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Slide 1EE40 Fall2006Prof. Chang-HasnainEE40Lecture 3Prof. Chang-Hasnain8/31/07Reading: Chap. 2Slide 2EE40 Fall2006Prof. Chang-HasnainChapter 2• Outline– Resistors in Series – Voltage Divider– Conductances in Parallel – Current Divider– Node-Voltage Analysis– Mesh-Current Analysis– Superposition– Thévenin equivalent circuits– Norton equivalent circuits– Maximum Power TransferSlide 3EE40 Fall2006Prof. Chang-HasnainConsider a circuit with multiple resistors connected in series.Find their “equivalent resistance”.• KCL tells us that the same current (I) flows through every resistor• KVL tells usEquivalent resistance of resistors in series is the sumR2R1VSSIR3R4−−−−+Resistors in SeriesSlide 4EE40 Fall2006Prof. Chang-HasnainI = VSS/ (R1+ R2+ R3+ R4)Voltage Divider+–V1+–V3R2R1VSSIR3R4−−−−+Slide 5EE40 Fall2006Prof. Chang-HasnainSS432122VRRRRRV⋅+++=Correct, if nothing elseis connected to nodesWhy? What is V2?SS432122VRRRRRV⋅+++≠When can the Voltage Divider Formula be Used?+–V2R2R1VSSIR3R4−−−−+R2R1VSSIR3R4−−−−+R5+–V2Slide 6EE40 Fall2006Prof. Chang-Hasnain• KVL tells us that thesame voltage is droppedacross each resistorVx= I1R1= I2R2• KCL tells usR2R1ISSI2I1xResistors in ParallelConsider a circuit with two resistors connected in parallel.Find their “equivalent resistance”.Slide 7EE40 Fall2006Prof. Chang-HasnainWhat single resistance Reqis equivalent to three resistors in parallel?+−VIV+−IR3R2R1Reqeq≡General Formula for Parallel ResistorsEquivalent conductance of resistors in parallel is the sumSlide 8EE40 Fall2006Prof. Chang-HasnainVx= I1R1= ISSReqCurrent DividerR2R1ISSI2I1xSlide 9EE40 Fall2006Prof. Chang-HasnainR2R1 II2I1I3R3+−V++=321R1R1R1IV++==3213331/R1/R1/R1/RIRVIGeneralized Current Divider FormulaConsider a current divider circuit with >2 resistors in parallel:Slide 10EE40 Fall2006Prof. Chang-HasnainTo measure the voltage drop across an element in a real circuit, insert a voltmeter (digital multimeter in voltage mode) in parallel with the element. Voltmeters are characterized by their “voltmeter input resistance” (Rin). Ideally, this should be very high (typical value 10 MΩ)Ideal VoltmeterRinMeasuring VoltageSlide 11EE40 Fall2006Prof. Chang-Hasnain+=212SS2RRRVV+=′1in2in2SS2RR||RR||RVVExample: V1VK900R ,K100R ,V10V212SS=⇒===VSSR1R2210 , ?inR M V′= =Effect of Voltmeterundisturbed circuitcircuit with voltmeter inserted_++–V2VSSR1R2Rin_++–V2′Compare to R2Slide 12EE40 Fall2006Prof. Chang-HasnainEE40Lecture 4Prof. Chang-Hasnain9/5/07Reading: Chap. 2Slide 13EE40 Fall2006Prof. Chang-HasnainTo measure the current flowing through an element in a real circuit, insert an ammeter (digital multimeter in current mode) in series with the element. Ammeters are characterized by their “ammeter input resistance” (Rin). Ideally, this should be very low (typical value 1Ω).Ideal AmmeterRinMeasuring CurrentSlide 14EE40 Fall2006Prof. Chang-HasnainRinV1ImeasR1R2ammetercircuit with ammeter inserted_+V1IR1R2undisturbed circuitExample: V1= 1 V, R1= R2= 500 Ω, Rin= 1Ω211RRVI+=in211measRRRVI++=11 , ?500 500measVI mA I= = =Ω + ΩEffect of AmmeterMeasurement error due to non-zero input resistance:_+Compare to R2+ R2Slide 15EE40 Fall2006Prof. Chang-HasnainSimplify a circuit before applying KCL and/or KVL:−+7 VUsing Equivalent ResistancesR1= R2= 3 kΩR3= 6 kΩR4= R5= 5 kΩR6= 10 kΩIR1R2R4R5R3R6Example: Find ISlide 16EE40 Fall2006Prof. Chang-Hasnain1. Choose a reference node (“ground”)Look for the one with the most connections!2. Define unknown node voltagesthose which are not fixed by voltage sources3. Write KCL at each unknown node, expressing current in terms of the node voltages (using the I-V relationships of branch elements)Special cases: floating voltage sources4. Solve the set of independent equationsN equations for N unknown node voltagesNode-Voltage Circuit Analysis MethodSlide 17EE40 Fall2006Prof. Chang-Hasnain1. Choose a reference node.2. Define the node voltages (except reference node and the one set by the voltage source).3. Apply KCL at the nodes with unknown voltage.4. Solve for unknown node voltages.R4V1R2+-ISR3R1Nodal Analysis: Example #1Slide 18EE40 Fall2006Prof. Chang-HasnainA “floating” voltage source is one for which neither side is connected to the reference node, e.g. VLLin 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 Va-Vb.Solution: Define a “supernode” – that chunk of the circuit containing nodes a andb. Express KCL for this supernode. Incorporate voltage source constraint into KCL equation. R4R2I2VaVb+-VLLI1Nodal Analysis w/ “Floating Voltage Source”Slide 19EE40 Fall2006Prof. Chang-HasnainsupernodeEq’n 1: KCL at supernodeSubstitute property of voltage source:R4R2I2VaVb+-VLLI1Nodal Analysis: Example #2Slide 20EE40 Fall2006Prof. Chang-HasnainV2V1R2R1R4R5R3I1VaNodal Analysis: Example #3Challenges: Determine number of nodes needed Deal with different types of sourcesSlide 21EE40 Fall2006Prof. Chang-HasnainNODAL ANALYSIS(“Node-Voltage Method”)0) Choose a reference node 1) Define unknown node voltages2) Apply KCL to each unknown node, expressing current in terms of the node voltages=> N equations forN unknown node voltages3) Solve for node voltages=> determine branch currentsMESH ANALYSIS(“Mesh-Current Method”)1) Select M independent mesh currents such that at least one mesh current passes through each branch*M = #branches - #nodes + 12) Apply KVL to each mesh, expressing voltages in terms of mesh currents=> M equations forM unknown mesh currents3) Solve for mesh currents=> determine node voltagesFormal Circuit Analysis Methods*Simple method for planar circuitsA mesh current is not necessarily identified with a branch current.Slide 22EE40 Fall2006Prof. Chang-Hasnain1. Select M mesh currents.2. Apply KVL to each mesh.3. Solve for mesh currents.Mesh Analysis: Example #1Slide 23EE40 Fall2006Prof. Chang-HasnainEq’n 1: KVL for supermeshEq’n 2: Constraint due to current source:Mesh Analysis: Example #2iaibSlide 24EE40 Fall2006Prof. Chang-HasnainMesh Analysis with Dependent Sources• Exactly analogous to Node Analysis• Dependent Voltage Source: (1) Formulate and write KVL mesh eqns. (2) Include and express


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Berkeley ELENG 40 - Lecture Notes 3

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