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THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Cardiovascular Dynamics Lab • Department of Biomedical Engineering
617 Bowser Road • Piscataway • New Jersey 08854-8014 •
732/445-3727 • FAX: 732/445-3753 • e-mail: [email protected]
RC Frequency Response
Aim: In this experiment, we will measure theMagnitude and Phase response of a
RC-electrical circuit to a sinusoid at various frequencies. Measured parameters
will be the time constant for the circuit and its cutoff frequency. These
parameters are then used to model the physical experiment using MATLABSIMULINK. The key parameters of the system will be determined by 2
methods and compared to derived values from the lecture theory.
1. Theoretical introduction
The subject of this experiment is the single branch RC circuit that is excited by a
sinusoidal voltage source applied across the branch. We will take the voltage
output of the circuit to be the voltage across the capacitor. The circuit schematic
Is shown in Figure 1. Note that this circuit is different than the RC branch
studied in earlier experiments since we are focusing on the Capacitor Voltage
Instead of the Branch current.
Figure 1.
Electrical RC branch circuit Set-up.
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Begin the study of this circuit by first is analyzing the relationship
between the voltage source and the voltage across the capacitor. The
capacitor voltage is simply the capacitive impedance multiplied by the
current phasor as,
VC ( j ) 
1
 I( j  )
jC
Since there is only one loop in the circuit, we apply Kirchoff's voltage law
to find I(j)as follows,
V( j )  I(j )[R 1/( jC] ,
Solving for I(j) and then replace I(j) to solve for VC(j) yields
VC ( j ) 
V( j )
(1  jRC)
If the magnitude of the VoltageMagnitude of the voltage source phasor is
A. Then, the magnitude of the capacitor voltage phasor is
| VC ( j  ) | A
1
1 (RC)2
Notice that the magnitude depends on the frequency applied to the
circuit. Using the real and imaginary components of the phasor we can
also find the phase from
 (j  )  tan 1 (RC) .
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To simplify the analysis let
(RC) 2  1
also,  c  2 f
fc 
1
2RC
A
and | VC |  2 for c 
1
RC
Also, define c as the cutoff frequency, for which case, if the voltage
source frequency is equal to the cutoff frequency, we have the magnitude
of the capacitive voltage reduced to 0.707A. Notice that the cutoff frequency
is related to the product RC. RC is also the time constant for a RC circuit.
In this experiment, we will use this relationship to find RC first from the
sinusoidal response of the circuit and second from its time dependent
response. Recall that the time dependent RC response is
Vc (t)  Vc (0)et / 
Where =RC
And where VC(0) is the initial condition or the starting value of the
capacitive voltage
2. Experiments
Equipment list:
1. Biopac MP30 analog to digital converter system
2. Hewlett- Packard frequency generator.
3. RC substitution box.
4. Biopac input cable.
5.
A.RC frequency response experiment [Biopac]
Procedure:
1.In this experiment you will need to measure voltage. So, begin by calibrating
two channels of the Biopac system for zero and +5 volts.
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2. After the Biopac is calibrated to read voltage, wire the RC circuit of figure 1.
Use the RC substitution Box for R and C. Use the clip leads cables to connect to
your circuit
3.Connect one channel of the Biopac to the input voltage of the RC circuit that is
the function generator output. Connect a second channel of the Biopac MP30 to
the output of the RC circuit. In this case, take the output voltage as the voltage
across the capacitor and the input voltage as the output of the signal generator.
4. Set the values on the RC box to be R=500K and C= 1.0F.
5.Adjust the frequency generator for the following:
a. free run mode.
b. frequency multiplier X1.
c. attenuation - 0dB.
d. frequency dial =1Hz.
Pull out the Power supply tab to the “on” position.
Launch the Biopac to open a window on your PC. Open the channel Menu and
click the box on channel one and two to record and display data.
Close that window and open the BP 100 Acquisition Menu. Set the acquisition
rate for 100 Hz.
Save to hard drive and record once.
Set recording duration for 30 seconds. Verify that you can record a 1Hz
sinewave from both channels.
Once you are able to make recordings, use the measurements menu on the
Biopac window and find the peak-to-peak voltage measurement.
Adjust the function generator output control so that you have a one volt peak to
peak voltage input to the RC circuit.
Now begin to collect data to measure the frequency response of the circuit by
sampling the circuit's capacitive voltage peak-to-peak output at various
frequencies.
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Since the generator is already set to 1Hz, measure the peak-to peak voltage on
the second channel of the Biopac window. This is the sinusoidal magnitude of
the circuit's output.
Next, repeat the measurements of the circuit 's input and output voltage at other
frequencies. It is a good idea to start by moving the generator frequency dial to 2
Hz. and then doubling again to 4 Hz, 8Hz, 16 Hz and so on. You will need to
increase the frequency generator multiplier switch to 10X in order to reach the
higher frequencies.
Record your data in a table containing Frequency, Input, Voltage, Output
voltage, and Phase. To measure Phase, return the frequency generator to 2 Hz.
