Slide 1EE100 Summer 2008 Bharathwaj MuthuswamyEE100Su08 Lecture #6 (July 7th2008)• Outline– Today:• Midterm on Monday, 07/14/08 from 2 – 4 pm– Second room location changed to 120 Latimer• Questions?• Chapter 4 wrap up– Thevenin and Norton– Source Transformations– Miscellaneous:» Maximum Power Transfer theorem• Chapter 6 wrap up– Capacitors (definition, series and parallel combination)– Inductors (definition, series and parallel combination)• Chapter 7: Intuitive IntroductionSlide 2EE100 Summer 2008 Bharathwaj MuthuswamyRecap: Thevenin EquivalentsSlide 3EE100 Summer 2008 Bharathwaj MuthuswamyRThCalculation Example #2Find the Thevenin equivalent with respect to the terminals a,b:Since there is no independent source and we cannot arbitrarily turn off the dependence source, we can add a voltage source Vxacross terminals a-b and measure the current through this terminal Ix. Rth= Vx/ IxVx+-IxSlide 4EE100 Summer 2008 Bharathwaj MuthuswamyNorton equivalent circuitNorton Equivalent Circuit•Any* linear 2-terminal (1-port) network of indep. voltage sources, indep. current sources, and linear resistors can be replaced by an equivalent circuit consisting of an independent current source in parallel with a resistorwithout affecting the operation of the rest of the circuit.networkofsourcesandresistors≡RLiL+vL–abaRLiL+vL–iNbRNSlide 5EE100 Summer 2008 Bharathwaj MuthuswamyI-V Characteristic of Norton Equivalent• The I-V characteristic for the parallel combination of elements is obtained by adding their currents:ii = IN-GvI-V characteristic of resistor: i=GvI-Vcharacteristic of current source: i = -INFor a given voltage vab, the current i is equal to the sum of the currents in each of the two branches:vi+vab–iNbRNaSlide 6EE100 Summer 2008 Bharathwaj MuthuswamyFinding INand RN =RThIN≡ isc= VTh/RThAnalogous to calculation of Thevenin Eq. Ckt:1) Find o.c voltage and s.c. current2) Or, find s.c. current and Norton (Thev) resistanceSlide 7EE100 Summer 2008 Bharathwaj MuthuswamySource Transforms: Finding INand RN• We can derive the Norton equivalent circuit from a Thévenin equivalent circuit simply by making a “source transformation”: RLRNiLiN+vL–ab–+RLiL+vL–vThRThscThThNscocThN ; iRviivRR ====abSlide 8EE100 Summer 2008 Bharathwaj MuthuswamySource TransformationsSlide 9EE100 Summer 2008 Bharathwaj MuthuswamySource TransformationsSlide 10EE100 Summer 2008 Bharathwaj MuthuswamySource TransformationsSlide 11EE100 Summer 2008 Bharathwaj MuthuswamySource TransformationsSlide 12EE100 Summer 2008 Bharathwaj MuthuswamyMaximum Power Transfer TheoremA resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit.L2LThThL2LRRRVRip⎟⎟⎠⎞⎜⎜⎝⎛+==–+VThRThRLiL+vL–Thévenin equivalent circuit()()()() ()LThLThL2LTh4LThLThL2LTh2Th 02 02RRRRRRRRRRRRRRVdRdpL=⇒=+×−+⇒=⎥⎦⎤⎢⎣⎡++×−+=To find the value of RLfor which p is maximum, set to 0:Power absorbed by load resistor:LdRdpSlide 13EE100 Summer 2008 Bharathwaj MuthuswamySummary of Techniques for Circuit Analysis -1• Resistor network (Chapter 3)– Parallel resistors– Series resistors– Voltage Divider– Current Divider– Voltmeters and Ammeters– Y-delta conversion: OPTIONALSlide 14EE100 Summer 2008 Bharathwaj MuthuswamySummary of Techniques for Circuit Analysis -2• Node Analysis (Chapter 4)– Node voltage is the unknown– Solve for KCL– Floating voltage source using super node • Superposition– Leave one independent source on at a time– Sum over all responses– Voltage off Æ SC– Current off Æ OC• Mesh Analysis: OPTIONAL– Loop current is the unknown– Solve for KVL– Current source using super mesh• Thevenin and Norton Equivalent Circuits– Solve for OC voltage– Solve for SC current• Source Transforms:– Voltage sources in series with a resistance can be converted to current source in parallel with a resistance.Slide 15EE100 Summer 2008 Bharathwaj MuthuswamyComments on Dependent Sources• Node-Voltage Method – Dependent current source: • treat as independent current source in organizing node eqns• substitute constraining dependency in terms of defined node voltages.– Dependent voltage source: • treat as independent voltage source in organizing node eqns• Substitute constraining dependency in terms of defined node voltages.Slide 16EE100 Summer 2008 Bharathwaj MuthuswamyComments on Dependent Sources (contd.)A dependent source establishes a voltage or current whose value depends on the value of a voltage or current at a specified location in the circuit.(device model, used to model behavior of transistors & amplifiers)To specify a dependent source, we must identify:1. the controlling voltage or current (must be calculated, in general)2. the relationship between the controlling voltage or current and the supplied voltage or current3. the reference direction for the supplied voltage or currentThe relationship between the dependent sourceand its reference cannot be broken!– Dependent sources cannot be turned off for various purposes (e.g. to find the Thévenin resistance, or in analysis using Superposition).Slide 17EE100 Summer 2008 Bharathwaj MuthuswamyChapters 6• Outline– The capacitor– The inductorSlide 18EE100 Summer 2008 Bharathwaj MuthuswamyThe CapacitorTwo conductors (a,b) separated by an insulator:difference in potential = Vab=> equal & opposite charge Q on conductorsQ = CVabwhere C is the capacitance of the structure, ¾ positive (+) charge is on the conductor at higher potentialParallel-plate capacitor:• area of the plates = A (m2)• separation between plates = d (m)• dielectric permittivity of insulator = ε(F/m)=> capacitancedACε=(stored charge in terms of voltage)FSlide 19EE100 Summer 2008 Bharathwaj MuthuswamyA note on circuit variablesSlide 20EE100 Summer 2008 Bharathwaj MuthuswamySymbol:Units: Farads (Coulombs/Volt)Current-Voltage relationship:orNote: Q (vc) must be a continuous function of timeCapacitor+vc–icdtdCvdtdvCdtdQiccc+==C C(typical range of values: 1 pF to 1 µF; for “supercapa-citors” up to a few F!)+Electrolytic (polarized)capacitorCIf C (geometry) is unchanging, iC= C dvC/dtSlide 21EE100 Summer 2008 Bharathwaj MuthuswamyVoltage in Terms of Current)0()(1)0()(1)()0()()(000ctctcctcvdttiCCQdttiCtvQdttitQ+=+=+=∫∫∫Uses: Capacitors are used to store energy for camera flashbulbs,in filters that separate various
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