UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE
Department of Electrical and Computer Engineering
Experiment No. 6 - Active Filters
Overview: The purpose of this experiment is to familiarize the student with active filter
networks. A filter is a device that passes electrical signals at certain frequencies or frequency
range while preventing the passage of others.
Filter circuits are used in a wide variety of applications. In the field of communication, band-pass
filters are used in audio frequency range (0 KHz to 20 KHz) for modem and speech processing.
High-frequency band-pass filters are used for channel selection in telephone central offices. Data
acquisition systems usually require anti-aliasing low-pass filters as well as low-pass noise filters
in their preceding signal conditioning stages. System power supplies often use band-rejection
filters to suppress the 60-Hz line frequency and high frequency transients. At high frequencies
(greater than 1 MHz), all of these filters usually consist of passive components such as inductors
(L), resistors (R), and capacitors (C). They are then called RLC filters. In the lower frequency
range (1 Hz to 1 MHz), however, the inductor value becomes very large and the inductor itself
gets quite bulky, making economical production difficult. In these cases, active filters become
important. An active filter is one that incorporates gain, and is often based on an op-amp as the
active device in combination with resistors and capacitors to provide an LRC-like filter
performance at low frequencies. The principle advantages over passive counterparts are:
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They are able to perform complex filtering functions without the need for inductors.
Because gain is available it is possible to implement filters with characteristics that
could not be obtained from passive filters, such as negative impedance conversion.
By including electronically adjustable gain blocks, filters with electronically variable
characteristics are possible. Active filters are used extensively for signal conditioning in
electronic circuits.
Poles and Zeros: In analog circuit theory, circuits and signals are often analyzed in the sdomain. This means that the Laplace transform is applied to signals and circuits, enabling them
to be analyzed algebraically rather than as time varying signal. Conversion to the s-domain is
achieved via. Laplace transforms. In essence, this means that instead of having a time-varying
signal y(t) and a circuit impulse response function h(t) that relates the output to the input, we
have a signal Y(s) and a transfer function H(s) that gives the output signal as H(s)Y(s). The
signal Y(s) is a complex entity, reflecting the frequency content of the signal. The domain for s is
given by, –j ∞ < s < +j ∞ (i.e., frequencies are mapped to the complex frequency domain). The
transfer function H(s) provides the complex amplitude gain for each frequency as the signal
passes from the input to the output. Poles and zeros determine the system function within a
constant. In the time domain, coefficients of a differential equation represent the poles and zeros
of a system.
The pole/zero method of analysis is useful for studying the behavior of linear, time-invariant
networks. In pole/zero analysis, a network is described by its network transfer function that, for
any linear time-invariant network, can be written in the general form
a m s m + a m −1 s m −1 + .... + a 0
T ( s) =
s n + bn −1 s n −1 + .... + b0
Where the coefficients a and b are real numbers and the order m of the numerator is smaller
than or equal to the order n of the denominator. The order of the denominator is called the order
of the network. In the factorized form the previous equation will be
T (s) = a m
( s − Z 1 )( s − Z 2 )...( s − Z m )
( s − P1 )( s − P2 )...( s − Pn )
Where am is a multiplicative constant, Z1, Z2 …. Zm are the roots of the numerator polynomial,
called the transfer-function zeros or transmission zeros of the network, and P1, P2…..Pn are
the roots of the denominator polynomial, called the transfer-function poles or natural modes
of the network.
The poles and zeros can be either real numbers or complex numbers. However, since the a and b
coefficients are real numbers, the complex poles or zeros must occur in conjugate pairs. That is,
if 5 + j3 is a zero, then 5 – j3 also must be a zero. A zero that is pure imaginary (+jz or –jz)
causes the transfer function T(jω) to be exactly zero at ω = +z or –z
First Order Filter: the general first-order transfer function is
a1 s + a 0
s + ω0
T (s) =
Where – ωo is the location of the real pole, and ωo is called the pole frequency.
The values of ao and a1 determine the type of the filter. When a1 is zero we have a first-order
low-pass network with the transfer function
T (s) =
a0
s + ω0
In this case the dc gain is ao/ωo, and ωo is the corner or 3-dB frequency. This transfer function
has one zero at s = ∞. On the other hand, when ao is zero we have a first-order high-pass network
with the transfer function
T (s) =
a1 s
s + ω0
Here we have one zero at s = 0 and a 3-dB or corner frequency of ω0.
