A
Fourier Series Problem
Problem
Freq4P00Int
This is a guided problem to help you with basic Fourier Series concepts.
If you click here you can get a
calculator for some Fourier Series calculations. Use that calculator
for the questions below, as needed.
Problems
-
Run the calculator for
a sine wave. The value of the sine component should be the amplitude
that you set. The value of the cosine component should be zero.
Be sure that you understand why that is so. Be able to explain why
that is true.
-
There should be only one
non-zero component in a sine wave. Be sure that you see that when
you run the calculator, and that you understand why that is so.
-
Run the calculator for
a cosine wave. The value of the cosine component should be the amplitude
that you set. The value of the sine component should be zero.
Be sure that you also understand why that is so.
-
There should be only one
non-zero component in a cosine wave. Be sure that you see that when
you run the calculator, and that you understand why that is so.
P1
Is there a second harmonic component in the sine wave?
-
Run the calculator for
a triangle wave. Leave the frequency and amplitude settings at the
default values. Answer the following questions for this signal.
P2
Is there a second harmonic component in the triangle wave?
P3
Is there a third harmonic component in the triangle wave?
Now, try running the calculator for some of the other signals and observe
the pattern of the harmonics. Be sure that you can answer these questions.
-
What is the common characteristic
of those signals that do not have even harmonics?
Now, here is a link to another
calculator. This calculator will let you examine two triangle waves:
-
Triangle signal #1 starts
(at t = 0) at a negative peak. The positive peak occurs half way
through a period. (This is the triangle signal in the multi-signal
calculator.)
-
Triangle signal #2 starts
(at t = 0) at a positive-going zero crossing. The positive peak occurs
a quarter way through a period.
A triangle wave that starts
at a positive zero crossing has a Fourier Series with the general term
given by:
-
bk =
[8Vp/p2k2]sin(pk/2).
-
The sin() term just changes
sign and makes the even coefficients equal to zero. Notice that:
-
sin(pk/2)
= 1 for k = 1
-
sin(pk/2)
= 0 for k = 2
-
sin(pk/2)
= -1 for k = 3
-
sin(pk/2)
= 0 for k = 4
-
etc.
-
Using the formula above,
calculate the fundamental term for a triangle wave that starts at a positive
zero crossing and which has a peak amplitude of 10.
Behind the scenes
in the calculators
The calculators used in this lesson do not really integrate the functions
mathematically. Rather, they compute a numerical
approximation to the integrals. Since they do the calculations
on the fly - using data as it is generated - they do not do the FFT (i.e.
the Fast Fourier Transform). They just plow through the calculations,
doing them brute force. You might want to think of the calculations
in the calculators as the SFT (i.e the Slow Fourier Transform).
While the calculations here do not take advantage of the speedier FFT algorithm,
the number of points is small enough that you will not notice any slowness.
(At least, our testing of the calculators has not shown them to be noticably
slow.) However, since the calculations are done numerically, there
will be times when the calculations will be slightly off from the theoretical
values. In our experience, they are always pretty close, but there
is the possibility of small errors on occasion.