1. Kevin D. Donohue, University of Kentucky1 Additional Analysis Techniques for Linear Circuits Models and Equivalent Circuits for Analysis and Design

2. Kevin D. Donohue, University of Kentucky2 Example - Superposition  Solve for I o in the 3 circuits. What is the relationship between these results. 44 66 66 IoIo 9V 22

3. Kevin D. Donohue, University of Kentucky3 Linearity and Superposition  If a linear circuit has multiple independent sources, then a voltage or current quantity anywhere in the circuit is the sum of the quantities produced by the individual sources (i.e. the result when all other sources are deactivated). This property is called superposition.  To deactivate a voltage source, set the voltage equal to zero (equivalent to replacing it with a short circuit).  To deactivate a current source, set the current equal to zero (equivalent to replacing it with an open circuit).

4. Kevin D. Donohue, University of Kentucky4 Examples  Solve for voltages and currents in circuits containing multiple sources using the principle of superposition.

5. Kevin D. Donohue, University of Kentucky5 Example - Equivalent Circuit  Find the voltage and currents generated in 3 different loads across terminals AB:  open circuit  resistor R L  short circuit R th VsVs IsIs A A B B

6. Kevin D. Donohue, University of Kentucky6 Results - Equivalent Circuit V AB I AB Open0 Short0 RLRL Current Source Circuit V AB I AB Open0 Short0 RLRL Voltage Source Circuit What would the value of the voltage source have to be so it is equivalent to the current source circuit? What would the value of the current source have to be so it is equivalent to the voltage source circuit?

7. Kevin D. Donohue, University of Kentucky7 Equivalent Circuits  Circuits containing different elements are equivalent, if their response with respect to (wrt) a pair of terminals is the same.  For the two previous circuits to be equivalent, what would have to be true about their source and resistance values?

8. Kevin D. Donohue, University of Kentucky8 Source Transformation  The following circuit pairs are equivalent wrt to terminals AB. Therefore, these source and resistor combinations can be swapped in a circuit without affecting the voltages and currents in other parts of the circuit R th I s R th IsIs R th A A B B VsVs A A B B IsIs A B VsVs A B VsVs A B A B IsIs

9. Kevin D. Donohue, University of Kentucky9 Source Transformation  Some equivalent circuits can be determined by transforming source and resistor combinations and combining parallel and serial elements around a terminal of interest.  This method can work well for simple circuits with source- resistor combinations as shown on the previous slide.  This method is limited, if dependent sources are present.

10. Kevin D. Donohue, University of Kentucky10 Examples - Source Transformation  For several circuits find voltages and currents in circuits with independent sources and resistors using the method of source transformation.

11. Kevin D. Donohue, University of Kentucky11 Thévenin Equivalent Circuits  Find the value for V th and R th so the two circuits will be equivalent at terminals AB. + V A - + Vo - 22 VsVs R th A B 2 V A 22 22 A B 12V

12. Kevin D. Donohue, University of Kentucky12 Norton Equivalent Circuits  Find the value for I n and R th so the two circuits will be equivalent at terminals AB. + V A - + Vo - 22 2 V A 22 22 A B R th InIn A B 12V

13. Kevin D. Donohue, University of Kentucky13 Finding Thévenin and Norton Equivalent Circuits  Identify terminal pair around which to find the equivalent circuit.  Find voltage across the terminal pair when no load is present (open-circuit voltage V oc )  Short the terminal and find the current in the short (short-circuit current I sc )  Compute equivalent resistance as: R th = V oc / I sc

14. Kevin D. Donohue, University of Kentucky14 Finding Thévenin and Norton Equivalent Circuits  The equivalent circuits can then be expressed in terms of these quantities R th I sc A B V oc R th A B

15. Kevin D. Donohue, University of Kentucky15 Examples -Finding Equivalent Circuits  Find the Thévenin and Norton equivalents for circuits containing independent and dependent sources and resistors.  Show that for a maximum power transfer from a circuit to a load resistor, it must equal the Thévenin resistance of the circuit.

16. Kevin D. Donohue, University of Kentucky16 SPICE Examples  Find the Thévenin and Norton equivalents for circuits containing independent and dependent sources and resistors.  Show that for a maximum power transfer from a circuit to a load resistor, it must equal the Thévenin resistance of the circuit.