EE100 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 EE100 Summer 2008 Slide 1 Bharathwaj Muthuswamy Generalization 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 i2 i3 This could be a big chunk of a circuit e g a black box EE100 Summer 2008 i4 i1 Slide 2 Bharathwaj Muthuswamy Generalized KCL Examples 50 mA 5 A 2 A i i EE100 Summer 2008 Slide 3 Bharathwaj Muthuswamy Using Kirchhoff s Voltage Law KVL Consider a branch which forms part of a loop One possibility for sign convention loop v1 loop v2 Moving from to We add V1 Moving from to We subtract V2 Use reference polarities to determine whether a voltage is dropped No concern about actual voltage polarities EE100 Summer 2008 Slide 4 Bharathwaj Muthuswamy Formulations of Kirchhoff s Voltage Law Conservation of energy Formulation 1 Sum of voltage drops around loop sum of voltage rises around loop Formulation 2 Algebraic sum of voltage drops around loop 0 Voltage rises are included with a minus sign Handy trick Look at the first sign you encounter on each element when tracing the loop Formulation 3 Algebraic sum of voltage rises around loop 0 Voltage drops are included with a minus sign EE100 Summer 2008 Slide 5 Bharathwaj Muthuswamy A 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 va vb Applying KVL in the clockwise direction starting at the top vb va 0 vb va EE100 Summer 2008 Slide 6 Bharathwaj Muthuswamy KVL Example Three closed paths 1 va b vb v2 v3 2 a c vc 3 Path 1 Path 2 Path 3 EE100 Summer 2008 Slide 7 Bharathwaj Muthuswamy I V Characteristic of Elements a Vab Find the I V characteristic i R i vs b v EE100 Summer 2008 Slide 8 Bharathwaj Muthuswamy More Examples Are these interconnections permissible EE100 Summer 2008 Slide 9 Bharathwaj Muthuswamy EE100 Summer 2008 Slide 10 Bharathwaj Muthuswamy Summary 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 energy EE100 Summer 2008 Slide 11 Bharathwaj Muthuswamy Chapters 3 and 4 Outline Resistors in Series Voltage Divider Conductances in Parallel Current Divider Node Voltage Analysis Mesh Current Analysis Superposition Th venin equivalent circuits Norton equivalent circuits Maximum Power Transfer EE100 Summer 2008 Slide 12 Bharathwaj Muthuswamy Resistors in Series Consider a circuit with multiple resistors connected in series Find their equivalent resistance I R1 VSS R2 KCL tells us that the same current I flows through every resistor KVL tells us R3 R4 Equivalent resistance of resistors in series is the sum EE100 Summer 2008 Slide 13 Bharathwaj Muthuswamy Voltage Divider I R1 VSS V1 I VSS R1 R2 R3 R4 R2 R3 V3 R4 EE100 Summer 2008 Slide 14 Bharathwaj Muthuswamy When can the Voltage Divider Formula be Used I I R1 VSS R2 R1 V2 VSS R3 R4 EE100 Summer 2008 V2 R3 R4 R 2 V V 2 SS R R R R 1 2 3 4 Correct if nothing else is connected to nodes R2 R5 R 2 V V 2 SS R R R R 1 2 3 4 Why What is V2 Slide 15 Bharathwaj Muthuswamy EE100 Summer 2008 Slide 16 Bharathwaj Muthuswamy Resistors in Parallel ISS Consider a circuit with two resistors connected in parallel Find their equivalent resistance x KVL tells us that the same voltage is dropped I1 I2 across each resistor R1 R2 Vx I1 R1 I2 R2 KCL tells us EE100 Summer 2008 Slide 17 Bharathwaj Muthuswamy General Formula for Parallel Resistors What single resistance Req is equivalent to three resistors in parallel I I V R1 R2 R3 eq V Req Equivalent conductance of resistors in parallel is the sum EE100 Summer 2008 Slide 18 Bharathwaj Muthuswamy EE100 Summer 2008 Slide 19 Bharathwaj Muthuswamy Current Divider x ISS EE100 Summer 2008 I1 I2 R1 R2 Slide 20 Vx I1 R1 ISS Req Bharathwaj Muthuswamy Generalized Current Divider Formula Consider a current divider circuit with 2 resistors in parallel I I1 R1 V I3 I2 R2 R3 V I 1 1 1 R1 R 2 R 3 1 R 3 V I I3 R3 1 R 1 R 1 R 1 2 3 EE100 Summer 2008 Slide 21 Bharathwaj Muthuswamy Measuring Voltage To 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 Voltmeter Rin EE100 Summer 2008 Slide 22 Bharathwaj Muthuswamy Effect of Voltmeter circuit with voltmeter inserted undisturbed circuit R1 VSS R1 R2 VSS V2 R2 Rin V2 Compare to R2 R2 V2 VSS R1 R 2 R 2 Rin V2 VSS R 2 Rin R1 Example VSS 10 V R 2 100 K R1 900 K V2 1V Rin 10 M V2 EE100 Summer 2008 Slide 23 Bharathwaj Muthuswamy EE100 Summer 2008 Slide 24 Bharathwaj Muthuswamy Measuring Current To 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 1 Ideal Ammeter Rin EE100 Summer 2008 Slide 25 Bharathwaj Muthuswamy Effect of Ammeter Measurement error due to non zero input resistance undisturbed circuit circuit with ammeter inserted Imeas I R1 R1 V1 V1 R2 V1 I R1 R 2 Rin R2 V1 Imeas R1 R 2 Rin Example V1 1 V R1 R2 500 Rin 1 1V I 1mA I meas 500 500 EE100 Summer 2008 ammeter Slide 26 Compare to R2 R2 Bharathwaj Muthuswamy Using Equivalent Resistances Simplify a circuit before applying KCL and or KVL Example Find I I R1 7V R2 R6 R4 R5 5 k R6 10 k R4 R5 EE100 Summer 2008 R3 R1 R2 3 k R3 6 k Slide 27 Bharathwaj Muthuswamy Wheatstone s Bridge Section 3 6 EE100 Summer 2008 Slide 28 Bharathwaj Muthuswamy Labs 1 and 2 COME ON TIME FOR THE LABS UNDERSTAND how to use the breadboard You need to get familiar with the instruments feel free to use TA office hours for extra help You will be
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