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ece159s_2002_exam - Page I of 12 UNIVERSITY OF TORONTO...

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Unformatted text preview: Page I of 12 UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION APRIL 23, 2002 FIRST YEAR PROGRAM: ENGINEERING SCIENCE ECE 1598: FUNDAMENTALS OF ELECTRICITY & ELECTRIC CIRCUITS EXAMINERS: M.L.G. Joy and S. Zukotynski (Coordinator) NAME: L25! First STUDENT NO.: INSTRUCTIONS: 0 This is a Type A examination; no aids are allowed. 0 Only non—programmable calculators are allowed. - Answer all parts of all five questions. 0 All five questions are of equal weight. 0 The weight of each of the individual parts of each question is stated in the margins. 0 All work is to be done on these pages. 0 Place your final answers in the provided boxes unless instructed otherwise. 0 When answering the questions include all the steps of your work on these pages. For additional Space, you may use the back of the preceding page. 0 Do not unstaple this exam. CONSTANTS: e = 1.6x 10"9C £0 = 8.85 x10_nF/m “o = 411x io‘7HJm Part A [6] (ii) Page 2 of I? Question 1: Electricity A non conducting cylindrical pipe made of material with a relative dielectric constant of one has an inside diameter of 2cm and an out- sf 1e diameter of 3cm. The axis of the pipe is aligned with the z-axis. as shown. The height of the pipe is 4m. A charge of-SmC has been deposited uniformly throughout the volume of the pipe (that is. between 2 cm and 3 cm diameters).. [2] (i) Find the charge density in the cylinder p. ”2 Cm—o 0—3 cm—» The axis of the pipe is aligned with the z~axis of the coordinate system and the centre of the pipe is _) at the origin, as shown. Find the electric field along the x-axis, E(x.0,0) , as a function of the distance from the z—axis for 0<x<6cm. -) For 0<x<1crn E(x,0,0) = _) For 1<x<l.5cm E(x.0,0)= .9 For l.5<x<6cm E(x,D,O) = Page 3 of 12 Question 1: Electricity Part B Two parallel conducting plates of area 0.5m2 each Curiaiicn'nt: Plum. 0“ are 3cm apart. One plate (B) lies in the xy plane, ' while the other (A) intersects the positive z-axis. The gap between the plates extends from 2:0 to NWCE‘im‘mK z=3cm. Between the plates and parallel to the 0 plates lies a non conducting slab of thickness 1cm and relative dielectric constant of one. It is posi- 1‘5 cm tioned between 2:0.5cm and 1:1.5cm, as shown in the figure. Ignore all edge effects, that is, treat l.0 cm y the electric field distribution as if the plates and slab were of infinitely large area. 0.5 cm Conducting Plate. 03 I [3] (i) Find the capacitance of the assembly. Now. a total charge of lSmC is uniformly distributed throughout the volume of the non conducting slab. [4] (ii) Find the surface charge densities 0A and 03 on the inner surfaces of plates A and B, respectively. A voltage source is used to force VAB=5V. Assume this does not affect the distribution of the lSmC charge in the slab. [5] (iii) Find the electric field between the conducting plates. For 0<z<0.5cm 312) = For l.5<z<3cm Eu) : Page 4 of I? Question 2: Magnetism A 15cm long cylindrical wire has a cross-section of 3mm2 and _ . . . —8 Loop m {-1 reststiwly of 2 x 10 9cm. [1] (i) What is the resistance of the wire. -) B _—p— l’ Lt“ k Loop :11 r=n The wire is bent into a circular loop, the ends connected, and placed in the xy-plane. with the centre of the loop at the origin. Loop :tt r=r The loop is made to perform very small sinusoidal oscillations about the x-axis. as shown in the figure. The angle a = 0.05 sin( lOt rad/s) rad. A uniform magnetic field .) a B 2 2j T induces a current, [(t), in the loop. Him: (a) ¢B(t = O) = O, (b) sinaza for small a. [6] (ii) Find the magnetic flux through the loop, ¢B(t) . as a function of time. Loop at r=U [3] (iii) Find the current in the loop, i(t). as a function of time. Page 5 of I2 Question 2: Magnetism [4] (it!) Find the magnitude of the current at time I=O.3s. Indi- cate the direction of the current in the figure on the left. Loop at 1:0 [6] (v) Assuming that the induced current in the loop is i(t) = A sin(u)t). find the total magnetic field at the h) center of the loop BToutli) as a function of time. Give the three components. Page 6 of 12 Question 3: DC Circuits Part A [7] (i) Consider the circuit shown below. Assuming that RL is a load resistor, find the Norton equivalent for the rest of the circuit (excluding RL). Sketch the Norton equivalent circuit and give the values of all of its parameters. VS=10V Norton equivalent circuit i [4] (ii) Using the circuit parameters from part m above, find the value of the load resistor RI, that maximizes the power transfer to the load and find this maximum power. Page '1' of l2 Question 3: DC Circuits Part B Consider the circuit shown below [2] (i) Label all of the currents in the figure above and write the equations required to find the voltages V]. V2, V3 and V4. using nodal analysis. Take V4 as the reference node. [7] (ii) You know from direct measurements the following: (a) For Vs=10 V and RL=10 kQ you measure V2-\’4=4.6V and V3-V4=3.8V, (b) For V5=5 V and RL=I k9 you measure V2'V4=2.2V and V3-V4=2.IV. Using the nodal analysis approach find the values of R2 and R4. Page 8 01'12 Question 4: Transient Analysis The switch is closed at time t=O in the circuit shown below and then opened again at time t=2ms. Assume that the switch was open for a long time prior to time zero. 1:0? IOV ll-LF 2kQ 21(3) [2] (i) Find the value of vC(l) for t = 0- and t = 0+. [5] (ii) Write the differential equation for vdt) as a function of time for O < t < 2ms. Page 9 of l2 Question 4: 'Ii-ansient Analysis [4] (iii) Find the analytical expression for vdt) as a function of time for 0 <.' t < 2ms; give the numerical values of all of the parameters. Find the value of vC(t) at time t=2ms-. vdt): vdt=2ms-)= [5] (iv) Find the analytical expression for vdt) as a function of time for t > Ems; give the numerical values of all of the parameters. vdt) = Page100f12 Question 4: Transient Analysis [4] (v) Sketch lid!) for time -2ms < t < 5ms; clearly label the axes and give the numerical values of all critical parameters. Page Hofl‘.J Question 5: AC Circuits Part A [10] Consider the RLC circuit shown below. The AC source v5(t) is lOOVmS at o) =1000rad/s. Assume that the value of the variable capacitor is set to 0.1 mF and that the phase of the voltage source at time t=0 is (punt/4. Find the current in the inductor and write it in phasor form. Give the value of the current in the inductor at time t=1ms and the power being delivered to the inductor at time [=1 ms. Page I? of IE Question 5: AC Circuits Part B Consider the RLC circuit shown below (the same circuit as in Part A). The AC source v50) is 100%,“s at to =1000radfs. [2] (i) Write the expressions for the source voltage in the time domain and in the frequency domain [8] (ii) Find the specific value of the variable capacitor for which the current in the source is maximum and give the expression for this current in the time domain. ...
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