PHYS 3322 Modern Laboratory Methods I AC R, RC, and RL Circuits

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1 Purpose PHYS 3322 Modern Laboratory Methods I AC, C, and L Circuits For a given frequency, doubling of the applied voltage to resistors, capacitors, and inductors doubles the current. Hence, each of these circuit elements obeys Ohm s Law and are called linear devices. esistors, capacitors, and inductors are passive devices in the sense that they do not have a built in power supply. The frequency response of these circuit elements renders them useful for many applications, a few of which will be studied in this and a subsequent experiment. This experiment focuses on increasing your understanding of the frequency response of resistive, capacitive, and inductive impedance in multi-component circuits both theoretically and experimentally by use of the transfer functions for each circuit element. Background The AC C circuit, as well as the AC L circuit, falls into the realm of circuits used to filter unwanted signals. The properties of these passive filters are dependent on the values of, C, and L, as well as the Figure 1a shows how an C filter might be inserted into a circuit and Figure 1b shows the magnitude of the transfer function V / versus the input V OUT =V V / (a) frequency (Hz) (b) Figure 1. a) Circuit diagram for the series C circuit as a high pass filter. b) The magnitude of the transfer function V / for the series C circuit with =1 kω and C=0.039 µf. Note that the magnitude of the transfer function is plotted versus a log frequency scale. With the setup shown in Figure 1a, low frequencies are filtered or attenuated while high frequencies are passed with little attenuation. Hence, this circuit arrangement is called a high pass filter. The range of frequencies that are filtered is typically called the stopband and the range of frequencies that are not filtered is typically called the passband. Even though the relative smoothness of the transfer function of Figure 1 does not allow identification of a single frequency that divides the stop band from the pass band, a meaningful cutoff frequency for this filter can still be defined. The cutoff frequency for this filter, as well as for the low pass filter, is the frequency for which the magnitude of the transfer function is decreased by the factor 1 / 2 from its maximum value. The factor 1 / 2 appears to an arbitrary choice, however, at the cutoff frequency the average power delivered by the circuit is one-half the maximum average power. Thus the cutoff frequency is also called the half power In the pass band, the average evised: 16 January /5

2 AC, C, and L Circuits power delivered by the circuit is at least 50% of the maximum average power. To be more specific, the pass band is defined as the range of frequencies in which the amplitude of the output voltage is at least 70.7% of the maximum possible amplitude. For the filter shown in Figure 1, the cutoff frequency and the 70.7% amplitude point is marked on the graph with straight lines. The cutoff frequency for the C circuit is, in the ideal case, Equipment ω = 2 πf c = (1) C c 1 Multimeter 47 Ω resistor Function generator 4.7 µf capacitor Oscilloscope 10 mh inductor Procedure The lab consists of three parts: the circuit, the C circuit, and the L circuit. The theoretical expressions for each circuit, assuming ideal circuit elements, were obtained in Transfer Functions of, C, and L Circuits. For the circuits you will experimentally determine the magnitude of the transfer functions and phase relationships as a function of frequency as a comparison to the theoretical expressions as well as experimental values for the magnitude of the current and impedance. For your convenience, questions regarding each circuit are included within each section to assist in making and entering your observations in the lab notebook. These questions are the bare minimum to consider and answer and should by no means be the minimum you, as the student should consider. The answers to the questions, in report form, must be included in the lab write up. It is also advisable to complete the data acquisition, plotting, etc., for each section before continuing on to the next. Obtain the appropriate circuit elements listed above and use the appropriate multimeter to determine the exact values of the resistor, capacitor, and inductor. THE CICUIT esistive impedance effects in an circuit: Measurement of V / function generator ch 1 V Figure 2. Set up for measuring V / and the phase of the current through the resistor Set up the circuit shown in Figure 2. All ground connections are made to the ground connection of the function generator. Set the function generator to output a sinusoidal voltage and set the amplitude to the midpoint of its range. In this circuit the output of the function generator is read from the via channel 2 and the voltage across the resistor is read from channel 1. evised: 16 January /5

