共 45 页 第 1 页东 南 大 学 考 试 卷（A 卷）课程名称电工学 考试学期 11-12-2得分适用专业 机械工程及自动化 考试形式 闭卷 考试时间长度120分钟1)Birds routinely land and relax on power lines whichcarry tens of thousands of volts of electricity. Explain why these birds do not get electrocuted. (6 pts)The resistance of the air is too large for current to flow from the line to the bird to the ground.Alternately, the resistance is too large for current to divert from the wire, into the bird, and then back into the wire. It is not because of wire insulation. Birds could land on uninsulated wires just fine.6 pts for right answer. 1 pt for something vaguely correct.2) Suppose your car battery is dead, and you need to charge it using another car’s battery. You have a pair of cables 电缆which you can use to connect the terminals终端of the batteries. In order to charge your car battery, should you connect the same terminals (positive of one battery to the positive of the other, and the same with the negative) or the opposite terminals (positive of one battery to the negative of the other battery)? Why did you choose this configuration配置；结构? (6 pts)First note that real batteries have internal内部的resistance, so the universe领域will not explode爆炸if you directly connect two car batteries.Next, if the opposite terminals are connected, then P=VI and common sense tells us that both batteries will be supplying power, and the internal resistances will be consuming消耗power, so clearly no power is being delivered 递送by either battery. If the same terminals are connected,共45 页第 2 页then we can see that if one battery is of a higher voltage than the other, it will provide power to theother battery. Key to this realization实现is to know that battery voltages drop下降；终止as the battery is depleted耗尽的.6 pts for right answer. 2 pts if you try to say there is no current if two batteries are connected + to+ because voltage sources are perfectly balanced.3) A standard procedure for testing the internal resistance of a battery is the “dual双重的pulse脉冲” test. We first attach an ideal 5 mA current source between the terminals of the battery, so that current flows in the usual direction (positive to negative), and measure the voltage across the battery terminals. We then remove the 5 mA source, and attach an ideal 505 mA current source instead,and again measure the battery terminals. (6 pts)共45 页第 3 页i.Suppose that we find a 1.485V voltage with the 5 mA source, and a 1.385 with the 505 mA source, what is the internal resistance?Easy way to solve these is to subtract减去；扣掉 2nd from first, giving, or .3 pts for right answer. 2 pts if sign error or algebra 代数学mistake.ii. Assuming the battery is perfectly linear 线型的(i.e. accurately modeled 模式化；被效仿by a Thevenin 戴维宁equivalent), is it possible to find the voltage provided by the battery with no load attachedusing the data above? If so, what is it? If not, why not?共45 页第 4 页Plug塞住；用插头将与电源接通 in to one of our equations above, e.g., giving , or finally. If you did it with the current source the other way, you should still have gotten 1.486V.3 pts for right answer.Find the Thevenin or Norton equivalent circuit model.(12 pts ).共45 页第 5 页(6 pts )(6 pts )Find the Voltage v and the currents i1 and i2 for the circuit shown below. (10 pts )共45 页第 6 页SolutionLabel标注 the meshes；网状物:Since i3=2 it is not a variable.Writing KVL around the first mesh:24*(i2-i3) + 12*i1 = 0Writing KVL around the second mesh:12*i2 + 6*(i2 –i3) = 0Substituting代替，取代 for i3 and rearranging 36*i1 = 4818*i2 = 12共45 页第7 页Namelyi1 = 4/3 Ai2 = 2/3A7. Find the current i for the circuit shown below (10 pts )SolutionZeroing the 1A source gives共45 页第8 页By the current division principle, the contribution of the 2A source to i1 is2*(-1/30) / (1/30+1/5) = -2/7A.Zeroing the 2A source givesBy the current division principle, the contribution of the 1A source to i1 is1*(1/15)/(1/15+1/20) = 4/7A.Hence i1 = 4/7-2/7 = 2/7A.