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Vectoring in on a Solution – Phasors

Graphical representation of phase and amplitude.

Graphical representation of phase and amplitude.

Mathematics is really a bag of tricks to try to represent real and actions a provides a way to calculate and predict things.  Over the next 2 or 3 posts I am going to arm you with a new set of tricks.  However, it will be necessary to learn a few things.   Those things are not hard concepts, but they will be different.  First, I want to sell you on what we will have after we go through the effort of learning the techniques.

What we really would like it the voltage and phase across each component and the current and phase through each component.   It would be really nice if we even handle complications such as the leakage resistance in capacitors and the resistance in inductors.  If we have several combinations of capacitors, inductors and resistors in series and parallel, it would be nice to be able to combine all of these to get the total impedance or the circuit and of each part of the circuit.  It would be nice to be able to handle multiple phase shifts and if the total is 360 degrees or more it simply starts back at zero because sine waves repeat every cycle.  It would be nice to use the rules we have already learned for Ohm’s law and resistors and be able to apply it and get all of the results.   We will be able to do all of that, but first we have to go through a little math theory.   Hopefully, now that you see what the results will be, you will be willing to take this trip with me.

Engineering and design is built upon the concept of a vector.  A vector is simply a line with a direction and a length.   When we were talking about statics and calculating the force on the Popsicle crane boom,  for example in the post, “Why I built the Crane Boom Hinge the way I did.” I was talking about force vectors although I did not use the term.  Later, I presented: “Episode 60 – The Cartesian Coordinate System (Graphing)“.  In that post I talked about creating a grid to locate points so lines can be drawn between the points.   These lines can be thought of as vectors.

The Cartesian coordinate system is not the only way to represent vectors.  Probably as common if not more so is the polar coordinate system.  Imagine a radar rotating and scanning the sky.  The radar can determine how far away something is, and the direction the antenna is pointing can determine the direction.  This can provide a graph that is a series of concentric circles and the direction is calculated with due east (3 o’clock) is zero and positive angles are in the counter-clockwise direction.   So far we have only talked about a two-dimension system.   It is possible to go into 3rd dimension, but we have no need to do that at this time and I have no desire to complicate any more.

We can convert from the polar coordinate system to the cartesian coordinate system by using the following trig. formulas.    X = r * cosine(Θ) and Y = r * sine(Θ)  where r is the length of the vector from 0 and Θ is the angle.   X is the direction on the horizontal axis, and Y is the direction on the vertical axis.   To convert from the Cartesian coordinate to the polar coordinates:  Θ = arctan(Y/X) and r = √(X2 + Y2).   it is much better to understand the why of those equations than it is to memorize the equations.   Those equations come from our old friends, right triangles, and Pythagoras’s equation.

Now that we can see how to draw vectors on a polar coordinate system, it is time to introduce the new concept, imaginary numbers.   If we take a number and multiply by itself we say we squared the number.   For example 2 * 2 = 4,  and also -2 * -2 = 4.  This means that all squares are positive.  It also means there are two possible answers for the √4: 2 and -2, sometimes written as +/- 2.   So what happens if we are in the situation of having to solve √-4?  One answer is “It isn’t possible”, but we really need an answer, so we will make one.  √-4 = 2i and -2i where i is an imaginary number and represents √-1.  (Please hang in there, there will be a very good reason for this nonsense before we are done.)   In electrical theory, the letter i is already used so the letter j is used instead of i to represent the imaginary value.

The value of using the imaginary value comes from multiplication and division of it.  √-1 * √-1  = -1 this is the same as    (j * j = -1) .  Also:  √-1 * √-1* √-1 = – √-1 or (j*j*j = -j. Finally j*j*j*j =-1*-1 = 1.

Now go back and look at the polar plot above.   If instead of calling the Y axis, Y we instead call it the j axis this means a 90 deg. shift, j  times a 90 deg.shift = -1 or 180 degree shift.

For example, we will now define ZR = R or R will be drawn along the positive axis.  ZL = jXL and ZC = -jXC.   This means XL will be drawn in an upward direction on the vertical axis, and XC will be drawn downward on the vertical axis.  Already we are showing the reactance of capacitors are 180 degrees from the reactance of inductors and both are 90 degrees different than a resistor.

When a real number and an imaginary number are combined the combination is called a complex number.  I avoided using that term until now because it sounds scary and “complex”.  It is not.  The rules are really fairly simple although it gets messy quickly when doing the calculations by hand.   However, we are all using computers so things are not that bad!.  If we have a 500Ω resistor in series with an inductive reactance of 100Ω our impedance is: Z = (500, j100).

In the next post I will explain the rules for addition, subtraction, multiplication, and division of complex numbers.  We will do that both by hand and using a spread sheet for the values used in our series RLC network..

Gary


 

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