Designing Information Devices and Systems I Spring 2018 Homework 7


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1 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Selfgrdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should consist of one file. hw7.pdf: A single PDF file tht contins ll of your nswers (ny hndwritten nswers should e scnned). Sumit the file to the pproprite ssignment on Grdescope. 1. (PRACTICE) Circuit Anlysis Solve the circuit given elow for currents nd voltges. 1 R 2 20Ω 2 1A R 1 10Ω R 3 50Ω 3A 2. (PRACTICE) Superposition Solve the circuits shown elow using superposition. 3Ω i 1.5Ω 5V 1A 3Ω 1.5Ω EECS 16A, Spring 2018, omework 7, Lst Updted: :32 1
2 3. Fruity Fred Fruity Fred just got ck from Berkeley Bowl with unch of mngoes, pinepples, nd coconuts. e wnts to sort his mngoes in order of weight, so he decides to use his knowledge from EE16A to uild scle. e finds two identicl rs of mteril (M 1 nd M 2 ) of length L (meters) nd crosssectionl re A c (meters 2 ), which re mde of mteril with resistivity ρ. e knows tht the length of these rs decreses y k meters per Newton of force pplied, while the crosssectionl re remins constnt. e uilds his scle s shown elow, where the top of the rs re connected with n idel electricl wire. The left side of the digrm shows the scle t rest (with no oject plced on it), nd the right side shows it when the pplied force is F (Newtons), cusing the length to decrese y kf meters. Fred s mngoes re not very hevy, so L kf. F L A M 1 M 2 M 1 M 2 B A B LkF () Let R AB e the resistnce etween nodes A nd B. Write n expression for R AB s function of A c, L, ρ, F, nd k. () Fred s scle design is such tht the resistnce R AB chnges depending on how much weight is plced on it. owever, he relly wnts to mesure voltge rther thn resistnce. Design circuit for Fred tht outputs voltge tht is some function of the weight. Your circuit should include R AB, nd you my use ny numer of voltge sources nd resistors in your design. Be sure to lel where the voltge should e mesured in your circuit. Also provide n expression relting the output voltge of your circuit to the force pplied on the scle. 4. Resistive Voltge Regultor In this prolem, we will design circuit tht provides n pproximtely constnt voltge divider cross rnge of lods. We will use resistive voltge divider circuit. 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 resistnce R would chieve voltge of 4V? 10kΩ R EECS 16A, Spring 2018, omework 7, Lst Updted: :32 2
3 () Now using the sme resistor R s clculted in prt (), consider loding the circuit with resistor of 1kΩ s depicted in the following circuit. Wht is the voltge now? 10kΩ R 1kΩ (c) Now using the sme resistor R s clculted in prt (), consider loding the circuit with resistor of 100kΩ, insted, s depicted in the following circuit. Wht is the voltge now? 10kΩ R 100kΩ (d) Now we would like to design divider tht would keep the voltge regulted for lods for rnge of lods R 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 R l. As first step, wht is the Norton equivlent of the circuit on the left? Write I No nd G e f f in terms of conductnce vlues G 1 = 1 R 1 nd G 2 = 1 R 2. R 1 = 1 G 1 R 2 = 1 G 2 I No R e f f = 1 G e f f (e) For 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 R e f f = 1 G e f f R 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. EECS 16A, Spring 2018, omework 7, Lst Updted: :32 3
