Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 002 Circuits and Electronics Spring 2003 Handout S03 013 Homework 2 Issued Wed Feb 12 Due Fri Feb 21 Note Do your work on problems 2 2 through 2 4 directly on these sheets and turn them in with your solutions to the remaining problems Problem 2 1 below Imagine a device called a widget which has the electrical characteristics shown a Input a b WIDGET Output b 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 and may 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 be approximated 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 voltages and or currents Voltages are in volts and currents are in amperes a 2 v1 5 v2 1 2 b 4 v1 v1 1 3 v3 v2 3 2 v1 v2 i1 v3 i2 v3 v2 v3 i1 i2 Name Section Problem 2 3 For each of the circuits below express the resistance R or the equivalent conductance G R1 at the terminals in terms of the element resistances Rn or conductances Gn R1n Write your answers next to the circuits a R1 R2 R3 b R1 c R2 R3 R1 R2 d R2 R1 R3 R3 Name Section e Can the same methods used in Parts a through d be used to find the resistance at the terminals of the circuit below If so express it if not explain why not R1 R4 R3 R2 R5 Problem 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 k a 3 k 12 V 1 k v 2 k b 6 mA 6 k v 1 k 2 k i Problem 2 5 The circuit show below has two independent sources All elements are assigned numerical values e1 G2 0 5 e2 G3 1 I1 2 mA G1 1 1 5 V G4 2 The units are volts V milliamperes mA and millimhos mmho 1 milliampere 10 3 amperes 1 millimho 10 3 mhos The conductance of a resistor of R ohms is R1 mhos The questions which follow illustrate the utility of the superposition principle in linear circuits 1 Assume that the 2mA current source I1 is 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 e 1 and e2 in terms of the conductances You should be able to do this without solving simultaneous equations by using 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 determine the values of e1 and e2 for I1 acting alone Specify units 4 Repeat parts 1 2 and 3 for the 1 5 V voltage source active and the other source dead Use series and parallel reductions 5 Verify that the node equations for the complete circuit are G1 G2 e1 G 2 e2 I 1 G2 e1 G2 G3 G4 e2 G3 V After turning the algebraic crank the symbolic forms of the two node equations for the original two source circuit shown above yield the following solutions e1 e2 I1 G2 G3 G4 V G2 G3 G1 G2 G3 G4 G2 G3 G4 I1 G2 V G1 G2 G3 G1 G2 G3 G4 G2 G3 G4 When the component values are inserted in these equations the results are e1 1 55V e2 0 65V Add together the two component parts of e1 and e2 calculated in parts 3 and 4 That is e1 e1 I1 Acting e1 V Acting Alone Alone and 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 solving the node equations simultaneously The amount of grind required is less when superposition is employed This will be true whenever the circuit topology is such that series parallel reductions can be employed
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