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					                                 AC circuits: RLC series circuit
                                        (by Dr. James Wheeler)

    1. Objective
The object of this experiment is to measure the phase shift and amplitude of the voltage across a
resistor in a series resistance, inductance, capacitance circuit subject to a sinusoidal driving voltage.
These measurements will be related to the physical parameters of the circuit.

Equipment:
Signal generator, an oscilloscope with leads, a several millihenry inductor, a capacitor, a collection of
labeled resistors ranging from 10 to 10000 Ohms, and an Ohmmeter.

    2. Background
In a previous experiment a resistor (R) and a capacitor (C) were connected in series and subjected to
a sinusoidal driving voltage (Vd) of amplitude D and angular frequency . The voltage across the
resistor (Vr) was observed to be sinusoidal with angular frequency  and amplitude A, and preceded
the driving voltage in phase (). Precisely:
           Vr = A cos(t-) with
           tan  = - 1/(RC) and A = D cos

In another previous experiment a resistor (R) and an inductor (L) were connected in series and
subjected to a sinusoidal driving voltage (Vd ) of amplitude D and angular frequency . The voltage
across the resistor (Vr ) was observed to be sinusoidal with amplitude A and angular frequency ,
and followed the driving voltage in phase (). Precisely:
          V = A cos(t-) with
          tan  = L/R and A = D cos
       .
Our purpose is to see how the effects of these components combine when the components are
connected in series.

   3. Procedure
Determine the Capacitance of the capacitor.
Determine the inductance of the inductor.
Use the Ohmmeter to determine the resistance of the inductor and resistor.
       (The resistor should have a value on the order of 100.)
Connect the output of the signal generator to the capacitor.
Connect the other end of the capacitor to the inductor.
Connect the free end of the inductor to the resistor.
Connect the free end of the resistor to ground.

Attach the A-Channel oscilloscope probe to the signal generator-capacitor junction and the
Channel-B probe to the inductor resistor junction. Connect the assorted grounds.

The signal generator should be producing a sinusoidal driving signal. Observe the resulting voltage
across the resistor as a function of the frequency of the driving signal. You will notice that its
amplitude is a function of the frequency, and that the response shifts phase relative to the driving
signal as you pass the frequency for which the maximum response occurs. This phenomenon is
called Resonance.

If the driving voltage is:                    Vd = D Cos( t)

we observe that the voltage across the resistor is     Vr = A() Cos[ t + ()].
Our task is to determine both functions A() and ().

Determine the resonant frequency  ().
For 20 frequencies ranging from less than /100 to greater than 100, measure the period,
amplitude, and time shift of the voltage across the resistor, as well as the period and amplitude of
the driving voltage. Notice that the time shift between peaks of the driving voltage and the voltage
across the resistor can be either positive (high frequency) or negative (low frequency).
Make sure that 10 of these measurements lie between /3 and 3

Be especially careful to keep track of the time and voltage scales on the
oscilloscope, and thereby record the measurements correctly.
Data (Keep careful records of the oscilloscope settings!)

Period of               Amplitude of            Period of          Amplitude of       Resistor Voltage
Driving Voltage         Driving Voltage         Resistor Voltage   Resistor Voltage   Time Lag
       T                       D                       T’                   A                t
    (             )      (                )      (            )     (             )   (             )
What is the resistance of the resistor (R)?
What is the inductance of the coil?
What is the capacitance of the capacitor?
What is the observed frequency with zero phase shift? (phase ).


Analysis
                   1
Let  theory            be the theoretical resonant frequency.
                 2 LC

For each frequency:
Check that the driving and resistor periods are equal.
Convert the measured periods to angular frequencies. ( = 2/T.)

        Take the ratio of the voltage across the resistor to the driving voltage (A/D), which is called
                   A
the response. r 
                   D
Let the frequency with maximum response be called response.
Does this agree with phase ?

Take the ratio of the time shift to the total period, and multiply by 2, this will be the phase angle.
         =2t/T=t (in RADIANS).


Calculate the inductive reactance XL = L.
Calculate the capacitive reactance XC = 1/(C) .
Calculate the Impedance Z  R 2  ( X L  X C )2 .
Calculate rth = R/Z. This is the theoretical value for the response.
Calculate tan (th )= (XL -XC)/R and th = tan-1((XL -XC)/R).
        This is the theoretical value for the phase shift.
Angular   XL          XC         Z         Response Response Phase    Phase
Frequency                                  observed theory   observed theory
=2/T                                     r        rth              th
Rad/s     ()         ()        ()                         (rad)    (rad)




On a single semilogarithmic graph, as a function of the frequency
       Plot the response (r).
       Plot the theoretical response (rth).

On a single semilogarithmic graph, as a function of the frequency:
       Plot the observed phase shift 
       Plot the theoretical phase shift th.

Do the theoretical responses and phases agree with those you observed?

				
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posted:10/3/2012
language:English
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