A. RC Filter CharacteristicsEquipmentExperiment Guide: RC Filters and Simulation/Instrumentation SoftwareDescription and BackgroundIn this lab you will (a) manipulate instruments manually to determine the input-output characteristics of RC filters, (b) use software systems to simulate the characteristics as functions of frequency, and (c) use a software system to control equipment to measure and plot the filter characteristics automatically. A. RC Filter CharacteristicsFigure 1 below shows an RC filter connected to a sinusoidal voltage source. This circuit is termed a two-port circuit (see Fig. 2) where the voltage source produces the input voltage Vin and the output voltage Vout appears across resistor R.Figure 1. RC filter with series capacitor and output resistor R.Figure 2. Two-port circuit.Recall that we customarily represent an AC voltage as a periodic function of time such as V(t) = V0cos(t) where V0 is the amplitude of the voltage, t is time, and is the so-called angular frequency, whose units are radians per second. The angular frequency is related to the “ordinary” frequency, f, measured in Hertz, by = 2f. For example, if the frequency, f, of the ordinary power line voltage in the U. S. is 60 Hz, then the associated angular frequency, , is 377 radians/s (260).Transfer FunctionA two-port circuit is characterized by its so-called transfer function, whose magnitude isdefined as |Vout/Vin|, where Vout and Vin are phasor voltages (as indicated by the boldfacetype). The variation of the transfer function with frequency characterizes the circuit, whether the circuit is an amplifier (does it amplify high frequencies more than low +-VoutVin+-RCvinvoutfrequencies?) or a filter (does the filter pass the low frequencies or the high frequencies better?).If you analyze the RC circuit of Fig. 1 using Kirchhoff’s voltage law, the phasor voltages Vout and Vin, the resistance R and the impedance of the capacitor ZC = 1/jC, you can show that the magnitude of the transfer function isFig. 3. Log-log plot of transfer function magnitude vs. angular frequency times RC . Note: the horizontal axis is really a frequency axis since R and C are constants.The filter characteristic has been simplified to appear as two lines that intersect at the angular frequency for which RC = 1, or = 1, where is the time constant RC for this circuit. If plotted precisely, the characteristic would transition smoothly from the upward sloping line to the horizontal line, but for many purposes the straight-line approximation is adequate. This circuit is called a high-pass filter, since for frequencies above = 1/RC the output voltage equals the input voltage.If we reverse the positions of R and C in the filter circuit (Figure 4), we obtain the transfer function and filter characteristic shown below:Fig. 4. Circuit with a series resistor R and the capacitor C as the output element.2)ωRC(1ωRCinoutVV (Eq. 1)An approximate log-log plot of transfer function magnitude vs. frequency is shown in Figure 3: 1|Vo ut/Vi n|RC1+-VoutVin+-RCFig. 5. Log-log plot of transfer function magnitude vs. frequency times RC for thecircuit of Fig. 4. VoutVin---------1RC 21+---------------------------------·=1|Vo ut/Vi n| =RCProceduresP1. Connect a 10k resistor and a (non-polarized) 0.1 F capacitor in series with a signal generator, making sure that your oscilloscope ground and the signal generator ground are connected together. Set the signal generator to output a 1-volt peak sine wave. Measure and plot the amplitude of the voltage between the components versus frequency on log-log graph paper. You can download log-log graph paper from the EECS 43/100 web site.P2. Reverse the order of the two components and repeat. P3. Observe the effects of filtering on square and triangular waves.P4. Using software SignalExpressReferences (on Reserve for EE 40 in Engineering Library)P. Horowitz and W. Hill, The Art of Electronics, 2nd ed. (Cambridge U. Press, 1989), pp. 35-8.R. White and R. Doering, Electrical Engineering Uncovered, 2nd ed. (Prentice Hall, 2001). See p. 27 ff.for explanation of decibels, and pp. 285-7 on transfer functions and Bode plots. Description and BackgroundGraphical circuit stimulation software, such as LabVIEW (National Instruments, Inc.), ispopular among engineers working in industry and researchers in universities because it reduces the tedium and cost of circuit and system testing. So far in this lab, you’ve used an analog function generator and an oscilloscope to get the graph that shows the transfer functions -- the ratios of the output and input voltages versus the frequency. Plotting the graph by hand is time-consuming and it may give inaccurate results. With LabVIEW, however, you can obtain accurate tabular and graphical results automatically after you program the system.A recent development from National Instruments is the application SignalExpress, whichenables you to set up and control the measurement equipment easily. SignalExpress actually creates the LabVIEW programs that run the equipment. Each piece of equipment you use here – the function generator and the oscilloscope, for instance, has a cable connected to it to provide control signals (“set the function generator frequency to 300 Hz”) and to collect data from a real circuit under test (here the actual voltages input to and output from the RC circuit are measured by the real oscilloscope, whose measurement go to the computer via the cable connected in the back of the oscilloscope). Each piece of equipment has a unique number so that the SignalExpress or LabVIEW program can recognize it. These cables are identified as GPIB buses (GPIB = General Purpose Interface Bus).Note that your EECS 40 text (A. R. Hambley, “Electrical Engineering: Principles and Applications”, 3rd Ed.) discusses LabVIEW on pages 425-437, and contains a LabVIEW CD-ROM in the envelope inside the back cover of the book.On the class web site you should find the brief document “Simulation and Instrumentation Software” that describes in simple terms what several popular and useful simulation and instrumentation applications do. Included are LabVIEW and SignalExpress, along with Orcad (the schematic capture program on a CD inside the backcover of your text), PSpice (a very popular and useful circuit simulation program – use it to check your homework!), and
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