Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Electronic CircuitsLab 3: Second-Order NetworksHandout F04-49Fall 2004IntroductionThe purpose of this lab is to give you experience with second-order networks, and to illustrate thatreal network elements do not always behave in an ideal manner. All exercises in this lab focus onthe behavior of the network and network elements shown in Figure 1. You should complete thepre-lab exercises in your lab notebook before coming to lab. Then, carry out the in-lab exercisesbetween November 15 and November 19. After completing the in-lab exercises, have a TA or LAcheck your work and sign your lab notebook. Finally, complete the post-lab exercises in your labnotebook, and turn in your lab notebook on or before November 24. There will also be labhours on Monday November 22, Tuesday November 23, and Wednesday November 24(1-3 pm). This is a good time to catch up on lab work or get started on lab 4, whichwill be handed out on November 23.Before asking to get checked off, make sure you meet all the requirements in thecheckoff list at the end of the In-Lab ExercisesBring in your favorite CD for In-Lab Exercise 3-5; it is meant to be a fun experimentand its results will not be needed for the post lab exercises.Pre-Lab ExercisesYou are strongly encouraged to use Matlab to generate the graphs for Exercises 3-2 and 3-5. Matlabwill not only save you time, but will also help you generate graphs that are extremely accurate andprecise. See the appendix for help with Matlab. There are also two Matlab scripts which you candownload from web.mit.edu/6.002 . By filling in a few relations and changing resistor values these+-vIN(t) CLRvOUT(t)+-RIN50Ω=Signal GeneratorFigure 1: second-order network.scripts will produce the graphs in the pre-lab excercises.(3-1) Assume that the network in Figure 1 is initially at rest. At t = 0, the input voltage vIN(t)steps from 0 V to VTI. Given this input, determine the transient response of vOUT(t). Notethat vOUT(t) takes the form vOUT(t)=VTOe−αTtsin(ωTt + φT). Hint: You are free to usethe results from Homework Problem 9-2 for this exercise.(3-2) Let L = 47 mH, C =0.0047 µF, R = 220 Ω and VTI= 10 V. Under these conditions,graph the transient response of vOUT(t) for 0 ≤ t ≤ 0.3 ms; graphing the peaks and zerocrossings of the response and a few points in between each peak and zero crossing should besufficient. On a separate graph, repeat this exercise for R = 1000 Ω. Hint: See the Matlabappendix and download the transient.m matlab script from web.mit.edu/6.002.(3-3) For both values of R, compute the voltage VTPat the first peak of the transient response,the frequency ωTat which the transient response oscillates, and the rate αTat which thetransient response decays. Note that peaks of the transient response occur at times suchthat tan(ωTt + φT)=ωT/αT; you should verify this.(3-4) Assume that the network is in sinusoidal steady state. Determine the response of vOUT(t)to the input vIN(t)=VSIcos(ωSt). Note that vOUT(t) will take the form vOUT(t)=VSO(ωS) cos(ωSt + φS(ωS)). Hint: You are free to use the results from Homework Prob-lem 10-1 for this exercise.(3-5) Let L = 47 mH, C =0.0047 µFandR = 220 Ω. On separate graphs, graph log |HS(ωS)|and φS(ωS) versus log(ωS/(2π × 10 kHz)) for 2π × 1kHz≤ ωS≤ 2π × 100 kHz whereHS(ωS) ≡ VSO(ωS)/VSI. Ten to fifteen points per graph should be sufficient to clearlyoutline HSif you space the points more closely near the peak of HS. Again on a separategraph, repeat this exercise for R = 1000 Ω. You may find it easiest to use log-log graphpaper for the graph of HSand linear-log graph paper for the graph of φS. Hint: See theMatlab appendix and download the forcedosc.m matlab script from web.mit.edu/6.002 .(3-6) For both values of R compute the peak value HSPof HS, the frequency ωSPat which thepeak occurs, and Q. Note that Q is defined as Q ≡ ωSP/2αT, and that HS(ωS) will havefallen from its peak value of HSPby a factor of√2atωS≈ ωSP± αT.In-lab ExercisesThe in-lab exercises involve measuring both the step response and sinusoidal response of the networkshown in Figure 1 for two values of R. Afterwards, you will use the same network to filter a signalfrom a CD player.Real network elements do not always behave the way we model them in 6.002. For example,a real inductor might be better modeled as an ideal inductor in series with a resistor as shown inFigure 2. The resistor is a parasitic element, meaning that it is undesired, but unavoidable. Theresistor accounts for the resistance of the wire used to wind the inductor. Yet more complex modelscould account for core losses and the capacitance between winding turns. For this reason, the modelshown in Figure 2 is not the only possible model. In a similar way, a real capacitor might be bettermodeled as an ideal capacitor in parallel with a parasitic conductance which models leakage throughthe dielectric of the capacitor. This is also shown in Figure 2.In the exercises which follow, the network in Figure 1 will be exposed to inputs that vary athigh enough frequencies that you can ignore the parasitic parallel conductance of the capacitor.Therefore, we need only be concerned with the parasitic series resistance of the inductor.(3-1) Take a 47 mH inductor, a 0.0047 µF capacitor, a 220 Ω resistor, and a 1000 Ω resistorfrom your lab kit to the instrument desk and use the GenRad impedance meter to measurethese elements and determine the parasitic resistance and conductance of the inductor andcapacitor, respectively.To measure the inductor, set the meter for 1 kHz, the series model, and the appropriateelement type and value range. The meter will directly read the inductor value. It will alsoread Q from which you can determine RPfrom Q = ωL/RP, where ω =2π× 1 kHz.To measure the capacitor, set the meter for 1 kHz, the parallel model, and the appropriateelement type and value range. The meter will now directly read the capacitor value. It willalso read D from which you can determine GPfrom D = GP/ωC, where ω =2π× 1 kHz.(3-2) Construct the second-order network shown in Figure 1 using the measured inductor, capac-itor and 220 Ω resistor.(3-3) Set the signal generator to produce a 10 V peak-to-peak square wave at 50 Hz with a 5 Voffset so that its open-circuit output voltage steps between 0 V and 10 V. Also, obtain aBNC to BNC cable
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