共45 页第9 页Find the Thevenin and Norton equivalent circuits across terminals a and b for the following circuitWith an open circuit across terminals a and b the current through the 20 resistor must be 0.5ix and the voltage across it is then 10ix. ByKVL we also get30V = 5ix + 10ixSo ix = 2A and we calculateVTH = 10ix = 20VTo find RTH we zero the independent独立的 source to get the circuit in Figure 4.We wish to determine what resistance this circuit isequivalent 等价的to across共45 页第10 页the terminals a and b. One way to do this is to imagine connecting anindependent voltage source of voltage VS across the terminals and findingthe current drawn from this voltage source. We therefore consider thecircuit in Figure 5.KVL gives−5Ωix = VS → ix = −VS/5Ω A (1)KVL also gives20Ωi1 = VS → i1 =VS/20 Ω (2)KCL givesi = i1 + 0.5ix − ix = i1 − 0.5ix (3)共45 页第11 页Plugging (1) and (2) into (3) yieldsi =VS/20Ω+VS/10Ω=3VS/20ΩEventually,RTH =VS/i=20/3 ΩThe Thevenin equivalent circuit is depicted描述； in Figure 6.From this we get the Norton equivalent circuit in Figure 7 where we usedWe could have also determined IN by applying a short circuit across theterminals a and b and computing 计算；处理the current through the short circuit短路.This involves analyzing the circuit in Figure 8. Here wehave by KVL共45 页第12 页5ix = 30V → ix = 6AThis gives, by KCLi = ix − 0.5ix = 0.5ix = 3Awhich matches the earlier calculation计算；估计.Suppose that v1(t) = 80 cos (ωt) and v2(t) = 60 sin (ωt). Use phasors相量 to reduce the sum vs(t) =v1(t) + v2(t) to a single 单一的term of the form Vm cos (ωt +θ). Draw a phasor diagram, showing V1, V2, and Vs. State the phase relationships between each pair of these phasors. (16共45 页第13 页pts )The phasor corresponding tov1(t) = 80 cos (ωt)isV1 = 80∠0 = 80Equivalently, we get forv2(t) = 60 sin (ωt) + 60 cos (ωt −π/2)the phasorV2 = 60∠−π/2 = −j60To find the sum using phasors we add the complex numbers which yieldsV = V1 + V2 = 80 − j60 = 100e−j arctan 3/4This corresponds符合，一致 to the time signalvs(t) = 100 cos (ωt − jarctan3/4)The phasor diagram is depicted描述 in Figure 2.Note that V1 leads V2 by π/2 and leads Vs by arctan反正切 ( 4/3 ).共45 页第14 页Find an expression 表达式for v(t) of the form Vm cos (ωt +θ) when v(t) = v1(t) + v2(t) + v3(t) + v4(t) withv1(t) = 20 sin (ωt)v2(t) = 20 cos (ωt +π/6)v3(t) = 20 sin (ωt +π/3)v4(t) = −10 cos (ωt)Use phasors. (6 pts )Solution:The phasors corresponding 相应的to each component 成分；组件；[电子] 元件signal are given in Table表格 1. The complex number equivalent复数等效to each phasor is given as well.共45 页第15 页Adding all complex numbers giveswhich corresponds to the phasorwhich can be transformed变形 into the time signalFind the complex impedance阻抗 in polar form 点斜式极坐标形式极形式of the network shown below for ω = 10001/s , ω= 20001/s ,and ω = 40001/s . (6 pts )共45 页第16 页(5 pts)Both capacitors电容器 and inductors电感器 can be used to store energy for use later. Capacitors, though, are far more popular for energy storage than inductors, because they are easier to keep charged带电荷的带电的. In lab, for example, you can easily connect up a capacitor to a battery, disconnect it from the source, carry it around in your pocket to grab lunch午餐, go back to lab, and then find that it's still holding a charge even after a long time. Explain why this is relatively difficult with inductors.共45 页第17 页(5 pts) If you've ever had the batteries on a flash li ght闪光灯；手电筒 run out,you know that as soon as the ligh t starts dimming调光；变暗,your battery will very soon be dead.This is because a battery supplies its maximum vo ltage峰点电压 until right before it is depleted耗尽的.On t he axis 坐标轴on the left,I have plotted标绘的 the brightn ess亮度辉度 of a battery powered flashlight over the lif e of the battery.We can also power a flashlight using共45 页第18 页a capacitor.Assuming the brightness of a bulb 电灯泡is p roportional比例的 to the voltage across it,on the right axis,draw (qualitatively定性地) the brightness vs. Time for a capacitor powered flashlight.Draw so that bright ness is 1 at time=0,and has dropped to at least 0.1 by time=1.We just want the shape形状,not necessarily the exact values.c) (5 pts) A student is not satisfied with the amount of共45 页第19 页energy stored in his one-capacitor circuit. He wants to store more energy, so puts 10 capacitors in series串联 and charges them with a single voltage source. Is this a good idea? Why or why not?Find the Thevenin equivalent 戴维南定理of the circuit电路below at the two terminals on the far right:共45 页第20 页e) Find the Thevenin equivalent of the circuit below between terminals and [Think about the various algorithms算法 we’ve used to find Thevenin equivalents.If you get stuck被困住了,complete the rest of these tricky 棘手的复杂的problem and come back to this one and maybe some inspiration will reach you]:共45 页第21 页共45 页第22 页For the circuits below, please find expressions for the specified voltage表达式指定的电压over the indicated time 表示时间ranges in terms of the circuit parameters电路参数. Plot 绘图 the waveform 波形图on the provided axes轴线, and clearly identify. 确定 the key parameters参数 in your graph图表. (3A) Consider the circuit of Fig. 4. The switch is open for t <0, closed for 0≤t < t1, and open for t ≥t1, where t1 = 3RC. Find and plot the voltage v C (t).共45 页第23 页(3B) Consider the circuit of Figure 5, in which > -1.共45 页第24 页The switch is open for t <0, and closed for t 0. vC(0-) = V0. Find and plot the voltage v1(t).共45 页第25 页The circuit shown in Figure 4 has been at rest with theswitch open for a long time. At t = O the switch is closed.共45 页第26 页Sketch 素描；略图；梗概the current ic集成电路 through the capacitor for t > O on the axes below. Using the circuit parameters R1, R2, C, and Vo,indicate on your sketch (i) the initial value初始值 ic(Ot), (ii) the final value ic(oo), and (iii) the time constant时间常数.共45 页第27 页Problem 1F: 5 pointsThe capacitor in the circuit in Figure 6 has an initial初始最初的voltage V O on its terminals at t =0−when a step of voltage Vu(t) is applied at t = 0. Find an expression for the voltage across the capacitor v C for t> 0.共45 页第28 页Consider the current divider circuit分流电路 shown below:共45 页第29 页1a. How many branches分支 are there in the circuit?1b. How many nodes节点 are there in the circuit?2. Derive an expression派生一个表达式 for the output current 输出电流(Iout, shown above) as a function of theinput current (Iin) and the two resistors (R1 and R2). Use any technique you’d like.3. How should the resistors be chosen such that twice as much current flows throughR1 compared to R2?4. Using the equation 方程等式you obtained in #2, show what happens to Iout输出电流 if R2 is replacedwith a short circuit (i.e. R2 = 0). Repeat if R2 is replaced with an open circuit.(think about what R2 should be replaced with here) Now suppose the following values are used: Iin = 3 A, R1= 5 , R2 = 20 , and we wish to find the Thevenin共45 页第30 页equivalent circuit across the two indicated terminals below:5. Find Voc and Isc across the two terminals indicated on the circuit.6. Draw the Thevenin equivalent circuit for the two terminals indicated above.7. Suppose a load is now attached across the two terminals of the Theveninequivalent circuit:a. Write an equation for I in terms of V.b. Sketch the IV characterstics伏安特性. Use I as the“y-axis Y轴” and V as the“x-axis”. Label标注all relevant points相关点.共45 页第31 页8. Suppose the load has an IV characteristic given as follows:a. What are the values of I and V if the load is attached to the Thevenin equivalent circuit ] 等效电路?b. Looking at the IV characteristic, what is the load? (describe the load as much as you can)For problems 9 through 12, consider the following circuit:9. Using KVL and mesh analysis, write down the equations needed to solve the circuit. Convert转变the resistor voltages into currents.10. The dependent source requires one additional equat共45 页第32 页ion附加方程to be generated产生. Write down that equation by looking at the 3 resistor.11. Solve the equations simultaneously同时地to determine the two mesh currents网孔电流.12. To check your answer, confirm确认that power is conserved 守恒in the circuit. Find the power dissipated浪费的in each resistor. Find the power generated生成的(or dissipated in each source). Be clear in your answers whether power is being generated or dissipated in each circuit element. For problems 1–7, consider the following circuit:1. Find the transfer转移function, i.e. Vout输出电压()/Vin(), in terms of R, L, and .2. What is the magnitude大小of the transfer function传递函数, i.e. | H() |, in dB?3a. What is | H() | equal to (in dB) for very low共45 页第33 页frequencies频率?3b. What happens to | H() | at high frequencies? What kind of a filter滤波器do we have here?4. Derive the equation for the break frequency推导出方程为截止频率. The break frequency is definedas the frequency where | H() | = –3.0103 dB.5. Sketch the magnitude Bode波特图plot for this filter滤波器. Label标注all critical points andslopes. Show how you obtained the value for any non-zero slopes零斜坡.6. Suppose R = 1 , L = 2 H, and Vin = 3 cos(5t + 30o). Find Vout(t).7. Draw a phasor diagram, indicating the voltage and current phasors of all thecircuit elements电路参量.For problems 8–12, consider the following circuit:共45 页第34 页8a. Find the current that appears on the output side ofthe transformer变压器.8b. What is the voltage that appears across the capacitor. (phasor form is acceptable,assume an angular frequency角频率of 1 rad/s)9. Find the voltage that appears on each side of the transformer.10. Find the voltage that appears across the current source.11. Replace the transformer and input circuitry with an equivalent current sourceand inductor. In other words, draw a circuit which will have a current source,an inductor, and a 3 F capacitor all in series. Label the values for the current source and the inductor.Check your answer to #11 by comparing the amount of power共45 页第35 页generated by the currentsource in each circuit.12. Find the amount of power generated by the current source in the new circuit.Compare this with the amount of power generated by the current source in theoriginal circuit原电路 (they should be the same)For problems 4–7, consider the following circuit:4. Find the transfer function, i.e. Vout ()/Vin (), in terms of R, L, and (or f).5. What is the magnitude of the transfer function, i.e. | H() |?共45 页第36 页6. What is | H() | equal to for very low frequencies? For very high frequencies?What happens at resonance谐振? (i.e. = o =LC1) What kind of a filter滤波器do we have here?7. Sketch the magnitude Bode plot for this filter. Label all critical points andslopes斜率. Use R = 2 , L =41H, and C =81F. Show how you obtained the valuefor any non-zero slopes.In order to measure the voltage across a resistor in a circuit, a voltmeter电压表is used.Although we assume an ideal voltmeter in most cases, a real voltmeter has a finite有限的resistance (R3) which can have an effect on your circuit. Consider the followingmeasurement on a voltage divider分配共45 页第37 页Vin, R1, and R2 are just the standard elements of a voltage divider.1a. How many nodes are there in the circuit?1b. How many meshes are there in the circuit?2. Write a KCL equation at node A. Convert the currents into voltages.3. Solve the equation for Vout, so that you have an equation for Vout based onVin, R1, R2, and R3.4. Show that as R3 approaches infinity无限大(i.e. R3 ), you get the standardequation for a voltage divider.For problems 5 through 11, consider the following circuit:共45 页第38 页Find the Norton equivalent circuit, as follows:5. Short circuit the two terminals and redraw the circuit without the 2 resistor.Explain why the 2 resistor can be dropped from the circuit.放弃了电路6. Use node analysis to find Isc. (You will need to come up with one additionalequation to eliminate消除ix and get a numerical数字的answer) 7. Now consider the circuit when the two terminals are open circuited. Using meshanalysis, find ix under these conditions.8. Using your results from #7 (or using node analysis, which will take more time),find Voc, the voltage across the two terminals in an open circuit condition.