Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits & ElectronicsSpring 2005Problem Set #2Issued 2/9/05 – Due 2/16/05Exercise 2.1: In each circuit shown below, three light bulbs are driven by a single source. Intwo of the circuits the source is a voltage s ou rce, and in the other it is a current source. Assumethat one of th e three light bulbs burns out and becomes an open circuit. Do the other two lightbulbs get brighter, get dimmer, or exhibit no change in intensity? Why?Exercise 2.2: Using the nod e m ethod, develop a set of simultaneous equations for the networkshown below that can be used to solve for the three unknown node voltages in the network. Expressthese equations in th e formGe1e2e3= Swhere G is a 3 × 3 matrix of conductance terms and S is a 3 × 1 vector of terms involving thesources. You need not solve the set of equations for the nod e voltages.R1e1R2R5e3R4e2R3IR6VProblem 2.1: This problem analyzes the n etwork shown below by two methods: superpositionand the direct app lication of the node method. You should compare for yourself the work r equiredto analyze the network by these two methods.(A) First, use superposition to determine e1and e2. That is, s uperpose the two partial nodevoltages obtained with only single sources active to find the total node voltages. Rememberthat a zero-valued voltage source is a short circuit, and a zero-valued current s ou rce is an opencircuit. Hint: rather than employing the node method twice, once for each partial analysis,consider employing alternative simpler analyses involving th e use of parallel and series resistorcombinations, and voltage and current dividers.(B) Second, use the node method to directly determine e1and e2in total.(C) Compare the solutions to Parts (A) and (B). The two solutions should be the same.VR1e1R3e2R4IR2Problem 2.2: Two networks, N1 and N2, are described graphically in terms of their i-vrelations, and connected together through a single resistor, as shown below.(A) Find the Thevenin and Norton equivalents of N1 and N2.(B) Find the currents i1and i2that result from the interconnection of N1 and N2.i2+v2−Ri1+v1−N 1 N 2ivN 1N 2−I1V1−V2I2Problem 2.3: Find the Thevenin and Norton equivalents of the following n etworks, and graphtheir i-v relations as viewed at their ports.IR1i+v−R2Network AVR2i+v−R1Network BVR2R1i+v−INetwork CProblem 2.4: This problem studies the network shown below. The network contains anonlinear resistor having the terminal relation iN= αv2Nfor vN≥ 0 and iN= 0 f or vN≤ 0, whereα is a constant with units A/V2. Assume that α and vSare both positive.(A) Analyze the network graphically to determine iNand vNin terms of vSand the networkparameters. To do so, note that the current source and linear resistor together constrain therelation between iNand vN, and that the nonlinear r esistor also constrains this relation. Statethe two constraints, and on a single graph sketch both constraints and identify the solutionfor iNand vN. Within what voltage range will vNlie?(B) Analytically solve for vNin terms of vS. Check that this solution is consistent with the graphicalsolution from Part (A).(C) Now let vS= VS+ vsand let vN= VN+ vn, where VSand VNare constant large-signal voltageswhich together form an operating point, and vsand vnare time-varying small-signal voltages.Using th e solution from Part (B), determine VNin terms of VS. Then , linearize the solutionfrom Part (B) around the operating point to determine vnapproximately in terms of
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