# Using Inhomogeneous Planar Layers as Impedance Matchers between Two Different Mediums

Authors: Mohammad Khalaj-Amirhosseini
Journal: International Journal of Microwave Science and Technology
 (2008-02-18)

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## Abstract

A method is proposed to synthesize optimally inhomogeneous planar layers (IPLs) as impedance matchers between two mediums. First, the electric permittivity function of the IPL is expanded in a truncated Fourier series. Then, the optimum values of the coefficients of the series are obtained through an optimization approach. The usefulness of the proposed method is verified using some examples.


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## Body

## 1. Introduction

Impedance matching is a very
important concept in microwave and antenna engineering.  For example, in ground
penetrating radars (GPRs), we need matching between the air and the soil (the
earth) [1, 2]. Usually the impedance matching is made into the structure of
antennas. On the other hand, inhomogeneous planar layers (IPLs) are widely used
in microwave and antenna engineering [3–5]. In this
paper, we propose utilizing IPLs as impedance matchers between two mediums and
introduce a method to synthesize them optimally in a desired frequency range.
In fact, the IPLs are located between a free space designed antenna and the
earth or walls. To optimally design IPLs, the electric permittivity function of
them is expanded in a truncated Fourier series, first. Then, the optimum values
of the coefficients of the series are obtained through an optimization
approach. The usefulness of the proposed method is verified using some
examples.


## 2. Analysis of IPLs

In this section, the frequency
domain equations of the IPLs are reviewed. Figure1 shows a typical IPL with
thickness d, whose left medium is the free space and whose right
(target) medium has the conductivity and electric permittivity σ2 and εr2, respectively. Also,
the electric permittivity and the magnetic permeability functions of IPL are εr(z) and 1 (for easy
implementation). One way to fabricate the IPLs is to place several thin
homogeneous dielectric layers beside each other. We are interested to utilize
IPLs as impedance matchers between their right and left mediums, that is,
between the target medium and free space. Two different polarizations are
possible, one is the TM and the other is the TE.  It is assumed that the incident plane
wave propagates obliquely towards positive x and z directions
with an angle of incidence θi and electric filed
strength Ei. Of course, we know that the wave radiated by an
antenna can be decomposed to many plane waves.A typical IPL as an
impedance matcher between two mediums: (a) TE polarization mode; (b) TM polarization mode.
(a)(b)The differential equations
describing IPLs have nonconstant coefficients and so except for a few special
cases no analytical solution exists for them. There are some methods to analyze
the IPLs such as finite difference [6], Taylor ’s
series expansion [7], Fourier series expansion [8], the equivalent sources method
[9], and the method of moments [10]. Of course, the most straightforward method
is subdividing IPLs into K uniform electrically short segments with
length
(1)Δz=dK≪λmin≅cfmaxεr2,
in which c is the velocity
of the light and fmax is the maximum frequency of the
analysis. The ABCD parameters [11] of the IPL are obtained from those of
the segments as follows:
(2)[ABCD]=[A1B1C1D1]…[AkBkCkDk]…[AKBKCKDK],
where the ABCD parameters of
the kth segment are as follows:
(3)Ak=Dk=cos(Δθk),(4)Bk=Zc2((k−0.5)Δz,θi)Ck=jZc((k−0.5)Δz,θi)sin(Δθk).
In (4) and (5),
(5)Δθk=2πfcεr((k−0.5)Δz)−sin2(θi)Δz
is the electrical length of the kth
segment and Zc(z,θi) is the characteristic impedance
of the IPL, defined as the ratio of the transverse electric field to the
transverse magnetic field given by
(6)Zc(z,θi)={μ0ε01εr(z)−sin2(θi),TE,μ0ε01εr(z)εr(z)−sin2(θi),TM.
Finally, the input impedance and
reflection coefficient of the matcher are determined as follows:(7)Zin(f,θi)=AZL(f,θi)+BCZL(f,θi)+D,Γin(f,θi)=Zin(f,θi)−ZS(θi)Zin(f,θi)+ZS(θi),where
(8)ZL(f,θi)={jωμ0σ2+jωε0εr21cos(θi),TE,jωμ0σ2+jωε0εr2cos(θi),TM,ZS(θi)={μ0ε01cos(θi),TE,μ0ε0cos(θi),TM,
are the equivalent load and source
impedances, respectively.


