1. Lecture 6, Slide 1EECS40, Fall 2004Prof. White Lecture #6 OUTLINE Complete Mesh Analysis Example(s) Superposition Thévenin and Norton equivalent circuits Maximum Power Transfer Reading Chapter 2

2. Lecture 6, Slide 2EECS40, Fall 2004Prof. White 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.

3. Lecture 6, Slide 3EECS40, Fall 2004Prof. White 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

4. Lecture 6, Slide 4EECS40, Fall 2004Prof. White Superposition Example Find V o –+–+ 24 V 2  4  4 A 4 V + – +Vo–+Vo–

5. Lecture 6, Slide 5EECS40, Fall 2004Prof. White 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 )

6. Lecture 6, Slide 6EECS40, Fall 2004Prof. White Voltage sources in series can be replaced by an equivalent voltage source: Current sources in parallel can be replaced by an equivalent current source: Source Combinations i1i1 i2i2 ≡ i 1 +i 2 –+–+ –+–+ v1v1 v2v2 ≡ –+–+ v 1 +v 2

7. Lecture 6, Slide 7EECS40, Fall 2004Prof. White 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

8. Lecture 6, Slide 8EECS40, Fall 2004Prof. White 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

9. Lecture 6, Slide 9EECS40, Fall 2004Prof. White Finding V Th and R Th Only two points are needed to define a line. Choose two convenient points: 1. Open circuit across terminals a,b i = 0, v ab ≡ v oc 2. Short circuit across terminals a,b v ab = 0, i ≡ -i sc = -V Th /R Th i v = V Th + iR v ab -i sc v oc – + V Th R i sc i i

10. Lecture 6, Slide 10EECS40, Fall 2004Prof. White Calculating a Thévenin Equivalent 1.Calculate the open-circuit voltage, v oc 2.Calculate the short-circuit current, i sc Note that i sc is in the direction of the open-circuit voltage drop across the terminals a,b ! network of sources and resistors a b + v oc – network of sources and resistors a b i sc

11. Lecture 6, Slide 11EECS40, Fall 2004Prof. White Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

12. Lecture 6, Slide 12EECS40, Fall 2004Prof. White Alternative Method of Calculating R Th For a network containing only independent sources and linear resistors: 1.Set all independent sources to zero voltage source  short circuit current source  open circuit 2.Find equivalent resistance R eq between the terminals by inspection Or, set all independent sources to zero 1.Apply a test voltage source V TEST 2.Calculate I TEST network of independent sources and resistors, with each source set to zero R eq network of independent sources and resistors, with each source set to zero I TEST –+–+ V TEST

13. Lecture 6, Slide 13EECS40, Fall 2004Prof. White R Th Calculation Example #1 Set all independent sources to 0:

14. Lecture 6, Slide 14EECS40, Fall 2004Prof. White Comments on Dependent Sources A dependent source establishes a voltage or current whose value depends on the value of a voltage or current at a specified location in the circuit. (device model, used to model behavior of transistors & amplifiers) To specify a dependent source, we must identify: 1.the controlling voltage or current (must be calculated, in general) 2.the relationship between the controlling voltage or current and the supplied voltage or current 3.the reference direction for the supplied voltage or current The relationship between the dependent source and its reference cannot be broken! –Dependent sources cannot be turned off for various purposes (e.g. to find the Thévenin resistance, or in analysis using Superposition).

15. Lecture 6, Slide 15EECS40, Fall 2004Prof. White R Th Calculation Example #2 Find the Thevenin equivalent with respect to the terminals a,b:

16. Lecture 6, Slide 16EECS40, Fall 2004Prof. White Networks Containing Time-Varying Sources Care must be taken in summing time-varying sources! Example: 20 cos (100t) 1 k  10 sin (100t) +–+– – +

17. Lecture 6, Slide 17EECS40, Fall 2004Prof. White

18. Lecture 6, Slide 18EECS40, Fall 2004Prof. White Norton equivalent circuit Norton 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 current source in parallel with a resistor without affecting the operation of the rest of the circuit. network of sources and resistors ≡ RLRL iLiL +vL–+vL– a b a RLRL iLiL +vL–+vL– iNiN b RNRN

19. Lecture 6, Slide 19EECS40, Fall 2004Prof. White I-V Characteristic of Norton Equivalent The I-V characteristic for the parallel combination of elements is obtained by adding their currents: i i = -I N + Gv I-V characteristic of resistor: i=Gv I-V characteristic of current source: i = -I N For a given voltage v ab, the current i is equal to the sum of the currents in each of the two branches: v i + v ab – iNiN b RNRN a

20. Lecture 6, Slide 20EECS40, Fall 2004Prof. White Finding I N and R N = R Th I N ≡ i sc = V Th /R Th Analogous to calculation of Thevenin Eq. Ckt: 1) Find o.c voltage and s.c. current 2) Or, find s.c. current and Norton (Thev) resistance

21. Lecture 6, Slide 21EECS40, Fall 2004Prof. White Finding I N and R N We can derive the Norton equivalent circuit from a Thévenin equivalent circuit simply by making a source transformation: RLRL RNRN iLiL iNiN +vL–+vL– a b –+–+ RLRL iLiL +vL–+vL– v Th R Th a b

22. Lecture 6, Slide 22EECS40, Fall 2004Prof. White Maximum Power Transfer Theorem A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit. –+–+ V Th R Th RLRL iLiL +vL–+vL– Thévenin equivalent circuit To find the value of R L for which p is maximum, set to 0: Power absorbed by load resistor: