1. Slide 1EE100 Summer 2008Bharathwaj Muthuswamy EE100Su08 Lecture #5 (July 2 nd 2008) Outline –Questions? –Lab notes: Labs 2 and 3 have been shortened Monday lab: go to your SECOND lab section next week. –Node-Voltage Analysis: wrap up –Mesh analysis: read it, OPTIONAL –Superposition –Thevenin’s Theorem

2. Slide 2EE100 Summer 2008Bharathwaj Muthuswamy V 2 V 1 R 2 R 1 R 4 R 5 R 3 I 1 VaVa Nodal Analysis: Example #2 Challenges: Determine number of nodes needed Deal with different types of sources

3. Slide 3EE100 Summer 2008Bharathwaj Muthuswamy A “floating” voltage source is one for which neither side is connected to the reference node, e.g. V LL in the circuit below: Problem: We cannot write KCL at nodes a or b because there is no way to express the current through the voltage source in terms of V a - V b. Solution: Define a “supernode” – that chunk of the circuit containing nodes a and b. Express KCL for this supernode. Incorporate voltage source constraint into KCL equation. R 4 R 2 I 2 V a V b + - V LL I 1 Nodal Analysis w/ “Floating Voltage Source”

4. Slide 4EE100 Summer 2008Bharathwaj Muthuswamy supernode Eq’n 1: KCL at supernode Substitute property of voltage source: R 4 R 2 I 2 V a V b + - V LL I 1 Nodal Analysis: Example #3

5. Slide 5EE100 Summer 2008Bharathwaj Muthuswamy Superposition A linear circuit is one constructed only of linear elements (linear resistors, and linear capacitors and inductors, linear dependent sources) and independent sources. Linear means I-V charcteristic of elements/sources are straight lines when plotted Principle of Superposition: In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone.

6. Slide 6EE100 Summer 2008Bharathwaj Muthuswamy Superposition Procedure: 1.Determine contribution due to one independent source Set all other sources to 0: Replace independent voltage source by short circuit, independent current source by open circuit 2.Repeat for each independent source 3.Sum individual contributions to obtain desired voltage or current

7. Slide 7EE100 Summer 2008Bharathwaj Muthuswamy Open Circuit and Short Circuit Open circuit  i=0 ; Cut off the branch Short circuit  v=0 ; replace the element by wire Turn off an independent voltage source means –V=0 –Replace by wire –Short circuit Turn off an independent current source means –i=0 –Cut off the branch –open circuit

8. Slide 8EE100 Summer 2008Bharathwaj Muthuswamy Superposition Example Find V o –+–+ 24 V 2  4  4 A 4 V + – +Vo–+Vo–

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11. Slide 11EE100 Summer 2008Bharathwaj Muthuswamy Equivalent Circuit Concept A network of voltage sources, current sources, and resistors can be replaced by an equivalent circuit which has identical terminal properties (I-V characteristics) without affecting the operation of the rest of the circuit. +vA_+vA_ network A of sources and resistors iAiA ≡ +vB_+vB_ network B of sources and resistors iBiB i A (v A ) = i B (v B )

12. Slide 12EE100 Summer 2008Bharathwaj Muthuswamy Thévenin Equivalent Circuit Any* linear 2-terminal (1-port) network of indep. voltage sources, indep. current sources, and linear resistors can be replaced by an equivalent circuit consisting of an independent voltage source in series with a resistor without affecting the operation of the rest of the circuit. network of sources and resistors ≡ –+–+ V Th R Th RLRL iLiL +vL–+vL– a b RLRL iLiL +vL–+vL– a b Thévenin equivalent circuit “load” resistor

13. Slide 13EE100 Summer 2008Bharathwaj Muthuswamy I-V Characteristic of Thévenin Equivalent The I-V characteristic for the series combination of elements is obtained by adding their voltage drops: –+–+ V Th R Th a b i i + v ab – v ab = V Th - iR I-V characteristic of resistor: v = iR I-V characteristic of voltage source: v = V Th For a given current i, the voltage drop v ab is equal to the sum of the voltages dropped across the source ( V Th ) and across the resistor ( iR Th ) v

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18. Slide 18EE100 Summer 2008Bharathwaj Muthuswamy Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

19. Slide 19EE100 Summer 2008Bharathwaj Muthuswamy Thévenin Equivalent Example (contd.)

20. Slide 20EE100 Summer 2008Bharathwaj Muthuswamy Aside: Prelab 1, question 3

21. Slide 21EE100 Summer 2008Bharathwaj Muthuswamy Thévenin Equivalent Example (contd.)

22. Slide 22EE100 Summer 2008Bharathwaj Muthuswamy Thévenin Equivalent Example (contd.)

23. Slide 23EE100 Summer 2008Bharathwaj Muthuswamy Thévenin Equivalent Example (contd.)

24. Slide 24EE100 Summer 2008Bharathwaj Muthuswamy R Th Calculation Example #1 Set all independent sources to 0: