Frequency Response and Filters
Objectives:
This experiment provides practical experiences with frequency responses of analog filters. Filters will be
constructed and graphs of gain magnitude and phase versus frequency will be created based on data
collected in the lab.
Pre Lab:
1.
Read and understand Hambley Chapter 6 Sections 6.1, 6.2 and 6.5 covering low
pass filters and high pass filters.
2.
The gain (also called the transfer function) of a filter is the ratio of the phasor output
voltage to the phasor input voltage. Using phasor analysis, the transfer function of a
first-order RC Low Pass filter as a function of frequency is given by the equations
below.
st
H (ω ) =
1
Vout
1/jωC
=
=
Vin R + 1/jωC 1 + jωRC
3.
1 order low pass transfer function:
4.
Evaluate the transfer function in item 3 above using the nominal values of your
resistor and capacitor components as shown in Figure 1 to determine the
magnitude and phase at frequencies of 50 Hz, 200 Hz, 1kHz, and 5 kHz. These
values will be used to determine if your circuit is correct in the lab. Show all
calculations in your lab report.
5.
Use the voltage divider rule to perform phasor analysis on the High Pass filter
circuit shown in Figure 2. Notice that the input signal is measured by Channel 1
(Vin) and the output voltage (Ch2) is measured across the 1 KΩ resistor. The
transfer function will be different from that shown in item 3.
Evaluate the transfer function you derived in item 5 using the nominal values of the
resistor and capacitor to determine theoretical gain and phase at frequencies of
500 Hz, 1kHz, 5kHz, and 50 kHz
Mark the four points calculated in the Prelabs (for the low pass and high pass
filters) on the graphs for Parts A and B.
6.
7.
Part A: Low Pass Filter Frequency Response
Procedure:
1. Measure the 1 kΩ resistor using the Fluke 45 as an Ohmmeter. Measure the 1 μF capacitor
using the universal bridge on the instructor’s table. Enter these values on the data sheet.
2. Construct the low pass filter circuit illustrated in Figure 1.
3. Set the function generator output to a sinusoid with amplitude of 1 V peak-to-peak and an
initial frequency of 50 Hz.
4. The oscilloscope display may have a natural coarseness, which can be reduced by selecting
the Average function. Try to use as few samples as possible to avoid long delays while
moving from one display to another.
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NOTE: If the displayed wave
is extremely jumpy, the
problem is generally faulty
banana leads or an
ungrounded probe. If you
cannot remedy the problem,
ask for assistance.
Figure 1: The Low Pass Filter Circuit
5. Push the AUTOSCALE button to display both channels. Reposition so that the 0V is at the
midline on the both channels and the waveforms appear to overlap. Adjust the
Volts/division to 500mV/division for both channels and the sec/division to display more then
one complete period. At high frequencies, the amplitude of the filtered signal may be too
small to be detected by the Autoscale feature. You may need to set the Volts/division scale
manually in order to see the signal.
6. Press the MEASURE button to determine the Voltage pk-pk amplitudes for both channels,
and record in the data sheet for the frequency of 50 Hz. At least one full period of the
waveform must be in view to for the oscilloscope measure functions to be accurate.
7. Press the CURSOR button and adjust the cursors to measure the change in time between
peaks. Record the Δt in the data sheet for the frequency of 50 Hz.
8. Calculate the Gain magnitude and Phase using the following formulas and enter in Table 1 of
the data sheet.
| Gain |=
VOut
VIn
Phase = Frequency x Δt x 360
9. Compare with what you would expect for this circuit based on theoretical calculations from
the prelab.
10. Repeat steps 6 through 8 to make measurements for all frequencies shown in Table 1 and
record these values in Table 1. At very high frequencies AUTOSCALE may not work
because the filtered signal is too small. Use manual adjustments of the volts/division to
reveal the signal.
11. Graph the Gain and Phase on the semi log paper provided. Do your results agree with what
you would expect for a low pass filter? Explain what is meant by a low pass filter.
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Part B: High Pass Filter Frequency Response
Procedure:
1. Measure the 1-kΩ resistor using the Fluke 45 as an Ohmmeter. Measure the 0.1-μF capacitor using
the universal bridge on the instructor’s table. Enter these values on the data sheet.
2. Construct the high pass filter circuit illustrated in Figure 2.
3. Using any method, set the waveform generator output to a sinusoid with an amplitude of 1 V
peak-to-peak and an initial frequency of 10 kHz.
4. The oscilloscope display may have a natural coarseness, which can be reduced by selecting the
Average function. Try to use as few samples as possible to avoid long delays while moving from
NOTE: If the displayed wave is
extremely jumpy, the problem is
generally faulty banana leads or
an ungrounded probe. If you
cannot remedy the problem, ask
for assistance.
one display to another.
Figure 2: The High Pass Filter Circuit
5. Push the AUTOSCALE button to display both channels. Reposition so that the 0V is at the
midline on the both channels and the waveforms appear to overlap. Adjust the Volts/division to
500mV/division for both channels and the sec/division to display more than one complete period.
At low frequencies, the amplitude of the filtered signal may be too small to be detected by the
Autoscale feature. You may need to set the Volts/division scale manually in order to see the
signal.
6.
7. Press the MEASURE button to determine the Voltage pk-pk amplitudes for both channels, and
record in the data sheet for the frequency of 10 kHz. At least one full period of the waveform
must be in view to for the oscilloscope measure functions to be accurate.
8. Press the CURSOR button and adjust the cursors to measure the change in time between peaks.
Record the Δt in the data sheet for the frequency of 10 kHz.
9. Calculate the Gain magnitude and Phase using the following formulas and enter in Table 2 of the
data sheet.
