Network Theorems in Electric Circuits

1. ENT 114: CIRCUIT THEORY Chapter 8: Network Theorems (dc circuits)

2. Constant-current and Constant Voltage Source • In an electric circuit, a source of electrical energy could be represented by a source of e.m.f in series with a resistance. • This is not, however, the only form of representation • Consider a source load resister RL as shown in Figure 8.1

3. Constant-current and Constant Voltage Source Is Where Is= E/Rs is the current which flow across output terminals of the source Figure 8.1

4. Constant-current and Constant Voltage Source

5. Removing a voltage source and a current source to permit the application of the superposition theorem.

6. Superposition Theorem • The superposition theorem state that, The current in any given branch of a multiple-source linear circuit can be found by determining the currents in that particular branch produced by each source acting alone, with all the other source replaced by their internal resistance. The total current in the branch is the algebraic sum of the individual source currents in that branch.

7. Superposition Theorem • Applying Superposition Theorem • Take one voltage source at a time and replace each of the other voltage source with a short (a short represent zero resistance) • Determine the current or voltage just as if there were was only one source in the circuit. • Take the next source in the circuit and repeat the previous two steps for each source. • To find actual current or voltage, add or substract the currents or voltages due to individual source. If the currents are in the same directions or the voltages are of the same polarity, add them and vice versa.

8. Example 9.1 I1 I3 I2

9. Example 9.1 I1a I3a • The network with the 20V source replaced by a short-circuit. I2a The current I2b is negative because opposite direction

10. Example 9.1 I1b I3b • Next replace 10V source with a short. I2b

11. Example 9.2 • Determine the current in the 8 Ω resistor (R1) in the circuit.

12. IR1’ • The network with the 3 A source replaced by an open circuit. • Then, replaced 6 V source by a short circuit • Apply current devider rule to determine IR1’’ IR1’’

13. Example 9.3 • Determine the current in the 12 Ω resistor

14. Used current devider rule to determine I’2

15. Used current devider rule to determine I’’2

16. To determine current I2 for the network: The net current therefore is the difference of the two and in the direction of the larger current:

17. Thevenin’s Theorem • Thevenin’s Theorem is a method for simplifying a circuit to a standard equivalent form.

18. Thevenin’s Theorem • Thevenin’s Theorem state that, • An active network having two terminals A and B can be replaced by a constant-voltage source having an e.m.f E and internal resistance r. The value of E is equal to the open-circuit potential difference between A and B with the load disconnected and the source of e.m.f replaced by their internal resistance.

19. Thevenin’s Theorem Procedure Step 1: Remove the load resistor RL. Step 2 : Mark the terminal as a and b. We have an open circuit across terminal a and b. Step 1 and 2 Original

20. Thevenin’s Theorem Procedure • Step 3: • Replace the voltage source with a short-circuit equivalent. • Calculate the RTH

21. Step 4: • Put back the voltage source. Apply voltage devider rule to find VTH • Or using Mesh current analysis: Loop 1 Loop2

22. Thevenin’s Theorem Procedure • Step 5: • Draw the Thevenin equivalent circuit. • Placed the RL. Across terminal a and b. • Addition: • If require to measure current IL,

23. Example 9.4 • Determine the current in the 12 Ω resistor

24. Apply KVL to Loop A Apply KVL to Loop B A B

25. The Thevenin equivalent circuit • Current across 12 Ω resistor is:

26. Norton’s Theorem • The theorem states that, • Any two-terminal linear bilateral dc network can be replaced by an equivalent circuit consisting of a current and a parallel resistor.

27. Norton’s Theorem Procedure • Step 1: Remove the portion of the network across which the Norton equivalent circuit is found. • Step 2: Mark the terminal as a and b. We have an open circuit across terminal a and b.

28. Norton’s Theorem Procedure • Step 3: • Replace the voltage source with a short-circuit equivalent. • Calculate the RN

29. Norton’s Theorem Procedure • Step 4: Indicate the short circuit connection between the terminal a and b.

30. Norton’s Theorem Procedure Step 5: Draw the norton equivalent circuit. Converting the Norton equivalent circuit to a Thevenin equivalent circuit.

31. Example 9.5 • Determine the current in the 12 Ω resistor using Norton Theorem

32. Maximum Power Transfer Theorem • The maximum power transfer theorem states the following: A load will receive maximum power from a network when its total resistive value is exactly equal to the Thévenin resistance of the network applied to the load. That is, RL = RTh

33. For the Thevenin equivalent circuit like the figure below, when the load is equal to the Thevenin resistance, the load will receive maximum power from the network

34. With RL= RTh, the maximum power delivered to the load can be determined by first finding the current:

35. PL versus RL for the network

36. Example 9.6 • For the circuit below, determine • The value of the load resistor, RL, which would give the maximum power transfer. • The maximum power transferred to the load

37. Example 9.4 • The Thevenin resistance is • Determine the Thevenin voltage

38. Example 9.4 • Applying KVL