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Transcript
Experiment 4 โ Series RLC Resonant Circuit Physics 242 โ Electronics Introduction RLC resonant circuits are useful as tuned filters where the resonant peak can serve as a narrow pass band of the filter. We will investigate the properties of a resonant bandpass filter in this experiment. Procedure and Questions 1. Set up the circuit shown above left, with L = 2.6 H (approx.), C = 0.5 nF (approx.), and R = 2 k๏. As usual, measure the components individually using the LCR meter; record both the inductance and series resistance values for the inductor. Apply an input sine wave of 1 V peakto-peak amplitude to the input (be careful not to exceed 1 V p-p), and measure the output voltage amplitude (peak-to-peak) as a function of frequency. Choose frequencies spaced closely enough to map out the shape of the peak near resonance, and also measure at more widely spaced frequencies across the full range of frequencies that you can measure. ๐ ๐ Plot |๐๐ | vs. f (Hz) using your data. Also plot the theoretical curve of |๐๐ | calculated from circuit ๐ ๐ theory (using your measured component values). Note the real inductor can be represented as a pure inductance in series with its intrinsic coil resistance ๐ ๐ฟ as shown above right; because it is not negligible, you will need to include the inductor's resistance ๐ ๐ฟ in your calculations. To display both the theory curve and your data points, it might be easiest to use Mathematica (KGraph will also work); see the appended example Mma code. Does your observed value of the resonant frequency ๐๐๐๐ agree closely with the predicted value 1 ๐๐๐๐ = 2 ๐ ๐ถ? What is the percent difference? โ๐ฟ If you see a noticeable discrepancy in the fit, particularly in the resonant frequency, see if adjusting the capacitance value will fix the problem. What addition to the capacitance is needed to give an optimal fit of the data? Can you speculate about where this additional capacitance could be coming from? 2. Measure the phase shift โ๐ = ๐๐๐ข๐ก๐๐ข๐ก โ ๐๐๐๐๐ข๐ก at a frequency 100 Hz below the resonant frequency. To make the measurement, you will need to display the voltage ๐๐ and the voltage ๐๐ at the same time on your oscilloscope, with leads connected to channel 1 and channel 2 inputs on the 'scope. Measure the time difference ฮ๐ก (horizontal scale) between the sine waves and the ฮ๐ก time period t of one of the waves; the phase difference โ๐ is given by โ๐ = 2๐ ๐ก . Be sure to note whether the output wave leads or lags behind the input wave. Compare your measured phase difference to the theoretical value, calculated from circuit theory using your measured component values. Note: Calculate the phase difference between ๐๐๐๐ and 1 ๐๐๐๐ โ 100 ๐ป๐ง, where ๐๐๐๐ = 2 ๐ ๐ถ . This way the phase difference will not depend on whether โ๐ฟ your experimental value of ๐๐๐๐ agrees exactly with the predicted value, which it might not. Find the percent difference between experimental and theoretical phase shifts. 3. Measure the resonant frequency ๐๐๐๐ and the bandwidth B using the different resistance values R from the following list (one at a time): 2 k๏, 5 k๏, 10 k๏, 20 k๏. To measure the resonant frequency, find the frequency at which the amplitude ๐๐ is maximum. To find the bandwidth B, measure the frequency difference between the half-power points, the frequencies ๐ where the amplitude is reduced by โ2, i.e. ๐๐ = ๐ . โ2 Plot theoretical Q vs. experimental Q for your measurements. Experimental Q is given by ๐ ๐๐๐ฅ๐ = ๐ต๐๐๐ . Theoretical Q is calculated from circuit theory and component values; we showed 1 ๐ฟ in class that it is given by ๐๐กโ๐๐๐๐ฆ = ๐ โ๐ถ where R represents the total resistance (the sum of the discrete resistor and the inductor's intrinsic resistance). If theory and experiment are in agreement, the slope of the plot should be unityโdoes your plot indicate that theory and experiment are in agreement? ๏ (* Example code to display data points and theory curve *) (* First the data set is defined. *) datalist = {{2000,.016}, {3000,.036}, {3500,.058}, {4000,.142}, {4200,.24}, {4400,.67}, {4600,.31}, {4800,.158}, {5000,.104}, {5200,.081}, {5400,.069}, {6000,.039}, {6500,.036}, {7500,.022}}; plot1=ListPlot[datalist,PlotMarkers๏ฎ{Automatic,10}]; (* Next the theory curve is defined. *) r = 2000.0;vi = 1.0; rL = 1000.0;L = 2.6; c = 500.0 10-12; r rL 2 2 fL 2 1 fc 2 g[f_]:=(vi r)/ ; plot2=Plot[g[f],{f,2000,7500},PlotRange๏ฎ{All,All}]; (* Finally, the data points and theory curve are displayed together. *) Show[plot1,plot2] 0.6 0.5 0.4 0.3 0.2 0.1 3000 4000 5000 6000 7000