3.2 Shielded microstrip tranmission linesBasic theory

Fig. 3.2A.1  Selected types od planar microwave transmission lines: a) open microstrip line, b) shielded slotline, c) finline, d) shielded coplanar line 

Shielded
microstrip transmission line
is a member of the family of
planar microwave transmission lines.
Open microstrip line (fig. 3.2A.1a), shielded slotline (fig. 3.2A.1b) shielded finline (fig.
3.2A.1c) or coplanar waveguide (fig. 3.2A.1d) belong to the most common planar transmission lines.
Using segments of planar transmission lines, microwave circuits can be composed. Planar conductors can serve for connecting circuit components or
can be used for creating passive circuit elements (capacitors and inductors, especially). Moreover, planar circuits can be relatively simply completed
by active elements (transistors and diodes, e.g.). Therefore, the planar technology is of wide use these days and analysis of planar structures is of
rising importance.
In this paragraph, we deal with the computation of the electromagnetic field distribution in shielded
microstrip transmission line
(fig. 3.2A.2), which parameters are assumed to be constant in the longitudinal direction. Then, only a twodimensional structure
has to be analyzed (crosssection of the transmission line), which simplifies computation
[20].

Fig. 3.2A.2  Shielded microstrip transmission line (longitudinally homogenous) 

Even if the abovedescribed
simplification is done, field distribution in the shielded
microstrip transmission line
cannot be computed analytically. Therefore, numerical methods or approximate ones have to be exploited. All the methods can be divided into two groups, to
quasistatic methods
and to
fullwave methods
[20].
Quasistatic methods
are based on the assumption that the dominant mode of the wave, which propagates along the transmission line, can be approximated (with good accuracy) by the
transversal electromagnetic wave
(TEM). Unfortunately, this assumption is valid on low microwave frequencies (typically by 5 GHz). If frequency is increased, then the value of
longitudinal components of electromagnetic field rises, and hence, it cannot be neglected.
Working on higher microwave frequencies,
fullwave methods
have to be exploited for the analysis. Fullwave methods are based on the direct solution of Maxwell equations. Since the transmission line is going
to be analyzed in the harmonic steady state, Maxwell equations are of the following form:

$\nabla \times \mathbf{E}=j\omega {\mu}_{0}{\mu}_{r}\mathbf{H}$,

( 3.2A.1a )


$\nabla \times \mathbf{H}=+j\omega {\epsilon}_{0}{\epsilon}_{r}\mathbf{E}+\sigma \mathbf{E}+{\mathbf{J}}_{s}$,

( 3.2A.1b )


$\nabla \cdot \left({\epsilon}_{0}{\epsilon}_{r}\mathbf{E}\right)=\rho $,

( 3.2A.1c )


$\nabla \cdot \left({\mu}_{0}{\mu}_{r}\mathbf{H}\right)=0$.

( 3.2A.1d )

Here, E denotes electricfield intensity vector, H is magneticfield intensity vector, operator nabla
is in Cartesian coordinate system of the form

,


J_{S} denotes density of currents, which are imposed to the analyzed system by sources, ω is angular frequency,
ε_{0} and μ_{0} are permittivity and permeability of vacuum, ε_{r} and
μ_{r} denote relative permittivity and permeability of dielectrics inside the analyzed structure, ρ is volume charge density
in dielectrics and σ denotes electric conductivity of dielectrics.
Assume that sources of electromagnetic field are in a large distance from the area where computations are performed (then, imposed currents
J_{s} are zero in this area). Next, zero charge density is supposed in the structure and the media are expected to be linear
and isotropic (permittivity and permeability are scalar quantities, which do not depend on the value of respective field intensities).
Further, dielectrics are assumed to be lossy (represented by electric conductivity σ) and all metallic parts (shielding waveguide, microstrip)
are expected to be perfect electric conductors.
All the assumptions are substituted to (3.2A.1). Then, both sides of (3.2A.1a) are multiplied
by the operator nabla from left (in vector way). That way, curl of the magneticfield intensity vector is obtained on the righthand side of
(3.2A.1a), which is replaced by the righthand side of (3.2A1b). Finally, the following equation is
obtained:

$\nabla \times \left(\nabla \times \mathbf{E}\right)=j\omega {\mu}_{0}{\mu}_{r}\left(\sigma +j\omega {\epsilon}_{0}{\epsilon}_{r}\right)\mathbf{E}$.