4 (g) Sy we wnt to support lods in the rnge 1kΩ R 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 R 2 in terms of R 1. (h) Note tht conductnce is lwys nonnegtive. From the ounds on G 2 you found in the previous prt, derive ound on G 1 tht ensures tht G 2 is lwys nonnegtive nd nonempty (tht is, the whole rnge of possile G 2 vlues is nonnegtive nd is not empty). Trnslte this rnge into rnge of possile R 1 vlues. int: In ddition to the conductnce eing nonnegtive, lso mke sure tht the rnge for G 2 is nonempty. (i) Pick the vlues of R 1 nd R 2 tht chieve 3.80V 4.20V for 1kΩ R 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 R 1 nd R 2? ow 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) Now using the sme vlues R 1 nd R 2 from the previous prt, lod the circuit with lod of 51kΩ. ow much power is consumed y ech of the three resistors, R 1, R 2 nd R l (use the originl circuit to compute the power)? 5. Multitouch Resistive Touchscreen In this prolem, we will look t simplified version of the multitouch resistive touchscreen. In prticulr, rther thn mesuring the position of two potentil touch points in oth dimensions (i.e. pir of coordintes (x 1, y 1 ) nd (x 2, y 2 ) corresponding to two touch positions), let s think out version where we re interested in mesuring only the verticl position of the two touch points (i.e. y 1 nd y 2 ). Therefore, unlike the touchscreens we looked t in clss, oth of the resistive pltes (i.e. oth the top nd the ottom plte) would hve conductive strips plced long their top nd ottom edges, s shown elow. Top View Side View (no touch) E 2 E 2 E 1 E 1 T W () Assuming tht oth of the pltes re mde out of mteril with ρ = 1Ωm nd tht the dimensions of the pltes re W = 3cm, = 12cm, nd T = 1mm, with no touches t ll, wht is the resistnce etween terminls E 1 nd E 2 (which would e the sme s the resistnce etween terminls nd )? EECS 16A, Spring 2018, omework 7, Lst Updted: :32 4
5 () Now let s look t wht hppens when we hve two touch points. Let s ssume tht t wherever height the touch occurs, perfect contct is mde etween the top plte nd the ottom plte long the entire width of the pltes (i.e. you don t hve to worry out ny lterl resistors), ut tht otherwise none of the electricl chrcteristics of the pltes chnge. Defining the ottom of the plte s eing y = 0 cm (i.e. touch t E 1 would e t y = 0cm), let s ssume tht the two touches hppen t y 1 = 3cm nd y 2 = 7cm nd tht your nswer to prt () ws 8kΩ (which my or my not e the right nswer). Drw model with 6 resistors tht cptures the electricl connections etween E 1, E 2,, nd nd clculte their resistnces. Note tht for clrity, the system hs een redrwn elow to depict this scenrio. Top View Side View (2 touches) E 2 E 2 E 1 y 2 T E 1 y 1 y 2 y 1 W (c) Using the sme ssumptions s prt (), if you drove terminls nd with 1 ma current source (s shown elow) ut left terminls E 1 nd E 2 opencircuited, wht is the voltge you would mesure cross (i.e. V E4E3 )? 1mA V E4E3 (d) Now let s try to generlize the sitution y ssuming tht the two touches cn hppen t ny two ritrry points y 1 nd y 2, ut with y 1 defined to lwys e less thn y 2 (i.e. y 1 is lwys the ottom touch point). Leving the setup the sme s in prt (c) except for the ritrry y 1 nd y 2, y mesuring only the voltge etween nd, wht informtion cn you extrct out the two touch positions? Plese e sure to provide n eqution relting V E4E3 to y 1 nd y 2 s prt of your nswer, nd note tht you my wnt to redrw the model from prt () to help you with this. (e) One of your collegues clims tht y mesuring the pproprite voltges, not only cn they extrct wht oth y 1 nd y 2 re in this system, ut they cn even do so y formulting system of three independent voltge equtions relted to y 1 nd y 2. As we will see lter, this will llow us to gin some roustness to noise in the voltge mesurements. EECS 16A, Spring 2018, omework 7, Lst Updted: :32 5
6 In order to fcilitte this, write equtions relting V E4E2 nd V E1E3 to y 1 nd y 2. (The third voltge we ll use is V E4E3, which you should hve lredy derived n eqution for in the previous prt of the prolem.) 6. omework Process nd Study Group Who else did you work with on this homework? List nmes nd student ID s. (In cse of homework prty, you cn lso just descrie the group.) ow did you work on this homework? EECS 16A, Spring 2018, omework 7, Lst Updted: :32 6
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