EE100 Su08 Lecture #3 (June 27th 2008)Generalization of KCLGeneralized KCL ExamplesUsing Kirchhoff’s Voltage Law (KVL)Formulations of Kirchhoff’s Voltage LawA Major Implication of KVLKVL ExampleI-V Characteristic of ElementsMore ExamplesSlide 10SummaryChapters 3 and 4Resistors in SeriesVoltage DividerWhen can the Voltage Divider Formula be Used?Slide 16Resistors in ParallelGeneral Formula for Parallel ResistorsSlide 19Current DividerGeneralized Current Divider FormulaMeasuring VoltageEffect of VoltmeterSlide 24Measuring CurrentEffect of AmmeterUsing Equivalent ResistancesWheatstone’s Bridge (Section 3.6)Labs #1 and #2Slide 1EE100 Summer 2008 Bharathwaj MuthuswamyEE100 Su08 Lecture #3 (June 27th 2008)•Administrivia–Videos for lectures 1 and 2 are up (WMV format). Quality is pretty good .•For today:–Questions?–Wrap up chapter 2.–Start chapter 3. Refer to pdf slides for week 2.–An outline of Labs #1 and #2.Slide 2EE100 Summer 2008 Bharathwaj MuthuswamyGeneralization of KCL•The sum of currents entering/leaving a closed surface is zero. Circuit branches can be inside this surface, i.e. the surface can enclose more than one node!This could be a big chunk of a circuit, e.g. a “black box”i1i2i3i4Slide 3EE100 Summer 2008 Bharathwaj MuthuswamyGeneralized KCL Examples5A2Ai50 mAiSlide 4EE100 Summer 2008 Bharathwaj Muthuswamy•Use reference polarities to determine whether a voltage is dropped •No concern about actual voltage polaritiesUsing Kirchhoff’s Voltage Law (KVL)Consider a branch which forms part of a loop. One possibility for sign convention: +v1_loopMoving from + to -We add V1–v2+loopMoving from - to +We subtract V2Slide 5EE100 Summer 2008 Bharathwaj MuthuswamyFormulations of Kirchhoff’s Voltage LawFormulation 1: Sum of voltage drops around loop = sum of voltage rises around loopFormulation 2:Algebraic sum of voltage drops around loop = 0•Voltage rises are included with a minus sign.Formulation 3:Algebraic sum of voltage rises around loop = 0•Voltage drops are included with a minus sign.(Conservation of energy)(Handy trick: Look at the first sign you encounter on each element when tracing the loop.)Slide 6EE100 Summer 2008 Bharathwaj MuthuswamyA Major Implication of KVL•KVL tells us that any set of elements which are connected at both ends carry the same voltage.•We say these elements are connected in parallel.Applying KVL in the clockwise direction, starting at the top:vb – va = 0 vb = va +va _+ vb_Slide 7EE100 Summer 2008 Bharathwaj MuthuswamyPath 1:Path 2:Path 3:vcva++321+ vbv3v2+ +-Three closed paths:abcKVL ExampleSlide 8EE100 Summer 2008 Bharathwaj MuthuswamyI-V Characteristic of ElementsFind the I-V characteristic.+_vsii+Vab_vabRSlide 9EE100 Summer 2008 Bharathwaj MuthuswamyMore Examples•Are these interconnections permissible?Slide 10EE100 Summer 2008 Bharathwaj MuthuswamySlide 11EE100 Summer 2008 Bharathwaj MuthuswamySummary•An electrical system can be modeled by an electric circuit (combination of paths, each containing 1 or more circuit elements) –Lumped model•The Current versus voltage characteristics (I-V plot) is a universal means of describing a circuit element.•Kirchhoff’s current law (KCL) states that the algebraic sum of all currents at any node in a circuit equals zero.–Comes from conservation of charge•Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around any closed path in a circuit equals zero.–Comes from conservation of potential energySlide 12EE100 Summer 2008 Bharathwaj MuthuswamyChapters 3 and 4•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 13EE100 Summer 2008 Bharathwaj MuthuswamyConsider 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 14EE100 Summer 2008 Bharathwaj MuthuswamyI = VSS / (R1 + R2 + R3 + R4)Voltage Divider+–V1+–V3R2R1VSSIR3R4 +Slide 15EE100 Summer 2008 Bharathwaj MuthuswamySS432122VRRRRRV Correct, if nothing elseis connected to nodesWhy? What is V2? SS432122VRRRRRV ≠When can the Voltage Divider Formula be Used?+–V2R2R1VSSIR3R4 +R2R1VSSIR3R4 +R5+–V2Slide 16EE100 Summer 2008 Bharathwaj MuthuswamySlide 17EE100 Summer 2008 Bharathwaj Muthuswamy•KVL tells us that the same voltage is dropped across each resistorVx = I1 R1 = I2 R2•KCL tells usR2R1 ISSI2I1xResistors in ParallelConsider a circuit with two resistors connected in parallel.Find their “equivalent resistance”.Slide 18EE100 Summer 2008 Bharathwaj MuthuswamyWhat single resistance Req is equivalent to three resistors in parallel?+VIV+IR3R2R1ReqeqGeneral Formula for Parallel ResistorsEquivalent conductance of resistors in parallel is the sumSlide 19EE100 Summer 2008 Bharathwaj MuthuswamySlide 20EE100 Summer 2008 Bharathwaj MuthuswamyVx = I1 R1 = ISS ReqCurrent DividerR2R1 ISSI2I1xSlide 21EE100 Summer 2008 Bharathwaj MuthuswamyR2R1 II2I1I3R3+V321R1R1R1IV3213331/R1/R1/R1/RIRVIGeneralized Current Divider FormulaConsider a current divider circuit with >2 resistors in parallel:Slide 22EE100 Summer 2008 Bharathwaj MuthuswamyTo 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 M)Ideal VoltmeterRinMeasuring VoltageSlide 23EE100 Summer 2008 Bharathwaj Muthuswamy212SS2RRRVV1in2in2SS2RR||RR||RVVExample: V1VK900R ,K100R ,V10V212SSVSSR1R2210 , ?inR M V�= =Effect of Voltmeterundisturbed circuitcircuit with voltmeter inserted_++–V2VSSR1R2Rin_++–V2′Compare to R2Slide 24EE100 Summer 2008 Bharathwaj MuthuswamySlide 25EE100 Summer 2008 Bharathwaj MuthuswamyTo measure the current flowing through an element in a real circuit, insert an ammeter (digital multimeter in current mode) in series
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