Ch. 4 – Circuit Theorems Linearity

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Presentation transcript:

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

Find the current in resistor R2.

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.

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.

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

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

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

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

 Use superposition: IRL = Iorig + IVnew only = 0 Iorig = -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 terminals RTh = 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 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.

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 terminals RN = 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.