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LABORATORY PROJECT NO. 2
FREQUENCY RESPONSE AND FILTERS
1. Introduction
In many engineering applications, it is important to be able to select signals of a
given frequency, or signals in a given band of frequencies. For example, the signal picked
up by a radio antenna consists of signals generated by thousands of transmitting stations
(plus noise signals) at frequencies that literally cover the spectrum. A device in the radio
receiver called a tuner selects one station by allowing signals at that station frequency
(actually a very narrow band of frequencies) to pass, while rejecting signals at other
frequencies. Another example is multiplexing in communications systems. In this method,
a unique band of frequencies is allocated to each information channel, and many channels
are sent over one transmission medium. At the receiving end, frequency selection
techniques are used to separate the channels. In touch-tone telephone systems, filters are
used to differentiate the tones produced by the telephone buttons. Frequency-selective
circuits are used in satellite communications, in television receivers, and in many other
devices.
In this project, you will design a frequency-selective circuit based on the circuit
shown in Fig. 1. This circuit will strongly reject one frequency and strongly pass another
frequency, with intermediate response for other frequencies. More specifically, you will
design this circuit to reject a specific first harmonic and pass a specific third harmonic. This
circuit is thus a combination of a bandpass and a bandreject filter.
Be sure that you measure the inductance, shunt capacitance, and dc resistance of the
inductor during the first lab period, as you will need these values for your calculations.
R
R
v (t)
v (t)
C
C
Inductor
b
a
L
C
s
o s
g
v (t)
g
R
s
(a) (b)
Fig 1. (a) A circuit designed to reject a specific fundamental and pass a specific third
harmonic. The combination of v s
(t) and R g
is a model of a generator, with v g
(t)
representing its terminal voltage. The inductor is modeled by a simple
inductance. (b) A more sophisticated model of an inductor than a simple
inductance; to be used in a later part of the design.
2. Measurement of Circuit Components
A. Inductance. Use either of the inductors mounted on the lab-bench console or
check out a similar one from the stockroom. Measure the inductance of the inductor using
a suitable meter.
B. Inductor Resistance. Measure the dc resistance of the inductor. This dc
resistance will have the same order of magnitude as R
s
in the inductor model, but will not be
equal to R
s
because R
s
changes with frequency. Since measuring R
s
as a function of
frequency is complicated and requires more time than you have to spend on it, use the dc
resistance of the inductor as a rough approximation to R
s
in the work that follows.
C. Distributed Capacitance of the Inductor. Measure the distributed
capacitance of the inductor. This corresponds to C s
in Fig. 1(b), in which the inductor is
modeled by the R
s
, L, and C s
combination as shown. One way to measure C s
is to connect
a large resistor in series with the inductor and a sinusoidal generator and adjust the
frequency until parallel resonance of L and C s
occurs. Then C s
can be calculated using the
measured values of the resonant frequency and the inductance (R
s
can usually be
neglected). The frequency at which the parallel LC resonance occurs is the frequency at
which the voltage across the large series resistor is minimum.
- Determine the frequency response of the circuit. The frequency response
is obtained by letting v s
(t) be a sinusoidal voltage source of radian
frequency ω and then calculating or measuring | V
o
/ V
g
|, the magnitude of
the transfer function, as a function of ω or f, where ω = 2 πf (f will
probably be more convenient). Calculate the frequency response as
follows:
a. Write a MATLAB program that will calculate | V
o
/ V
g
| and plot it versus
f for any set of circuit parameters.
b. Using this MATLAB program, plot the frequency response
(| V
o
/ V
g
|) for the same set of circuit parameters you used in Part 3B
above.
- On the basis of the frequency response you plotted and the harmonics of
the triangular waveform v g
(t), explain the shape of the v o
(t) you plotted in
Part 3B2 above.
- Plot v o
(t) as in Part 3B2 above but for the period of v g
(t) equal to 0.6 ms
instead of 1 ms. On the basis of the frequency response and the
harmonics of v g
(t), explain the shape of this v o
(t). Now let the period of
v g
(t) be 0.834 ms, plot v o
(t) and explain its shape on the basis of the
frequency response.
4. Circuit Design
A. Approximate Equations for Frequency Response. Neglecting R
s
and C
s
transform the circuit of Fig. 1 to the frequency domain and write an expression for V
o
/ V
g
Since R
s
and C
s
are neglected, this expression is approximate. Make consistency checks on
the expression.
B. Using these approximate equations, choose component values that will reject a
fundamental at 1 kHz ( V o
= 0) and pass a third harmonic at 3 kHz ( V o
= max). Using the
MATLAB program you wrote, plot v o
(t) vs t and | V o
/ V
g
| for these component values (use
your measured values of L, R s
, and C s
) when v g
(t) is a triangular wave with a fundamental
of l kHz. Explain the shape of v o
(t) on the basis of the frequency-response plot. Note that
even though the fundamental is suppressed and the third harmonic is maximized, the higher
harmonics will all be present in v o
(t).
C. Using these approximate equations, choose component values that will reject a
fundamental at 9 kHz and pass a third harmonic at 27 kHz. Plot v o
(t) vs t and | V o
/ V
g
| for
these component values (use your measured values of L, R s
, and C s
). Explain why the
approximate design equations don't give good results for rejecting a fundamental at 9 kHz,
and why R s
and C s
affect the circuit response more at the higher frequencies than at the
lower frequencies.
5. Measurements
Find circuit components with values as close as you can to those you calculated in
4B above to reject a fundamental near 1 kHz and pass the corresponding third harmonic.
You may vary the frequency of the fundamental slightly to get the best fundamental
suppression and third harmonic enhancement with available capacitors. Measure the values
of all the circuit components you select.
A. Frequency Response. With these measured circuit components, measure the
frequency response by measuring | V o
(f)/ V g
(f)| with a sinusoidal generator voltage. Take
the data carefully so you can compare it with calculated data. Be sure you have measured
values of all components to use in your calculations. Be sure to get values for f
corresponding to the fundamental and third harmonic.
B. Time-Domain Reponse. Change the generator voltage to a triangular wave
with a fundamental frequency of 1 kHz and display the output voltage on an oscilloscope.
Vary the frequency slightly to get the best third harmonic on the output. Measure the
frequency of the fundamental. Record the waveform of v o
(t) for comparison with calculated
results.
8. Your Grade
Your report will be graded according to the following:
Category Percentage
Communication 30
Technical Content:
- Component Measurements 5
- Preliminary Work 20
- Circuit Design 20
- Measurements 15
- Comparison of Calculated and
Measured Results 5
- Conclusions 5
According to the items listed on p. 4 in "Course Procedure."
