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so the stage is set ... now let’s fire up the TI-36X and work through our poster’s example problem ... Figure 2 shows the keystrokes of my shortcut method ...


qanda/uploads/Y=fig02.JPG



pressing [ON/AC] clears out the calculator ... using the [3rd] [STAT2] key combination puts the calculator into the “2-variable statistics” mode ... statistics for scaling? ... oh, yeah ... just hide in the bushes and watch ...



0 ... [flip/flop] ... 0 ... [sum] ...



32767 ... [flip/flop] ... 3000 ... [sum] ...



and that’s all there is to setting up the problem ... now I just need to ask the calculator for the “Rate” and the “Offset” values and we’re just about done ...



the [2nd] [SLP] key combination tells the calculator that I want to know the “slope of the line” ... (we’ll soon cover all of these terms in gruesome detail) ... and the calculator displays 0.091555528 ... this is the “Rate” entry for my SCL - except for one little problem which we’ll cover in just a minute ... now on with the calculator shortcut ...



the [2nd] [ITC] key combination tells the calculator that I want to know the “intercept” ... and the calculator displays 0 ... this is the “Offset” entry for my SCL ...



now notice that so far all I’ve done is crank in four values to tell the calculator the range of my input signal and the range of my desired scaled values ... then I punched a few more keys and got the values that I need to set up my SCL ... NO MATH! ... no multiplication ... no addition ... no division ... no subtraction ... and believe me there’s a lot more “good stuff” to come ... but now back to that pesky little problem with the “Rate” entry that I mentioned earlier ...



remember that the SLC-5/02 processor that our friend is working with won’t handle floating point math ... specifically, no decimal point values are allowed ... so how do we enter the 0.09155 that we need for our “Rate” entry? ... now we’ve come to the reason that the SCL “Rate” entry uses the “/10000” system that confuses so many people ... specifically, it’s a “work around” for the SLC-5/02’s “no decimal point” limitations ... so we take our original 0.09155 value and multiply it by 10000 and that gives us 915.5 and change ... oops, we still have a decimal point ... so next we round off to 916 - and that’s as close as we can come to a perfect answer while using the hardware that we’re working with ... now each time the processor executes the SCL instruction, it will automatically divide the 916 “Rate” entry by 10000 ... (specifically the “/10000” means “divide by 10000”) ... the processor will come up with 0.0916 for a “Rate” value to use for its internal calculations ... secret handshake: the SCL is a “work around” which allows us to get reasonable data resolution with a processor that can’t support floating point numbers ...



quick summary ... we had an input signal which ranged from 0 to 32767 ... we wanted a scaled value which ranged from 0 to 3000 ... we had to use an SCL to do the scaling conversion ... we needed a value for the “Rate” and a value for the “Offset” in order to program the SCL ... I whipped out my trusty TI-36X calculator and cranked in four numbers ... then a few more keystrokes and I provided the “Rate” and the “Offset” values that we needed ... with NO MATH! ... now in the neighborhood that I grew up in, that’s considered some pretty powerful kung-fu ...



on the downside, I’ve been throwing around a lot of weird terms and giving little or no explanation of what they mean ... things like “Rate” ... “Offset” ... “Slope” ... “Intercept” ... and there’s still more weirdness to come before we finish skinning this “scaling” cat ...



next we’re going to tackle the “equation of a straight line” which is usually written “y=mx+b” ... this is a VERY handy math tool and in my personal opinion it’s well worth the time it takes to gain a full understanding of it ... and the good news is that once we understand how it works, the TI-36X calculator will make using it practically painless ...

here’s a little bit of history on how I was introduced to “y=mx+b” ... about 15 years ago I went back to school (technical college) as an adult with bad knees ... a nice fat man was my very first math instructor ... one night he announced that we were about to study the “formula for a straight line” ... “this will be a piece of cake”, I figured ... how hard can a simple straight line be? ... well, I’ll let you decide that for yourself after we’re finished here ... but first I’m going to do something for you that the nice fat man didn’t do for me ... I’m going to try to give you the “big picture” idea of what the heck this thing is - and what it can do for us in the PLC programming business ... when I first ran into “y=mx+b” it was presented purely as a mathematical exercise ... there was no explanation of why I needed to learn it - and thus no reason for me to remember it once I’d taken the final exam ... but now I know better ... this thing is a very useful tool ...



so for the “big picture” idea behind how we’re going to use “y=mx+b” take a look at the roadmap type “mileage” chart at the top of Figure 3 ...