( 3.2A.2 )

If the square of
wave number
in vacuum is denoted as

${k}_{0}^{2}=j\omega {\mu}_{0}j\omega {\epsilon}_{0}$

( 3.2A.3 )

and complex relative permittivity of dielectrics as

$\tilde{{\epsilon}_{r}}=\frac{\sigma}{j\omega {\epsilon}_{0}}+{\epsilon}_{r}$,

( 3.2A.4 )

eqn. (3.2A.2) can be rewritten to

$\nabla \times \left(\nabla \times \mathbf{E}\right)\tilde{{\epsilon}_{r}}{\mu}_{r}{k}_{0}^{2}E=0$.

( 3.2A.5 )

Analyzed
microstrip transmission line
is situated to the
Cartesian coordinate system
(coordinates x and y in transversal directions, coordinate z longitudinal). Hence, electromagnetic wave propagates along the axis z (along microstrip)
and electricfield intensity vector depends on the longitudinal coordinate according to

$\mathbf{E}\left(x,y,z\right)=\mathbf{E}\left(x,y\right)\mathrm{exp}\left(\gamma z\right)$,

( 3.2A.6 )

where γ is
propagation constant

$\gamma =\beta +j\alpha $;

( 3.2A.7 )

β is
attenuation constant
and α denotes
phase constant.
Mathematical description of electromagnetic wave propagating in the direction z (3.2A.6), is substituted
to vector equation (3.2A.5). Then, all the partial derivatives according to z can be evaluated (respective terms are multiplied
by the
propagation constant
γ). Rewriting all vectors in (3.2A.5) as a sum of the transversal component and the longitudinal one

$\mathbf{E}={\mathbf{E}}_{t}+{z}_{0}{\mathbf{E}}_{z}$,

( 3.2A.8a )


$\nabla ={\nabla}_{t}+{\mathbf{z}}_{0}\frac{\partial}{\partial z}={\nabla}_{t}\gamma {\mathbf{z}}_{0}$,

( 3.2A.8b )

we get

${\nabla}_{t}\times \left({\nabla}_{t}\times {\mathbf{E}}_{t}\right)\gamma \left({\nabla}_{t}{E}_{z}+\gamma {\mathbf{E}}_{t}\right)={k}_{0}^{2}{\mu}_{r}\tilde{{\epsilon}_{r}}{\mathbf{E}}_{t}$,

( 3.2A.9a )


${\nabla}_{t}\times \left[\left({\nabla}_{t}{E}_{z}+\gamma {\mathbf{E}}_{t}\right)\times {\mathbf{z}}_{0}\right]={k}_{0}^{2}{\mu}_{r}\tilde{{\epsilon}_{r}}{E}_{z}{\mathbf{z}}_{0}$.

( 3.2A.9b )

Where as (3.2A.9a) is vector equation of transversal components, (3.2A.9b) is scalar equation of
a longitudinal component.
The set of differential equations (3.2A.9) has to be completed by
boundary conditions,
which have to be met by the solution of (3.2A.9)

$\begin{array}{r}{\mathbf{n}}_{0}\times {\mathbf{E}}_{t}=0\\ {E}_{z}=0\end{array}\}na\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\Gamma}_{1}$,

( 3.2A.10a )


$\begin{array}{r}\left[{\nabla}_{t}{E}_{z}+\gamma {\mathbf{E}}_{t}\right]\cdot {\mathbf{n}}_{0}=0\\ {\nabla}_{t}\times {\mathbf{E}}_{t}=0\end{array}\}na\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\Gamma}_{2}$.

( 3.2A.10b )

