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Good ole boy Calculus

Our Car with no air friction

Our Car with no air friction

Sometimes I get stuck. I have turned this blog into somewhere between a technician type of discussion and full-blown engineering. Many very good technicians I know are very good a fundamental understanding and have almost a magical sense of understanding a problem. Those same guys give you a glassy eyed look if you mention math. That is why they chose to work with their hands.  Sitting here in front of a computer screen and trying to reach an imaginary person like that is sometimes a tough row to hoe,.

I had intended to be talking about at least one of the final two basic op-amp circuits called an integrator, and differentiator.  The more I thought about describing the operation of those circuits the more I got stuck.   I realized I was getting the cart before the horse, so tonight I will talk more about what those two terms mean, but I will do my best to keep it at the conceptual level with some physical examples.

The first example is filling a bucket with water.  Imagine I have a 5 gallon bucket and I place a garden hose in the bucket and crack the faucet open so the water flows at a rate of 1 gallon per minute.   (For you people that are not in the USA, replace gallon with liters.and the discussion will still work. Yeah, I know it is litres, we don’t even spell it “correctly” here. )  If I walk away for 30 seconds I can expect to have 0.5 gallons in my bucket.  If I come back after 2 minutes I would expect to have 2 gallons in the bucket.  Guess what?  The integral of flow is volume.  It really is that simple.

Imagine I need to drive on a trip of 500 miles, or 500 km and I drive at a constant speed of 50 mph (or 50 kph, you choose the units). After 10 hours on the road I will have travelled 500 miles.  Again, the integral of speed is distance.

In both of these examples I used a constant value so we could do simple multiplication to calculate the integral.   What happens if the 500 mile trip is more like my normal trips?  I drive for an hour at about 70 mph and then have to stop and get a cup of coffee because I need to stretch.   Then after that short rest period, I again drive about 70 mph and then about an hour later take another break.  This one is a little longer because I have to get rid of the first cup of coffee and purchase a second one.  Then on the road, I again get up to 70 mph, but I have to go through a city with some traffic and for about 30 minutes I am only able to reach 40 mph.   And on it goes throughout the trip.   All the time in math class is spent dealing with “functions” that have these variable curves.

I guess about now I need to define what a function is.  A function is a formula with one or more inputs that gives one and only one answer for each combinations of those inputs.   Most of the time we talk about a function with one input and usually that input is given the value of X and the answer is given the value of Y.   So we write Y = ƒ(X).   By this definition: Y = X2 is a valid function because each value of X returns one and only one answer.   However, Y = √X is not a function.   For example, if X = 4 we have two answers, Y = 2 and Y = -2.  However, we could say: Y = only the positive values of √X and everything is happy.

Often our function has an input of time instead of X.  For example:  Y = sine(2*Π*f*t).
Now that we have that term past us, I will continue on.

After it is all said and done we end up with some general rules and about a half-dozen basic functions we can integrate.   Then it is on to the computer classes where we learn to slice up the function up into tiny slices just like I did in the PID controller and car model.   And that is where we will be on anything we calculate in this blog.  So, when I throw out something like I will in the next post that looks like Vout= Gain * ∫ Vin(t), Just think of the simple case where Vin(t) is a constant voltage and the equation becomes Gain*Vin*t.

The inverse of Integration is the derivative.   It the integral tells us how much and how long, the derivative tells us how fast it is happening.  The simple way of writing this is:  Change of Y / Change of X and this is written as.  ΔY/ΔX and the Greek symbol delta, Δ, means “change of”.   Calculus takes the Δ to the infinitesimal small value.  This is the slope of the curve and if the slope is constant like it is in the speed value on the graph it really does not matter if I use the delta or the derivative.  However, imagine we are wanting to take the derivative of the sine wave which continuously changes.  We have a problem unless we know some rules… or we have a computer program!

I will provide you with simple examples so you can learn the rules for the easy stuff and you already know somewhat how to do the computer program version from the PID spread sheets.   So, all in all, it is a simple concept and nothing to go into panic mode about.

For the final example I will talk about a capacitor.   A capacitor is exactly the same thing as the bucket example, except instead of filling the bucket with water, we will be filling it with charge.   The exact equation is Vc = C * Q where Q is the charge in Coulombs.   This is exactly equivalent to the bucket equation where h = K *  V  where h is the height of the water in the bucket, V is the Volume of water in the bucket and K is some constant that converts V in gallons to height in inches.

Now if we look at the bucket receiving a constant flow the equation becomes h = K * f * t.   The flow of current, I, in Amperes is Q/t.  If we have a constant flow of current into the capacitor, we can write Vc=C * I * t.   In both cases we can simply multiply the constant flow by the time to come up with the volume in the case of the bucket or the charge in the case of the capacitor.

Now imagine the flow is not constant but changes with time and is expressed as a function of time, ƒ(t).  Now our more general equations become:for the bucket:  h = K * ∫f(t)  — the f here means flow with respect to time.
for the capacitor:  Vc = C * ∫I(t).

It is my sincere goal here to make you a little more comfortable with some math terms and remove some of the mystery and misery associated with those.   Hopefully I have been successful, but if you have some questions or concerns please e-mail me or comment in this post.

Gary

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