Why MathCAD?
The Engineer’s Scratch Pad.
How it Works!
Why MathCAD?
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A design tool
A mathematical problem solver
Unit converter
A good way to present your results.
A way to visualize the mathematical Idea
A method to verify your result
Started!
• The desire to introduce mathematics not
just by using the blackboard and
calculator.
• Why not a graphing calculator?
• In the working place you have computer
and not a GC.
• One of the idea was to use Excel.
• As a compromise MathCAD was
introduced.
Approach!
• To overlap or not to overlap? This is the
Question!
• How to combine the lectures with
MathCAD Lab component?
• Why not supplement and overlap and
introduce something new, different from
regular class.
• To do different Tests for lecture and for
Lab.
Evaluation (‘,’)
• Examination- include part from lectures
and part from Lab.
• Student reaction-who cares!
• To be serious : they love it! (not every one)
• Statistical analyses – in process of
developing.
MathCAD Lab#9
Frequency of Periodic Functions
Sample
• Part #1 Mathematical Background-brief
description of Frequency
• Part #2 Exercises – couple of examples
• Part #3 Assignment – set of exercises
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Mathematical Background:
In electronics, frequency is expressed as the number of cycles per second.
The unit of frequency is 1 Hertz  1 cycle per second.
eg. 3.2 Hz  3.2 cycles per second, and 7.7KHz  7.7×1000 Hz  7700 cycles per second.
The frequency can be found by counting the number of cycles on the graph in a given time.
For example, 4.5 cycles in 7 seconds gives 4.5/7 Hz = 0.643Hz.
There is another method to find the frequency: change the X values to show only one
cycle, and read off the period from the graph.
The frequency is the reciprocal of the period: frequency = 1/period; or
where f
is the frequency and T is the time required to complete one cycle.
Note that if T is in seconds then f is in Hz ; If T is in milliseconds (ms) then f is in kilohertz
(kHz) ;
If T is in microseconds (μs) then f is in megahertz (MHz) ; etc . . .
For example, if the period is 3.8 milliseconds, then the frequency is 1/(3.8х10-3) = 263 Hz.
Part #2 Exercises – couple of
examples
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1.
In this exercise we will not use degrees, so we will use lower-case
sine . The X-axis will represent time, therefore the variable t will be used
instead of x.
Define f(t) := 3.5sin(5.7t), and graph it for 0 ≤ t ≤ 8 seconds.
Count the number of cycles during the first 8 seconds. There were 7
complete cycles, plus 1/4 of a cycle. The frequency is 7.25 cycles in 8
seconds. f 7.25 / 8  0.91 Hz.
Another way to see this: Change the upper limit of t until you see exactly
one cycle. The period is 1.1 seconds, and the frequency is 1 cycle / 1.1 s =
0.91Hz.
2.
Define g(t):= 2sin(3t)+3sin(5t)4sin(4t) and graph it for 0 ≤ t ≤ 15 μs. It
is difficult to determine the number of cycles, but you can see that the end
of the first complete cycle is between 5 and 7. After some experimentation
you will find that by changing the time to 0 ≤ t ≤ 6.3, you can count 1
complete cycle. The period is T6.3 μs, and the frequency is its reciprocal:
f1/T. There is 1 cycle / 6.3 μs, Since 1μs106s, 1 cycle / 6.3 μs  1cycle /
(6.3×10-6sec)  106cycles/6.3 s  158730 cycles per second  159×103
cycles per second. (rounding to a whole number). Therefore f  159KHz.
f(t)=3.5sin(5.7t)
7 and ¼ complete cycles/8sec=0.91Hz
3.5
4
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f ( x)
0
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 3.5
4
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0
1
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x
5
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8
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Another method- resize the graph
3.5
4
2
f ( x)
0
2
 3.5
4
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0
0.2
0.4
0.6
0.8
x
1
1.2
1.4
1.5
until one cycle is displayed
the period is seen from the graph and
f=1/1.1s=0.91Hz
3.5
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f ( x)
0
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 3.5
4
0
0
0.2
0.4
0.6
x
0.8
1
1.1
Assignment:
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Assignment:
Graph each function, and determine its amplitude, period (in s, ms, or μs) and
frequency (in Hz, or KHz). Change the values on the X-axis so that your graph shows
exactly one cycle. Type all the answers in a text box region below each graph.
Note: Do NOT redefine the sine function (as SIN) for degrees the way we did in
Lab # 8 !
1.
y  3sin(1.575t) when t is measured in seconds.
Amplitude____
Period  ________ Frequency  __________
2.
y  2.7sin(4.189t) when t is measured in ms.
Amplitude____
Period  ________ Frequency  __________
3.
y  sin(0.766t0.383) when t is measured in μs.
Amplitude____
Period  ________ Frequency  __________ Phase
shift  ____
4.
y  2sin(3t) + 4 with t in ms.
Amplitude____
Period  ________ Frequency  __________ Vertical
shift: ____
5.
Challenge question: y  sin(t)+sin(3t)+sin(5t)+sin(7t)+sin(9t) with t in μs.
Amplitude____
Period  ________ Frequency  __________
Summary
• Ten Labs
• Two review labs
• One Test
• And complete understanding on Mathematics
for many years in advance
(‘,’)