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					PHYSC 3322                                  Experiment 1.2                       3 November, 2012


                                          AC Circuits
Purpose      The frequency response of a circuit containing reactive elements (inductors and
             capacitors) illustrates the importance of relative phase in the electrical behavior of
             multi-component circuits. The net current or voltage is the sum of the magnitudes and
             phases developed by the individual components. This experiment uses different
             combinations of the same components to illustrate the importance of relative phase in
             resonant AC circuits.


Background   The time-varying voltage V t  in an AC circuit may be expressed as

                   V t   V0 cos t ,                                                          (1)

             where V0 is the magnitude of the voltage and  is the angular frequency (   2f ,
             where f is the frequency). The current I t  is given by

                   I t   I 0 cost   ,                                                    (2)

             where I 0 is the magnitude of the current and  is the relative phase between the
             voltage and the current. (If the circuit contains reactive components, the voltage and
             current are generally not in phase.)
             The voltage and current can also be expressed in phasor notation, in which the
             quantities are expressed as vectors, V and I , which rotate around the origin of the
             coordinate system at frequency f .
             The impedance of a circuit (here, circuit is defined as any combination of resistors,
             capacitors and inductors) can also be represented as a phasor by
                        V
                   Z     ,                                                                      (3)
                        I
             where the vector division is governed by the rules of complex arithmetic. Review the
             phasor diagrams for each type of reactive and resistive component in Chapter 36,
             sections 3 and 4, of Halliday and Resnick.


Procedure    Select a capacitor (~ 10 nF) and inductor (~ 1 mH) to form a resonant circuit. Determine
             the value of each component using the Elenco multimeter. To insure the greatest
             accuracy, plug the component leads directly into the wire clips (not the test lead jacks)
             on the front of the meter, rather than using test leads. Calculate the resonance
             frequency ( f 0 ) of series and parallel resonant circuits composed of these two
             components, using Halliday and Resnick as a reference.
             Use an oscillator (function generator) and oscilloscope to measure the current as a
             function of frequency for a 100  resistor over the frequency range f 0 / 2 to 2 f 0 (see
             Figure 1). Fix the voltage supplied by the oscillator and vary the frequency. The
             magnitude of the frequency is read from the frequency counter and the voltage drop
             across the resistor is determined from the peak-to-peak value of the waveform
             displayed on the oscilloscope. Use Sigma Plot to graph the peak current as a function of
             frequency. (This part of the procedure is intended to verify that the current, and
             therefore the voltage of the waveform, provided by the function generator, is
             independent of frequency.)



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PHYSC 3322                             Experiment 1.2                             3 November, 2012


             Series resonant circuit (current): Use the circuit shown in Figure 2 to determine the
             magnitude of the current as a function of frequency over the range f 0 / 2 to 2 f 0 . Plot
             the results.




             Figure 1. Current vs. frequency.                  Figure 2.     Series   resonant    circuit
                                                               (current).




             Figure 3. Series       resonant    circuit        Figure 4. Series       resonant    circuit
             (capacitor voltage).                              (inductor voltage).




             Figure 5. Series resonant          circuit        Figure 6. Parallel resonant circuit
             (combined voltage).                               (current).




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PHYSC 3322                             Experiment 1.2                            3 November, 2012


             Series resonant circuit (voltage): Determine the frequency dependence of the voltage
             across each of the reactive elements in the circuit, as shown in Figures 3–5. Plot the
             results.
             Parallel resonant circuit (current): Determine the current as a function of frequency for
             the parallel resonant circuit shown in Figure 6. Plot the results.


Questions    Compare the results of your measurements with theoretical predictions. Discuss any
             discrepancies and consider possible sources of error.
             For the series resonant circuit, discuss how the individual voltages across the reactive
             components ( VC , VL ) combine to obtain the total voltage ( VT ) across the two
             components. Use Sigma Plot to verify your method.
             Plot the frequency response of the current for both the series and parallel resonant
             circuits on one graph. Discuss the features of the curves in the context of destructive
             interference.




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