Theory of impedance networks: A new formulation

1. Theory of impedance networks: A new formulation F. Y. Wu FYW, J. Phys. A 37 (2004) 6653-6673

2. R a b Resistor network

3. Ohm’s law R I V Combination of resistors

4. D-Y transformation: (1899) = Star-triangle relation: (1944) Ising model = =

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

8. r1 3 2 r2 r1 r1 4 1 r1

9. r1 3 2 r2 r1 r1 4 1 r1 3 2 3 3 1 1 1

10. r1 r1 3 3 2 2 r2 r2 r1 r1 r1 r1 4 4 1 1 r1 r1 I I/2 I/2 I I/2 I/2 3 2 3 3 1 1 1

11. r 1 r r r r r r r r r r 2 r

12. I r r I/3 1 1 I/6 r r I/6 I/3 r r r r r r r r r r r r r r I/3 r r r r I/3 2 2 r I/3 r I

13. I I/4 I/4 I/4 I/4 Infinite square network

14. I I I/4 I/4 I/4 I/4 I/4 V01=(I/4+I/4)r

15. Infinite square network

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

17. Kirchhoff’s law 2 I0 r02 3 r03 1 r01 r04 4 Generally, in a network of N nodes, Solve for Vi Then set

18. 2D grid, all r=1, I(0,0)=I0, all I(m,n)=0 otherwise I0 (1,1) (0,1) (0,0) (1,0) Define Then Laplacian

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

20. General I1 I2 I3 N nodes

21. Properties of the Laplacian matrix All cofactors are equal and equal to the spanning tree generating function G of the lattice (Kirchhoff). 1 Example c3 c2 G=c1c2+c2c3+c3c1 3 2 c1

22. I1 I2 network IN Problem: L is singular so it can not be inverted. Day is saved: Kirchhoff’s law says Hence only N-1 equations are independent →no need to invert L

23. Solve Vi for a given I Kirchhoff solution Since only N-1 equations are independent, we can set VN=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.

25. Writing b a I I We find Where La is the determinant of the Laplacian with the a-th row & column removed Lab= the determinant of the Laplacian with the a-th and b-th rows & columns removed

26. 1 Example c3 c2 3 2 c1 or The evaluation of La & Lab in general is not straightforward!

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

28. x 1 2 x x x y y y y + y + y y + y G(x,y)= x x x 4 3 x =2xy2+2x2y 1 2 3 4 1 2 3 4 N=4

29. Consider instead Solve Vi (e) for given Iiand set e=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.

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

31. Consider where Let This gives

32. Let = orthonormal Theorem:

33. r1 3 2 r2 r1 r1 4 1 r1 Example

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

35. r r N-1 r N r 2 3 1 a b

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

37. New summation identities New product identity

38. M×N network M=5 s s r r s r r N=6 IN unit matrix

39. M, N→∞

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

41. Moebius strip Klein bottle

42. Orientable surface Non-orientable surface: Moebius strip

43. Orientable surface Non-orientable surface: Moebius strip

44. Free Cylinder

45. Moebius strip Klein bottle

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

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