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Linear scaling with offset.

A line represented with one of the standard formulas

The thermistor circuit we will be working with will require us to do a lot of scaling and manipulation. We will have to try to make a curvy thing becomes as straight as possible. As I thought about that, I realized I never really talked about how to fit one linear equation into another one. Luckily, temperature gives us a very good example of how to do that and it still keeps us on the same path!

We are going to have to deal with some fairly simple math at the start of this, but I promise I will try to keep it short and to the point. After it is all done, I will link a video that should provide a laugh for you. Just eat your meat before you go for the pudding. (Pink Floyd fan here.)

Independent, Dependent variables, and X and Y:
When thinking about equations or functions there are two types of variables.  First, the independent variables.  These are the variables that can be set by the person using the equation.  Another way of thinking of them are inputs.
Dependent variables on the other hand are the outputs.  These are the result of following the equation and plugging in the inputs.
Often, but not always, the independent variable is assigned the name x and the dependent variable is assigned the name y.  When a graph is created showing these two the independent variable is shown along the horizontal or X axis and the dependant variable is shown along the vertical or Y axis.

Because we are starting off talking about a general equation form we will use X and Y for our variables.  Later as I talk about temperature conversion we will be using other letters.
There are many equations for a line, but the main one I remember is Y=mX+b.  If the formula is graphed as shown in the picture m describes how much the output rises per unit increase in X.  If m = 1 and the X scale and Y scale are the same the graph line will be at 45 degrees.  The smaller the m value, the less rise until at 0 the line would be flat.  The larger the m value, the more upright the line is with an m of infinity giving a perfectly vertical line.

The value b in the equation describes the Y value when X=0.

Temperature Measurement:
Here in the USA we are still using the imperial measurement for temperature, degrees Fahrenheit,  for normal day-to-day use.  However, almost everything technical is described in degrees Celsius.   (Many of us still say degrees centigrade, but the official term is Celsius.)   The problem for us “yanks” is while we can deal in deg C, we don’t really feel deg C.  If someone tells us it is 35 deg outside we think near freezing and wearing a coat, while the rest of the world is thinking stay inside in the air conditioning or going swimming.

Thermistors are described in deg C, but we will hopefully be using them to measure solar energy projects.  Some of the measurements will probably be the outdoor  temperature and temperature of our storage unit.  Us “yanks” just will not be able to feel what is really going on in deg C.  (The rest of the world can ignore those calculations, but this exercise in converting will get us back to Y = mx+b.)  We now have 3 choices: memorize the conversion equation, look up the conversion equation, or develop in ourselves.  Actually the last option is easiest for me in most cases.

There is two physical facts that we all know of:  Freezing of water, and boiling of water.   In the imperial, US, system freezing occurs at 32 deg F and boiling occurs at 212 deg F.  For the rest of the world freezing occurs at 0 deg C and boiling occurs at 100 deg C.   With these two points, and 4 numbers, we can calculate the conversion equation.   The temperature difference between the two points is: (100 – 0) or 100 deg C, and (212-32) or 180 deg F.   This means m in our equation when converting from deg C to deg F is 180/100 and if you want to reduce the faction 9/5.  When converting from deg F to deg C  it becomes 100/180 or 5/9.

Now we need to deal with the offset.  Since deg C starts at 0 at freezing when converting from deg C to deg F our b value will be 32.  So our standard equation form becomes:  Deg F = ((9/5) x Deg C) + 32.  Our standard equation does not work so well in the opposite direction, because 0 deg F is a somewhat random number  of -17.78 deg C.  In this case it is easier to do the algebra and convert the Deg C to Deg F into Deg F to Deg C.  Deg C = (Deg F – 32) * 5/9.

Now if our Asian, European, South American, Middle Eastern, African, Australian, and even Canadian friends have bloated egos laughing at us in the USA it is time for me to pop that bubble.   Actually neither measurement starts at true zero.  True zero is when there is no energy available and is called absolute zero.  We may have to deal with the temperature scale that uses that for a reference and there are two of those also.   For thermistors, it is stated in degrees Kelvin.  The Kelvin scale uses units of the same size as the Celsius scale and the conversion from deg C to deg K is:  deg K = deg C – 273.15.

Just of completeness there is also an absolute zero scale that uses units the same size as the Fahrenheit scale.  It is used in a lot of thermodynamic calculations to find the power needed in horse power units.  It is called the Rankine scale and deg R = deg F – 459.67.

We will be using the deg C to deg F conversion quite a bit.  I doubt if I will be using the equation that uses the deg K, because there is a shorter version that is good enough.

Since I took you out into a hike into nerd land tonight, the least I can do is provide you a laugh,  The following short video describes how the variable X came about… the punchline is at the end.  Enjoy!

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