LAMINATED TRANSMISSION LINES. I 



901 



wlicro 



= (cr' - y'V, Vp = ^(1 - y~/o-')\ 



m 



aiul we have iisetl tlie ahhreN'iatioiis 



Irs = Ir(lips), Krs = /^^(/vp,). 



70) 



It may be verified that the dehM'niinaut .1/ of the sciuare matrix a{)- 

 l)eariiig iti ((18) is simply 



M = p./ 



Pi/ pi 



(71) 



III i)riii('iple equation ((38) permits us to determine by matrix multi- 

 phcation the relation between the tangential fields at the inner and outer 

 surfaces of a coaxial double layer, or of a laminated stack of any number 

 of double layers, such as is shown in Fig. 4. The difficulty is that the 

 elements of the matrix of a single layer are not functions only of the 

 electrical properties of the layer and its thickness, but depend in a more 

 complicated way on the inner and outer radii separately. Whereas in 

 the plane case we had merely to take the ?ith powder of a single matrix, 

 we are now faced with the problem of multiplying together n matrices, 

 each of which (Uffers more or less from all the others. An exact expres- 

 sion for the result is practically out of the ciuestion; but we can make 

 some reasonable approximations if we assume that each individual layer 

 is thin compared to its mean radius, so that the matrix elements do not 

 change much from one layer to the next. 



If the thickness t {= p-i — pi) of a single layer is small compared to pi , 

 then the Bessel function combinations appearing in (68) may be ex- 

 panded in series, as shown in Appendix I, and the circuit parameter 

 matrix takes the following approximate form, 



' + 2^J 



ch Kt — - — sh d rjp 

 2k pi 



.' + 2., 



sh kI 



Vp L 



'^^u 



sh Kt 



' + 2.J 



, (72) 



ch Kt + - — sh Kti 

 2/vpi 



wliere terms of the order of t/pi represent the first-order curvature cor- 

 rections. If we use the same value of pi , say p, for both parts of a double 

 layer, then up to first order the elements of the matrix of the double 

 laver become 



