volume | PIER Journals

Abstract

A wave-based inversion algorithm for the recovery of deviation in he values of elements of discrete lossless inductance-capacitance and capacitance-inductance ladder networks from their nominal values is formulated. The algorithm uses ultra wideband source excitation over the frequency range where forward and backward voltage and current waves propagate along the network. Employing a weak type scattering formulation renders the voltage wave reflection coefficient to be a Z transform of the sequence of perturbation in the value of the elements. Inversion of the reflected date from the transformed domain to the spatial domain by Fourier type integration yields the element's perturbations and consequently, the actual elements of the network. Demonstrations of the algorithm performance on several test cases show its efficacy as a non-destructive testing tool.

1. Ramo, J. S. and T. V. Duzer, Fields and Waves in Communication Electronics, 3rd Ed., John Wiley & Sons, Inc., 1994.

2. Collin, R. E., Foundations for Microwave Engineering, 2nd Ed., IEEE Press Series on Electromagnetic Wave Theory, IEEE Press, 2001.

3. Caloz, C. and T. Itho, Electromagnetic Metamaterials, Transmission Line Theory and Microwave Applications, IEEE Press, Hoboken, New Jersey, 2006.

4. Jaulent, M., "The inverse scattering problem for lcrg transmission lines," J. Math. Phys., Vol. 23, No. 12, 2286-2290, 1982.
doi:10.1063/1.525307

5. Zhang, Q., M. Sorine, and M. Admane, "Inverse scattering for soft fault diagnosis in electric transmission lines," IEEE Transactions on Antennas and Propagation, Vol. 59, 141-148, Jan. 2011.
doi:10.1109/TAP.2010.2090462

6. Tang, H. and Q. Zhang, "An inverse scattering approach to soft fault diagnosis in lossy electric transmission lines," IEEE Transactions on Antennas and Propagation, Vol. 59, 3730-3737, Oct. 2011.

7. Bruckstein, A. M. and T. Kailath, "Inverse scattering for discrete transmissionline models," SIAM Review, Vol. 29, No. 3, 359-389, 1987.
doi:10.1137/1029075

8. Frolik, J. and A. Yagle, "Forward and inverse scattering for discrete layered lossy and absorbing media," IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 44, 710-722, Sep. 1997.
doi:10.1109/82.624998

9. Case, K. M. and M. Kac, "A discrete version of the inverse scattering problem," J. Math. Phys., Vol. 14, No. 5, 594-603, 1973.
doi:10.1063/1.1666364

10. Berryman, J. G. and R. R. Greene, "Discrete inverse methods for elastic waves in layered media," Geophysics, Vol. 45, No. 2, 213-233, 1980.
doi:10.1190/1.1441078

11. Godin, Y. A. and B. Vainberg, "A simple method for solving the inverse scattering problem for the difference helmholtz equation," Inverse Problems, Vol. 24, No. 2, 025007, 2008.
doi:10.1088/0266-5611/24/2/025007

12. Noda, S., "Wave propagation and reflection on the ladder-type circuit," Electrical Engineering in Japan, Vol. 130, No. 3, 9-18, 2000.
doi:10.1002/(SICI)1520-6416(200002)130:3<9::AID-EEJ2>3.0.CO;2-S

13. Ucak, C. and K. Yegin, "Understanding the behaviour of infinite ladder circuits," European Journal of Physics, Vol. 29, No. 6, 1201, 2008.
doi:10.1088/0143-0807/29/6/009

14. Parthasarathy, P. R. and S. Feldman, "On an inverse problem in cauer networks," Inverse Problems, Vol. 16, No. 1, 49, 2000.
doi:10.1088/0266-5611/16/1/305

15. Dana, S. and D. Patranabis, "Single shunt fault diagnosis in ladder structures 22 Shlivinski with a new series of numbers," Circuits, Devices and Systems, IEE Proceedings G, Vol. 138, 38-44, Feb. 1991.
doi:10.1049/ip-g-2.1991.0008

16. Doshi, K., "Discrete inverse scattering," University of California, Santa Barbara, 2008.

17. Desoer, C. A. and E. S. Kuh, Basic Circuit Theory, McGraw-Hill, 1969.

18. Jirari, A., Second-order Sturm-Liouville Difference Equations and Orthogonal Polynomials, Memoirs of the American Mathematical Society, American Mathematical Society, 1995.

19. Felsen, L. B. and N. Marcuvitz, "Radiation and Scattering of Waves," IEEE Press Series on Electromagnetic Waves, The Institute of Electrical and Electronics Engineers, New York, 1994.

20. Elaydi, S. N., An Introduction to Difference Equations, Springer-Verlag New York, Inc., Secaucus, NJ, USA, 1996.

21. Langenberg, K. J., "Linear scalar inverse scattering," Scattering: Scattering and Inverse Scattering in Pure and Applied Science, 121-141, Academic Press, London, 2002.

22. Tsihrintzis, G. and A. Devaney, "Higher-order (nonlinear) diffraction tomography: Reconstruction algorithms and computer simulation," Processing of IEEE Transactions on Image, Vol. 9, 1572, Sep. 2000.

23. Tsihrintzis, G. and A. Devaney, "Higher order (nonlinear) di®raction tomography: Inversion of the Rytov series," IEEE Transactions on Information Theory, Vol. 46, 1748-1761, Aug. 2000.
doi:10.1109/18.857788

24. Marks, D. L., "A family of approximations spanning the Born and Rytov scattering series," Opt. Express, Vol. 14, 8848, Sep. 2006.
doi:10.1364/OE.14.008837

25. Markel, V. A., J. A. O'Sullivan, and J. C. Schotland, "Inverse problem in optical diffusion tomography. IV. Nonlinear inversion formulas," J. Opt. Soc. Am. A, Vol. 20, 903-912, May 2003.
doi:10.1364/JOSAA.20.000903

26. Oppenheim, A. V., R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd Edition, Prentice-Hall Signal Processing Series, Prentice Hall, 1999.

27. Devaney, A. J., "A filtered backpropagation algorithm for diffraction tomography," Ultrasonic Imaging, Vol. 4, No. 4, 336-350, 1982.
doi:10.1016/0161-7346(82)90017-7

Dr. Amir Shlivinski