LAB 1: FOURIER SERIES
1. INTRODUCTION
This lab has four objectives:
• to help you make a connection between the mathematics of Fourier series and the signals you measure
• to familiarize you with the Panasonic FFT Spectrum Analyser
• to have you design a resonant RLC circuit
• to develop your ability to explain a circuit response in both time and frequency domains
The lab writeup should not be a long one (10 pages maximum ), although it must be clearly written. Essentially, you should record and explain what you observed. Some questions are included in the text below as guidelines. If there were differences between observed responses and those you expected from idealized mathematical models, explain the differences.
2. FOURIER COMPONENTS OBSERVED ON THE SPECTRUM ANALYZER
Use a function generator and the Panasonic FFT Spectrum Analyzer for this part. The
Panasonic unit displays, among other things, the distribution of signal energy in frequency.
It is also useful as sort of storage scope, from which you can print out a waveform. There are also a couple of workstations available. For convenience, there is a copy of Mathcad
6.0 on each one for your calculations, although you can use any calculation package or computer language you want.
(a) Set the function generator to a sine wave at about 10 kHz. Display the input signal on the Panasonic to make sure you’re not clipping it - if you are, then reduce the voltage. Now check that the spectrum consists of a single tone
(b) Set the function generator to a symmetric (i.e., 50% duty cycle) square wave. Does the distribution of power among the harmonics as measured by the Panasonic unit follow your expectations from theory?
(c) The triangle wave can be obtained from the square wave by integration. Do its harmonics follow the expected pattern?
(d) Change to a square wave with about 10% duty cycle. Comment on the distribution of compared with the 50% duty cycle square wave. When we get through the part of the
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course that deals with Fourier transforms, you will be able to derive the spectrum. At that point, have a look at your measured result and see if it corresponds to your expectation
(but don’t bother to hand it in).
3. DESIGN AND BUILD A RESONANT CIRCUIT
In this section, you’ll design and construct a resonant circuit. “Design” here means analysis of the circuit’s frequency response and selection of appropriate component values.
This is all in preparation for Part 4, where you pick out separate Fourier components, or harmonics, of a signal using this circuit.
The circuit is shown below. You can make it more resonant - i.e., responsive to a narrower range of frequencies - by increasing the resistance. Mathematically, there is no limit to how selective you can make the frequency response, but,.in reality, energy losses in the LC pair put a lower limit on the selectivity.
(a) Calculate the frequency response of the circuit using as the impedances Z
L and Z
C
=
1 / j
ω
C . Alternatively, work in Hertz with
Plot its frequency response, magnitude and phase, for
Z
C
L
=
= j
0 .
1
2
π f
µ
F
L and
and
Z
C
L
=
1
=
1 /
= mH j
ω j 2
π f
L
C and a
.
few R values.
(b) Choose one of the wire-wound inductors with L = 1 mH (you can sign them out for a day at a time), then choose C to give resonance at about 16 kHz. The exact resonant frequency doesn’t matter. Then choose a value of R to give high selectivity. Build the circuit and check its frequency response. Is the resonant frequency where you calculated it to be? Does its selectivity improve indefinitely as R increases?
4. RESONANT RESPONSE TO SQUARE WAVE INPUT
In this section, you’ll put together what you gained in Sections 2 and 3, and use your resonant circuit to pick out the harmonics of a square wave.
(a) If the input to the circuit were a fixed frequency square wave and you had a tunable resonant circuit, then you would be able to observe resonance at every harmonic, or
Fourier component, of the input simply by sweeping the resonant frequency of the circuit.
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Unfortunately, your circuit is not tunable, so you’ll have to take the opposite tack and reduce the frequency of the function generator.
Input a 50% duty cycle square wave at 16 kHz, or wherever your circuit resonates, and observe the output. Is it close to a sine wave? Now reduce the input frequency. Does the circuit respond to harmonics of the input? Is its output close to a sine wave? Do the amplitudes follow the expected ratios?
(b) Now explain this behaviour in the time domain. With a low frequency square wave, observe the step response of the circuit. Now slowly increase the frequency and watch what happens to the output as you slide through each submultiple, odd or even, of the resonant frequency. Is it clear why the square wave has only odd harmonics?
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