Proof for Geometric Mean Radius (GMR)

Discussion in 'General Electronics Chat' started by YoGMan, Oct 25, 2017.

  1. YoGMan

    Thread Starter Member

    Sep 20, 2017
    67
    1
    Hello friends, I've always been using GMR but I can't find a good simple proof or analogy on how we get the formula .Can you provide me with a good proof or analogy to understand GMR better please?

    The+geometric+mean+radius+(GMR)+of+two-conductor+bundle+is+given+by.jpg
     
  2. MrAl

    AAC Fanatic!

    Jun 17, 2014
    5,953
    1,278
    Hi,

    Since this concept has been used in magnetics for so long you can start by looking up some papers on inductance for example.

    One proof i did some time back in the past was a numerical proof which was interesting too. The proof goes something like this...

    Generate two random points x1,y1 and x2,y2 where the two points are inside either one conductor or two conductors, depending on what GMD you are trying to prove.
    Calculate the distance between those two points.
    Repeat this millions or billions of times, summing the distances and the number of times you did this.
    Calculate the average distance.

    This works out pretty well for some shape conductors but some take many more ponts to get close to the exact solution. One of the problems is that when generating points x,y we have to ensure that the point is inside the conductor and that takes an extra calculation, and if the point turns out to not be inside the conductor we have to reject it. Square conductors work out a little easier because if we confine the space to within the max x and max y we always get a point inside the conductor, while with a round conductor we have to also satisfy:
    point (x,y) such that R^2>=x^2+y^2
    with R the conductor radius.

    The proof using this method of course comes in the form of matching numbers using either method. The more random points generated the higher the precision, and it starts to become obvious that as we increase the number of points the solution number gets closer and closer to the exact value.
     
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