1 The Coupling of Perfectly Conducting and Impedance Coaxial Waveguides Alinur Büyükaksoy (1) ,  I. Hakk Tayyar (1) , Feray Hacvelio glu (1) and Gkhan Uzgren (2) Abstract In the present work a matrix Wiener-Hopf equation connected with the scattering of the dominant TEM mode at the junction of a two-part coaxial waveguide is solved and the inuence of the inner and outer cylinders radii and the surface impedances on the reection and transmission coefcients are presented. I. I NTRODUCTION It is well known that cable impedance discontinuities are effective on the performance of coaxial systems and have been considered by several authors. For example, in [1] and [2] the scattering of a shielded surface wave in a coaxial waveguide by a wall impedance discontinuity in the inner cylinder has been analyzed. In the present work a matrix Wiener-Hopf equation con- nected with a new canonical scattering problem is solved. We consider the scattering of the dominant TEM mode at the junction of a two-part coaxial waveguide whose left part is perfectly conducting while the surface impedances of the inner and outer cylinders of the right part are different from each other (see Fig. 1). The representation of the solution to the two-part mixed boundary-value problem in terms of Fourier integrals leads to two simultaneous Wiener-Hopf equations which are uncoupled by using the analytical properties of the functions that occur and by introducing some innite sums over certain poles with unknown expansion coefcients The solution involves two innite sets of unknown coefcients satisfying two innite systems of linear algebraic equations. These systems are solved numerically and some graphical results showing the inuence of the inner and outer cylinders radii and the surface impedances on the reection and transmission coefcients are presented. A time dependence exp(i!t) with ! being the angular frequency is assumed and supressed throughout. (1) The authors are with the Gebze Institute of Technology, 41400, Gebze, Kocaeli, TURKEY (2) The author is with the Istanbul Culture University, Ataky Campus, 34510 Istanbul, TURKEY Fig.1 Geometry of the two-part coaxial waveguide II. ANALYSIS Consider a coaxial waveguide whose inner cylinder is of radius  = a, while the radius of the outer cylinder is  = b with (; ; z) being the usual cylindrical coordinates. We assume that both the inner and outer cylinders of the left part (z< 0) are perfectly conducting while the surface impedances of the inner and outer cylinders of the right part (z> 0) are denoted by Z 1 =  1 Z 0 and Z 2 =  2 Z 0 , respectively. Here, Z 0 stands for the impedance of the free space. Assume that aTEM mode u i = e ikz (1a) with wavenumber k; is travelling towards the junction at z = 0, from the left. For the sake of analytical convenience, the total eld in the region a<<b;  2 (;) ; z 2 (1; 1) will be expressed as follows: u T (; z)= u i + u(; z): (1b) For the unknown eld u which satises the Helmholtz equa- tion  1  @ @   @ @  + @ 2 @z 2 + k 2  u (; z)=0; (2a) it is appropriate to use the following Fourier integral repre- sentations: ; u(; z)= Z L h A()J 0 (K)+ B()H (1) 0 (K) i e iz d (2b) Here J 0 and H (1) 0 are the Bessel and Hankel functions.The square-root function K()= p k 2   2 ; (2c) is dened in the complex -plane, cut along  = k to  = k + i1 and  = k to  = k  i1, such that K (0) = k.