Designing Information Devices and Systems I Anant Sahai, Ali Niknejad. This homework is due October 19, 2015, at Noon.
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1 EECS 16A Designing Informtion Devices nd Systems I Fll 2015 Annt Shi, Ali Niknejd Homework 7 This homework is due Octoer 19, 2015, t Noon. 1. Circuits with cpcitors nd resistors () Find the voltges cross nd currents flowing through ll the cpcitors. () Find the voltges cross nd currents flowing through ll the resistors nd cpcitors. 2. Thévenin nd Norton equivlent circuits () Find the Thévenin nd Norton equivlent circuits seen from the outside the ox. EECS 16A, Fll 2015, Homework 7 1
2 () Find the Thévenin nd Norton equivlent circuits seen from the outside the ox. 3. Nodl Anlysis Using techniques presented in clss, lel ll unknown node voltges nd pply KCL to ech node to find ll the node voltges. You my use the concept of super nodes" to simplify clcultions. () Solve for ll node voltges using nodl nlysis. Verify with superposition. 20Ω 1 1A 10Ω 2 2A 50Ω () Solve for ll node voltges using nodl nlysis. 50Ω 10V 10Ω 1V 55Ω 10 Vx 20Ω Vx 60Ω (c) Setup mtrix of equtions in the form Av = s using nodl nlysis. The vector v = (v1 v2 )T (use the node numers given in the schemtic). EECS 16A, Fll 2015, Homework 7 2
3 i s 7 r 2 1 r 4 3 r 8 4 V s i 1 r 1 r 3 2 r 5 r 6 r 7 V x r 9 5i V x 5 4. (OPTIONAL) Wr of the Currents In the lte 19th century, the world wtched with ited reth s Thoms Edison s Electric Light Compny (now GE) nd George Westinghouse s Electric Compny ttled over how electronic power would e distriuted cross the United Sttes. The wr of the currents - s we now cll it - ws etween Edison s low-voltge DC distriution strtegy nd Westinghouse compny s high voltge AC distriution which took dvntge of pivotl inventions like the trnsformer nd the induction motor. Instrumentl to AC s ultimte victory were the inventions nd discoveries of Yugoslv-Americn engineer Nikol Tesl. () A mjor dvntge of AC trnsmission ws the ese with which high voltges could e generted compred to DC. In the U.S. typicl overhed power line cn hve upwrds of 750,000V! From power perspective, why is high voltge trnsmission dvntgeous? Assume ech house (represented s h elow) hs fixed lod power P L (sy 1kW), the power trnsmission lines hve resistnce w, nd clculte the power loss in the wire resistnce s function of the mplitude of the line voltge. w " " V s h V h #" #" () Your Stnford friend offers counterrgument. He sttes tht the power dissipted in the line is ctully P = V 2 / w where V is mplitude of the line voltge nd w is the resistnce of the wire. Where hs your misguided friend gone wrong? (c) Let s suppose, for moment, tht you mesure the voltge nd current on the power line leding up to your house while no pplinces re running (plese do not ctully do this - live power lines cn e very dngerous!). You find tht the voltge is given y V h (t) = V o sin(ωt) nd the current into h is given y I h (t) = I o cos(ωt). Give n expression for the power P(t) = I h (t) v(t). Wht is the verge power? (d) Now you turn on your refrigertor, your lundry mchine, nd your television. The line voltge is still V h (t) = V o sin(ωt), ut the current is now I h (t) = I o sin(ω 45 ). Once gin, give n expression for power nd find the verge power. (e) Upon hering the result of your first verge power clcultion, your Stnford cquintnce decides to test your result y plcing 1kΩ resistor (much like the one from your l) etween the power line EECS 16A, Fll 2015, Homework 7 3
4 nd ground in socket (gin... plese do not do this t home). This resistor cn hndle 1/4W. For the rief instnt efore the poor resistor lets out finl puff of smoke, how much power is eing dissipted through it? Suppose the mplitude of the AC voltge in your outlet is 170V. (f) AC trnsmission systems often use wht is referred to s 3-phse power. In the network given elow, find the sum of the power provided y the three sources V 1, V 2, nd V 3. How does this compre to single AC source driving resistor? V 0 sin(ωt) V 0 sin(ωt 120 ) V 0 sin(ωt 240 ) (g) With the development of high power semiconductors, the concept of High-Voltge DC (HVDC) power lines hs ecome more relistic. Over line with resistnce w nd totl power plnt power P, how much verge power is lost y three-phse AC voltge with mplitude V o. How much verge power is lost y DC voltge with voltge V o. L V 0 sin(ωt) V 0 sin(ωt 120 ) V 0 sin(ωt 240 ) L L EECS 16A, Fll 2015, Homework 7 4
