Transformer Impedance Matching Equation

MrAl

Joined Jun 17, 2014
7,196
Hello again,

Here is a graph of L and R vs phase and current.
The first pic is the graph, the second (-1a) is the same graph with an example.

The example goes like this...
1. Find the current 3 amps (two heavy green curves)
2. Find the phase 0.6 rads.
3. Draw a vertical line up from 0.6 rads until it hits the first heavy green line (thick light gray vertical line).
4. Draw a horizontal line from the intersection of that vertical line and the first heavy green line until it hits the y axis, read off the inductance (1.9H approximately).
5. Continue the vertical line (heavy black vertical line) until it hits the next heavy green line.
6. Draw a horizontal line from that intersection to the right hand side y axis, read off the resistance (2.8 ohms).
So the result for 3 amps and 0.6 rads came out to L=1.9 Henries and R=2.8 ohms.
Note the test voltage is 10 volts and normalized for angular frequency (w=1).

The graph scales up or down also.

For current:
If we had 30 amps then L=0.19 Henries and R=0.28 ohms (inversely scaled).
If we had 0.3 amps then L=19 Henries and R=28 ohms (inversely scaled).

For voltage:
If we used a 1v test voltage (I=3 amps) then L=0.19 H and R=0.28 ohms (proportionally scaled).
If we used a 100v test voltage (I=3 amps) then L=19 H and R=28 ohms (proportionally scaled).

For frequency (w=10):
v=10v, I=3 amps, L=0.19 (inversely proportional) R=2.8 (does not change)
 

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Thread Starter

SamR

Joined Mar 19, 2019
1,843
Very interesting graph and I can interpret a couple of things from it, More change at lower current, higher inductance, and resistance. Not sure where the 0.6 radians came from or the 0's at 1&1.6 radians for the span. It has been a long time since I worked with radian polar notation since almost all of my engineering was done in English notation since college long ago. At this point Al, Grobb is not going into a great deal of depth and trying to at least give a good introduction to AC/DC theory. Once I complete this I have a couple of other books to go through. I do want a good grounding in BJT, FET, and MOSFET and will be working toward that. It is strictly to give me something to keep me occupied in my retirement and something I have always had an interest in. My career was as an Integrator of Control systems and not at the device design level. Again, thanks for the info and help.
Sam
 

MrAl

Joined Jun 17, 2014
7,196
Hi,

That 0.6 rads is the phase angle measured in radians, however i remade the graph in degrees also.
That span is the span we could possibly measure but it wont be outside of that range unless there is also capacitance.
I made the new graphs in degrees though rather than radians.

For the new example we measure 30 degrees and 3 amps. The inductance and resistance are as shown.
Note that the upper L and R are now also normalized in the new graphs (1 each).

I might do R and C next. It should look similar but with different phase angles.
 

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MrAl

Joined Jun 17, 2014
7,196
Here is one more graph. Both ratios L/R and R/L vs phase. Also the simple expressions that produce them.

RL_20190807_003151.gif
 

MrAl

Joined Jun 17, 2014
7,196
@MrAl What are you using to generate the plots?
Hi,

It is an symbolic algebra like program that is free to download called "Maxima".
I recommend downloading and learning how to use if you have to do a lot of mathematics and dont want to pay for something.
It does algebra and calculus and various ways of plotting graphs.
Before i downloaded that some years ago i wrote a program to do that also, but because it takes a long time to include everything you need i instead went for the download where much of the stuff you need is in there already.
It has some limitations of course but they are not too bad at all. It works for a lot of problems we encounter. For example i think it is limited to 2nd order differential equations, except in the form of ODE's and then i think it may go higher.
There are various versions one for Windows for example and one for console but i think others too.
 

Thread Starter

SamR

Joined Mar 19, 2019
1,843
Thanks,

That is a new one for me, I'll have to check it out. I've been using WolframAlpha primarily for solving dual simultaneous equations and like it because it will also show the graphical expressions in addition the step by step expressed solution. https://www.wolframalpha.com/
 

Thread Starter

SamR

Joined Mar 19, 2019
1,843
Thx Bertus and MrAl,

I found and installed the win64 version. Briefly looked at it and, like LTS, it will take some using to get comfortable with it. Like most well-made programs it apparently has everybody's pet bells and whistles included in it.

Sam
 
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