1Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003EECS 42 Introduction to Electronics for Computer ScienceAndrew R. NeureutherLecture #9 Node Equations• Recap and Checking Solutions• Applications to parallel and bridge • Midterm Exam Topics• Thevenin/Norton Eq. Cir. Reviewhttp://inst.EECS.Berkeley.EDU/~ee42/Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003Game Plan 02/24/03Monday 02/24/03 Node Equations: S&O 2.3, 2.5,2.6; Exam Topics; Thevenin ReviewWednesday 02/26/03: Sheila Ross instructor Quiz on Basic Circuit Analysis and Transients Logic – Functions, Tables, Circuit Symbols 391-406Next (7th) Week: Monday 3/3: Brief Exam Review; Logic Synthesis Monday 3/3: TA Exam Review Session (247 Cory?) Wednesday: Midterm In Class, Closed BookProblem Set #5 – Out 2/19/03 - Due 2/26/03 2:30 in box in 240 Cory; Node Analysis: basic, supernode, advanced; review: circuit analysis, transientsNo Problem Set Due 7thweek, Problem set #6 out Monday 3/3 and due at 2:30 3/10 in box in 240 Cory2Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003FORMAL CIRCUIT ANALYSIS USING KCL:NODAL ANALYSIS2 Define unknown node voltages (those not fixed by voltage sources)1 Choose a Reference Node4 Solve the set of equations (N equations for N unknown node voltages)(Memorize these steps and apply them rigorously!)3 Write KCL at each unknown node, expressing current in terms of the node voltages (using the constitutive relationships of branch elements*)* With inductors or floating voltages we will use a modified Step 3: The Supernode Method – see slide 10Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003FLOATING VOLTAGE SOURCES (cont.)Use a Gaussian surface to enclose the floating voltage source; write KCL for that surfaceR4R2I2VaVb+-VLLI1supernodeWe have two unknowns: Vaand Vb.We obtain one equation from KCL at supernode:0IRVRVI24b2a1=+−−⇒ 2 Equations & 2 UnknownsWe obtain a second “auxiliary” equation from the property of thevoltage source:abLLVVV−=(often called the “constraint”)3Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003ANOTHER EXAMPLE1 Choose reference node (can it be chosen to avoid floating voltage source?)+−K10R2=K20R4=+−1VV6K10R1=20KR3=V12V2=ab2 Label unknowns Vaand Vb4 Auxiliary equation: 2abVVV=−Solve:12V 0V :ba==SOLUTION2a4b1a1RVRVRVV+=−3 Equation at supernode:RVRV)R1R1(V114b21a=++-VVV2ba=−RV+1)RR(V1121a=+R4R4R4-V2+VVV2ba=Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003NODAL ANALYSIS EXAMPLEFind Va, Vbif R1= R2= R3= R4= 1MΩ, and V1= V4=1.5V with VLL= 1VSolution: At supernode enclosing nodes a and b :R3R2V 4VaVb+-VLLV1R4R1+-+-44b3b2a1a1)/RV-(V/R V/RV)/RV-(V+=−andVb= Va+ VLLThus:Va= 0.25Vb= 1.25Be sure to check answer with KCL!4Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003CHECK ANSWER WITH KCLIs Va= 1.25 and Vb= 0.25 if R1= R2= R3= R4= 1MΩ, and V1= V4=1.5V with VLL= 1V ????KCL at the Supernode: 0.25 -1.25 + 1.25 - 0.25 =0Clearly the current into the supernode is zero and we have verified that the solution is correct. :R3R2V 4VaVb+-VLLV1R4R1+-+-0.25 1.251.01.0M1.0M 1.0M1.0M1. 5 1. 51.25µA0.25µA0.25µA 1.25µACopyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003RESISTORS IN PARALLELR2R1ISSI2I12 Define unknown node voltagesVX1 Select Reference Node21SSXR1R11IV+⋅=⇒Note: Iss= I1+ I2, i.e.,2X1XSSRVRVI +=2121SSRRRRI+⋅=RESULT 1 EQUIVALENT RESISTANCE:212121||RRRRR||RR+=≡RESULT 2 CURRENT DIVIDER:211SS2X2212SS1X1RRRI RV IRRRIRVI+×==+×==5Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003IDENTIFYING SERIES AND PARALLEL COMBINATIONSUse series/parallel equivalents to simplify a circuit before starting KVL/KCL1R2R3ReqR−+VIRRRR214R3R65K 10RR21== K20R3= K55R4== K10R6=−+IRX ?654321XR||)R(RR||)R(RR+++=K15=21RR +3Rparalleleq321RR||)RR(=+Please note the order of math operators here!Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003IDENTIFYING SERIES AND PARALLEL COMBINATIONS(cont.)Some circuits must be analyzed (not amenable to simple inspection) -+R2R1VIR4R3R5Special cases:R3= 0 OR R3= ∞R1and R5are notin seriesR1and R2are not in ||OR IF R3= ∞⇒(R1+ R5) || (R2+ R4)R1−+R4R5R2VR3Req= R1|| R2+ R4|| R5Example: R3= 0 ⇒ R1|| R2; R4|| R5in series;6Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003First Midterm Exam: Topics• Basic Circuit Analysis (KVL, KCL)• Equivalent Circuits and Graphical Solutions for Nonlinear Loads• Transients in Single Capacitor Circuits• Node Analysis Technique and Checking SolutionsExam is in class 3:10-4:03 PM, Closed book, Closed notes, Bring a calculator, Paper providedCopyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003I-V CHARACTERISTICS OF LINEAR TWO-TERMINAL NETWORKS5.51v(V)-.5i(mA)-1Unassociated(i defined out)If V = 2.5VIf R = 2.5Ki+−v+−5VApply v, measure i, or vice versaUnassociatedConsider how the graph changes with differences in V and R.First consider change in V, eg V= 2.5V, not 5V5KNow consider change in R (with V back at 5V)Clearly by varying V and R we can produce an arbitrary linear graph … in other words this circuit can produce anylinear graph7Copyright 2001, Regents of University of CaliforniaLecture 9: 02/24/03 A.R. NeureutherVersion Date 02/24/03EECS 42 Intro. electronics for CS Spring 2003FINDING VT, RTBY
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