3.2 Shielded microstrip tranmission linesAdvanced theory
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Fig. 3.2B.1 | Shielded microstrip transmission line (longitudinally homogenous) |
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In this paragraph,
we turn our attention to the computational details of the analysis of electromagnetic field distribution in a shielded
microstrip transmission line
(fig. 3.2B.1) assuming constant parameters along the longitudinal axis. Then, only a two-dimensional structure has to be analyzed
(cross-section of the transmission line), which simplifies computations [20].
For the analysis, a
full-wave method,
a
finite-element method
is exploited.
As already described in layer A, the analysis is based on Maxwell equations in differential form. Sources
of electromagnetic waves are assumed to be in long distance from the region of analysis (then, imposed currents Js are zero in this region).
Charge density r in dielectrics, which surround the microstrip, is supposed to be zero. Dielectrics is expected to be isotropic and linear
(permittivity and permeability are scalar quantities, which values do not depend on the value of respective intensity) and to exhibit electric losses (represented
by electric conductivity s). All the metallic parts (shielding waveguide, microstrip) are assumed to be perfect electric conductors.
The analyzed
microstrip transmission line
is placed to
Cartesian coordinate system
(coordinates x and y in transversal directions, coordinate z in longitudinal one). Then, the wave can be said to propagate in the longitudinal direction
z (along the microstrip), and the electric-field intensity vector depends on the longitudinal coordinate by the following way:
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( 3.2B.1 )
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Here, γ is the
propagation constant.
Expressing all vectors as a sum of the transversal vector (index t) and the longitudinal one (index z), we get
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( 3.2B.2a )
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( 3.2B.2b )
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Whereas (3.2B.2a) is vector equation for transversal components, (3.2B.2b) is scalar equation for
longitudinal components. In those relations, ∇t is a transversal operator nabla, Et
is transversal electric-field intensity vector, γ is propagation constant, Ez denotes longitudinal component of electric-field
intensity, k0 is
wave number
in vacuum, μr denotes relative permeability inside structure, ε~r
is complex relative permittivity inside structure and z0 denotes unitary vector in the longitudinal direction.
The set of differential equations (3.2B.2) has to be completed by boundary conditions, which have to be met by the solution
of (3.2B.2)
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(3.2B.3a )
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( 3.2B.3b )
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Eqns. (3.2B.2) completed by
boundary conditions
(3.2B.3) are initial relations for the full-wave analysis of the shielded
microstrip transmission line.
Unfortunately, the set (3.2B.2), (3.2B.3) includes the first Maxwell equation and the second one only.
In order to meet the third Maxwell equation and the fourth one, the analysis has to be based on
hybrid finite elements.
If the shielded microstrip transmission line is analyzed exploiting
hybrid finite elements,
all components of electric-field intensity vector or all components of magnetic-field one have to be included in computations. Eqn.
(3.2B.2) stays the initial relation of the analysis.
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Fig. 3.2B.2 | Mesh examples of rectangular bi-elements for the analysis of a shielded microstrip transmission line |
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The matter of
hybrid finite elements
consists in modeling the longitudinal component of field intensity exploiting
nodal approximation
and in modeling transversal components using
edge vectors.
The first step of the
finite-element method
consists in dividing the analyzed structure (cross-section of the transmission line) to
finite elements
(non-overlapping sub-regions, which contain all the points of the analyzed structure). In the area of a finite element, parameters of the analyzed
structure (permittivity, permeability, conductivity) have to be constant. There are no restrictions to shape and dimensions of finite elements.
Examples of finite-element meshes are depicted in fig. 3.2B.2.
In the second step of the solution, the distribution of a computed quantity is approximated over each
finite element
in a formal way. The approximation is expressed as a linear combination of elected partial approximation functions and unknown approximation coefficients.
Analyzing the shielded microstrip transmission line, a formal approximation of a scalar function Ez =
Ez(x, y) and a vector one Et = Et(x, y)
have to be expressed. Let us start with the scalar function.
The global approximation of the scalar function Ez over the whole cross-section of the transmission line is composed of local
approximations over single
finite elements.
Local approximation of the longitudinal component of electric-field intensity vector over a
finite element
is expressed as a
linear combination
of elected partial approximation functions and unknown approximation coefficients. Considering linear approximation, the approximation plane
over a finite element is composed of three partial approximation planes. Each partial approximation plane is unitary in a unique vertex of the
triangle and is zero in the other two vertexes (see fig. 3.2B.3). Coefficients cn at partial functions
in the linear combination play the role of spatial samples of the computed function in vertexes of the finite element (fig. 3.2B.3).
Vertexes of the finite element are called
nodes
and respective functional values are called
nodal values.
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Fig. 3.2B.3 | Linear approximation of E over finite element. Composed of three shape functions |
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Partial approximation functions are called shape functions. All the
shape functions,
which are unitary in the same node (see fig. 3.2B.4), compose together a
basis function.
