MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 002 Electronic Circuits Fall 2002 Quiz 1 Solutions Name Recitation Section Recitation Instructor Teaching Assistant Enter all your work and your answers directly in the spaces provided on the printed pages Make sure that your name is on all sheets Use the backs of the printed pages as scratch paper but we will only grade the work that you neatly transfer to the spaces on the printed pages Answers must be derived or explained not just simply written down The quiz is closed book but calculators are allowed This quiz contains 9 pages including the cover sheet Make sure that your quiz contains all 9 pages and that you hand in all 9 pages Problem Points Grade Grader 1 30 2 40 3 30 Total 100 Name Solutions Problem 1 Figure 1 2 30 points For parts A C use the associated branch variables as de ned in Figure 1 Circuit for Problem 1 A 1 B and 1 C A Write element laws for the resistor R1 the current source IA and the voltage source VB v1 R1 i1 i7 IA v6 VB B Write a complete set of independent KVL equations expressed only in terms of the branch voltages v1 v2 and v8 There are several possible answers one set of independent KVL equations is v1 v2 v6 v5 0 v3 v7 v2 v6 0 v3 v4 v8 0 C Write a complete set of independent KCL equations expressed only in terms of the branch currents i1 i2 and i8 Any ve of the following is a complete set of independent KCL equations i1 i5 0 i6 i2 0 i2 i1 i7 0 i4 i7 i3 0 i8 i4 0 i5 i6 i3 i8 0 Name Solutions 3 D The same circuit is shown in Figure 2 labeled for node analysis Write out the node equations necessary to solve for the three unknown node voltages e1 e2 and e3 DO NOT SOLVE these equations Figure 2 Circuit for Problem 1 D The node equations are e1 VA e1 VB IA 0 R1 R2 e2 e3 e2 IA 0 R3 R4 e3 e2 IB 0 R4 In matrix form we have 1 R1 0 0 1 R2 0 1 1 R3 R4 R14 VA VB 0 e1 R1 R2 IA R14 e2 IA 1 e3 IB R4 Name Solutions 4 E The current i1 in Figure 3 can be written in the form i1 aVA bVB cIA dIB Determine the coe cients a b c and d in terms of the resistor variables in the circuit Figure 3 Circuit for Problem 1 E Here superposition is used to determine each constant The four gures below depict the circuit when only one source is active i1 i1 R 1 R 1 R 2 A R 2 V B IA i1 i1 R 1 R 1 R 2 R 2 R 3 When VA is the only source active i1 R 4 VA R1 R2 R 3 IB Name Solutions When only VB is active i1 VB R1 R2 i1 R2 IA R1 R2 When only IA is active From the gures above IB does not contribute to i1 a 1 R1 R2 b 1 R1 R2 c R2 R1 R2 d 0 5 Name Solutions 6 Problem 2 40 points Network 1 shown in Figure 4 is described by its v i relationship measured at the terminals Network 2 shown in Figure 5 is described by a schematic diagram of its components A Find the The venin and Norton equivalent circuits that have the same v i relationship as Network 1 Figure 4 Network for Problem 2 A vOC 1 6V iSC 8mA vOC 0 2k RT H iSC i 0 2 k 1 6 V i v 0 2 k 8 m A T h e v e n in v N o r to n Name Solutions 7 B Find the The venin and Norton equivalent circuits that have the same v i relationship as Network 2 Figure 5 Network for Problem 2 B Suppressing all the sources and looking at the resistance into the port gives RT H 6 RT H 2k 3k k 1 2k 5 Using superposition vOC and iSC can be determined 2 3 2 2V 2V vOC V 5 2 3 5 1 iSC 2mA mA mA 3 3 i 1 2 k 2 V i v 5 m A 3 1 2 k v T h e v e n in N o r to n Name Solutions 8 C Suppose that the two networks are connected together through a resistor as shown in Figure 6 Find the current i1 and the voltage v1 Figure 6 Network for Problem 2 C First redraw the circuit using the The venin equivalents of each network 0 2 k 1 6 V 0 6 k i1 1 2 k v1 i1 2V 1 6V 0 2mA 2k Using superposition and voltage dividers 0 2 1 8 1 6 2 2 2 1 44 0 2 1 64V v1 i1 0 2mA v1 1 64V 2 V Name Solutions 9 D Suppose that the two networks are connected together through a resistor as shown in Figure 7 Find the current i2 and the voltage v2 Figure 7 Network for Problem 2 D First redraw the circuit using the Norton equivalents of each network 0 2 k 8 m A v2 1 2 k 1 2 k i2 5 mA 8mA 0 2k 1 2k 1 2k v2 3 29 mA 0 15k 1 45V 3 29 1 45V mA 1 208mA i2 1 2k 24 i2 1 208mA v2 1 45V 5 m A 3 Name Solutions 10 Problem 3 30 points Determine all node potentials in the network shown in Figure 8 in terms of the conductances of the resistors GA GB GC GD GE and GF the current sources IC and IF and the voltage sources VA VB VD and VE Figure 8 Circuit for Problem 3 The potentials e1 and e2 can be determined immediately from the circuit e1 VA e2 VA VB Noting that e3 e4 and e5 are related in the following way e5 e3 VD VE e4 e3 VD e5 e4 VE the circuit can be simpli ed as shown below Name Solutions 11 I C e2 e3 G C VA VB VD VE e5 G F S u p e rn o d e I F Now we write a node equation for the supernode encompassing nodes 3 4 and 5 e3 VA VB GC e5 GF IC IF Using e5 e3 VD VE in the equation above gives e3 GC e3 GF GC VA VB GF VD VE IC IF GC VA VB GF VD VE IC IF e3 GC GF Using the relationship e4 e3 VD gives an expression for e4 GC VA VB GF VD VE IC IF VD GC GF GC VA VB VD GF VE IC IF e4 GC GF e4 Using the relationship e5 e4 VE gives an expression for e5 GC VA VB VD GF VE IC IF VE GC …
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