Why MathCAD?

1. Why MathCAD? The Engineer’s Scratch Pad.

2. How it Works!

3. Why MathCAD? • 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

4. 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.

5. 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.

6. 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.

7. MathCAD Lab#9Frequency of Periodic Functions Sample • Part #1 Mathematical Background-brief description of Frequency • Part #2 Exercises – couple of examples • Part #3 Assignment – set of exercises

8. 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.

9. Part #2 Exercises – couple of examples • 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.

10. f(t)=3.5sin(5.7t)7 and ¼ complete cycles/8sec=0.91Hz

11. Another method- resize the graph

12. until one cycle is displayedthe period is seen from the graph and f=1/1.1s=0.91Hz

13. Assignment: • 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  __________

14. Summary • Ten Labs • Two review labs • One Test • And complete understanding on Mathematics for many years in advance (‘,’)