1. Superposition, Thevenin / Norton Equivalents, Maximum Power Transfer Circuits 1 Fall 2005 Harding University Jonathan White

2. Outline – Ch. 4  Superposition  Method of analyzing a circuit by turning off all sources but 1 and then finding their contributions individually. End by summing up all the contributions.  Thevenin Equivalent Circuits  A circuit at a given 2 terminals can be replaced by a voltage source with a resistor in series.  Norton Equivalent Circuits  A circuit at a given 2 terminals can also be replaced with a current source and a parallel resistor.  Maximum Power Transfer  When you have a load, when does it receive the maximum power? We’ve already answered this in lab.

3. Superposition  Resistors are linear elements, meaning that the output is linearly related to the input.  Voltages around a loop can simply be added up – no non linear math is required.  Instead of analyzing circuits like we did in Ch. 2 and Ch. 3, we can analyze them using Superposition.  Definition: The voltage across (or current through) a resistor is the algebraic sum of all the contributions due to each source acting alone.  So, another way to analyze a circuit is to find the contribution of each source individually and them add them up at the end to get the total.

4. Superposition 2  We only consider 1 independent source at a time when we use superposition. This means that we:  Replace voltage sources with a wire (0 V).  Replace current sources with an open circuit (no current can flow).  Dependent sources are left intact since they are controlled by circuit variables.

5. Superposition 3  To solve a circuit using superposition:  Turn off all independent sources but 1. Use the techniques of Ch. 2 and Ch. 3 to solve for the desired voltage or current.  Repeat for each independent source.  Find the total voltage or contribution by taking the algebraic sum.

6. Superposition – Exp. 1 Find the voltage over the 2 Ohm resistor using superposition.

7. Superposition Exp. 2 + V - Find the voltage over the 5 ohm resistor using superposition.

8. Equivalent Circuits  A model of the real thing.  Used to capture only the necessary details of a potentially complex circuit.  Examples of various models:  Battery  OSI network layer  Function calls  You (as a user), don’t really care how the function operates, just that it does.

9. Thevenin Equivalent Circuits  Consists of a voltage source and a resistor in series.  Used to provide a “black box” picture from the view of a load. The load, looking back in to the circuit, only wants to know the voltage and current that is provided to it.

10. Finding a TEC  Steps:  Find the open circuit voltage – disconnect the load from the circuit and calculate the voltage looking in to the circuit.  Find the open circuit equivalent resistance looking back in to the circuit  Remove all independent current sources  Replace all independent voltage sources with wires.  R th is then that equivalent resistance and V th is just the voltage that you found.

11. TEC Example - 1 Find the Thevenin Equivalent Circuit: a b

12. TEC Example - 2 Find the Thevenin Equivalent Circuit: a b

13. Norton Equivalent Circuits  Consists of a current source with a resistor in parallel.  Electrically equivalent to the Thevenin model  R th is the same  I n is equal to V th / R th  When finding Norton equivalents, I often recommend just finding the Thevenin equivalent and then just switch at the end.

14. Norton Example Find the Norton Equiv. Circuit

15. Source Transformations  Like the Wye-Delta transformation, we can transform a voltage source with a resistor in series into a current source with a resistor in parallel without changing the rest of the circuit and vice versa.  Like superposition, however, this is often more work than just using mesh currents to solve the problem.

16. Source Transformation Exp. i0 +V-+V- Find i0 and the voltage over the 3 ohm resistor using source transformations.

17. Maximum Power Transfer  When does the load receive maximum power? – see notes  When R L = R th

18. Maximum Power Example Find the RL that achieves maximum power transfer. Find the power it absorbs. Note: You must find V th to calculate the power.