Slide 1

  • Topic: Equivalent Circuits
  • Definition: An equivalent circuit is a simplified representation of a more complex circuit that has the same electrical behavior.
  • Purpose: To simplify complex circuits for analysis and calculations.

Slide 2

  • Need for Equivalent Circuits:
    • Complex circuits can be difficult to analyze.
    • Equivalent circuits help in simplifying the calculations and understanding the behavior.
    • Allows us to replace a complex circuit with a simpler one without changing the overall behavior.

Slide 3

  • Series and Parallel Connections:
    • Series Connection:
      • Definition: Components connected one after the other.
      • Current remains the same throughout the components.
      • Voltage divides across the components.
    • Parallel Connection:
      • Definition: Components connected across the same two points.
      • Voltage remains the same across the components.
      • Current divides among the components.

Slide 4

  • Series and Parallel Resistance:
    • Series Resistance:
      • Total resistance (Rt) is the sum of individual resistances (R1, R2, etc.): Rt = R1 + R2 + …
    • Parallel Resistance:
      • Reciprocal of total resistance (1/Rt) is the sum of reciprocals of individual resistances (1/R1 + 1/R2 + …): 1/Rt = 1/R1 + 1/R2 + …

Slide 5

  • Series and Parallel Capacitance:
    • Series Capacitance:
      • Reciprocal of total capacitance (1/Ct) is the sum of reciprocals of individual capacitances (1/C1 + 1/C2 + …): 1/Ct = 1/C1 + 1/C2 + …
    • Parallel Capacitance:
      • Total capacitance (Ct) is the sum of individual capacitances (C1, C2, etc.): Ct = C1 + C2 + …

Slide 6

  • Series and Parallel Inductance:
    • Series Inductance:
      • Total inductance (Lt) is the sum of individual inductances (L1, L2, etc.): Lt = L1 + L2 + …
    • Parallel Inductance:
      • Reciprocal of total inductance (1/Lt) is the sum of reciprocals of individual inductances (1/L1 + 1/L2 + …): 1/Lt = 1/L1 + 1/L2 + …

Slide 7

  • Example 1:
    • Given resistors R1 = 2 Ω, R2 = 4 Ω, and R3 = 6 Ω connected in series.
    • Calculate the total resistance.
    • Solution: Rt = R1 + R2 + R3 = 2 + 4 + 6 = 12 Ω.

Slide 8

  • Example 2:
    • Given capacitors C1 = 2 μF, C2 = 4 μF, and C3 = 6 μF connected in parallel.
    • Calculate the total capacitance.
    • Solution: Ct = C1 + C2 + C3 = 2 + 4 + 6 = 12 μF.

Slide 9

  • Equivalent Circuit for Series Connection:
    • For resistors: Replace with a single resistor equal to the sum of the individual resistances.
    • For capacitors: Replace with a single capacitor equal to the sum of the individual capacitances.
    • For inductors: Replace with a single inductor equal to the sum of the individual inductances.

Slide 10

  • Equivalent Circuit for Parallel Connection:
    • For resistors: Replace with a single resistor equal to the reciprocal of the sum of the reciprocals of individual resistances.
    • For capacitors: Replace with a single capacitor equal to the sum of the individual capacitances.
    • For inductors: Replace with a single inductor equal to the reciprocal of the sum of the reciprocals of individual inductances.

Slide 11

  • Circuit Analysis Techniques:
    • Kirchhoff’s Laws:
      • Kirchhoff’s Current Law (KCL): The sum of currents entering a node is equal to the sum of currents leaving the node.
      • Kirchhoff’s Voltage Law (KVL): The sum of voltages around a closed loop is equal to zero.
    • Superposition Theorem:
      • States that in a linear circuit with multiple sources, the total response is the sum of individual responses caused by each source independently.

Slide 12

  • Thevenin’s Theorem:
    • Statement: Any linear circuit containing resistors and voltage/current sources can be replaced by an equivalent circuit consisting of a voltage source in series with a single resistor.
    • Thevenin’s Equivalent Voltage (Vth): The voltage across the load terminals when the load is disconnected.
    • Thevenin’s Equivalent Resistance (Rth): The resistance seen by the load terminals when all independent sources are turned off.

Slide 13

  • Thevenin’s Theorem (contd.):
    • Steps to find Vth and Rth:
      1. Consider the load terminals as open circuit.
      2. Find Vth by calculating the voltage across the load terminals.
      3. Consider the load terminals as short circuit.
      4. Find Rth by calculating the resistance between the load terminals.

Slide 14

  • Norton’s Theorem:
    • Statement: Any linear circuit containing resistors and voltage/current sources can be replaced by an equivalent circuit consisting of a current source in parallel with a single resistor.
    • Norton’s Equivalent Current (In): The current entering the load terminals when the load is connected.
    • Norton’s Equivalent Resistance (Rn): The resistance seen by the load terminals when all independent sources are turned off.

