I guess I really don't understand that model. I'm not sure what the resistors connecting different rows represented.
In the new model that you proposed in this post, I'm not sure how the length between nodes would be changed, what this represents, and what the different values of the resistors represent.
Hi again,
Here's the thing...
If you connect four resistors into a square so that they form four nodes,
you have the beginning of a sheet resistance. It is so far a very crude
model, but still a starting point. The resistance of each resistor
represents a hypothetical incremental resistance along one dimension,
and that's a dimension that has length (or width). Thus, the starting
resistance (say 1 ohm) is the starting incremental resistance and the
length (say 1 inch) is the starting length or width. By calling the lower
left node the origin, we now have a method to not only measure
resistance, but also a method to relate that resistance to actual position
of the four nodes. Thus, resistance becomes a function of position.
Next, we would increase the number of those squares (using more resistors
of course) until we had 4 squares of resistors of squares, except that
all branches are still made from one resistor (not two in parallel).
The other thing we do though, because we are working up to an
infinitely accurate model, is to halve the value of each resistor.
Because we halve the value of each resistor though, we also halve
the length of each resistor. This keeps the same distance between the
four corners of the entire square and so the position still remains relative
to the resistance. The only difference then is that we have now increased
the resolution of the grid, making it more like a sheet.
The next step would again be to double the number of resistors,
halve the resistance and length of each one again, which would increase
the resolution two fold again.
If we keep doing this procedure, eventually we get a really good model
even if we dont use an infinite number of resistors. The resistance
between two points would be related to the position of those points,
and a formula can be developed.
This could be interesting to do even if you dont ever use it.
BTW, what is your goal with this?