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\begin{document}


\title{Analysis of Sonnet and Electronic Modeling of a Thick Metal Microstrip}


\author{Wayne Chan}

\maketitle

\lyxaddress{\centering Department of Physics, University of Cincinnati, Cincinnati, Ohio
45221}

\begin{abstract}
Expanding the voltage equation pertaining to a transmission line and collecting
certain terms, we have equations of transmission and reflection. Computational
analysis of the electronic program, Sonnet, is based on this approximation principle.
Application of Sonnet on a thick metal microstrip provides information on energy
loss and current density.
\end{abstract}

\section{INTRODUCTION}

Transmission lines are a type of waveguide, which guide waves of any frequency
from one point to another. It is desired that the energy be conveyed with maximum
efficiency and losses in heat or radiation to be as small as possible. In addition,
they only can accommodate transverse electromagnetic modes (TEM modes). If we
assume the propagation of the wave is in the \textbf{z} direction, then there
are no longitudinal fields (H\( _{z} \) and D\( _{z} \) are zero). One of
the characteristics of this type of conductive waveguide is that it must have
guiding walls that consist at least of two isolated pieces, e.g. a coaxial-cylinder
or two parallel plates. The integral form of the curl \textbf{H} Maxwell equation,

{\raggedright \( \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \)\( \oint _{C} \)d\( \overrightarrow{l} \)\( \cdot  \)\( \overrightarrow{H} \)=\( \int _{S} \)d\( \overrightarrow{S} \)\( \cdot  \)(\( \overrightarrow{J} \)+\( \frac{\partial \overrightarrow{D}}{\partial T} \))=0\( \quad \quad \quad \quad \quad \quad \quad \quad (1) \)\par}

{\raggedright supports this property. We shall prove by contradiction. Suppose
we have a single-conductor waveguide such as a rectangular or cylindrical waveguide.
We take the closed contour C lying perpendicular to the \textbf{z} direction
to be completely inside the waveguide. Hence, \textbf{J}=0 and d\( \overrightarrow{S} \)\( \cdot  \)\( \partial  \)\( \overrightarrow{D} \)/\( \partial  \)z=dS\( \partial  \)D\( _{z} \)/\( \partial  \)z=0.\( ^{1} \)
Let the contour encompass the magnetic field. Then \textbf{H} is parallel to
d\textbf{l} and this integral is not zero. Thus, this contradiction rules out
the existence of a TEM mode.\par}

In this paper, we study the expansion series of the voltage equation pertaining
to a transmission line based on the above characteristics of a TEM mode. Then
we do a computational simulation of a thick microstrip with certain parameters
to obtain energy loss and current density graphs.


\section{FORMALISM}

In the lossy transmission line consisting of two parallel plates of length z=0
to z=l, we assume the propagation of the wave is in the \textbf{z} direction,
so

{\centering \( \overrightarrow{E} \)(\( \overrightarrow{r} \))=-E\( _{i0} \)\( \overrightarrow{y} \)(e\( ^{-i\beta z} \)-\( \Re  \)e\( ^{i\beta z} \)),\par}

{\raggedright \( \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \)\( \overrightarrow{H} \)(\( \overrightarrow{r} \))=\( \frac{1}{\eta _{0}} \)E\( _{i0} \)\( \overrightarrow{x} \)(e\( ^{-i\beta z} \)-\( \Re  \)e\( ^{i\beta z} \)),\( \quad \quad \quad \quad \quad \quad \quad \quad \quad (2) \)\par}

{\raggedright where \( \Re  \)=(\( \eta _{0} \)-\( \eta _{1} \))/(\( \eta _{0} \)+\( \eta _{1} \)).
In addition,\par}

{\centering V(z)=\( \frac{I_{L}}{2} \)(Z\( _{L} \)+Z\( _{0} \)){[}e\( ^{\gamma (l-z)} \)+e\( ^{-\gamma (l-z)} \){]},\par}

