Lecture 3a, Prof. WhiteEE 42/100, Spring EE 42/100 Discussion sections SectionDay/TimeRoomGSI Dis101M 3-4pm241 CoryLiu, Vincent Dis102W 4-5pm241 CoryLi, Li Dis103F 9-10am3108 EtcheverryLiu, Vincent Dis104F 10-11am71 EvansLiu, Vincent Dis?105Th 4-5pm3107 EtcheverryLi, Li Dis106Th 12-1pm45 EvansLi, Li

Lecture 3a, Prof. WhiteEE 42/100, Spring Lecture Week 3a OUTLINE Superposition: Analysis method for circuits with sources and linear elements Thévenin and Norton equivalent circuits Maximum Power Transfer

Lecture 3a, Prof. WhiteEE 42/100, Spring Superposition A linear circuit is constructed only of linear elements (linear resistors, linear dependent sources*) and independent sources. * We’ll discuss dependent sources a bit later 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. Procedure: 1.Determine contribution due to an independent source Set all other independent sources to 0 2.Repeat for each independent source 3.Sum individual contributions to obtain desired voltage or current

Lecture 3a, Prof. WhiteEE 42/100, Spring Superposition Example Find V o –+–+ 24 V 2  4  4 A 4 V + – +Vo–+Vo–

Lecture 3a, Prof. WhiteEE 42/100, Spring 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 )

Lecture 3a, Prof. WhiteEE 42/100, Spring 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

Lecture 3a, Prof. WhiteEE 42/100, Spring Thévenin Equivalent Circuit Any linear 2-terminal (1-port) network of independent voltage sources, independent 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 Actual circuit

Lecture 3a, Prof. WhiteEE 42/100, Spring Why use such equivalent circuits? They may be much easier to use than the actual circuits when doing circuit analysis Example: We can reduce the entire telephone network or the entire power system that delivers energy to an AC outlet to a Thevenin equivalent containing just one voltage source (V th ) and one resistor (R th ) [or one impedance Z th, which we’ll see a little later]

Lecture 3a, Prof. WhiteEE 42/100, Spring 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

Lecture 3a, Prof. WhiteEE 42/100, Spring Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

Lecture 3a, Prof. WhiteEE 42/100, Spring 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

Lecture 3a, Prof. WhiteEE 42/100, Spring R Th Calculation Example #1 Set all independent sources to 0:

Lecture 3a, Prof. WhiteEE 42/100, Spring Norton equivalent circuit Norton Equivalent Circuit Any linear 2-terminal (1-port) network of independent voltage sources, independent 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

Lecture 3a, Prof. WhiteEE 42/100, Spring 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 open-circuit voltage and short-circuit current 2) Or, find short-circuit current and Norton (Thevenin) resistance

Lecture 3a, Prof. WhiteEE 42/100, Spring 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

Lecture 3a, Prof. WhiteEE 42/100, Spring Maximum Power Transfer Theorem A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit. Example: Maximizing power to speakers from music system –+–+ 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:

Lecture 3a, Prof. WhiteEE 42/100, Spring