

Solution

  • When resistors are connected in series, their equivalent resistance (Rs) is given by: Rs = R4 + R5 + R6
  • Substituting the given values, we have: Rs = 2 + 3 + 5

Equivalent Circuits - Example 5- Composite Resistor

  • In some circuits, resistors are connected in different combinations.
  • When resistors are connected in series or parallel, they can be replaced by an equivalent resistor.
  • The equivalent resistor has the same resistance as the combination of resistors.
  • Let’s see an example of a composite resistor.

Example 5

  • Consider three resistors connected in parallel: R1, R2, and R3.
  • The resistor values are: R1 = 4 Ω, R2 = 6 Ω, and R3 = 8 Ω.
  • We need to find the equivalent resistance of this combination.

Solution

  • When resistors are connected in parallel, their equivalent resistance (Rp) is given by:
    • Rp = (1/R1) + (1/R2) + (1/R3)
  • Substituting the given values, we have:
    • Rp = (1/4) + (1/6) + (1/8)

Solution (contd.)

  • Evaluating the equation:
    • Rp = (1/4) + (1/6) + (1/8) = 0.25 + 0.1667 + 0.125
  • Adding the values gives:
    • Rp ≈ 0.5417

Solution (contd.)

  • Therefore, the equivalent resistance (Rp) of the combination is approximately 0.5417 Ω.
  • This means that this combination can be replaced by a single resistor of 0.5417 Ω.

Equivalent Circuit

  • The equivalent circuit for the given combination is shown below: [Image of circuit diagram]

Equivalent Circuits - Composite Resistor

  • The equivalent resistor represents the combination of resistors in a circuit.
  • It simplifies the circuit analysis by reducing complex configurations.
  • Equivalent resistors are determined based on the type of connection (series or parallel) and their respective formulas.
  • Composite resistors can be replaced by their equivalent resistance for easier calculations.

Equivalent Circuits - Example 6- Series Resistor

  • Similar to parallel combinations, resistors can also be connected in series.
  • The equivalent resistance and circuit simplification for series resistors will now be discussed.

Example 6

  • Consider three resistors connected in series: R4, R5, and R6.
  • The resistor values are: R4 = 2 Ω, R5 = 3 Ω, and R6 = 5 Ω.
  • We need to find the equivalent resistance of this series combination.

Solution

  • When resistors are connected in series, their equivalent resistance (Rs) is given by:
    • Rs = R4 + R5 + R6
  • Substituting the given values, we have:
    • Rs = 2 + 3 + 5

Solution (contd.)

  • Evaluating the equation gives:
    • Rs = 2 + 3 + 5 = 10 Ω

Solution (contd.)

  • Therefore, the equivalent resistance (Rs) of the combination is 10 Ω.
  • This means that this combination can be replaced by a single resistor of 10 Ω.

Equivalent Circuit

  • The equivalent circuit for the given combination is shown below: Equivalent Circuit

Equivalent Circuits - Series Resistor

  • When resistors are connected in series, their equivalent resistance is the sum of individual resistances.
  • Series resistors increase the total resistance of the circuit.
  • The equivalent resistance of series resistors can be found by adding up their values.

Key Concepts - Equivalent Circuits

  • Equivalent circuits simplify circuit analysis by replacing complex combinations of resistors with a single equivalent resistor.
  • For parallel resistors:
    • Rp = (1/R1) + (1/R2) + (1/R3) + …
    • The equivalent resistance is the reciprocal of the sum of the reciprocals of individual resistances.
  • For series resistors:
    • Rs = R1 + R2 + R3 + …
    • The equivalent resistance is the sum of individual resistances.

Summary

  • Equivalent circuits are used to simplify complex resistor combinations.
  • Parallel resistors have an equivalent resistance given by: Rp = (1/R1) + (1/R2) + (1/R3) + …
  • Series resistors have an equivalent resistance given by: Rs = R1 + R2 + R3 + …
  • The equivalent resistor represents the combination of resistors in a circuit.
  • It simplifies circuit analysis and calculations.

Example Problems

  1. Three resistors (2 Ω, 3 Ω, and 4 Ω) are connected in parallel. Find the equivalent resistance.
  1. Two resistors (10 Ω and 15 Ω) are connected in series. Find the equivalent resistance.
  1. A parallel combination of resistors has an equivalent resistance of 6 Ω. If one of the resistors is 2 Ω, find the other resistors in the combination.

Example Problems (contd.)

  1. A series combination of resistors has an equivalent resistance of 20 Ω. If one of the resistors is 10 Ω, find the other resistors in the combination.
  1. A circuit contains a combination of parallel and series resistors. Find the equivalent resistance of the entire circuit.

Practice Questions

  1. Three resistors, R1 = 8 Ω, R2 = 12 Ω, and R3 = 20 Ω, are connected in parallel. Find the equivalent resistance.
  1. Two resistors, R4 = 4 Ω and R5 = 6 Ω, are connected in series. Find the equivalent resistance.
  1. A parallel combination of resistors has an equivalent resistance of 15 Ω. If one of the resistors is 5 Ω, find the other resistors in the combination.

Practice Questions (contd.)

  1. A series combination of resistors has an equivalent resistance of 30 Ω. If one of the resistors is 15 Ω, find the other resistors in the combination.
  1. A circuit contains a combination of parallel and series resistors. Find the equivalent resistance of the entire circuit.