qanda/uploads/y=fig03.JPG





suppose that I’m planning to drive from Charleston to Fayetteville ... I find Charleston in the “where-you-are-now” list at the bottom of the chart ... I find Fayetteville in the “where-you-want-to-go” list at the left side of the chart ... I run one finger along the column and one finger along the row ... I find a number where the two lines cross ... bingo! ... 211 miles ... well “y=mx+b” works “sort of – kind of” like the mileage chart ... but not exactly ...



once you get this thing set up, it’s basically like a straight line drawn on a piece of graph paper ... you find the “incoming” number that you’re interested in somewhere along the bottom (on the x-axis) ... then you go straight up until you hit the line ... then you move straight across until you find a new number at the left side (on the y-axis) ... so the “big picture” is that this thing converts from one number (such as a PLC’s raw input data value) into another number (such as a scaled data value for an operator’s display) ... or said another way, when you shove a “raw” number into a slot at the bottom of the “math box”, then “y=mx+b” spits out a new corresponding “scaled” number on the left side ... this can be quite handy for a lot of different applications ...

now we’re ready to look at the simple “y=mx+b” graph shown in Figure 4 ...



qanda/uploads/Y=fig04.JPG




here we’ve taken a piece of graph paper and marked it up with a horizontal “x-axis” and a vertical “y-axis” ... the best way to think about this type of graph is something like the lines of latitude and longitude on a map of the earth ... we can specify any location on the planet by giving its coordinates of latitude and longitude ... in the same way, we can represent any specific point on this chart by giving its value of “x” and its value of “y” ... usually points like these are specified by giving the “x” value first, then a comma, and then the “y” value ... for example: 4,6 and -8,-3 would adequately define two points shown on our graph ...



suppose that we pick a point located at x=4 and y=6 and mark it with a dot ... suppose that we pick a second point at x=-8 and y=-3 and mark it with another dot ... then once those two points have been marked, suppose that we draw a straight line which passes through both dots and then continues on at each end right across the graph ... as shown at the bottom of the figure, the mathematical formula which defines this particular straight line is “y = 0.75x + 3” ... let’s see how that formula works ...



first of all, we can easily see that the line is “sloped” across the graph ... specifically, it’s not perfectly horizontal ... and it’s not perfectly vertical ... so it’s “sloped” at some specific angle ... let’s see if we can come up with a number which will perfectly define exactly what angle is involved here ...



first look at the point located at x=4, y=6 ... notice that I’ve shown a small triangle which starts at this point ... the base (or bottom) of the triangle is 4 squares wide ... now notice that the far side of the triangle is 3 squares high ... this little triangle is all that we need to adequately specify the “slope” of our line ... the “rule” that we’ll use to mathematically calculate the “slope” is often expressed as “the RISE over the RUN” ... as shown in the figure, the “rise” is the height of the little triangle ... the “run” is the length of its base ... so in our example, the “slope” of the line can be calculated by saying “3 OVER 4” ... or in other words, “3 divided by 4” ... the answer is, of course, “0.75” ... and that’s exactly why the number 0.75 appears in the formula for this particular straight line ...



some people have a hard time with the concept of how just one single number can specify a slope ... but the fact is that the number that we’re talking about is a “ratio” (in other words, a “fraction”) that takes into account TWO numbers – one number divided by the other ... the same concept is often used by carpenters and roofers when discussing the “pitch” of a roof ... for example, the roof on my new workshop has a pitch of “five-in-eleven” ... this means that for a horizontal “run” of 11 feet, the roof “rises” 5 feet in height ... we could divide 5 by 11 and express the pitch of the roof with just one single number: 0.4545 ...



and common sense tells us that the “slope” of a straight line is always constant throughout its entire length ... that’s demonstrated in Figure 4 by the second triangle which starts at point -8,-3 ... this one is exactly the same size and shape of the first triangle ... in fact, you could pick ANY point – ANY where – on the straight line and draw a triangle of exactly the same shape ... the only reason that I chose these two was because they work out to even numbers which are easy to recognize on the graph ... in a little while we’ll use my shortcut method on the TI-36X to calculate the “slope” of the line without even having to draw or measure the little triangles at all ...



now one more thing about the little “slope” triangles ... some people wonder “how do you know how big to make the triangle?” ... in other words, how did you know to use a “run” of 4? ... why not some other number? ... well, the truth is you can use ANY number for the “run” and you’ll still get exactly the same “slope” of the line ... the only thing is, it’s a lot easier to see the “slope” when the sides of the triangle work out to even whole numbers ... but try it if you don’t believe it ... for specific examples, start out at our original data point located at -8,-3 ... now go to the right eight squares (for a “run” of 8 ... now count straight up until you hit the line ... you’ll find that the “rise” is 6 squares ... and 6 over 8 (6 divided by 8) equals the same 0.75 “slope” that we’ve already determined for our sample straight line ...


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