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Pre-Lab – Active Filters
1. Draw the pin connections for the LM741 OpAmp.
2. Write a voltage transfer function T(s) for the circuit of Fig. 1. Use the transfer
function for this circuit to construct a plot of the gain versus frequency.
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3. Filter design: Select a number to look up the cutoff frequency fC in the table below.
Design a high- pass first-order filter with a pass-band gain of –1.5, and a cutoff
frequency as specified. Any standard components available in your laboratory may be
used in the design. After the design is complete, derive the transfer function for the
circuit and construct a plot of gain vs. frequency.
fc
0
1
2
3
4
5
1 KHZ
2 KHZ
3 KHZ
4 KHZ
5 KHZ
6 KHZ
6
7 KHZ
7
8
9
8 KHZ
9 KHZ
10KHZ
(INSTRUCTOR’S SIGNATURE_____________________________DATE
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)
Lab Session – Active Filters
On a logarithmic grid, the numbers 1, 2, 5, and 10 (in any decade) are about equal
distances from each other. Since gain plots are usually drawn on log-log grids, it is
common practice to acquire data in frequency increments of 1, 2, 5, 10. For example, if the
desired frequency range starts at 10 Hz, gain information would be acquired at 10 Hz, 20 Hz, 50
Hz, 100 Hz, 200 Hz . . . until the upper end of the desired frequency range is reached.
Normally, data is collected over a range of frequencies starting 2 decades below the cutoff
frequency, up through a frequency that is 2 decades above fc. This method should be used for
collecting frequency response data in this exercise.
1. Low-pass filter
a. Build the circuit of Figure 1 on a breadboard.
b. Create a table in your report with four columns to record the data. Label the first row
with the column titles: Frequency, vin, vout, and Gain (AV=vout/vin).
c. Set the function generator to produce a sine wave with a frequency of approximately
10 Hz, and an amplitude of 2 Vp-p. Enter this value in the first column, second row of
the data table.
d. Verify that vin is still near 2 Vp-p. If vin is off by more than ±10%, adjust the waveform
generator to bring the vin closer to 2 Vp-p. Measure the actual vin, and record this
value in the second column. Measure and record the peak-to-peak voltage of vout.
e. Increase the frequency of the signal produced by the waveform generator to the next
increment. Repeat the procedures outlined in step d above. Repeat procedures d and e
until you have collected data up to an input frequency of 1 MHz.
f. Collect one more data point at the cutoff frequency (-3 dB point) of this filter. To find
the “3 dB down” point, compute the gain in the pass band from the data contained in
your data table. Multiply this gain by 0.707 (3 dB down). While observing both vin
and vout, adjust the frequency until the gain is equal to the 3 dB down gain. Record
the signal frequency f, vin, and vout at this cutoff frequency.
2. High-pass filter design
a. Build your high-pass filter design.
b. Prepare a data table in your report similar to the one used in part 1, above.
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c. Start collecting frequency, vin, and vout data at a frequency of 10 Hz, using the
method described at the beginning of the Procedure section. Continue collecting and
recording data up to and including 1 MHz.
d. Collect one more data point at the cutoff frequency of the high-pass filter, as
described in part 1f.
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Lab Session – Active Filters (Data Sheet)
INSTRUCTOR'S INITIALS
DATE:
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Post Lab – Active Filters
The gain calculations and the plots requested (below) are performed most easily in a
spreadsheet program. However, if you prefer, these calculations and plots may be done by
hand. If you do the calculations by hand, enter the calculated gain into the data tables in
parts 1 and 2 in the Procedure section.
1.
For the low-pass filter perform the following:
a. Enter your data into a spreadsheet program, including the data collected at
the cutoff frequency. Sort the data by frequency so that the cutoff data
appears in the correct location in the spreadsheet.
b. In the gain column, create an equation to compute the gain (vout/vin).
b. Using a log-log grid, plot gain on the vertical axis and frequency on the
horizontal axis. Take the spreadsheet plot and paste it into
your lab report.
d. On the plot of the frequency response, label the pass band, the pass band
gain, the stop band and the cut-off frequency.
2- Repeat a through d for the high pass filter.
For both the high-pass and the low-pass filters, compute the percent error between the
cut-off frequency predicted from the transfer function T(s), and the actual cut-off
frequency measured in the lab. Compute the error between the theoretical and measured
pass band gain. What could be the cause of this error?
At high frequencies the gain curve of the high-pass filter data may roll off or even turn
back up. This behavior is not predicted by the transfer function for this circuit. Can you
explain what might cause this behavior?
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