An ABCD -Matrix Representation for the Dielectric Ring

In order to investigate the impact of a coating on a circular port, we derive anABCD-matrix representation for the modal components that allows to transform impedances from the reference plane of the outer port to the one of the inner and vice versa. Starting with the case of an impedance transformed from the inner port to the outer, we are interested in a matrix of the form

This is a generalized version of the well known ABCD-matrix [113], where the scalar entries A, B, C, and D are replaced by matricesA, B, C, and D, respectively. The submatrices can be derived directly from the entries of the matrices Uand H and are diagonal as well.

The detailed derivation of the entries of the submatrices of the generalized ABCD-matrix is outlined in Appendix A.5. For convenience, the notation of cross products¹⁴ is used.

As the submatrices of the generalized ABCD-matrix are diagonal, an ordinary ABCD -matrix can be written connecting the voltages and currents of a specific circular harmonic with the index n.

The inverse representation for the voltages and currents on the inner port as a function of the voltages and currents on the outer ports is found by inverting the ABCD-matrix, see Appendix A.5

In [124, p. 32], analytic expressions for voltages and currents of the fundamental mode in a radial waveguide are derived. These expressions are equal to (2.82) and (2.83) for n = 0 which validates the result to a certain extent.

14For the properties ofcross products of Bessel functions, see Appendix A.1.3 and [1, Section 10.6]

The impedance measured at the inner port as a function of the boundary impedance at the outer port can be written as

Z_nⁱⁿ = −^πka₂^o rn(ka_o, ka_i)Z_n^out+j^ωµd₄ pn(ka_o, ka_i) j^πka_ωµd^iπkao sn(ka_o, ka_i)Z_n^out+ ^πka₂ ⁱqn(ka_o, ka_i)

=j ωµd 2πka_i

j^2πka_ωµd^o qn(kao, kai)Z_n^out+ pn(kao, kai) j^2πka_ωµd^o sn(kao, kai)Z_n^out+ rn(kao, kai).

(2.84)

With thisABCD-matrix, we can write the impedance seen at the outer contour as a function of the impedance used as a boundary condition at the inner contour

Z_n^out = ^πka2i qn(ka_o, ka_i)Z_nⁱⁿ−j^ωµd₄ pn(ka_o, ka_i)

−j^πka_ωµd^iπkao sn(ka_o, ka_i)Z_nⁱⁿ+^πka₂^o rn(ka_o, ka_i)

=j ωµd 2πka_o

j^2πka_ωµdⁱqn(ka_o, ka_i)Z_nⁱⁿ+ pn(ka_o, ka_i) j^2πka_ωµdⁱ sn(ka_o, ka_i)Z_nⁱⁿ−rn(ka_o, ka_i).

(2.85)

For the case of a PEC boundary on the inner contour, the equation simplifies to Z_n^{out Z}

nin→0

= j ωµd 2πkao

pn(ka_o, ka_i)

rn(kao, kai). (2.86) This impedance may be used to model dielectrically coated ground vias. Instead of modeling the ground via and the coating separately, the coated ground via is modeled as an inclusion of the radius of the coating, with an impedance given by (2.86). The frequency behavior of the impedance is shown in Figures 2.14 and 2.15. At DC, the impedance is zero as it is for the non-coated ground via. For increasing frequencies up to the first resonance, the impedance is inductive. For the fundamental mode, the slope at DC is non-zero and an approximation for low frequencies or thin coatings can be found in form of an inductor.

Using (A.33) and (A.34) yields

L_o = µd 2π lna_o

a_i. (2.87)

This formula shows that the inductance increases with the thickness of the coating. Moreover, only the permeability and not the permittivity have an impact on the low frequency behavior.

To clarify what is meant bylow frequency, let us take a look at the resonance of the impedance Z₀^out. Comparing Figure 2.14 and 2.15 shows that both, the thickness of the layer and the dielectric constant have an influence on the resonance. The first resonance frequency is given by the first zero of rn(ka_o, ka_i). This zero was evaluated using the Newton-Raphson Method [125] as a function of the ratio of a_o anda_i, see Figure 2.16. Here, the radius of the inner contour is assumed to be 10 mil. The resonance frequency rapidly decreases for an

2.5 Extension to Concentric Circular Contours

0 10 20 30 40 50

Frequency (GHz) -40

-20 0 20 40

Inputreactance(jΩ)

Z₀^out Z₁^out

jωL₀ =jω42.3pF

εr = 100

d= 12 mil PEC

10 mil 20 mil

Figure 2.14: Comparison of imaginary part of the input impedance of a dielectrically coated ground via with εr= 100 using the analytical formula (2.86) and the low frequency approximation with the inductance (2.87). The reference plane is on the circumference of the coating (dashed line in the drawing) looking in the direction of the via.

0 10 20 30 40 50

Frequency (GHz) -2.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Inputreactance(jΩ)

Z₀^out

jωL₀ =jω5.8pF ε_r = 11.6

d= 12 mil PEC

10 mil 11 mil

Figure 2.15: Comparison of imaginary part of the input impedance of a dielectrically coated ground via with ε_r = 11.6 using the analytical formula (2.86) and the low frequency approximation with the inductance (2.87). The reference plane is on the circumference of the coating (dashed line in the drawing) looking in the direction of the via.

1 2 3 4 5 6 7 8 9 10 ao/ai

1 10 100 1000

Resonancefrequency(GHz)

ε_r = 1 εr = 11.6 ε_r = 100

Figure 2.16: Imaginary part of the input impedance of a dielectrically coated ground via.

increasing ratio of a_o toa_i. Nevertheless, it is very high for practical applications. E.g. a silicon coating layer with ε_r = 11.6 with a thickness of the coating of 10% of the radius of the via (as in Figure 2.15) has the first resonance at about 848 GHz. For a cavity height of 10 mil, the equivalent inductance is L_o = 4.8 pH. This justifies that, in practice, the dielectric coating can be ignored very often.

2.5.3 Validation

The ABCD-matrix (2.82) can be found in literature for n= 0, which partially validates the underlying formulas. To further validate the formulas, the validation example used for the circular inclusion in Section 2.4.4 is extended. The geometry is the same, but inside the circular inclusion of ε_r = 100 a second circular inclusion exists.

First, we consider a PEC inclusion of radius a_i = 10 mil inside the dielectric inclusion.

Figure 2.17 shows the transfer impedance obtained with different choices of boundary conditions and basis functions using the CIM. The same geometry is computed in five different ways: discretization of the inner contour 2 and the outer contour1 with linear contour segments, discretization of the outer contour1with linear contour segments and the inner contour is modeled using a circular port, modeling of both circles with circular ports, and impedance boundary conditions on the outer circular port taking the inner into account.

The results obtained with the linear contour segments and the circular boundary are in excellent agreement. That validates the entries of the matrices Uand Hfor the concentric