{\raggedright \( \quad \quad \quad \quad \quad \quad \quad \quad \quad \:  \)I(z)=\( \frac{I_{L}}{2Z_{0}} \)(Z\( _{L} \)+Z\( _{0} \)){[}e\( ^{\gamma (l-z)} \)+e\( ^{-\gamma (l-z)} \){]},\( \quad \quad \quad \quad \quad \quad \quad \:  \)\( \:  \)(3)\par}

{\raggedright where \( \Gamma  \)=(Z\( _{L} \)-Z\( _{0} \))/(Z\( _{L} \)+Z\( _{0} \))
is the output reflection coefficient and \( \gamma = \)\( \alpha +i \)\( \beta = \)\( \sqrt{(r+i\omega L)(G+i\omega C)} \).\par}

{\raggedright Also, \( \Gamma _{S} \)=(Z\( _{S} \)-Z\( _{0} \))/(Z\( _{S} \)+Z\( _{0} \))is
the input reflection coefficient. Note that the subscript 0 denotes vacuum,
1 denotes dielectric medium, i denotes initial, L denotes load, and S denotes
source. Applying boundary conditions and Kirchhoff's voltage equation, we obtain\par}

{\raggedright \( \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \:  \)V(z)=V\( _{S} \)e\( ^{-\gamma z} \)\( \frac{1+\Gamma e^{-2\gamma (l-z)}}{1-\Gamma _{S}\Gamma e^{-2\gamma l}} \),\( \quad \quad \quad \quad \quad \quad \quad \: \quad \: \quad (4) \)\par}

{\raggedright where V\( _{S} \)=Z\( _{0} \)V\( _{S} \)/(Z\( _{S} \)+Z\( _{0} \)).
By expanding the denominator (assuming |1-\( \Gamma _{S}\Gamma e^{-2\gamma l} \)|<1),
we find\par}

{\centering V(z)=\( \sum ^{\infty }_{n=0} \)V\( _{n} \)(z)\par}

{\centering V\( _{1} \)(z)=V\( _{S} \)e\( ^{-\gamma z} \),\par}

{\raggedright \( \quad \quad \quad \quad \quad \quad \quad \quad \:  \)V\( _{2} \)(z)=V\( _{S} \)\( \Gamma  \)e\( ^{-\gamma (2l-z)} \)=V\( _{S} \)e\( ^{-\gamma l} \)\( \Gamma  \)e\( ^{-\gamma (l-z)} \),\( \: \: \quad \quad \quad \quad \quad \: \, \quad (5) \)\par}

{\centering V\( _{3} \)(z)=V\( _{S} \)\( \Gamma \Gamma _{S} \)e\( ^{-2\gamma l-\gamma z} \)=V\( _{S} \)e\( ^{-\gamma l} \)\( \Gamma  \)e\( ^{-\gamma l} \)\( \Gamma _{S} \)e\( ^{-\gamma z} \),\par}

{\centering V\( _{4} \)(z)=V\( _{S} \)\( \Gamma ^{2}\Gamma _{S} \)e\( ^{-2\gamma l} \)e\( ^{-2\gamma (l-z)} \)=V\( _{S} \)e\( ^{-\gamma l} \)\( \Gamma  \)e\( ^{-\gamma l} \)\( \Gamma _{S} \)e\( ^{-\gamma l} \)\( \Gamma  \)e\( ^{-\gamma (l-z)} \),\par}


\paragraph{\textmd{etc. Interestingly, the odd terms pertain to forward propagation while
the even terms pertain to reflection. For instance, the V\protect\( _{3}\protect \)(z)
term has two forward propagations and one reflection. Namely, the V\protect\( _{3}\protect \)(z)
term indicates a change in V\protect\( _{S}\protect \) by means of propagation
over the entire length z=l of the transmission line (i.e. the factor e\protect\( ^{-\gamma l}\protect \)),
then reflection off the load, characterized by the factor \protect\( \Gamma \protect \)
. Then propagation all the way back to the source (another factor e\protect\( ^{-\gamma l}\protect \))
and reflection off the generator (the factor \protect\( \Gamma _{S}\protect \)),
and finally forward proagation to z=l (the factor e\protect\( ^{-\gamma z}\protect \)
).}}