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Fig. 3.2B.4 | Linear basis function related to m-th node |
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In many cases, approximation functions of higher order are more suitable than the linear function. Although the triangular element has to contain
more nodes (6 for quadratic approximation, 10 for cubic one, etc.), the same error comparing to linear approximation is reached even if significantly
lower number of
finite elements
is exploited. Approximation functions of higher order are smoother, and therefore, they represent better natural quantities.
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Fig. 3.2B.5 | Two-dimensional simplex coordinates |
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Now, we are familiar with
shape functions,
and therefore, we can find their proper mathematical representation. For this purpose,
Lagrange polynomials
expressed in
simplex coordinates
[21] are usually used.
What does the term simplex coordinates mean? Considering triangular
finite elements,
simplex coordinate axes are of the direction of heights of the triangle. Simplex coordinates are unitary in the vertex and are zero on the opposite edge.
Simplex coordinate do not depend neither on the shape nor on the dimensions of the finite element,
and therefore, all the computations are sufficient to be performed once for a single finite element, and the results are recomputed for the other elements only.
Dealing with physical matter of simplex coordinates, a general point P inside a triangular finite element divides
its surface to three partial triangles (fig. 3.2B.6). The ratio of the surface of a triangle, which is positioned in front of
the first node, to the surface of the whole finite element equals to the simplex coordinate of P on the first simplex axis
For other simplex coordinate axes, the situation is similar. In eqn. (3.2B.4), σ(S1)
denotes surface of the partial triangle, which is positioned in front of the first node, and
σ(S) is surface of the whole finite element. Obviously, addition of all three simplex coordinates in an arbitrary point is unitary
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Fig. 3.2B.6 | Matter of simplex coordinates |
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As shown in [21], this conclusion can be generalized for an arbitrary dimension
and for an arbitrary order of an approximation polynomial.
Now, we turn our attention to
Lagrange interpolation polynomials.
Lagrange polynomial of nthcan be expressed (using simplex coordinate x) as
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( 3.2B.6 )
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Here, n is order of the approximation polynomial. Eqn. (3.2B.6) describes the whole family of polynomials: family
members differ in the index m, which can vary from zero to the order of polynomial n.
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Fig. 3.2B.7 | Family of Lagrange polynomials of 2nd order |
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Nulls of polynomials Rm(n) are equidistantly placed on coordinates
ξ= 0, 1/n till (m-1)/n, the polynomial is of unitary value in ξ = m/n. Hence, Rm(n)
is of m equidistantly placed nulls at the left from the coordinate ξ = m/n and of zero nulls at the right.
The above-given statement is illustrated by fig. 3.2B.7, where all the members of the family of quadratic polynomials
R(2) are depicted. Figure demonstrates the above-described equidistant distribution of nulls. The family member of index 0, i.e.
R0(2), does not have any null at the left from the coordinate 0 and is of unitary value at the coordinate 0. The family member of
index 1, i.e. R1(2), is of single null at the coordinate 0 and is unitary at 1/2. Finally, the family member of index 2, i.e.
R2(2), is of nulls at coordinates 0 and 1/2 and is unitary at the coordinate 1.
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Fig. 3.2B.8 | One-dimensional finite element, its simplex coordinates and indexes of Lagrange polynomials |
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Using
Lagrange polynomials,
we compose quadratic shape functions for one-dimensional finite element.
Simplex coordinate
ξ1 is oriented from left to right on this element, the coordinate ξ2 goes from the right to the left
(fig. 3.2B.8). The shape function related to the node 1 (unitary value in the node 1, zero value in the nodes 2 and 3) is
then composed by multiplying Lagrange polynomial of variable ξ1 and index 0 (constant function of value 1) by Lagrange polynomial
of variable ξ2 and index 2 (since the coordinate ξ2 is oriented fro the right to the left, the course of
the function R2(2) from fig. 3.2B.7 has to be reverted).
Similarly, shape functions for nodes 2 and 3 can be composed. For the node 2,
Lagrange polynomials
of variables ξ1 and ξ2and of index 1 are mutually multiplied. For the node 3, Lagrange polynomial of variable
ξ1 and index 2 is multiplied by Lagrange polynomial of variable ξ2 and index 0. Indexes of Lagrange polynomials,
which form shape functions of respective nodes, are written at these nodes (fig. 3.2B.8) in the form of a fraction; numerator
is an index of Lagrange polynomial of the coordinate ξ1 and denominator is an index of Lagrange polynomial of the coordinate
ξ2. Adding numerator and denominator, order of approximation polynomial n has to be obtained.
In general, the shape function of the node (i, j) of a one-dimensional finite element can be expressed as
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( 3.2B.7a )
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here n is order of an approximation polynomial, R denotes
Lagrange polynomials
defined by eqn. (3.2B.5) and ξ are
simplex coordinates.
In the next step, we turn our attention to a two-dimensional
finite element.
The only change, which has to be done, is adding a new simplex coordinate ξ3 to two existing coordinates
ξ1 and ξ2. Two multiplicands, which appear in relations for
shape functions
of a one-dimensional finite element, are completed by the third multiplicand, corresponding to
Lagrange polynomial
of a new
simplex coordinate
ξ3
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( 3.2B.7b )
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Here, ξ1, ξ2 and ξ3 denote
simplex coordinates
of a two-dimensional
finite element,
n is order of an approximation polynomial and R are Lagrange polynomials.