Then locate the time interval (∆t) measurement tool on the Biopac window. Use
the cursor to highlight one full sinewave on the output voltage channel. Record
the reading in msec. Then observe the time delay between the input and output
voltage channels. It will be a small amount but use the cursor to highlight the
time shift between input and output. This is the phase delay.
Using the following formula, you can convert from time shift to phase using
your readings

t
 360 degrees
T
Now repeat the measurement for phase at all of the other frequencies that
you measured the peak to peak output and add this result to your data
collection table.ADD MORE DATA TO YOUR TABLE BY ADJUSTING
THE FREQUENCY GENERATO TO Frequencies BELOW 1 Hz for
frequencies such as 0.5, 0.125, and lower if you still detect changes.
B.Time Response Experiment
Next, we perform a measurement on the same RC circuit to observe its
response in the time domain. Leaving everything else unchanged, return
the frequency generator to 1 Hz and push in the square wave button.
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Start a single recording on the Biopac. You should observe a square
wave on the input channel. The output channel should also have a signal
that looks like a charging capacitor voltage.
Verify that the input square wave has a peak to peak voltage of one volt.
If not, adjust the generator output control until you confirm that you have
one volt.
Next, using the cursor, measure voltage. Use it on the output wave
response to find the peak to peak voltage again. Then, move the cursor
until the voltage drops to 0.368 of the initialV(peak to Peak)
Record the time interval for which it takes the wave to fall to .368 of the
maximum value. This is the time constant () for the circuit.
Next, Measure the value of the cutoff frequency. Leaving everything
unchanged, return the waveform generator back to a sinusoidal output.
Start the Biopac to record and observe that you can record a sinwave on
both chanels. Adjust the wave generator to 1 Hz,p-p. Continue to record
the capacitive voltage sinwave and slowly reduce the frequency of the
wave generator until you find that the capacitive voltage p-p has reduced
to 0.707 volt p-p. Note that you may need to change the frequency
Multiplier control to achieve low enough frequencies
Record the frequency at which this occurs. This is the Cutoff frequency
for your circuit.
FinaLLY,Input a squarewave to your RC circuit for any frequency Above
the cutoff. Measure the capacitor voltage response and print your data.
Do the same for a triangle waveform. Can you determine what calculus
operation the circuit is perfoming? Repeat this measurement for one
frequency Below the cutoff frequency. Print yoyuur resuilts. Does the
circuit accurately perform the calculus operation below cutoff frequency?
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SIMULINK EXPERIMENT
In this design experiment, the Simulink functional blocks will be used to model
the RC circuit. The approach that we will use here is to utilize the information
gained from our Biopac experiment. Since SSIMUKLINk does not provde
circuit analysis features we will use the Transfer Function block to Model the
RC circuit.
Open the transfer function block in the Simulink program and find the
parameters necessary for input to this block. Then refer to your data and insert
the proper values.You will need to refer to the theory for the transfer function of
this circuit. N ote,The SIMULINK For our purposes it will be a good
approximation to assume that S=j,In which case you can use the Fourier
transfer function that was derived in the theory section.
Input a square wave source to the transferfunction block like you did in the
Biopac experiment.
Compare your Simulink output graph to your Biopac recorded square wave
response. If necessary, adjust the square wave signal of the filter block
parameters until you are able to model your data as best as possible.
Once your square wave response matches the data, replace the square wave
signal source with a sinusoidal source.
Then, as in your Biopac experiment, Find the cutoff frequency, Start with the
value that you measured from you r RC circuit and verify that the transfer
function provides a 0.707 reduction in the sinusoidal amplitude
3. REPORT:
1.Design an alternative Simulink model solves the loop equation for the
RC circuit.
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2.Compare your results for the time square wave response data and
Simulink model. find the % error in amplitudes
4.Submit graphs for all trials performed.
5. Provide a Bode plot using the frequency - amplitude data that you
measured from the RC circuit.
6. Use the time constant obtained from the RC experiment to calculate
the cutoff frequency.
7. Find the cutoff frequency from your Bode plot. Compare values with
question 6and your measured cutoff frequency. Do an error analysis
between each value obtained
8. Repeat the questions 5, 6, 7 for the data that was obtained from the
Simulink experiment.
11. Derive the capacitive voltage for the RC circuit in the frequency
domain using the phasor notation.
12.Prepare a written summary of your experiment. Discuss what you
learned relative to your lecture course.
13. What did you observe about the Phase shift for the RC circuit?
14. Does the measured Phase shift agree with the theoretical value? If
not discuss the error.
15. What kind of filter is the RC circuit?
16. Design an RC filter circuit that reduces the amplitude of a 3 Hz
sinusoidal input wave to .25 of the input amplitude.
17. At what frequency does the RC circuit provide the least output
amplitude. What frequency does it provide the greatest output?
18. Bandwidth =?
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19. What calculus operation did you find that this circuit provides?
Prove it using circuit analysis. Why did it only work for frequency
above the cutoff?
20. Provide a Bode graph of Phase versus frequency.
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