Microstrip T-junction: unphysical results and NaNs at high frequencies

[smaslovski]

Hi,

As was initially communicated by @aum on t.me/precisionanalog group, qucsator may produce unphysical results when modeling microstrip T-junctions at frequencies approaching the cutoff frequency for the higher-order modes, although commercial simulators such as ADS may still produce reasonable results at these frequencies.

The following figure illustrates the problem:

In this example, the unphysical results appear in the frequency region from about 8.5 to 10.2 GHz. It is also seen that in this frequency range the values of S-parameters are NaNs.

Investigation of the T-junction analytical model formulation used in qucsator (https://qucs.sourceforge.net/tech/node81.html) reveals that the source of this problem is their formula (11.216), which has a term $\sqrt{d_a\cdot d_b}$. As can be seen from Eq. (11.210) on the same page, either $d_a$ or $d_b$ can become negative at high enough frequencies, in which case the product $d_a\cdot d_b$ can also become negative. This results in NaNs in Eq. (11.216).

Apparently, Eq. (11.216) is "original art" by the authors of qucs, although the T-junction model itself can be traced back to the article by E. Hammerstad, ``Computer-Aided Design of Microstrip Couplers with Accurate Discontinuity Models,'' Symposium on Microwave Theory and Techniques, pp. 54-56, June 1981. In this article, Hammerstad gives a "recipe" on how to generalize his formula for the shunt susceptance $B_T$ of a symmetric T-junction (Eq. 14) to the same for a nonsymmetric junction (check the highlighted text):

Note that Hammerstad writes there: "For the side arm reference plane displacements and the shunt susceptance use the main line impedance equal to the geometric mean of the two actual ones." Evidently, Eq. (11.216) in qucs has resulted from an incorrect interpretation of this paragraph, since, besides taking the geometric mean of the line impedances (Eq. 11.214), the qucs authors decided also to take a geometric mean of the plane displacements $d_a$ and $d_b$, which does not have any physical meaning.

Indeed, it is evident from physical reasoning that, under first-order approximation, the expression for $B_T$ must be linear in $d_a$ and $d_b$, similarly to how it appears in Eq. (14) in Hammertstad's paper. In the following comments I will present a corrected formula and also provide a patch to fix this problem in qucs (now my wife needs me and I have to switch my attention to her).

[smaslovski]

Returning back to writing: As I mentioned above, a physically sound generalization of Eq. (14) from the Hammerstad's article to the case on nonsymmetric T-junction must be, to the first approximation, linear in $d_a$ and $d_b$, as the formula (14) itself suggests. Furthermore, the contributions from these two displacements to $B_T$ must be additive. Thus, to generalize Hammerstad's Eq. (14) for the case of nonsymmetric T-junction, we may note that for the symmetric T-junction with $d_a = d_b$, $D_a = D_b$, $Z_a = Z_b$, etc., we may identically rewrite Eq. (14) as

$\frac{B_T}{Y_2} = 5.5 \cdot \frac{\varepsilon_r+2}{\varepsilon_r}\left[1 + 0.9\cdot\log R + 4.5 \cdot R\cdot Q - 4.4\cdot e^{-1.3\cdot R} -20 \left(\frac{Z_2}{\eta_0}\right)^2 \right]\cdot\frac{1}{2D_2}\cdot\left(\frac{d_a D_a}{n_a^2\lambda_a}+\frac{d_b D_b}{n_b^2\lambda_b}\right),$

where $R = \frac{\sqrt{Z_a Z_b}}{Z_2}$ and $Q = \frac{f^2}{f_{p,a} f_{p,b}}$.

For a symmetric T-junction, the above formula gives numerically identical result for $B_T$ as Hammerstad's Eq. (14). On the other hand, when the lines "a" and "b" are not identical, this formula provides the necessary generalization for the nonsymmetric T-junction case, as it is 1) in line with the "recipe" given by Hammerstad in his original article; 2) to the first approximation, linear with respect to $d_a$ and $d_b$; and 3) the contributions to $B_T$ from the two lines are in complete agreement with the form of Hammerstad's Eq. (14).

[smaslovski]

Here is the patch for the file mstee.cpp that implements the necessary change in the formula for $B_T$:

diff --git a/src/components/microstrip/mstee.cpp b/src/components/microstrip/mstee.cpp
index 47aa2845..151095ad 100644
--- a/src/components/microstrip/mstee.cpp
+++ b/src/components/microstrip/mstee.cpp
@@ -173,10 +173,8 @@ void mstee::calcPropagation (nr_double_t f) {
   Tb2 = std::max (Tb2, NR_TINY);
 
   // shunt susceptance
-  Bt = 5.5 * std::sqrt (Da * Db / lda / ldb) * (er + 2) / er /
-    Zl2 / std::sqrt (Ta2 * Tb2) * std::sqrt (da * db) / D2 *
-    (1 + 0.9 * std::log (r) + 4.5 * r * q - 4.4 * std::exp (-1.3 * r) -
-     20 * sqr (Zl2 / Z0));
+  Bt = 5.5 * (er + 2) / er * 0.5 * (da * Da / (Ta2 * lda) + db * Db / (Tb2 * ldb)) / (D2 * Zl2) *
+     (1 + 0.9 * std::log (r) + 4.5 * r * q - 4.4 * std::exp (-1.3 * r) - 20 * sqr (Zl2 / Z0));
 }
 
 /* This function can be used to create an extra microstrip circuit.

After applying this patch, the example T-junction simulation from above produces the following result:

There are no NaNs anymore in the frequency range from 8.5 to 10.2 GHz. Moreover, after such a correction the results in qucs are similar to the results obtained for the same structure in a commercial simulator (ADS):

Note that ADS uses a closed-source proprietary model for the microstrip T-junction. The precise analytical formulation of their model, as far as I can tell, was never published anywhere. The only information that I have about it, is from the ADS manual:

[ra3xdh]

@smaslovski , See also ra3xdh/qucsator_rf#39