final solutions 2009

final solutions 2009 - ECE l34 Final Exam Fall 2009 FINAL...

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Unformatted text preview: ECE l34 Final Exam Fall 2009 FINAL EXAM You have 3 HOURS to complete this exam. Two pages of notes are allowed, otherwise there should be no books or other materials in your purview. You may use a calculator if necessary. When you are finished, check that your name appears prominently and leginy on the front page. The exam is worth 100 points total. IMPORTANT POINTS FOR GRADING: SHOW ALL WORK. ANSWERS GIVEN WITHOUT CLEAR SUPPORTING EVIDENCE OF INDEPENDENT WORK WILL BE IGNORED. CIRCLE ANSWERS. THOSE NOT CLEARLY MARKED AS YOUR FINAL ANSWER WILL BE IGNORED. BE NEAT. ILLEGIBLE OR SLOPPY WORK WILL BE IGNORED. Sometimes I think we’re alone in the universe, and sometimes I think we’re not. In either case the idea is quite staggering. Arthur C. Clarke Page 1 ofll ECE 134 Final Exam Fall 2009 Problem 1 (10 points) Coaxial cables for high-power transmission typically use a cross—linked polyethelyne (XLPE) insulator which has a high breakdown field. For this problem we will assume a maximum safe operating field of Emax : l OkV/mm. a) Using Gauss' law, find an expression for the radial electric field E,,(r) between the conductors (a<r<b) assuming a surface charge density ,05 on the inner V conductor: §§<E ‘Ag 3‘ glam/£05911. C“; éfi‘ilgfi/e 1 ‘65 {ix/((5 ~—/ b) The field is largest near the surface of the inner conductor, i.e. E fact to eliminate ,05 and express Er(r) in terms of E = E,(a). Use this max max, a , and r only: c) Now use your result of part (b) to relate Emax to the voltage V that is applied between the inner and outer conductors as shown in the figure: \I: «a may ~_- Ema lyfi/Q 1 d) It can be shown that cable designs satisfying ln(b / a) =1 will support the largest possible voltage for a given Emax. Combine this constraint and your result from part (c) to find the optimal dimensions of the cable (a and b) that will allow for safe operation at V = l 00kV: gar lmbfltyl (/ \f I loO cV ‘3 a: E; “’ lo léVf/nlm "1 1342i c) lotqel-EQWQ _/ Page 2 of 11 .63...- \ 2 (- “xi ~74 \l r l \ ECE 134 Final Exam 2' Fall 2009 Problem 2 (10 points) Rubbing a balloon over one's hair transfers a certain amount of charge Q onto the surface. Assume that the balloon is perfectly spherical with a radius a, centered on the origin, and that the total charge Q is distributed uniformly over the surface. a) Find the electric field inside and outside the balloon: Evy (4“ . G7 gr : 0 am draws; 6M“: (0553 \ . gm, :20. {Mam : .. ( Q . - p A ( C7 .. A J;\ u C) E‘r " ( ' e“ {3 “9 l J.“ij j b) Using your result from (b), find the electrostatic energy associated with this charge distribution. Your answer should be expressed in terms of the total charge Q, NOT the surface charge density: ‘ CQ 2 , l. . Z r q pflwr I by"? CL _ ‘\ Uta lélclv = {’13 um) ‘er la A, r @{Z "6? I $7 Tl’éb ‘1 f x c) The charge on the balloon causes it to expand. Find the outward pressure (pressure 2 force per unit area) on the surface ofthe balloon under the influence ofthe electrostatic forces. ,. Q u; a? ( a» W my A __.._. L /_ f A 2 (— r I ’2 k“ 73% C“ f 2 -2 c“ Q web 4;. "‘ (32 fl 0K /\ : 011 " ' ° Page3 ofll p (13m aw i ECE 134 _ Final Exam Fall 2009 Problem 3 (10 points) A large parallel plate capacitor is made using a thin Mylar film, with a thickness d = l,um and a dielectric constant of 5,. z3.2. The Mylar film is not a perfect insulator; there are some free charges in the material that “EMA; give rise to a conductivity 0'. The plates are square with a side a} 7 _ at V length of 10 cm. A measurement J M\ of DC current versus applied ' voltage on the capacitor yields the curve shown below. Below «3;» I 100 Volts the current—versus- ' voltage is roughly linear, passing ' through I uA at 100 Volts. But 100V léUV for applied bias ~160 Volts a large current begins to flow, limited only by the external circuit. ”" LJ U) «Cl; \vn a) Estimate the breakdown field strength in the material (s’hoqunits); c) Compute the conductivity of the film below breakdown (show units). 100v 3‘ b 3- : l0 .Jl j ' - - « er fly, 7) OT- yu’l go {out} Page4 of 11 ECE 134 Final Exam Fall 2009 Problem 4 (10 points) A UCSB student buys some 32—gauge copper wire for a low—power lighting project. The wire has a diameter d:0.2 mm. The wire is wound on a plastic spool and has an unknown length. The student finds information on-line that suggests a resistivity of 1.76 uQ-cm (or 1.76 ><10'8 Q - m ) for copper at room temperature. a) She applies a voltage of 100 mV between the ends of the wire, and observes a current of about 5 mA. What is the approximate length of wire on the spool? 