3 AC, C, and L Circuits Using both inputs, the phase shift between the input voltage and the voltage across the resistor can be determined. To view the frequency on the, press [Time] on the front panel followed by [Freq] on the display menu. To view the phase shift, press [Time], then on the display menu, press [Next Menu] twice, followed by [Phase]. Determine the magnitude of the transfer function V / as well as the phase difference between the input signal and signal across the resistor as a function of frequency over the range 20 Hz to 2500 Hz. On the same graph, plot the measured transfer function as well as a theoretical curve for the transfer function as a function of On a separate graph, plot the phase difference between the input voltage and the voltage drop across the resistor. Questions: resistive impedance Is a resistor a reactive (i.e. frequency dependent) component in an AC circuit? Looking at the phase differences, how much does the current lead or lag the applied voltage? THE C CICUIT: eactive impedance effects in an C circuit: Measurement of V / Set up the circuit shown in Figure 3. With this setup the input voltage is read via channel 2 and V is read via channel 1. C ch 1 V Figure 3. Setup for measuring V /, and the phase difference between V and. Over the range 20 Hz to 2500 Hz obtain the magnitude of V / and the phase difference between V and. Plot the measured impedance of the circuit along with the calculated impedance on linear scales as a function of Plot the measured current and the calculated current on linear scales as function of For both the measured and theoretical values plot V / versus log f. evised: 16 January /5

4 AC, C, and L Circuits difference on linear scales. eactive impedance effects in an C circuit: Measurement of V C / Set up the circuit shown in Figure 4. With this setup the input voltage is read via channel 2 and V C is read via channel 1. C ch 1 V C Figure 4. Setup for measuring V C /, and the phase difference between V C and. Over the range 20 Hz to 2500 Hz obtain the magnitude of V C / and the phase difference between V C and. For both the measured and theoretical values plot V C / versus log f. difference on linear scales as function of Questions: The C circuit Is a capacitor a reactive (i.e. frequency dependent) component in an AC circuit? Looking at the phase differences, does the current lead or lag the applied voltage? Does the voltage across the capacitor lead or lag the applied voltage? Does the voltage across the capacitor lead or lad the voltage across the resistor? THE L CICUIT: eactive impedance effects in an L circuit: Measurement of V / Set up the circuit shown in Figure 5. With this setup the input voltage is read via channel 2 and V is read via channel 1. L ch 1 V evised: 16 January /5

5 AC, C, and L Circuits Figure 5. Setup for measuring V /, and the phase difference between V and. Measure V and and the phase difference between V and over the frequency range 20 Hz to 2500 Hz. Plot the measured impedance along with the calculated impedance on linear scales as function of Plot the measured current and the calculated current on linear scales as function of For both the measured and theoretical values plot V / versus log f. Plot the measured phase difference along with the theoretical values for the phase difference on linear scales as function of eactive impedance effects in an L circuit - Measurement of V L / Set up the circuit shown in Figure 6. With this setup the input voltage is read via channel 2 and V L is read via channel 1. L ch 1 V L Figure 6. Setup for measuring V L /, and the phase difference between V L and. Measure V L and and the phase difference between V L and over the frequency range 20 Hz to 2500 Hz. For both the measured and theoretical values plot V L / versus log f. difference on linear scales as function of Questions: The L circuit Is an inductor a reactive (i.e. frequency dependent) component in an AC circuit? Does the current lead or lag the applied voltage? Does the voltage across the capacitor lead or lag the applied voltage? Does the voltage across the capacitor lead or lad the voltage across the resistor? Other Questions Three sinusoidal signals at100 Hz, 500 Hz, and 1000 Hz are superimposed. You wish to measure the 1000 Hz signal. Explicitly, how would you do this? evised: 16 January /5