Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits and ElectronicsSpring 2003Handout S03-047 - Lab #3: Second-Order NetworksIntroductionThe 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 exercises onyour assigned lab day between April 14 and April 18. After completing the in-lab exercises, havea TA or LA check your work and sign your lab notebook. Finally, complete the post-lab exercisesin your lab notebook, and turn it in to your TA on or before April 28.Pre-Lab Exercises(3-1) Assume that the network is initially at rest. At t = 0, the input voltage vIN(t) steps from0 V to VTI. Given this input, determine the transient response of vOUT(t). Note thatvOUT(t) takes the form vOUT(t) = VTOsin(ωTt + φT)e−αTt. Hint: see Homework Problem8.3, Spring 2002 at http://web.mit.edu/6.002/www/spring02/hw8sol.pdf.(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 shouldbe sufficient. On separate graphs, repeat this exercise for R = 560 Ω and R = 1000 Ω. Tosave time, you may wish to use MatLab on Athena as discussed below.(3-3) For all three 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.+-vIN(t) CLRvOUT(t)+-RIN50Ω=Signal GeneratorFigure 1: second-order network.(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: see Homework Problem 9.1, Spring 2002 athttp://web.mit.edu/6.002/www/spring02/hw9sol.pdf.(3-5) Let L = 47 mH, C = 0.0047 µF and R = 220 Ω. On separate graphs, graph log |HS(ωS)|and φS(ωS) versus log(ωS/(2π × 10 kHz)) for 2π × 1 kHz ≤ 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 separategraphs, repeat this exercise for R = 560 Ω and R = 1000 Ω. You may find it easiest touse log-log graph paper for the graph of HSand linear-log graph paper for the graph of φS.Alternatively, to save time, you may wish to use MatLab on Athena as discussed below.(3-6) For all three values of R compute the peak value HSPof HS, the frequency ωSPat whichthe peak occurs, and Q. Note that Q is defined as Q ≡ ωSP/2αT, and that HS(ωS) willhave fallen from its peak value of HSPby a factor of√2 at ωS≈ ωSP± αT.You are strongly encouraged, although not required, to use MatLab to plot the graphs outlinedabove. To use MatLab, you must first type “add matlab” at the Athena prompt, and then invokeMatLab by typing the command “matlab” at the Athena prompt. The MatLab commands, “step”and “bode” can then be used to plot the desired graphs. You can learn how to use these commandsby typing “help step” and “help bode” at the MatLab prompt. The “step” command acceptsan optional time-vector argument to specify the time range over which the step response is to beplotted. To define an appropriate time vector T, type “T = linspace(0,3e-4,100);” at the MatLabprompt. The vector T will then contain 100 evenly spaced points between 0 ms and 0.3 ms. The“bode” command accepts an optional frequency-vector argument to specify the frequency rangeover which the frequency response is to be plotted. To define an appropriate frequency vectorW, type “W = 2*pi*logspace(3,5,100);” at the MatLab prompt. The vector W will then contain100 logarithmically spaced points between 103and 105Hz. Additionally, note that the “bode”command in MatLab uses the frequency variable s, where s = jω. Finally, figures plotted byMatLab may be printed on an Athena printer using the MatLab “print” command. To learn more,type “help print” at the MatLab prompt.In-lab ExercisesThe in-lab exercises involve measuring both the step response and sinusoidal response of the networkshown in Figure 1 for three values of R. You should feel free to measure these responses for onlyone value of R, and then share your measurements with two other partners who have measured thenetwork responses for the other two values of R. However, should you take this team approach,all team members must use the same inductor and capacitor. You should also indicate in your labnotebook which responses you measured, and which responses you have taken from another teammember. Finally, you are advised to look at all responses on the oscilloscope before you leave thelab so that you see for yourself how they vary as R varies.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, a 560 Ω resistor and a1000 Ω resistor from your lab kit to the
View Full Document
Unlocking...