Theory of electric networks: The two-point resistance and impedance F. Y. Wu Northeastern University Boston, Massachusetts USA

Z Impedance network  

Ohm’s law Z V I Combination of impedances

In the phasor notation, impedance for inductance L is Impedance for capacitance C is where. Impedances

=  -Y transformation: (1899) Star-triangle relation: (1944) Ising model = =

 -Y relation (Star-triangle, Yang-Baxter relation) A.E. Kenelly, Elec. World & Eng. 34, 413 (1899)

z1z1 z1z1 z1z1 z1z1 z2z2

z1z1 z1z1 z1z1 z1z1 z2z

r1r1 r1r1 r1r1 r1r1 r2r I I/2 I r1r1 r1r1 r1r1 r1r1 r2r2

1 2 r r r r r r r r r r r r

1 2 r r r r r r r r r r r r I I/3 I 1 2 r r r r r r r r r r r r I/6

Infinite square network I/4 I

V 01 =(I/4+I/4)r I/4 I I

Infinite square network

Problems: Finite networks Tedious to use Y-  relation 1 2 (a) (b) Resistance between (0,0,0) & (3,3,3) on a 5×5×4 network is

I0I Kirchhoff’s law z 01 z 04 z 02 z 03 Generally, in a network of N nodes, Then set Solve for V i

2D grid, all r=1, I(0,0)=I 0, all I(m,n)=0 otherwise I0I0 (0,0) (0,1) (1,1) (1,0) Define Then Laplacian

Harmonic functions Random walks Lattice Green’s function First passage time Related to: Solution to Laplace equation is unique For infinite square net one finds For finite networks, the solution is not straightforward.

General I1I1 I2I2 I3I3 N nodes The sum of each row or column is zero !

Properties of the Laplacian matrix All cofactors are equal and equal to the spanning tree generating function G of the lattice (Kirchhoff). Example y3y3 y1y1 y2y2 G=y 1 y 2 +y 2 y 3 +y 3 y 1

Spanning Trees: x x x x xx yy y y y y y y x G(1,1) = # of spanning trees Solved by Kirchhoff (1847) Brooks/Smith/Stone/Tutte (1940)

x x yy G(x,y)= + x x x xx x ++ yyyyyy =2xy 2 +2x 2 y N=4

General I1I1 I2I2 I3I3 N nodes The sum of each row or column is zero !

I2I2 I1I1 ININ network Problem: L is singular so it cannot be inverted. Day is saved: Kirchhoff’s law says Hence only N-1 equations are independent → no need to invert L

Solve V i for a given I Kirchhoff solution Since only N-1 equations are independent, we can set V N =0 & consider the first N-1 equations! The reduced (N-1)×(N-1) matrix, the tree matrix, now has an inverse and the equation can be solved.

Example y3y3 y1y1 y2y2 or The evaluation of L  & L  in general is not straightforward!

II   Kirchhoff result: Writing Where L  is the determinant of the Laplacian with the  -th row & column removed. L  = the determinant of the Laplacian with the  -th and  -th rows & columns removed. But the evaluation of L  for general network is involved.

For resistors, z and y are real so L is Hermitian, we can then consider instead the eigenvalue equation Solve V i (  ) for given I i and set  =0 at the end. This can be done by applying the arsenal of linear algebra and deriving at a very simple result for 2-point resistance.

Eigenvectors and eigenvalues of L 0 is an eigenvalue with eigenvector L is Hermitian L has real eigenvalues Eigenvectors are orthonormal

Consider where Let This gives

Example r1r1 r1r1 r1r1 r1r1 r2r2

For resistors let = orthonormal Theorem for resistor networks: This is the main result of FYW, J. Phys. A37 (2004) which makes use of the fact that L is hermitian and is orthonormal

Corollary :

Example: complete graphs N=3 N=2 N=4

1 23 N-1 r rrr N  

If nodes 1 & N are connected with r (periodic boundary condition)

New summation identities New product identity

M×N network N=6 M=5 r s s I N unit matrix s r rr

M, N →∞

Resistance between two corners of an N x N square net with unit resistance on each edge whereEuler constant N=30 (Essam, 1997)

Finite lattices Free boundary condition Cylindrical boundary condition Moebius strip boundary condition Klein bottle boundary condition

Klein bottle Moebius strip

Free Cylinder

Klein bottle Moebius strip

Klein bottle Moebius strip Free Cylinder Torus on a 5×4 network embedded as shown

Resistance between (0,0,0) and (3,3,3) in a 5×5 × 4 network with free boundary

In the phasor notation, impedance for inductance L is Impedance for capacitance C is where. Impedances

For impedances, Y are generally complex and the matrix L is not hermitian and its eigenvectors are not orthonormal; the resistor result does not apply. But L^*L is hermitian and has real eigenvelues. We have with

Theorem Let L be an N x N symmetric matrix with complex Matrix elements and Then, there exist N orthnormal vectors satisfying the relation where * denotes complex conjugate and

Remarks: For nondegenerate one has simply For degenerateone can construct as linear combinations of

For impedances let = orthonormal Theorem for impedance networks: This is the result of WJT and FYW, J. Phys. A39 (2006)

The physical interpretation of is the occurrence of a resonance such as in a parallel combination of inductance L and capacitance C the impedance is Z = =

Generally in an LC circuit there can exist multiple resonances at frequencies where In the circuit shown, 15 resonance frequencies at M=6 N=4

Summary An elegant formulation of computing two- point impedances in a network, a problem lingering since the Kirchhoff time. Prediction of the occurrence of multi- resonances in a network consisting of reactances L and C, a prediction which may have practical relevance.

FYW, J. Phys. A 37 (2004) W-J Tzeng and FYW, J. Phys. A 39 (2006),