Substituting to (3.2B.6) a (3.2B.7), we get for linear approximation the following
shape functions
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( 3.2B.8 )
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Now, we are familiar with basis functions for the approximation of a scalar function Ez. Therefore, we can turn our attention
to the approximation of vector function Et. The approximation of a vector function formally corresponds to the approximation
of a scalar function; only basis functions are of vector nature
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( 3.2B.9 )
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Fig. 3.2B.9 | To the explanation of behavior of vector shape function; for simplicity, superscripts (n) are missed |
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In eqn. (3.2B.9) et,01(n) denotes the
edge approximation coefficient
for the approximation of the distribution of transversal components of electric-field intensity vector, which is related to the edge 0-1 of
nth
finite element. Next,
Nt,01(n) denotes vector shape function, which is multiplied by the edge coefficient 0-1 in order
to evaluate submission of this coefficient to the approximation of the distribution of transversal electric-field intensity vector over
nth finite element. Similar situation can be observed at the other two vector shape functions.
The vector
shape function
can be expressed as
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( 3.2B.10 )
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Here, A(n) denotes surface of nth finite element, lk,i(n)
is the length of the edge k-i of nth
finite element,
nk,i(n) is the normal to the edge k-i of nth finite element and
ξi is a classic
simplex coordinate,
which is unitary in node i and which is zero in the opposite edge. The meaning of the rest of symbols is similar.
Let us observe behavior of the shape function (2.3B.10) in node 0. Here, the
simplex coordinate
ξ0 is of unitary value and ξ1 is zero. In node 0, the shape function (2.3B.10)
is perpendicular to the edge 2-0 and is oriented inside the
finite element
(due to the negative sign). Its magnitude equals to the reverse value of the height v20.In node 1, shape function is of the direction
of the normal to the edge 1-2 and its magnitude equals to the reverse value of the height v01. Moving from node 0 to node 1 along
the edge 0-1, the direction of the shape function (2.3B.10) continuously changes from –
n01(n) to +n12(n) and its magnitude changes from the value
(1/v20) to (1/v01). Since the shape function (2.3B.10) does not depend on the coordinate
ξ2, the described behavior is the same along all the parallels of the edge 0-1.
Now, all the components of the electric-field intensity vector are approximated in a formal way. Therefore, the next step of the
finite-element method
can be done: the formal approximation is substituted to the solved equation, and an approximation error (the difference between the approximate solution
and the exact one) is minimized. Performing these steps, we obtain the final matrix equation:
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where
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( 3.2B.11a )
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( 3.2B.11b )
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( 3.2B.11c )
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( 3.2B.11d )
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( 3.2B.11e )
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In the above-given relations, Et(n) is column vector of three unknown
edge approximation coefficient
(related to the approximation of transversal components of electric-field intensity vector) over nth
finite element
and Ez(n) denotes column vector of three unknown
nodal approximation coefficients
(related to the approximation of longitudinal component of electric-field intensity vector) over nth finite element. Next,
γ denotes complex
propagation constant
propagation constant, k0 is
wave number
in vacuum, μr(n) is relative permeability of nth
finite element
and ε~r(n) is complex relative permittivity of the same element. Symbol dS denotes
elementary facet for the integration over nth finite element and S(n) is the total surface of nth
finite element. Summation over the index mrepresents addition over all nodes of the finite element (i.e., m = 0, 1, 2) and summation over
indexes i, j represents addition over all edges of the element (i.e., i, j = 0-1, 1-2, 2-0). Symbols
et, ij(n) represent edge approximation coefficients, symbols ez, m(n)
represent nodal approximation coefficients.
Matrices Tt(n), G(n), Sz(n),
Tz(n) and Tt(n) are matrices of coefficients of
nth
finite element
of the size 3 x 3. Elements of the above-described matrices were computed by the integration of the product of
basic functions
and weighting ones (or their derivatives) over nth
finite element
(in
simplex coordinates,
of course). Those matrices can be evaluated exploiting the following relations:
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( 3.2B.12a )
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( 3.2B.12b )
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( 3.2B.12c )
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( 3.2B.12d )
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( 3.2B.12e )
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where
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A(n) is surface of nth finite element and θi(n) is angle at
ith vertex of nth
finite element.
The above-given relations are valid for the following organization of nodes and edges:
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( 3.2B.12f )
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Solving the matrix equation for the vector of unknown approximation coefficients, the solution of the problem is obtained. Substituting approximation
coefficients to the formal approximation, a real approximation of a sought function in every point of nth
finite element
is obtained. Associating approximations over all finite elements, a global approximation of the solution of the partial differential equation in all
the points of the analyzed structure is found.
In layer C, a matlab program analyzing a shielded microstrip transmission line by the described finite-element
method is introduced. A practical programmer's description is given in the layer D.
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