1. Basic Theory of Circuits, SJTU
Chapter 4
Circuit Theorems
Basic Theory of Circuits, SJTU

2. Basic Theory of Circuits, SJTU
Linearity
Substitution
Maximum
Power Transfer
Superposition
Tellegen
Theorems
Source
transformation
Reciprocity
Thevenin’s
Source
Transfer
Norton’s
Basic Theory of Circuits, SJTU

3. Basic Theory of Circuits, SJTU
Linearity Property
Linearity is the property of an element describing a linear relationship between cause and effect.
A linear circuit is one whose output is linearly ( or directly proportional) to its input.
Basic Theory of Circuits, SJTU

4. Basic Theory of Circuits, SJTU
Example 4.2
Basic Theory of Circuits, SJTU

5. Basic Theory of Circuits, SJTU
Superposition(1)
The superposition principle states that voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.
Basic Theory of Circuits, SJTU

6. Basic Theory of Circuits, SJTU
Superposition(2)
Steps to Apply Superposition Principle:
Turn off all independent source except one source. Find the output(voltage or current) due to that active source using nodal or mesh analysis.
Repeat step 1 for each of the other independent sources.
Find the total contribution by adding algebraically all the contributions due to the independent sources.
Basic Theory of Circuits, SJTU

7. Basic Theory of Circuits, SJTU

8. Basic Theory of Circuits, SJTU
Fig For Example 4.3
Basic Theory of Circuits, SJTU

9. Basic Theory of Circuits, SJTU
Substitution Theorem 
Basic Theory of Circuits, SJTU

10. Basic Theory of Circuits, SJTU
Substitution Theorem 
Basic Theory of Circuits, SJTU

11. Basic Theory of Circuits, SJTU
Substitution Theorem 
Basic Theory of Circuits, SJTU

12. Basic Theory of Circuits, SJTU
Substitution Theorem   
Basic Theory of Circuits, SJTU

13. Basic Theory of Circuits, SJTU
Substitution Theorem
If the voltage across and current through a branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.
Basic Theory of Circuits, SJTU

14. Basic Theory of Circuits, SJTU
Substitution Theorem OR .
Basic Theory of Circuits, SJTU

15. Basic Theory of Circuits, SJTU
Thevenin’s Theorem
A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source Vth in series with a resistor Rth (accompanied voltage source), where Vth is the open-circuit voltage at the terminals and Rth is the input or equivalent resistance at the terminals when the independent source are turned off.
Basic Theory of Circuits, SJTU

16. Basic Theory of Circuits, SJTU
(a) original circuit, (b) the Thevenin equivalent circuit c d
Proof by figures.
Basic Theory of Circuits, SJTU

17. Basic Theory of Circuits, SJTU
Simple Proof by figures
Applying superposition theorem +
V=Voc-RoI
Basic Theory of Circuits, SJTU

18. Basic Theory of Circuits, SJTU
Thevenin’s Theorem
Consider 2 cases in finding Rth:
Case 1 If the network has no dependent sources, just turn off all independent sources, calculate the equivalent resistance of those resistors left.
Basic Theory of Circuits, SJTU

19. Basic Theory of Circuits, SJTU
Case 2 If the network has dependent sources, there are two methods to get Rth:
Turn off all the independent sources, apply a voltage source v0 (or current source i0) at terminals a and b and determine the resulting current i0 (or resulting voltage v0), then Rth= v0/ i0
Basic Theory of Circuits, SJTU

20. Basic Theory of Circuits, SJTU
Case 2 If the network has dependent sources, there are two methods to get Rth:
2. Calculate the open-circuit voltage Voc and short-circuit current Isc at the terminal of the original circuit, then Rth=Voc/Isc
Rth=Voc/Isc
Basic Theory of Circuits, SJTU

21. Basic Theory of Circuits, SJTU
Examples
Basic Theory of Circuits, SJTU

22. Basic Theory of Circuits, SJTU
Norton’s Theorem
A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the short-circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent sources are turned off.
Basic Theory of Circuits, SJTU

23. Basic Theory of Circuits, SJTU
(a) Original circuit, (b) Norton equivalent circuit d
(c) N
Basic Theory of Circuits, SJTU

24. Basic Theory of Circuits, SJTU
Examples
Basic Theory of Circuits, SJTU

25. Maximum Power TransferRL a b
Basic Theory of Circuits, SJTU

26. Basic Theory of Circuits, SJTU

27. Maximum Power Transfer (several questions)
If the load RL is invariable, and RTh is variable, then what should RTh be to make RL get maximum power?
If using Norton equivalent to replace the original circuit, under what condition does the maximum transfer occur?
Is it true that the efficiency of the power transfer is always 50% when the maximum power transfer occurs?
Basic Theory of Circuits, SJTU

28. Basic Theory of Circuits, SJTU
Examples
Basic Theory of Circuits, SJTU

29. Basic Theory of Circuits, SJTU
Tellegen Theorem
If there are b branches in a lumped circuit, and the voltage uk, current ik of each branch apply passive sign convention, then we have
Basic Theory of Circuits, SJTU

30. Inference of Tellegen Theorem
If two lumped circuits and have the same topological graph with b branches, and the voltage, current of each branch apply passive sign convention, then we have not only
Basic Theory of Circuits, SJTU

31. Basic Theory of Circuits, SJTU
Example
Basic Theory of Circuits, SJTU

32. Basic Theory of Circuits, SJTU

33. Basic Theory of Circuits, SJTU
Reciprocity Theorem
2 
6 
3 
3 
6 
2 
Basic Theory of Circuits, SJTU

34. Reciprocity Theorem (only applicable to reciprocity networks)
Case 1 The current in any branch of a network, due to a single voltage source E anywhere else in the network, will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured.
Basic Theory of Circuits, SJTU

35. Reciprocity Theorem (only applicable to reciprocity networks)
Case 2
Basic Theory of Circuits, SJTU

36. Reciprocity Theorem (only applicable to reciprocity networks)
Case 3
Basic Theory of Circuits, SJTU

37. Basic Theory of Circuits, SJTU
Example
All resistors are 1 , find out i. i
+ E --
+ E -- i
Basic Theory of Circuits, SJTU

38. Source Transfer Property
Voltage source transfer
Basic Theory of Circuits, SJTU

39. Basic Theory of Circuits, SJTU
Source Transfer
Current source transfer
Basic Theory of Circuits, SJTU

40. Basic Theory of Circuits, SJTU
Summary
Maximum Power Transfer
Tellegen Theorem
Inference of Tellegen Theorem
Reciprocity Theorem
Source Transfer
Linearity Property
Superposition
Source Transformation
Substitution Theorem
Thevenin’s Theorem
Norton’s Theorem
Basic Theory of Circuits, SJTU