Gain =
VOut
VIn
Phase = Frequency x Δt x 360
10. Repeat steps 6 through 8 to make measurements for all frequencies shown in Table 2 and record
these values in Table 2. At low frequencies, the amplitude of the filtered signal may be too small
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to be detected by the Autoscale feature. You may need to set the Volts/division scale manually in
order to see the signal.
11. Graph the Gain and Phase on the semi log paper provided. Do your results agree with what you
would expect for a high pass filter? Explain what is meant by a high pass filter.
Part C: Using Filters To Isolate Frequencies In An AM Signal
Procedure:
1. For this part, you will use the function generator to create an amplitude modulated (AM) signal to
simulate the sound of an engine running at 6000 RPM with an associated high-frequency vibration.
The AM signal is given by
v AM (t) = A[1 + μ cos(ω m t)]cos(ω c t)
Expanding the equation and using the trigonometric identity for the sum of cosines, we have:
v AM (t ) = A cos(ω ct ) +
Aμ
Aμ
cos[(ω m + ω c )t ] +
cos[(ω m − ω c )t ]
2
2
in which A is the carrier amplitude which we will set at 1 V, μ is the modulation index which we
choose as 0.3, cos(ωmt)is the modulating waveform which represents a high-frequency (5 kHz)
vibration, and cos(ωct) is the carrier waveform that results from engine rotation. For an engine
speed of 6000 rpm, the corresponding carrier frequency is 100 Hz.
Notice that the AM signal has three components with frequencies of 100 Hz, 4900 Hz and 5100
Hz. The 100 Hz component results from engine rotation while the 4900 and 5100 Hz
components result from vibration. We will use a low pass filter to pass the 100 Hz component
and reject the vibration components. Then we will use a high pass filter to pass the vibration
components and reject the 100 Hz component.
2. Set the waveform generator to sinusoidal mode as shown in Figure 3 (A).
Switch to the AM mode by pressing the MOD button (Fig 3B) and selecting AM using the soft
keys shown in (Fig 3C).
To return to the main menu press the MOD button (Fig 3B).
Select FREQ using the soft keys shown in (Fig 3C) and enter a frequency of 100 Hz for the carrier
waveform. The numbers can be set by using the knob or by pressing the buttons (Fig 3D).
Select AMPL using the soft keys shown in (Fig 3C) and enter 1 Vpk-pk for the amplitude of the
carrier waveform. This step is done by pressing the number one and selecting Vpp on the screen
Press the MOD (Fig 3B) then AMPL button (Fig 3C) to set the DEPTH to 30%. This corresponds
to μ = 0.3.
Press FREQ button (Fig 3C) to set the frequency of the modulating waveform and enter 5 kHz.
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3. Connect the low pass filter and oscilloscope as illustrated in Figure 1. Turn off Channel 2 of the
oscilloscope and examine the AM signal as the input to the filter. Adjust the volts/division and
seconds/division so that one period fills the screen.
4.
Turn on Channel 2 so that one complete waveform is viewed on the oscilloscope. At least one full
period of the waveform must be in view to for the oscilloscope measure functions to be accurate.
Examine the phase, amplitude, and appearance of the signal that is the output of the low pass
filter. Have your TA check your display and sign the data sheet. Explain in your lab report
the effect that the low pass filter has on the AM signal. Could this filtered signal be used to
determine the engine RPM?
5.
Disconnect the low pass filter and then connect the high pass filter and oscilloscope as illustrated
in Figure 2. Examine the resulting waveforms for the filter input and output. Examine the
appearance of the signal that is the output of the high pass filter. Have your TA check your
display and sign the data sheet. Explain in your lab report the effect that the high pass filter has
on the AM signal.
Log-Linear Plotting
In many applications, the ability to strongly reject signals in a given frequency band is of primary
importance. Comparing the performance of various filters, it is helpful to express the magnitudes of the transfer
function in decibels. To convert a transfer-function magnitude to decibels, the common logarithm of the
transfer-function needs to be multiply by 20
|H(ƒ)|DB=20log|H(ƒ)|
One of the best ways of plotting transfer functions is using a logarithmic scale. In this scale, the
variable is multiplied by a given factor for equal increments of length along the axis. On the linear scale ,equal
lengths on the scale correspond to adding a given amount to the variable.
The advantage of a logarithmic frequency scale compared with a linear scale is that the variations in
the magnitude or phase of a transfer function for a low range of frequency, as well as the variations in a high
range, can be clearly shown on a single plot. With the linear scale, either the low range would be severely
compressed or the high range would be off scale.
Figure 4 and 5 shows examples of Linear-Linear and Linear-Log graphs using the following formula:
|H(ƒ)|DB= 1/(1+j(f/fb), where fb is a fix constant (100) and the frequency f goes from 1 to 10000,
incrementing by one.
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Graphs for Part A
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Graphs for Part B
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Part A: Low Pass Filter Measurements
Measured 1kΩ resistor value
Measured 1μF capacitor value
Table 1
f (Hz)
CH-1 pk-pk
CH-2 pk-pk
gain
Δt
H ( f ) dB
phase
"+" if CH-2 leads
50
100
200
500
800
1k
2k
5k
10k
Part B: High Pass Filter Measurements
Measured 1kΩ resistor value
Measured 0.1 μF capacitor value
Table 2
f (Hz)
CH-1 pk-pk
CH-2 pk-pk
Δt
gain
50
100
200
500
800
1k
2k
5k
10k
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H ( f ) dB
phase
"+" if CH-2 leads
Part C: Using Filters To Isolate Frequencies In An AM Signal
TA Low Pass filter Verification
TA High Pass filter Verification
Your Name
Lab Instructor
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