4. Extension using Analytical Solutions
4.5. Modeling of Anisotropic Modes
4.5.2. Infinite Plane
The derivation starts by first considering infinite planes and neglecting the reflections from the outer boundary C, and thus the integral equation (4.5) is retained. However, (4.6) is no longer valid since the assumption of constant electric and magnetic fields and current definition (4.26), equation (4.37) is expressed as
50 Extension using Analytical Solutions
Substituting (4.39) into (4.38) yields the following equation system,
( )
It can be seen that the equation system (4.9) for the isotropic port is a subset of (4.40) assuming m = n = 0, which is related to the fundamental mode of the fields on circular ports. An “infinite matrix” expression can be formed from (4.40) as
pp p pp p,
U V =H I (4.41) where the identifier p denotes the circular ports, each of which has an infinite number of modes. Although (4.41) is identical to (4.10) in their form, the sizes of the vector and matrices are different. The vectors Vp and Ip in (4.41) contain infinite numbers of modal voltages and currents, which are written as
1 , , , , Appendix A.7. They are expressed as
Extension using Analytical Solutions 51
Here, a prime denotes a differentiation with respect to the argument of the Bessel and Hankel functions (Appendix A.1). The indices m and n signify the mode orders of the observation port i and source port j, respectively. It can be recognized that (4.44) and (4.45) are reduced to its isotropic form as in (4.12) and (4.11) when considering m=n=0 only. series is truncated up to the Kth mode for both source and observation ports, i.e., both m and n take integer values from –K to K. The determination of the truncation number K depends on frequency and port size, which will be discussed later.
For a single-port infinite plane pair, (4.46) can be evaluated analytically,
( ) ( ( ) )
It is clear in (4.47) that the modes are independent of each other and mode conversion does not happen for a single-port case, as indicated by zero transfer impedance between modes. One can recognize that the input impedance of the fundamental mode (m=n=0) in (4.47) is identical to the radial waveguide formula [63]. The input impedances of
52 Extension using Analytical Solutions
higher order modes for a single-port with 10-mil radius are shown in Fig. 4.11, whose magnitudes are smaller than the fundamental mode at low frequencies, however, develop with increasing frequency and can be even larger than the fundamental mode.
For more ports, mode conversions and multiple scattering take place and can alter the behavior of the fundamental mode. To show the effect, a two-port case is analyzed for a 10-mil thick infinite plane pair. The port radii of both ports are 10 mil and the distance between their centers ρ12 is 40 mil. The input and transfer impedances of the 0-th mode are shown in Fig. 4.12, with a different number of azimuthal modes (K = 0, 1, and 2) included in the computation. The results simulated by discretizing the port boundaries into 32 segments are also plotted for validation. The comparison indicates that only considering the isotropic mode (K = 0) is not sufficiently accurate for frequencies above 20 GHz, whereas adding one higher order mode in the calculation significantly improves the accuracy for both the input and transfer impedance. For K = 2, the modal solution obtained using (4.46) becomes indistinguishable to the one using the conventional CIM. Since the convergence can be achieved with only a few higher orders using the modal solution, the number of unknowns per port is much less, resulting in a much better computational efficiency than the conventional CIM. The
0 20 40 60 80 100
0 5 10 15 20 25 30
Magnitude of Input Impedance [ Ω]
Frequency [GHz]
(Z11)11 (Z11)00
(Z11)22
Figure 4.11 Magnitude of input impedances of different modes of a one-port infinite copper plane pair with port radius 10 mil and d = 10 mil (1 mil ≈ 25.4·10-6 m ), obtained with modal solutions (4.46). The relative permittivity and loss tangent of the dielectric are εr = 3.8 and tanδ = 0.02, respectively.
Extension using Analytical Solutions 53
0 20 40 60 80 100
0 5 10 15 20 25 30
Magnitude of Input Impedance (Z 11) 00 [Ω]
Frequency [GHz]
Modal K = 0 Modal K = 1 Modal K = 2
Numerical with 32 segments
{
On top of each other
0 20 40 60 80 100
0 2 4 6 8 10 12
Magnitude of Transfer Impedance (Z 12 ) 00 [Ω]
Frequency [GHz]
Modal K = 0 Modal K = 1 Modal K = 2
Numerical with 32 segments
On top of
{
each other
Figure 4.12 Magnitude of input and transfer impedance of the 0th mode of a two-port infinite copper plane pair with port radius 10 mil and d = 10 mil (1 mil ≈ 25.4·10-6 m ), obtained with modal solutions and the segmented approach. The distance between port centers is 40 mil. The relative permittivity and loss tangent of the dielectric are εr = 3.8 and tanδ = 0.02, respectively. The indices inside the bracket indicate port numbers, whereas the indices outside the bracket refer to mode numbers correspondingly.
54 Extension using Analytical Solutions
Magnitude of Transfer Impedance (Z 12 ) 00 [Ω]
Frequency [GHz]
K = 0 K = 1 K = 3 K = 10
Figure 4.14 Magnitude of transfer impedance of the 0th mode of a two-port infinite copper plane pair with port radius of 25 mil and d = 10 mil, obtained with modal solutions of different K. The distance between port centers is 100 mil. The relative permittivity and loss tangent of the dielectric are εr = 3.8 and tanδ = 0.02, respectively. The impedance curves converge by increasing the truncation number K.