共45 页第39 页9. Confirm that power is conserved in the circuit under open-circuit conditions.Find the power dissipated in each resistor. Find the power generated (ordissipated in each source). Be clear in your answers whether power is beinggenerated or dissipated in each circuit element.10a. Draw the Norton equivalent circuit. Label values for all circuit elements.10b. Draw the Thevenin equivalent circuit. (Note: you already have everything youneed)11a. Assume that a circuit element (that draws some current I and has somevoltage drop V) is attached to the terminals of the Norton equivalent circuit.Using KCL, write an equation for I in terms of V.11b. Graph the IV relationship (i.e. the load line). Use I as the “y-axis” and V as the共45 页第40 页“x-axis”. Label all relevant points.For problems 1–5, consider the following circuit:Use Vin = 100 v, R1 = 1000 , R2 = 250 , R3 = 1000 , and C = 1 F. At t = 0, theswitch is moved down, disconnecting R1 and connecting R2 to the circuit. We want to find the voltage v(t), and current i(t), across the capacitor as a function of time. 1a. Redraw the circuit, showing how the circuit looks just before the switch ismoved. Assume a long time has already elapsed before the switch is moved.1b. What is the continuity连续性condition for this circuit共45 页第41 页at t = 0?2. Using the circuit generated in #1a, find v(0–) and i(0–)3a. Find v(). Explain how you get this result.3b. Write a KCL equation for the circuit after the switch as been moved at t = 0.Convert the currents into voltages. (you should end up with a first-orderdifferential equation for v(t))4a. What form should the solution for v(t) take?4b. Substitute替代this solution into the differential equation generated in #3b. Use theresults of the substitution, along with the results for #1b and #2 to find all theconstants defined for v(t). (Clearly show your work) 5. Sketch curves of i(t) and v(t) as a function of time (t ). Indicate allcritical and asymptotic values.For problems 6–10, consider the following second-order共45 页第42 页circuit:The switch is opened at t = 0. The goal is to find the current through the inductor, i(t).6a. Assume that the switch has been closed for a long period of time. Find i(0–).6b. What should i(0+) be? Explain.7a. Find i(). Explain how you get this result. Draw the corresponding相应的circuitat this steady state condition.7b. Since this is a second-order circuit, another initial 初始/boundary condition is needed.Find that other value. Show your work.8. Derive导出the second-order differential equation for this circuit. Start by writing aKVL equation for the circuit after the switch has been opened. Then convert共45 页第43 页voltages in each passive element into currents.9a. What form does the solution for i(t) take?9b. Substitute your solution into the differential equation. Solve the correspondingcharacteristic equation.相应的特征方程10. Finish determining i(t) by using the results in #6b and #7b to find any constantsyou have defined in your solution for i(t).For problems 1–5, consider the following circuit:Use Vs = 5 v, R1 = 5 , R2 = 10 , and L = 1 H. After a long time has elapsed, theswitch is opened at t = 0. We want to find the voltage v(t), and current i(t), across共45 页第44 页R2 as a function of time.1a. Redraw the circuit, showing how the circuit looks just before the switch isopened.1b. Using this result, find v(0-).2a. What is the continuity/boundary condition for this circuit at t = 0?2b. Find the numerical value of the boundary condition.3. Write a KVL equation for the circuit after the switch as been opened at t = 0.(note: do not use the circuit you drew in #1a). Convert the voltages into currents.4. Solve the differential equation generated in #3 for i(t). Clearly show how youobtained all of the numerical values in i(t).5. Sketch curves of i(t) and v(t) as a function of time (t ). Indicate allcritical and asymptotic渐近的values.共45 页第45 页。