5 5. esistive Voltge egultor" In this prolem, we will design circuit tht provides n pproximtely constnt voltge divider cross rnge of lods. We will use resistor divider circuit s seen in discussion. The gol is to design circuit tht from source voltge of would yield n output voltge within 5% of 4V for lods in the rnge of 1kΩ to 100kΩ. () First, consider the resistive voltge divider in the following circuit. Wht vlue of the resistor would chieve voltge of 4V? 12kΩ () Now consider loding the circuit with resistor of 1kΩ s depicted in the following circuit with the sme vlue of the resistor s clculted in prt (). Wht is the voltge now? 12kΩ 1kΩ (c) Now consider loding the circuit with resistor of 100kΩ, insted, s depicted in the following circuit with the sme vlue of the resistor s clculted in prt (). Wht is the voltge now? 12kΩ 100kΩ (d) Now we would like to design divider tht would keep the voltge regulted for lods for rnge of lods l. By tht, we would like the voltge to remin within 5% window of 4V. Tht is, we would like to design the following circuit such tht 3.80V 4.20V for rnge of lods l. As first step, wht is the Norton equivlent of the circuit on the left? Write I No nd G e f f it in terms of conductnce vlues G 1 = 1 1 nd G 2 = 1 2. EECS 16A, Fll 2015, Homework 7 5
6 1 = 1 G 1 2 = 1 G 2 I No e f f = 1 G e f f (e) The second step, using the Norton equivlent circuit you found in prt (d), wht is the rnge of G e f f tht chieves 3.80V 4.20V in terms of I No nd G l? I No e f f = 1 G e f f l = 1 G l (f) Trnslte the rnge of G e f f in terms of I No nd G l (tht you found in prt (e)) into rnge on G 2 in terms of G 1 nd G l. (g) Sy we wnt to support lods in the rnge 1kΩ l 100kΩ with pproximtely constnt voltge s descried ove (tht is, 3.80V 4.20V). Wht is the rnge of G 2 in terms of G 1 now? Trnslte the rnge of G 2 in terms of G 1 into rnge of 2 in terms of 1. (h) Note tht conductnce is lwys non-negtive. From the ounds on G 2 you found in the previous prt, derive ound on G 1 tht ensures tht G 2 is lwys non-negtive nd non-empty (tht is, the whole rnge of possile G 2 vlues is non-negtive nd is not empty). Trnslte this rnge into rnge of possile 1 vlues. (Hint: In ddition to the conductnce eing non-negtive, lso mke sure tht the rnge for G 2 is non-empty.) (i) Pick the vlues of 1 nd 2 tht chieve 3.80V 4.20V for 1kΩ l 100kΩ while minimizing the power consumed y the voltge divider circuit in open circuit (when there is no lod ttched to the output). Wht re these vlues 1 nd 2? How much power is consumed in this cse? Clculte nd report this power consumption using oth the originl circuit nd the Norton equivlent circuit. Are the power you clculted using the originl circuit nd the power you clculted using the Norton equivlent circuit equl? (j) (OPTIONAL) Now using the sme vlues 1 nd 2 from the previous prt, lod the circuit with lod of 51kΩ, how much is consumed y ech of the three resistors, 1, 2 nd l (use the originl circuit to compute the power)? 6. DC-DC Voltge Divider As we hve lerned in clss, one of the resons for using AC voltges is tht we cn esily trnsform the voltge (step up or step down) using trnsformers. Unfortuntely, such circuits do not work t DC nd we need to come up with other wys of dividing DC voltges. We hve lerned out resistive dividers, ut we found issues such s inefficiencies. An lterntive circuit, cpcitive chrge pump, is shown elow. It relies on two switches which re ctivted in sequence, first switch Φ 1 is closed (during this period Φ 2 switches re open), nd next Φ 2 closes nd Φ 1 is opened. In prctice this is done periodiclly ut for this prolem we will nlyze ech phse seprtely. Note tht V in is DC voltge. EECS 16A, Fll 2015, Homework 7 6
7 Vin C1 Vout C2 () During phse, clculte the voltge cross nd chrge stored y ech cpcitor C1 nd C2. () During phse, clculte the output voltge Vout nd show tht it is frction of the input voltge Vin. (c) For the specil cse of C1 = C2, clculte the output voltge nd the efficiency of the system. To clculte the efficiency, clculte the energy stored in the cpcitors during the end of phse nd. (d) Assume tht this circuit is used with lod represented y the current source IL = 10mA. Suppose tht the cycle descried ove repets periodiclly t rte of 10 khz, or 10,000 times per second, with ech phse nd exctly 50% of ech cycle. During phse, which lsts 50µs, we wish the output voltge to not droop y more thn 5mV. Specify the size of C1 nd C2 to stisfy this constrint. Vout Vin C1 C2 IL 7. Your Own Prolem Write your own prolem relted to this week s mteril nd solve it. You my still work in groups to rinstorm prolems, ut ech student should sumit unique prolem. Wht is the prolem? How to formulte it? How to solve it? Wht is the solution? EECS 16A, Fll 2015, Homework 7 7
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