1. CHAPTER 3 NETWORK THEOREM

2. NETWORK THEOREM Superposition Theorem Source Transformation
Thevenin & Norton Equivalent

3. SUPERPOSITION THEOREM

4. Principle
If a circuit has two or more independent sources, the voltage across or current through an element in a linear circuit, is the algebraic sum of voltages across or current through that element due to each independent source acting alone.

5. Steps to apply superposition principle
Turn off all independent sources except one sources.
For voltage source replace by short circuit
For current source replace by open circuit.
Find voltage or current due to that active source using any technique.
Repeat the procedure for each of the other source
Find total contribution by adding algebraically all the contribution due to the independent source

6. Practice Problem 10.5 Find current in the circuit using the superposition theorem

7. Solution
Let Io = Io’ + Io” where Io’ and Io” are due to voltage source and current source respectively.
For Io’ consider circuit beside where the current source is open circuit
For mesh 1
…(1)
For mesh 2
…(2)

8. Solution
Substitute equation (1) into equation (2)

9. Solution
For Io” consider circuit beside where the voltage source is short circuit
Let

10. Solution
Therefore

11. Practice Problem 10.6 Calculate Vo in the circuit using superposition theorem

12. Solution
Let vo = vo’ + vo” where vo’ is due to the voltage source and vo” is due to the current source.
For vo’ we remove current source which is now open circuit
Transfer the circuit to frequency domain ( = 5)
By voltage division

13. Solution
For vo” we remove voltage source and replace with short circuit
Transfer the circuit to frequency domain ( = 10)
By current division
Let

14. Solution
Thus
Therefore

15. SOURCE TRANSFORMATION

16. Source Transformation
Source transformation in the frequency domain involves transforming a voltage source in series with an impedance to a current source in parallel with an impedance, or vice versa.

17. Example 10.7 Calculate Vx in the circuit by using source transformation

18. Solution
If we transform the voltage source to a current source, we obtain the circuit as shown.
The parallel combination of 5 resistance and (3+j4) impedance gives a new equivalent impedance Z1

19. Solution
Convert the current source to a voltage source yields the circuit as below
We then could solve for VX by using voltage division

20. Practice Problem 10.7 Find Io in the circuit by using the concept of source transformation

21. Solution
If we transform the current source to a voltage source, we obtain the circuit as shown.

22. Solution
We transform the voltage source to a current source as shown below
Note that
Let
Then

23. Solution
By current division

24. THEVENIN & NORTON EQUIVALENT CIRCUIT

25. Equivalent Circuit Thevenin Equivalent Circuit
Norton Equivalent Circuit

26. Relationship
Keep in mind that the two equivalent circuit are related as
and
Vth = VOC = open circuit voltage
IN = ISC = short circuit current

27. Steps to determine equivalent circuit
Zth or ZN
Equivalent impedance looking from the terminals when the independence sources are turn off. For voltage source replace by short circuit and current source replace by open circuit.
Vth
Voltage across terminals when the terminals is open circuit IN
Current through the terminals when the terminals is short circuit
Note
When there is dependent source or sources with difference frequencies, the step to find the equivalent is not straight forward.

28. Practice Problem 10.8 Find the Thevenin equivalent at terminals a-b of the circuit

29. Solution
To find Zth, set voltage source to zero

30. Solution
To find Vth, open circuit at terminals a-b

31. Example 10.9 Find Thevenin equivalent of the circuit as seen from terminals a-b

32. Solution To find Vth, we apply KCL at node 1
Applying KVL to the loop on the right-hand side
Thus the Thevenin voltage is or

33. Solution
To find Zth, we remove the independent source. Due to the presence of the dependent current source, connect 3A current source to terminals a-b
At the node apply KCL
Applying KVL to the outer loop
The Thevenin impedance is

34. Practice Problem 10.9 Determine the Thevenin equivalent of the circuit as seen from the terminals a-b

35. Solution To find Vth, consider circuit beside At node 1, …(1)
Thus node 2 become
…(2)
but

36. Solution
Substitute equation (2) into equation (1)

37. Solution
To find Zth, we remove the independent source and insert 1V voltage source between terminals a-b
At node a,
But
And So

38. Solution
Therefore

39. Example 10.10 Obtain current Io by using Norton’s Theorem

40. Solution To find ZN, i) Short circuit voltage source
ii) Open circuit source
As a result, the (8-j2) and (10+j4) impedances are short circuit.

41. Solution To find IN, i) Short circuit terminal a-b
ii) Apply mesh analysis
Notice that mesh 2 and 3 form a supermesh.
Mesh 1
…(1)
Supermesh
…(2)

42. Solution At node a, due to the current source between mesh 2 and 3
…(3)
Adding equation (1) and (2) gives
From equation (3)
The Norton current is

43. Solution
By using Norton’s equivalent circuit along with the impedance at terminal a-b, we could solve for Io.
By using current division

44. Practice Problem Determine the Norton equivalent circuit as seen from terminal a-b. Use the equivalent to find Io

45. Solution To find ZN, i) Short circuit voltage source
ii) Open circuit source
ZN = = =

46. Solution To find IN, i) Short circuit terminal a-b
ii) Solve for IN using mesh
analysis

47. Solution
Supermesh
…(1)
…(2)
Mesh 3
…(3)
Solving for IN

48. Solution
Using Norton equivalent, we
could find Io

49. Problem 11.14
Determine Thevenin equivalent circuit looking from the load, Z.
Determine load, Z that will produce maximum power transfer.
Value of the maximum power.
(is = 5 cos 40t A)

50. Problem 10.36 (Buku Electric Circuit by Nilsson & Riedel)
Determine Thevenin equivalent circuit looking from the load, Z.
Determine load, Z that will produce maximum power transfer.
Value of the maximum power.

51. Problem 11.15
Determine Thevenin equivalent circuit looking from the load, ZL.
Determine load, ZL that will produce maximum power transfer.
Value of the maximum power.