Eqn. (3.2A.10a) describes the fact that components of electricfield intensity vector, which are tangential to the perfectly
electricalconductive surfaces Γ_{1}, have to be zero on those surfaces. Eqn. (3.2A.10b) expresses the
fact that variation of components of electricfield intensity vector in the normal direction with respect to perfectly electricconductive surface
Γ_{2} has to be zero on this surface.
Eqn. (3.2A.9) completed by
boundary conditions
(3.2A.10) are the initial relations to the fullwave analysis of
microstrip transmission line.
We have to keep in mind, that eqns. (3.2A.9), (3.2A.10) cover the first Maxwell equation and the second
one only, and therefore, the solution has to be checked to meet the third Maxwell equation and the fourth one. In the opposite case, the solution
would be physically nonexisting (spurious one).
In the following paragraphs, we describe the way of implementing the fullwave analysis of a
shielded microstrip line,
which is based on (3.2A.9) and (3.2A.10), by
finiteelement method.
First, we briefly comment physical phenomena in the investigated structure because success of the analysis is conditioned by their correct modeling.
First, consider a hollow rectangular waveguide (inside, vacuum is assumed). In such waveguide, a transversal electric wave (longitudinal component
of magnetic intensity is present) or a transversal magnetic one (longitudinal component of electric intensity is present) can propagate. If a dielectric
substrate is inserted into the waveguide (filling becomes inhomogeneous), longitudinal components of both the electricfield intensity and the magneticfield
one appear in the structure at the same time. In the opposite case, boundary conditions could not be met at the interface between dielectric and vacuum.
Wave propagation in a waveguide of inhomogeneous filling can be described by hybrid modes LSM (Longitudinal Section Magnetic) and LSE
(Longitudinal Section Electric). Moreover, in inhomogeneous filling is completed by a metallic microstrip, transversal and longitudinal currents
are induced there, which mutually couple LSE and LSM modes.
Numerical analysis of wave propagation in a metallic waveguide (vacuum inside) is relatively simple because homogeneous wave equation is needed to
be solved for the longitudinal component of electricfield intensity (modes TM) or magneticfield one (modes TE). If microstrip transmission line is
going to be analyzed, attention has to be paid to two potential sources of spurious solutions.
First, production of spurious solutions can be caused by incorrect modeling of electromagnetic conditions on the boundary between the substrate and
vacuum. Second, incorrectly modeled influence of metallic strips can cause appearance spurious solutions. Exploiting
hybrid finite elements
can eliminate both the causes.
Analyzing a shielded microstrip line by
hybrid finite elements,
all the components of electricfield intensity or magneticfield one have to be included into the computations. Eqn. (3.2A.9)
is the initial relation of the analysis.
Matter of
hybrid finite elements
consists in modeling a longitudinal intensity component using
nodal approximation,
and transversal intensity components using an approximation, which is based
edge vectors.
Turn our attention to the general solution of (3.2A.9) under conditions (3.2A.10) by
finiteelement method.
At every step of the general approach, we discuss the operation of the method in our situation.
First, an analyzed structure is subdivided into subspaces
(finite elements),
which do not overlap and which cover all the points of the structure. In the space of a finite element, parameters of the analyzed structure
(permittivity, permeability, conductivity) have to be constant. There are no restrictions to size and shape of finite elements. Finiteelement mesh
can be denser in areas where details of the solution are of our interest, and can be relatively sparse in areas where details are out of our interest.
Finite elements can be curvilinear, and therefore, an arbitrary geometry can be modeled.
Analyzing longitudinally homogeneous microstrip transmission line, twodimensional problem is solved. Therefore, we turn our attention to twodimensional
finite elements.
Leaving piecewise constant approximation, a linear function is the simplest approximation function.
In two dimensions, the linear function is a plane over a finite element (the computed quantity can be imagined to be drawn above the element).

Fig. 3.2A.3  Mesh examples of rectangular bielements for the analysis of a shielded microstrip transmission line 

If three points uniquely determine the approximation plane, the analyzed
structure is subdivided to triangular
finite elements.
Coordinates of vertexes of triangular finite elements are independent variables of the twodimensional
linear approximation. Samples of sought field distribution are dependent functional values.
Interlaying the approximation plane through the dependent functional values in triangle vertexes,
a unique approximation plane is obtained.
The analyzed structure has to be divided to
finite elements
very carefully because both the final error and CPUtime demands of the analysis strongly depend on the mesh geometry.
In fig. 3.2A.3, two examples of triangular finiteelement meshes for the analysis of
shielded microstrip line
are depicted. For simplicity, two triangular finite elements create a rectangular bielement together, which is the basic building block of our mesh.
Mesh A is homogeneous (consists of identical finite elements). Since electromagnetic field can be expected to quickly
decrease with the distance from the microstrip, distant areas can be covered by a sparser mesh B. As shown later, the mesh B
negligibly decreases accuracy of the analysis, but significantly reduces CPUtime demands (lower number of
finite elements
corresponds to lower number of mathematical operations).
In the second step of the solution, an approximation of a sought function over each finite element is expressed in a formal way. Usually, the unknown
solution is approximated by a linear combination of elected approximation functions and unknown approximation coefficients. A linear function is the
simplest approximation. As an example, we consider a general plane, which is uniquely determined by functional values of the approximated function in three points of a
finite element.
This general plane can be composed of three subplanes (a linear combination), each of which is unitary in a single points of a finite element and
which is zero in other two points of a finite element. Unknown approximation coefficients, which appear in the linear combination at partial planes,
play the role of spatial samples of the spatial distribution of computed quantity in points, where partial functions are of unitary value (other
functions are zero in this point, and therefore, they do not participate in sampling). For the longitudinal component of electricfield intensity,
we get therefore

