Theory of electric networks: The two-point resistance and impedance

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

2. Z a b Impedance network

3. Ohm’s law Z I V Combination of impedances

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

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

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

9. z1 3 2 z2 z1 z1 4 1 z1

10. z1 3 2 z2 z1 z1 4 1 z1 3 2 3 3 1 1 1

11. 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

12. r 1 r r r r r r r r r r 2 r

13. 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

14. I I/4 I/4 I/4 I/4 Infinite square network

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

16. Infinite square network

17. 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

18. Kirchhoff’s law 2 I0 z02 3 z03 1 z01 z04 4 Generally, in a network of N nodes, Solve for Vi Then set

19. 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

20. 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.

21. General I1 I2 I3 N nodes The sum of each row or column is zero !

22. Properties of the Laplacian matrix All cofactors are equal and equal to the spanning tree generating function G of the lattice (Kirchhoff). 1 Example y3 y2 G=y1y2+y2y3+y3y1 3 2 y1

23. 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)

24. 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

25. General I1 I2 I3 N nodes The sum of each row or column is zero !

26. I1 I2 network IN 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

27. 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.

29. 1 Example y3 y2 3 2 y1 or The evaluation of La & Lab in general is not straightforward!

30. Writing b Kirchhoff result: a I I 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. But the evaluation of Lab for general network is involved.

31. For resistors, z and y are real so L is Hermitian, we can then consider instead the eigenvalue equation 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.

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

33. Consider where Let This gives

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

35. For resistors let = orthonormal Theorem for resistor networks: This is the main result of FYW, J. Phys. A37 (2004) 6653-6679 which makes use of the fact that L is hermitian and is orthonormal

36. Corollary:

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

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

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

40. New summation identities New product identity

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

42. M, N→∞

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

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

46. Moebius strip Klein bottle

47. Free Cylinder

48. Moebius strip Klein bottle

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

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