1. Ch. 4 – Circuit Theorems Linearity
- Linear relationship between cause and effect
Linear Circuit
- output is linearly related (proportional to) the input.
Ex.  = Ri
Homogeneity (scaling)
B. Additivity

2. Find the current in resistor R2.

3. Superposition
The response of a circuit due to multiple sources can be taken as the sum of the effects of each source acting alone (all others off)
Turning Off Sources:
Vs  replace with short circuit
Is  replace with open circuit
Note: Only independent sources can be turned off. Dependent sources should be left on.

4. Example 1. Apply superposition to find the current in the 3- resistor.
Ans: 5/6 A

5. Example 2. Find the terminal voltage Vab using superposition.
Ans: 6 V

6. Source Transformation
- Should be equivalent for any load attached to a-b.
Extreme cases:

7. For an arbitrary RL:

8. For Independent Sources:
For Dependent Sources:

9. Example 3. Find the current in RL=8 using source transformation.

10. Example 4. Find vx using source transformation.

11. Given the circuit.
Find the v-i characteristic of the circuit as seen by RL.
a) Remove RL (replace by open terminals a-b)

12. b) Determine (“measure”) the open-circuit voltage vab.

13. c) Connect a voltage source Vnew equal to vab at b.
Vnew = 16 V
Vab = 16 V
Vaa’ = 0 V

14. d. Reconnect RL.
What is the current through RL?
Vaa’ = 0 V
IRL = 0 A

15.  Use superposition: IRL = Iorig + IVnew only = 0
Iorig = -IVnew only = I(-Vnew)
Current in the original circuit = current produced by Voc = -Vnew 

16. 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, where:
VTh = Voc = open-circuit voltage at the terminals
RTh = equivalent resistance at the terminals with all the independent sources turned off.

17. Example 5. Find the Thevenin equivalent circuit with respect to terminals a-b. Find the current through RL = 6, 16, and 36 .
Ans: 30V, 4

18. Thevenin Equiv. Circuit: with dependent sources
VTh = Voc = open-circuit voltage at the terminals
RTh = equivalent resistance at the terminals with all the independent sources turned off.
Finding RTh:
Turn off all independent sources.
Apply either:
A test voltage source Vo  Find Io.
A test current source Io  Find Vo.
c) RTh = Vo/Io
Note: Easier to use test voltage or test current of value ONE.

19. Example 6. Find the Thevenin equivalent circuit.

21.  Norton’s Theorem - A corollary of to Thevenin’s Theorem
RN = RTh, and IN = VTh/Rth
Finding Norton Current:

22. Norton Equivalent Circuit
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 = Isc = short-circuit current through the terminals
RN = equivalent resistance at the terminals with all the independent sources turned off.

23. Example 7. Find the Norton equivalent circuit.
Answer: 1A, 4 ohms

24. Maximum Power Transfer
Find the value of the load resistor that will receive maximum power from the circuit.