Chapter 3 A Useful Multilayered Microstrip Pole Extraction Technique
3.2 Pole Extraction for Two Layered Structures
3.3.5 Numerical Results and Discussions
Based on the above algorithm, a Matlab program running on Pentium 4 platform has been written to extract all the various types of poles.
(i) Surface Wave Poles
The selected permittivity for the various microstrip configurations were selected as and
, 9 .
1 =5
εr εr2 =10.9,εr3 =12.8, εr4 =17.3. The adopted dielectric thickness are and
1 o 0.4,
h k = h k2 o =0.4, h k3 o =0.4, h k4 o =0.4 . Our algorithm with the new functional expressions is compared with the Davidenko’s method [126] with the original characteristic equations. Fig. 11 (a) and Fig. 11 (b) show the numerical comparisons of the two methods in terms of the residue of the function and the number of iterations for the evaluation of the first and second TM root of the four-layered microstrip geometry under the case εr1 <εr2 <εr3 <εr4 and ko ≤kρ ≤ko εr1 . With the exception of the functional expression, the results obtained in Fig. 4 are subject to the same constraints, namely, (a) the function variation, = 1− +1 <10−9
n n
f
df f , (b) the
step size variation, = 1− +1 <10−9
n n
x
dx x , and (c) the same initial guesses are used. As
shown in Fig. 11 (a) and Fig. 11 (b), our proposed algorithm achieves better results in terms of accuracy and number of iterations compared to the conventional functional expression. On average, an accuracy of after an average of 6 iterations is achieved for the determination of all of roots under our proposed new method. As
10−16
shown from the figure, the conventional method can not converge sometimes which leads to failure of finding the actual roots of the Greens’ function.
(a)
Fig. 11 A numerical comparisons of the two methods in terms of the residue of the function and the number of iterations. (a)-(b) For the first TM root of the four-layered microstrip geometry under the case of εr1 <εr2 <εr3 <εr4 and ko ≤kρ ≤ko εr1 .
(b)
(ii) Leaky Wave Poles
To validate our method for the leaky-wave pole extraction, a three-layered structure is first selected for comparison. The respective comparison results of using Halley’s method with the new functional expressions for leaky wave poles extraction and Davidenko’s method with classical equations [126] are shown in Table 4. These two methods are using same initial guess and the conventional function expression fails to converge as shown in the shadow area.
Leaky Poles
Halley’s method with Eqn.
(3.64)
Davidenko's method with Eqn.
(3.62)
No. of Iterations 6 100
1st root 1.583226666 -1.424774824
df 10−32 -0.948361489
dx 10−32 481.9477339
2nd root 3.575614406 3.575614406
df 10−32 10−32
dx 10−32 10−32
Table 4 Comparison of the proposed approach and the Davidenko’s method [126]
for leaky-wave poles extraction in TM mode. The parameters adopted are
1 =15
εr ,εr2 =59 , 5εr3 =37. , h k1 o =0.8, h k2 o =0.9, h k3 o =0.95 , and
1 3
r ko kρ r ko
ε < < ε .
As noted from the tables, our proposed method truly outperforms the Davidenko’s method with classical equations [126] for all cases in terms of accuracy and the number of iterations required. In addition, it is noted that for some cases in Table 4
(indicated by the shaded region), the Davidenko’s method, at times, fails to converge to the proper roots. On average, an accuracy of is obtained for our newly proposed method.
10−32
(iii) Lossy Improper Poles
To validate the proposed approach for the lossy improper pole extraction, a three- layered structure is first selected. Table 5 presents the comparison results between the proposed method and Davidenko’s method [126]. To clearly illustrate the advantages of our proposed method, the values of dielectric permittivities are selected in a random order. To test the algorithm robustness, we have selected a very highly lossy dielectric of εr1 =40+i25,εr2 =75+i25, 25εr3 =40+i ,εr4 =80+i25, h k1 o =0.4, h k2 o =0.45,
, and for our comparison and Table 6 gives the comparison results.
3 o 0.4
h k = h k4 o =0.45
As noted from the tables, our proposed method can indeed extract the roots accurately and within a minimum number of 14 iterations. It is noted that the Davidenko’s method using classical function expression can at times give wrong results (as indicated by the shaded box).
In this Chapter, a unified approach for the fast evaluation of the locations of the surface wave poles, leaky wave poles and lossy improper poles in multilayered microstrip technology has been presented for the first time. Using a simple re- organization of the classical characteristic equations, together with the Halley’s method, a third order convergence is observed for the multilayered microstrip pole
Lossy poles
Halley’s method with Eqn.
(3.66)
Davidenko's method with Eqn.(3.47)
No. of Iterations 14 100
1st root
1.01182227081261 - 0.00043924163301i
1.23730715979912 + 0.31064812216076i
df 10−32 -3.23468912
dx 10−32 5.752047286
2nd root
2.97282693490741 + 0.01579544532283i
2.97282693490741 + 0.01579544532283i
df 10−32 10−32
dx 10−32 10−32
3rd root
6.38008671592520 + 0.00807064377279i
6.13059487401265 + 0.01884096006964i
df 10−32 2.309324782
dx 10−32 2.623262689
Table 5 Comparison of the proposed approach and the Davidenko’s method [126]
for lossy improper poles extraction in TM mode. The parameters adopted are εr1 =35.5+i0.1 , εr2 =37.5+i0.1 , 1εr3 =59+i0. , h k1 o =0.4 ,
2 o 0.4
h k = , and h k3 o =0.5.
Lossy Poles
Halley’s method with Eqn.
(3.66)
Davidenko's method with Eqn. .(3.47)
No. of Iterations 14 100
1st root
2.46919934276706 - 0.07871013515017i
2.46919934276706 - 0.07871013515017i
df 10−32 10−32
dx 10−32 10−32
2nd root
3.60204381679136 + 0.02588984853330i
3.60204381679136 + 0.02588984853330i
df 10−32 -0.04405944499115
dx 10−32 10−32
3rd root
4.96220186733286 + 0.04776474037665i
4.96220186733286 + 0.04776474037665i
df 10−32 10−32
dx 10−32 10−32
Table 6 Comparison of the proposed approach and the Davidenko’s method [126]
lossy improper poles extraction in TM mode. The parameters adopted are 25
1 40 i
r = +
ε ,εr2 =75+i25 , 25εr3 =40+i ,εr4 =80+i25, h k1 o =0.4 ,
2 o 0.45
h k = , h k3 o =0.4, andh k4 o =0.45.