Slide 15

  • Norton’s Theorem (contd.):
    • Steps to find In and Rn:
      1. Consider the load terminals as short circuit.
      2. Find In by calculating the current entering the load terminals.
      3. Consider the load terminals as open circuit.
      4. Find Rn by calculating the resistance between the load terminals.

Slide 16

  • Example 3: Thevenin’s Theorem
    • Given the circuit below with RL as the load resistance.
    • Find the Thevenin’s equivalent voltage (Vth) and resistance (Rth).
    • Solution: Step 1: Open circuit the load terminals. V1 -R1-|--/\/\/\-| RL | | | |-R2-|-||

Slide 17

  • Example 3: Thevenin’s Theorem (contd.)
    • Solution: Step 2: Find Vth by calculating the voltage across the open load terminals.
    • Apply KVL to the outer loop:
      • V1 - (R1 + R2) * I - R2 * I = 0
      • V1 - (R1 + 2R2) * I = 0
      • Vth = (R1 + 2R2) * I
      • I = V1 / (R1 + 2R2)
      • Vth = (R1 + 2R2) * (V1 / (R1 + 2R2))
      • Vth = V1

Slide 18

  • Example 3: Thevenin’s Theorem (contd.)
    • Solution: Step 3: Short circuit the load terminals. V1 -R1-|--/\/\/\-|--| | | |-R2-|--|--|

Slide 19

  • Example 3: Thevenin’s Theorem (contd.)
    • Solution: Step 4: Find Rth by calculating the resistance between the shorted load terminals.
    • Rth = R1 || R2
    • Rth = (R1 * R2) / (R1 + R2)

Slide 20

  • Example 4: Norton’s Theorem
    • Given the circuit below with RL as the load resistance.
    • Find the Norton’s equivalent current (In) and resistance (Rn). V1 -R1-|--/\/\/\-| RL | | | |-R2-|-||
  • To be continued…

Slide 21

  • Example 4: Norton’s Theorem (contd.)
    • Solution: Step 1: Short circuit the load terminals. V1 -R1-|--/\/\/\-|--| | | |-R2-|--|--|

Slide 22

  • Example 4: Norton’s Theorem (contd.)
    • Solution: Step 2: Find In by calculating the current entering the shorted load terminals.
    • Apply KCL at the node connecting R1 and R2:
      • (V1 - 0) / R1 + (V1 - 0) / R2 = In
      • V1 (1/R1 + 1/R2) = In
      • In = V1 / (R1 || R2)
      • In = V1 / ((R1 * R2) / (R1 + R2))

Slide 23

  • Example 4: Norton’s Theorem (contd.)
    • Solution: Step 3: Open circuit the load terminals. V1 -R1-|--/\/\/\-|| | | |-R2-|--||

Slide 24

  • Example 4: Norton’s Theorem (contd.)
    • Solution: Step 4: Find Rn by calculating the resistance between the open load terminals.
    • Rn = R1 || R2
    • Rn = (R1 * R2) / (R1 + R2)

Slide 25

  • Network Theorems Summary:
    • Superposition Theorem:
      • Apply each source separately and then add the individual responses.
    • Thevenin’s Theorem:
      • Replace the given network with an equivalent circuit consisting of a voltage source in series with a single resistor.
    • Norton’s Theorem:
      • Replace the given network with an equivalent circuit consisting of a current source in parallel with a single resistor.

Slide 26

  • Application of Equivalent Circuits:
    • Circuit Simplification:
      • Reduce complex circuits into simpler equivalent circuits for analysis and calculations.
    • Circuit Design:
      • Develop and optimize the circuit using the equivalent circuit models, making it easier to modify and improve.
    • Troubleshooting:
      • Use equivalent circuit models to identify faulty components or analyze circuit behavior.

Slide 27

  • Limitations of Equivalent Circuits:
    • Nonlinear Elements:
      • Equivalent circuits are specifically designed for linear circuit elements.
      • Nonlinear elements like diodes and transistors require more advanced models.
    • Frequency Dependency:
      • Equivalent circuits assume the circuit behavior is independent of frequency.
      • At high frequencies, more advanced models such as impedance and admittance can be used.

Slide 28

  • Recap:
    • Equivalent circuits provide a simplified representation of complex circuits.
    • Series and parallel connections can be simplified using resistance, capacitance, and inductance equations.
    • Thevenin’s and Norton’s theorems provide equivalent circuits for linear circuits.
    • Equivalent circuit models facilitate circuit analysis, design, and troubleshooting.

Slide 29

  • References:
    • H.C. Verma, “Concepts of Physics”
    • Resnick, Halliday, Walker, “Fundamentals of Physics”
    • R.G. Gupta, “Understanding Physics”

Slide 30

  • Thank you for your attention!
  • Let’s summarize the key points covered in this lecture.
  • If you have any questions, feel free to ask.