Wed Nov 5 00:05:11 EST 2014
Symmetric conductivity matrix
Trying for a maximally connected 3-node network 0,1,2. Node 0 is the
ground node, giving two variables V10, V20. The effect of a current
source connected between node 1 and 2 can be absorbed into current
sources from ground to nodes 1,2. This gives the following equations:
G V = I
(G10 + G12) -G12 V10 I01
-G12 (G20 + G12) V20 I02
With Gxy conductivity between node x and y.
Ixy current flowing from x to y
Vxy voltage drop from x->y
Where does this symmetry come from?
Essentially from the symmetric relation "connected to".
I.e. a resistor doesn't have any sense of direction.
This generalizes to
a_xx = Gx0 + \sum_y!=x Gxy
a_xy = -Gxy (x != y)