${E}_{z}^{\left(n\right)}\left(x,y\right)\approx {\displaystyle \sum _{m=1}^{{M}_{n}}{c}_{m}{N}_{m}^{\left(n\right)}\left(x,y\right)}$,

( 3.2A.11 )

where E_{z}^{(n)} is an approximation of a sought function over n^{th}
finite element,
c_{m} are unknown approximation coefficients and N_{m}^{(n)} are elected partial approximation functions
on n^{th} finite element, M_{n} denotes the number of partial approximation functions exploited for composition of the
total approximation of sought function over n^{th} finite element.
Collecting approximations over all N finite elements, a global approximation over the whole analyzed structure is obtained

$\tilde{{E}_{z}}\left(x,y\right)\approx {\displaystyle \sum _{n=1}^{N}{E}_{z}^{\left(n\right)}\left(x,y\right)}$.

( 3.2A.12 )

Eqn. (3.2A.12) is a single equation for M unknown approximation coefficients c_{m}. If these coefficients
are found, approximation of the sought quantity is obtained.
More information about abovedescribed approximation can be found in the layer B.
In the third step, the formal approximation of the solution E_{z}^{~} is substituted to the solved partial differential equation.
Since the approximation differs from the exact solution, the initial equation is not met perfectly. This fact is respected by introducing a
residual function,
which equals to the difference between the exact solution and the approximation

$R\left(x,y\right)=\tilde{E}\left(x,y\right)E\left(x,y\right)$.

( 3.2A.13 )

The approximation is as accurate as low values of the
residual function are.
Therefore, we are going to minimize the residual function over the whole analyzed structure. The residual function is minimized exploiting the
method of weighted residuals.
The method of weighted residuals consists in multiplying the residual function R(x, y) by a
weighting function
W(x, y). The product is integrated over the whole analyzed space S and the result is set to equal zero

$\underset{S}{\int}R\left(x,y\right)W\left(x,y\right)dS}=0$,

( 3.2A.14 )

where dS = dx dy.
If as many properly elected
weighting functions
W_{n} are used as many unknown approximation coefficients are computed, a set of linear equations (as many equations as unknown
approximation coefficients) is obtained. Solving this set of equations, unknown approximation coefficients are obtained. If partial approximation
functions of nodes, where nodal quantity is unknown, are elected as
weighting functions
the method of weighted residuals comes to
Galerkin method.
For n^{th} finite element, the final matrix equation is of the form
where

${\mathbf{T}}_{t}^{\left(n\right)}{\mathbf{E}}_{t}^{\left(n\right)}={\displaystyle \sum _{i,j}\left\{{e}_{t,ij}^{\left(n\right)}{\displaystyle \underset{{S}^{\left(n\right)}}{\iint}\left[{\mathbf{N}}_{t,rs}^{\left(n\right)}\cdot {\mathbf{N}}_{t,ij}^{\left(n\right)}\right]}dS\right\}}$,

( 3.2A.15a )


${\mathbf{G}}^{\left(n\right)}{\mathbf{E}}_{z}^{\left(n\right)}={\displaystyle \sum _{m}\left\{{e}_{z,m}^{\left(n\right)}{\displaystyle \underset{{S}^{\left(n\right)}}{\iint}\left[{\mathbf{N}}_{t,rs}^{\left(n\right)}\cdot \left({\nabla}_{t}{N}_{z,m}^{\left(n\right)}\right)\right]dS}\right\}}$,

( 3.2A.15b )


${\mathbf{S}}_{z}^{\left(n\right)}{\mathbf{E}}_{z}^{\left(n\right)}={\displaystyle \sum _{m}\left\{{e}_{z,m}^{\left(n\right)}{\displaystyle \underset{{S}^{\left(n\right)}}{\iint}\left[\left({\nabla}_{t}{N}_{z,q}^{\left(n\right)}\right)\cdot \left({\nabla}_{t}{N}_{z,m}^{\left(n\right)}\right)\right]}\text{\hspace{0.05em}}dS\right\}}$,

( 3.2A.15c )


${\mathbf{T}}_{z}^{\left(n\right)}{\mathbf{E}}_{z}^{\left(n\right)}={\displaystyle \sum _{m}\left\{{e}_{z,m}^{\left(n\right)}{\displaystyle \underset{{S}^{\left(n\right)}}{\iint}\left[{N}_{z,q}^{\left(n\right)}{N}_{z,m}^{\left(n\right)}\right]}\text{\hspace{0.05em}}dS\right\}}$,

