Linearity- Linear relationship between cause and effect

Linear Circuit- output is linearly related (proportional to) the input.

Ex. = Ri

A. Homogeneity (scaling)

B. Additivity

Ch. 4 – Circuit Theorems

Find the current in resistor R2.

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)

Superposition

Turning Off Sources:a) Vs replace with short circuitb) Is replace with open circuit

Note: Only independent sources can be turned off. Dependent sources should be left on.

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

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

Source Transformation

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

For an arbitrary RL:

For Independent Sources:

For Dependent Sources:

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

Example 4. Find vx using source transformation.

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

Given the circuit.

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

c) Connect a voltage source Vnew equal to vab at b.

Vnew = 16 VVab = 16 VVaa’ = 0 V

d. Reconnect RL.

What is the current through RL?Vaa’ = 0 VIRL = 0 A

e. Use superposition:

IRL = Iorig + IVnew only = 0Iorig = -IVnew only = I(-Vnew)

Current in the original circuit = current produced by Voc = -Vnew

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 terminalsRTh = equivalent resistance at the terminals with all the independent sources turned off.

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

Thevenin Equiv. Circuit: with dependent sources

VTh = Voc = open-circuit voltage at the terminalsRTh = equivalent resistance at the terminals with all the independent sources turned off.

Finding RTh:a) Turn off all independent sources.b) 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.

Example 6. Find the Thevenin equivalent circuit.

Norton’s Theorem

- A corollary of to Thevenin’s Theorem

RN = RTh, and IN = VTh/Rth

Finding Norton Current:

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 terminalsRN = equivalent resistance at the terminals with all the independent sources turned off.

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

Maximum Power Transfer

Find the value of the load resistor that will receive maximum power from the circuit.