ENGINEERING 11, 2004

                                              ELECTRICAL CIRCUIT ANALYSIS

 

                    Laboratory 5: Frequency Response of First and Second Order Circuits

 

 

Objective

 

            To characterize the behavior of a variety of first and second order circuits as a function of frequency, and to relate these results to theoretical predictions.

 

 

Procedure

 

1.  First Order RC Circuit

 

a.  Connect the circuit shown below and drive it with a sinusoidal source.

 

 

b.  Measure the V_out, V_in, and the phase shift between the input and output as the frequency is varied from about 50 Hz to about 10 kHz (Take between 10 and 20 measurements, more when the measurements are changing rapidly).

 

 

2.  First Order RC Circuit Using an Op Amp

 

a.  Obtain the frequency response for the circuit shown below, using R₁ = 1 MW;  R₂ = 100 kW;  ^{C = 0.01 mF}.  Measure V_out, V_in, and the phase shift between the input and output vs. frequency.  Repeat using the dynamic signal analyzer.

 

 

 

^{b.  What is the purpose of the 1 MW resistor?  Hint:  consider the gain for a dc (or very low frequency) input signal.}

3.  Second Order RLC Circuit

 

a.  Connect the circuit shown below with L = 112 mH and C = 1 mF.

 

 

b.  Choose two values of R, one that will result in an overdamped circuit, and one that will result in an underdamped circuit, and measure V_out, V_in, and the phase shift between the input and output vs. frequency.  You may use the signal analyzer for this circuit.

 

Report

 

1.  First Order RC Circuit

 

Plot both the theoretical and calculated attenuation (V_out/V_in) and phase response as a function of frequency (use a log scale for both amplitude and frequency).  Include your derivations.

 

2.  First Order RC Circuit with Op Amp:

 

Plot both the theoretical and calculated attenuation and phase response as a function of frequency (use a log scale for both amplitude and frequency).  Include your derivations.

 

3.  Second Order RLC Circuit:

 

•  Present the derivation  attenuation and phase response as a function of frequency when the output is underdamped, and overdamped.  What value of resistance is needed for critically damped response?

 

•  Plot the theoretical and calculated attenuation and phase response as a function of frequency.

 

•  State how you can tell the difference between over-, under- and critically-damped from the plots.

   Note the frequency at which the phase is 0.  This is the resonance frequency.

   Simulate the RLC with Multisim or Matlab and compare the computed amplitude and phase response with the measured one.

 

4.  General questions:

 

•  Why isn’t the internal resistance of the function generator an issue in this lab?

 

•  Sketch the output of circuit 1 if the resistor and capacitor or interchanged, and for circuit 2 if the resistor and inductor are interchanged.  Support your answer with a brief written explanation.

 

•  Derive the expression for phase shift given in the notes on the following page.

   Compare the computed resonance frequency with the measured one.

 

Note:  Measuring attenuation and phase shift:

 

            If V_i and V_o are as shown below, the attenuation is given by:

 

                                                                       

and the phase shift is given by (in degrees):

 

                                                                     ,

 

where T is the period (1/f) of the sine wave.  For a radian measure, replace 360 by 2p.  Please be careful about the sign of the phase shift.  In the Figure below, f is negative, because the output follows the input.  If the output led the input, f would be positive.

 

           

 

 

            Make time measurements when the signal crosses zero volts.  At this time, the signal is changing most rapidly, which provides the most accurate measurements.

 

Resonance occurs when the apparent impedance is the resistance.  For an RLC circuit, this happens at a frequency