0 20 40 60 80 100
Magnitude of Mode Conversion Transfer Impedance [Ω]
Frequency [GHz]
Figure 4.13 Mode conversion transfer impedance of the same two-port system, as described in the caption of Fig. 4.12. The modal solution (solid line) overlaps with the numerical solution (dashed line).
Extension using Analytical Solutions 55
simulation time in this case is 0.2 ms per frequency point for the modal solution with K
= 2 and 7.8 ms per frequency point for the discretized one using a 64-bit PC with 2.8 GHz CPU.
Fig. 4.13 shows the mode conversion transfer impedances between the two ports. The finite value of this impedance implies that a uniform current source can excite anisotropic modes on nearby ports, and vice versa. It is also obvious that the mode conversion increases with frequency and is stronger between lower order modes, as the magnitude of the transfer impedance between the mode 1 and mode 0 is much higher than the one between mode 2 and mode 0.
E
maxE
minE
maxE
min
E
maxE
minE
maxE
min(a) K = 0 (b) K = 2
E
maxE
minE
maxE
min
E
maxE
minE
maxE
min(c) 32 segments per port (d) full-wave
Figure 4.15 Normalized electric field distribution (magnitude in linear scale) at 50 GHz for the same two-port system, as described in the caption of Fig. 4.12, obtained with modal solution (a) K = 0, (b) K = 2, (c) numerical solution with 32 segments, and (d) full-wave solver. Port 1 (left) is excited with an isotropic current source and port 2 (right) is left open.
56 Extension using Analytical Solutions
It has been observed that the mode conversion effects increase with frequency. Thus more modes have to be included in the computation for higher frequencies to obtain an accurate result. Fig. 4.14 shows the transfer impedance of mode 0 between two ports with a radius of 25 mil each and a distance of 100 mil between their centers, obtained with different truncation numbers K. The impedance curves converge by increasing K, whereas for each increment of K, the frequency range of accurate results is expanded by about 10 GHz in this case. As a rule of thumb for determining the mode truncation number K, one can say that K must be larger than the ratio of the port perimeter to a quarter wavelength.
The CIM can be used to generate an electric field distribution between the planes, as shown in Fig. 4.15. Here, port 1 (left) is excited with a 1-mA uniform current source and port 2 (right) is open. The same anisotropic field distribution around port 1 is observed at 50 GHz for the modal solution including higher order modes (b) and the numerical solution segmenting circular ports (c). Fig. 4.15(d) shows the electric field distribution obtained with a commercial full-wave finite element method based solver [117]. Although a minor difference from CIM results (b, c) exists due to the presence of via near fields, the same anisotropy is identified in the full-wave result. This anisotropy is attributed to the reflection and mode conversion caused by the open port 2. In contrast, only using mode 0 fails to account for this non-uniformity, as demonstrated in Fig. 4.15(a).
4.5.3.
Finite PlanesFor modeling of finite planes, equation (4.14) is recalled where the solution is formed by combining circular ports with boundary line ports. The difference from Section 4.2.2 is that the circular ports, denoted by superscript p, are represented by modal expressions.
Similar to the procedure in Section 4.2.2, the expansion to finite planes is carried out by assuming boundary line ports as point sources. Expressions for the matrices Upp determined by (3.14.a) and (3.14.b), respectively. To find elements of the off-diagonal matrices detailed in Appendix A.8. The coefficients of the off-diagonal matrices are written as
( )
Uijqp n = −k aπj jJn′( )
ka Hj n(2)(
kρij)
ejn(
φij+π)
, (4.48)Extension using Analytical Solutions 57
A one-port rectangular plane pair, as shown in Fig. 4.16, is analyzed using the extended CIM. A circular port is placed close to the upper boundary of the plane with a distance of b. The input impedance results of the mode 0 for b = 50 mil is shown in Fig. 4.17, compared to that obtained by the conventional CIM with port perimeter discretized into 32 segments. Similar to the two-port infinite plane case, the extended CIM agrees with the conventional one, when higher order modes are included in the calculation.
The influence of mode conversions on mode 0 is relatively small in this case since the dominant effect here is the plane resonances.
The mode conversion appears when the reflection from the plane boundary interferes with the local circular port field distributions. It depends on the distance of the port to
(0, 0)
58 Extension using Analytical Solutions
Figure 4.18 Magnitude of mode conversion of the case in Fig. 4.17 with b = 50 mil and 100 mil. Mode conversion is stronger when the port is closer to the boundary.
0 20 40 60 80 100
Magnitude of Input Impedance (Z 11) 00 [Ω]
Frequency [GHz]
Figure 4.17 Magnitude of input impedance of the 0th mode of the case in Fig. 4.16 with b = 50 mil.
Extension using Analytical Solutions 59
the boundary. The closer a port lies to the boundary, the more mode conversion is produced. The transfer impedance between the 0th and the 1st mode with b = 50 mil is compared to that with b = 100 mil, as shown in Fig. 4.18. As the port is moved further away from the boundary, the mode conversion becomes less significant.