( 3.2A.15d )


${\mathbf{S}}_{t}^{\left(n\right)}{\mathbf{E}}_{t}^{\left(n\right)}={\displaystyle \sum _{i,j}\left\{{e}_{t,ij}^{\left(n\right)}{\displaystyle \underset{{S}^{\left(n\right)}}{\iint}\left[\left({\nabla}_{t}\times {\mathbf{N}}_{t,rs}^{\left(n\right)}\right)\cdot \left({\nabla}_{t}\times {\mathbf{N}}_{t,ij}^{\left(n\right)}\right)\right]}dS\right\}}$.

( 3.2A.15e )

In the above relations, E_{t}^{(n)} is unknown column vector of edge approximation
coefficients (approximation of transversal components of electricfield intensity) over n^{th} finite element and
E_{z}^{(n)} denotes column vector of three unknown nodal approximation coefficients
(approximation of longitudinal component of electricfield intensity) over n^{th} finite element. Next,
γ denotes complex
propagation constant,
k_{0} is
wave number
in vacuum, μ_{r}^{(n)} is relative permeability of n^{th} finite element and
ε^{~}_{r}^{(n)} is complex relative permittivity of the same element. Symbol dS denotes an
elementary facet for the integration over n^{th} finite element and symbol S^{(n)} give the total surface of
n^{th}
finite element.
Summation including index m symbolizes addition over all nodes of a finite element (i.e., m = 0, 1, 2) a summation including indexes
i, j symbolized addition over all edged of an element (i.e., i, j = 01, 12, 20). Symbols
e_{t, ij}^{(n)} are edge approximation coefficients, symbols e_{z, m}^{(n)}
are nodal approximation coefficients.
Matrices T_{t}^{(n)}, G^{(n)}, S_{z}^{(n)},
T_{z}^{(n)} and S_{t}^{(n)} matrices of coefficients of n^{th}
finite element
of the size 3 x 3. Elements of the above matrices are computed by integration of the product of partial approximation functions and
weighting functions
(or their derivatives) over n^{th} finite element. Matrices can be evaluated using the following relations

${\mathbf{S}}_{t}^{\left(n\right)}=\frac{1}{{A}^{\left(n\right)}}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$,

( 3.2A.16a )


${\mathbf{T}}_{t}^{\left(n\right)}=\frac{1}{12}{\displaystyle \sum _{i=0}^{2}{\mathbf{Q}}_{i}cotg\left[{\theta}_{i}^{\left(n\right)}\right]}$,

( 3.2A.16b )


${\mathbf{G}}^{\left(n\right)}=\frac{1}{6}{\displaystyle \sum _{i=0}^{2}{\mathbf{C}}_{i}cotg\left[{\theta}_{i}^{\left(n\right)}\right]}$,

( 3.2A.16c )


${\mathbf{S}}_{z}^{\left(n\right)}=\frac{1}{2}{\displaystyle \sum _{i=0}^{2}{\mathbf{D}}_{i}cotg\left[{\theta}_{i}^{\left(n\right)}\right]}$,

( 3.2A.16d )


${\mathbf{T}}_{z}^{\left(n\right)}=\frac{{A}^{\left(n\right)}}{12}\left[\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right]$,

( 3.2A.16e )

where
A^{(n)} is surface of n^{th} finite element and θ_{i}^{(n)}
is angle at i^{th} vertex of n^{th} finite element. Given relations are valid for the following organization of nodes and edges

${\mathbf{E}}^{\left(n\right)}={\left[\begin{array}{cccccc}{e}_{z\mathrm{,0}}^{\left(n\right)}& {e}_{z\mathrm{,1}}^{\left(n\right)}& {e}_{z\mathrm{,2}}^{\left(n\right)}& {e}_{t\mathrm{,12}}^{\left(n\right)}& {e}_{t\mathrm{,20}}^{\left(n\right)}& {e}_{t\mathrm{,01}}^{\left(n\right)}\end{array}\right]}^{T}$.

( 3.2A.16f )

If the matrix equation is solved out for the vector of unknown approximation coefficients, then solution of the problem is obtained. Substituting
approximation coefficients to the formal approximation, a real approximation of a sought function in each point of n^{th} finite
element is obtained. Joining approximations over all finite elements, the global solution is found.
In the layer C, the reader can find a matlab program for the finiteelement analysis of a shielded microstrip
transmission line. Prom the programmer's point of view, the program is described in the layer D.