\paragraph{\textmd{\protect\( \quad \: \protect \)More specifically, pertaining to a
lossless parallel plate transmission line, we have}}

{\centering \( \gamma \rightarrow i\beta =i\omega  \)\( \sqrt{LC} \)\par}

{\raggedright \( \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \)\( \Gamma \rightarrow  \)(Z\( _{L} \)-R\( _{0} \))/(Z\( _{L} \)+R\( _{0} \)).\( ^{2} \)\( \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \: \: (6) \)\par}

{\raggedright Therefore the absolute values of voltage and current are\par}

{\centering |V(z)|=|\( \widetilde{V} \)|\( \sqrt{(1-|\Gamma |)^{2}+4|\Gamma |cos^{2}[\frac{1}{2}\phi (z)]} \),\par}

{\raggedright \( \quad \quad \quad \quad \quad \quad \quad \quad \:  \)|I(z)|=|\( \widetilde{I} \)|\( \sqrt{(1-|\Gamma |)^{2}+4|\Gamma |sin^{2}[\frac{1}{2}\phi (z)]} \),\( \quad \quad  \)\( \quad \quad \quad \quad \: \, (7) \)\par}

{\raggedright where \( \widetilde{V} \)=\( \frac{I_{L}}{2} \)(Z\( _{L} \)+Z\( _{0} \))
and \( \widetilde{I} \)=\( \widetilde{V} \)/Z\( _{0} \).\par}

{\raggedright In addition, \( \phi  \)(z)=\( \theta -2i \)\( \beta  \)(l-z)
and \( \theta  \) is the phase angle of the reflection coefficient. Summing
up all the odd V(z) terms, we can obtain P\( _{incident} \), and similarly,
summing up all the even V(z) terms, we can obtain P\( _{reflection} \).\par}


\section{RESULTS}

By using the high frequency simulation software, Sonnet, we can analyze a transmission
line with certain parameters based on the above principles. In this model, we
investigate the properties of a thick metal microstrip or two parallel plates
with unequal widths. The top layer is a conductor with thickness of 1000 mm,
while the bottom layer is a dielectric (\( \varepsilon =9.8 \), porcelain has
\( \varepsilon =6-8 \)) with thickness of 25 mm. Also, a 3 mm layer of air
is placed between the two layers. Identical ports are placed on the left and
right walls.

\vspace{0.3cm}
{\centering \resizebox*{4in}{5in}{\includegraphics{Ex2.ps}} \par}
\vspace{0.3cm}

\vspace{0.3cm}
{\centering \resizebox*{4in}{7in}{\includegraphics{Ex1.ps}} \par}
\vspace{0.3cm}


\section{CONCLUSION}

We have performed a response and a current density simulation of a thick metal
microstrip. Using Sonnet, we can determine the frequency that is most suitable
for an efficient transmission line. Then we can view the current density of
the geometry and establish its application in circuitry.

\begin{thebibliography}{}
\bibitem{1}R.E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Co., Inc., New York,
1960).
\bibitem{2}M. Salazar-Palma, Iterative and Self-Adaptive Finite-Elements in Electromagnetic
Modeling (Artech House, Boston, 1998).
\bibitem{3}T. Tsao, A Concise Course in Electromagnetism for Electrical Engineering (World
Scientific Publishing Co., Singapore, 1994).
\bibitem{4}A.Bonnet-BenDhia, K. Ramdani, MMET'98 Proceedings, 792 (1998).
\bibitem{5}L.N. Dworsky, Modern Transmission Line Theory and Applications (John Wiley and
Sons, New York, 1979).
\bibitem{6}D.H. Staelin, Electromagnetic Waves (Prentice Hall, Englewood Cliffs, 1994).
\bibitem{7}J.C. Slater, Microwave Transmission (McGraw-Hill Book Co., Inc., New York, 1942).
\end{thebibliography}
\end{document}
