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MIT 6 002 - Homework #2

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Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits and ElectronicsSpring 2003Handout S03-013 - Homework #2Issued: Wed. Feb 12Due: Fri. Feb 21Note: Do your work on problems 2.2 through 2.4 directly on these sheets and turn them in withyour solutions to the remaining problems.Problem 2.1: Imagine a device called a “widget” which has the electrical characteristics shownbelow:WIDGETaa’InputOutputbb’There is no electrical connection between the input terminals and the output terminals.When a signal of 0 Volts is applied to aa’, the resistance between b and b’ is negligibly small, andmay be approximated by a short circuit.When a ±5 V signal is applied to aa’, the resistance between b and b’ is very large, and may beapproximated by an open circuit.In the following problems, you are free to use constant voltage sources and resistors.(A) Can widgets be combined to create a NAND gate? Demonstrate.(B) Can widgets be combined to create a AND gate? Demonstrate.(C) Can widgets be combined to create a NOR gate? Demonstrate.(D) Can widgets be combined to create a OR gate? Demonstrate.Name:Section:Problem 2.2: In each of the following circuits, determine the values of the indicated voltagesand/or currents.. (Voltages are in volts and currents are in amperes.)(a)-v1v2v3++ +++-----+152v1=v2=v3=(b)-v2+-v3+3+-1+--+-+4v13i1i2-22v1=v2=v3=i1=i2=Name:Section:Problem 2.3: For each of the circuits below express the resistance R, or the equivalentconductance G =1R, at the terminals in terms of the element resistances Rn(or conductancesGn=1Rn). Write your answers next to the circuits.(a)R1R2R3(b)R2R1R3(c)R2R1R3(d)R2R1R3Name:Section:(e) Can the same methods used in Parts (a) through (d) be used to find the resistance at theterminals of the circuit below? If so, express it; if not, explain why not.R3R4R1R5R2Problem 2.4: In each of the following circuits determine the voltages and/or currents indicated.The units are volts (V), milliamperes (mA), and kilo-ohms (kΩ).(a)+−12V3kΩ1kΩ2kΩ+-v=?(b)1kΩ6kΩ 2kΩi=?+-v=?6mAProblem 2.5: The circuit show below has two independent sources. All elements are assignednumerical values.+−e1e2G1=1G2=0.5G3=1G4=21.5VI1=2mAThe units are volts (V), milliamperes (mA), and millimhos (mmho) (1 milliampere = 10−3amperes,1 millimho = 10−3mhos. The conductance of a resistor of R ohms is1Rmhos).The questions which follow illustrate the utility of the superposition principle in linear circuits.1) Assume that the 2mA current source I1is active and the voltage source is inactive or dead.Redraw the circuit on your solution pages under these conditions.2) Analyze this reduced circuit to obtain symbolic expressions for the voltages e1and e2in termsof the conductances. You should be able to do this withoutsolving simultaneous equations byusing parallel and series reductions to find e1, and then working to the right to find e2.3) Substitute numbers for the symbolic element values in your results for part 2 and determinethe values of e1and e2for I1acting alone. Specify units!4) Repeat parts 1, 2, and 3 for the 1.5 V voltage source active and the other source dead. Useseries and parallel reductions.5) Verify that the node equations for the complete circuit are:(G1+ G2)e1− G2e2= I1−G2e1+ (G2+ G3+ G4)e2= G3VAfter turning the algebraic crank, the symbolic forms of the two node equations for the originaltwo-source circuit shown above yield the following solutions:e1=I1(G2+ G3+ G4) + V G2G3G1(G2+ G3+ G4) + G2(G3+ G4)e2=I1G2+ V (G1+ G2)G3G1(G2+ G3+ G4) + G2(G3+ G4)When the component values are inserted in these equations, the results are:e1= 1.55Ve2= 0.65VAdd together the two component parts of e1and e2calculated in parts 3) and 4). That is:e1= e1¯¯¯¯I1ActingAlone+ e1¯¯¯¯V ActingAloneand similiarly for e2. If your analysis is correct, the values of the voltages obtained in this manner– that is, by employing the superposition principle – will be the same as those obtained by solvingthe node equations simultaneously.The amount of grind required is less when superposition is employed. This will be true wheneverthe circuit topology is such that series/parallel reductions can be


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MIT 6 002 - Homework #2

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