0) 11km” A 3/ A «(4%); e ._ 3: 2 Ti (l0 m £1 _-. , <3 T b) The wire is used to connect a 5V wall transformer to a high—intensity DC lamp. The lamp can be modeled by a resistance of 369. What is the maximum length of wire that can be used ifthe maximum allowable voltage droop is 10%? {LZL f: ii 2 a Page 5 ofll ECE 134 Final Exam Fall 2009 Problem 5 (10 points) In our derivation of fusing current for bare wires we assumed that thermal radiation is the dominant cooling mechanism because the melting point of most metals is quite high (ag. T ~1083°C for copper). However, some metals have relatively low melting points (6g. HIE/I T z 230°C for tin), and hence convection could be the dominant cooling mechanism. Hie/I a) Ifthe wire has a resistance R at temperatures near the melting point, find an expression for the steady—state fusing current when convection cooling dominates. Assume the wire has a surface area AS , a convection coefficient h, and is suspended in air at temperature To: b) Now express A: and R in terms of the physical parameters such as the wire diameter d, length (7, and conductivity 0. If h = 0.01/d [W/m2 °C], show that the fusing current will depend linearly on the wire diameter such that I‘m =Kd, and find the coefficient K in terms of a, T and To: A5 3 7T l/t “ A l _ A. ‘ {LI 0/ we)? ° D¢Ol z Gaol 5T1 c) Evaluate K numerically for a tin wire suspended in air at T0 = 25°C . The conductivity of tin is approximately a z 7.3 X 106 S/m near the melting temperature. Page6 of ll ECE 134 Final Exam Fall 2009 Problem 6 (10 points) A certain coaxial cable is constructed from a pair of hollow coaxial conducting cylinders as shown below, with radii of a and b for the inner and outer conductors, respectively. In operation, each conductor carries equal and opposite currents I as shown, forming an infinitely long loop: a) Using Ampere's law, find an expression for the magnetic field H¢(r) between the conductors (a < r < b ), assuming L —> oo : . _ $5 f; A E eye/Qua} 'i T C; ‘Z‘Trr/Li; Ayah/1;: \Hy «l; 27:: b) Find the total magnetic flux linked by the currents over a finite length L (i.e. the flux through the lined rectangular region in the figur ). x \L r: We? w M- c) Find an expression for the inductance per unit length for the structure. Page 7 of II ECE 134 Final Exam Fall 2009 Problem 7 (10 points) Two coils of N. and N2 turns are wound concentrically on a straight cylindrical core of radius aand permeability y. The windings have length El and £2 respectively. a) Using the long solenoid amtion, find the magnetic flux in the core if a current 1] flows in the long winding as shown. . - v w i“ N( 'A C; feted w‘\ W; gala” " {Ell/“L 7“ CAL" ,_..._—— c) Find the mutual inductance between the coils Page 8 ofll ECE 134 Final Exam Fall 2009 Problem 8 (10 points) Large AC currents are often measured in silu using an inductive pickup as shown (this is sometimes called a “current—sensing” transformer). A high— permeability torus of mean radius re and cross— sectional area A is clamped around the current to be measured. A secondary winding of N turns is wrapped on this core to measure the induced emf (voltage), the strength of which can be related to the AC current magnitude. a) Assuming the current passes through the center of the torus, find the B—field inside the core at the mean radius r M . Amp—MAJ) ' b) lfthe AC current to be measured is expressed as [(1) : IO cos(a)z), find an expression for the induced secondary voltage V(1.‘). Use the result of (a) and assume that the magnetic field is approximately constant over the cross section ofthe core, as we often did in the homework. c) How many turns in the secondary winding would be required to insure that the induced AC voltage amplitude is at least 10 mV for 60 Hz currents of >1 A? Assume core dimensions of r0 2 2 cm and A : 0.25 cm2, and a relative permeability of yr : 500. (43%} V3 ‘> lOMV “69‘” ijocpl l4 Page9of1] ECE I34 Final Exam Problem 9 i Fall 2009 (10 points) The figure shows an electromechanical device that is sometimes just referred to as a “solenoid”. A winding ofN turns is wrapped on the central arm of the high-permeability core. A similarly high—u “plunger” slides freely in a non-magnetic sleeve between arms of the core as shown. The cross section of all segments of the core and plunger is A. When the coil is energized with a magnetizing current I, there is a force on the plunger causing it to pull inward (to the left in the figure). High-p core High-p plunger a) Using the magnetic circuit method, find the total flux in the air gap as a function ofx and g (assume that the reluctance of the core and plunger are negligible, i.e. /,1 a co) r—VW ‘7 WM nealuifmla emf s5 {Mm/“2&6 ‘2‘} \ i r._‘ “C; It PM? hf]ch { . N: oh 6; V L? ’ ‘ x7 b) Find an expression for the magnetic stored energy (hint: find the self—inductance first). v“? MHZ/4 Lg— H U I x + 77/2, 7 2?.” U \ L “z. '\_ filo/U ff c) Find an expression for the retort-thepl‘ungerr 2/4 7 (A [B — \ V/UO Page 10 of 11 ECE 134 Final Exam Fall 2009 Problem 10 (10 points) A plane wave propagates in a non-magnetic material with 5r =9 and ,ur =1, and the electric field vector is given by E: 2395mm 0% —flz) Win a) Find the wave velocity Mfg“; ' Q 5 b) Find the propagation constant ,8 a W A quflVJ/s Rm G V/ row/s ’ \. M c) Find the magnitude and direction ofthe magnetic field (show units) h f fifidy§ A. 7 'Pagellofll ...
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final solutions 2009 - ECE l34 Final Exam Fall 2009 FINAL...

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