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19.4
[60 pts] |
- From Figure P19.4, T = 1.
- Perform the integration for bk
by hand (using integration by parts) AND by using MathCAD. If you haven't
had integration by parts yet in calculus, please look it up in your
calculus book and check out the examples. It requires only basic derivative and
integral skills. Be sure to simplify you answer as much as possible. If you need help, please stop by during office hours.
- You
can integrate over any range that includes a full period of the wave. So if
you define the function and integrate over -T/2 to T/2,which includes a full period
of the wave, the equation of the line is simple and it applies over the whole
range. If you integrate from 0 to T instead, you have to define the function and
integrate over two separate ranges (0 to T/2, and T/2 to T). The function (line
equation) is different for each range in this case. You will get the same answer
with both methods, and both approaches are valid and correct, but the first is
much easier.
- Write out the first three nonzero terms (harmonics) of the
Fourier Series.
- Generate all plots over two cycles (periods) of the function.
- Plot
the first three terms (harmonics) individually on the same plot.
- Plot
the sum of the first 3 terms and the sum of the first 50 terms on the same plot.
- Plot
the amplitude magnitude spectrum for the first 10 harmonics (including
the fundamental) as a histogram plot (bar chart).
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21.1a,c,e,g
for the
integrals in 21.1,
21.2,
and
21.3
[100 pts] |
- Parts b, d, and f are not required.
- You can use MathCAD for the
integral for part a.
- For part c, write a MathCAD program to perform the
trapezoid rule integration for any function. Define the program as a function
so you can evaluate it for different functions, numbers of intervals (n), and
different ranges (a and b).
- NOTE: MathCAD does not allow range variables
(other than in for loops) in programs.
- Use the Simpson's 1/3 rule with n=6 (not n=5) for part g.
- Also compute the true percent relative
error for each integral approximation.
- For part c, for problem 21.1, with
n = 4, integral = 16.515, error = 0.311%
- For part g, for problem 21.1, with
n = 6, integral = 16.566, error = 0.000635%
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