## 3. Synthesis of IPLs

In this section, a general method
is proposed to optimally design the IPLs as impedance matchers. First, we
consider the following truncated Fourier series expansion for the electric permittivity
function(9)ln(εr(z)−1)=C0+∑n=1N(Cncos(2πnzd)+Snsin(2πnzd))+Ezd.
The latter term has been considered
to prepare the possibility of having unequal values at z=0 and z=d. Also, the reason to use logarithm function at the left of (9) is
to keep εr(z)≥1.
Moreover, utilizing truncated Fourier series expansion (9) does not create any
discontinuity in the resultant IPL. An optimum-designed matcher has to have the
input reflection coefficient as small as possible in a desired frequency range.
Therefore, the optimum values of the coefficients Cn, Sn, and E in (9) can be obtained through minimizing the following
error function corresponding to M frequencies f1<f2<…<fM inside the desired matching bandwidth
and at zero angle of incidence.
(10)Error=1M∑m=1M|Γin(fm,0)|2.Moreover, the defined error
function should be restricted by some constraints such as having a minimum
matching at the largest desired angle of incidence, θi,max, as the following:
(11)|Γin(fm,θi,max)|≤ρmax,∀m=1,2,…,M.
It is noticeable that in contrary
to condition (11), the error function (10) is independent of the wave
polarization because TE and TM polarizations are identical for the zero angle
of incidence. Also, one can consider the following truncated Fourier series
expansion for the electric permittivity function, instead of (9),
(12)ln(εr(z)−1)=∑n=0NCn′cos(πnzd).


## 4. Examples and Results

We would like to design an IPL with thicknessd=10 cm (a practical and feasible chosen) as an impedance matcher in a frequency
range 2.0 to 4.0 GHz (an octave bandwidth), considering θi,max=60° and ρmax=0.316=−10 dB. We
consider two cases for the target medium; dry soil (εr2=4 and σ2=0.001) and moist soil
(εr2=10 and σ2=0.01). Using the
proposed optimization approach, considering N=5 spatial harmonics and M=21 frequencies with equal distance of 100 MHz, four IPLs were synthesized.
The unknown coefficients of the truncated Fourier series related to the
synthesized IPLs are written in Table 1. Figures 2 and 3 illustrate the
electric permittivity function εr(z) to match to
either the dry or moist soils, respectively, assuming TE or TM polarizations.
Figures 4 to 7 illustrate the magnitude of the input reflection coefficient  |Γin(f)| corresponding
to the dry or moist soils illuminated by TE or TM polarizations waves. It is observed
that the designed IPLs yield a good impedance matching in the desired frequency
band and angular range. Meanwhile, the reflection coefficient degrades with
increasing the angle of incidence. The variation of the permittivity of
designed IPLs in TM mode is slightly more than that in TE mode. Also, the
resulted matching efficiency for the moist soil is less than that for the dry
soil. To show the effect of the thickness of IPLs, we reduce d from 10 cm to 7 cm. The unknown coefficients of the truncated Fourier series, the
electric permittivity function εr(z) and the
magnitude of the input reflection coefficient |Γin(f)| corresponding
to the dry soil with d=7 cm are shown in Table 2 and Figures 8–10, respectively.
Comparing Figures 9, 10 with Figures 4, 5 results that as the thickness of the
IPL is chosen larger, its matching efficiency can be increased. Also, comparing
Figure 8 with Figure 2 results that as the thickness of the IPL is chosen
smaller, the necessary variations of the permittivity function is increased and
consequently its implementation becomes more difficult.Table 1
The unknown coefficients of the truncated Fourier
series related to the synthesized IPLs withd=10 cm.
C0C1C2C3C4C5S1S2S3S4S5EDry/TE−2.2846−0.4614−0.02790.26420.73250.3623−1.4137−0.9213−0.7610−0.40060.41352.0057Dry/TM−2.3059−0.5146−0.05640.22130.70820.4512−1.4236−0.9258−0.7737−0.49230.41742.0643Moist/TE−3.0211−0.13100.06850.16391.08530.5526−1.7717−1.0275−0.9858−0.82110.78962.8266Moist/TM−5.00000.12600.0578−0.75450.15641.8839−2.2487−0.5031−0.5168−2.41560.66784.4385Table 2
The unknown coefficients of the truncated Fourier
series related to the synthesized IPLs withd=7 cm.
C0C1C2C3C4C5S1S2S3S4S5EDry/TE−1.6962−0.87800.28441.90030.33550.6701−2.5874−2.2971−1.00160.59890.45480.0789Dry/TM−1.5613−0.37150.20271.36881.1148−0.4685−2.9285−1.6279−1.36311.2224−0.4454−0.3434Figure 2
The electric
permittivity functionεr(z) to match to
the dry soil with d= 10 cm.Figure 3
The electric
permittivity functionεr(z) to match to
the moist soil with d= 10 cm.Figure 4
The magnitude of|Γin(f)| corresponding
to the dry soil, d= 10 cm and TE polarization.Figure 5
The magnitude of|Γin(f)| corresponding
to the dry soil, d= 10 cm and TM polarization.Figure 6
The magnitude of|Γin(f)| corresponding
to the moist soil, d= 10 cm and TE polarization.Figure 7
The magnitude of|Γin(f)| corresponding
to the moist soil, d= 10 cm and TM polarization.Figure 8
The electric
permittivity functionεr(z) to match to
the dry soil with d= 7 cm.Figure 9
The magnitude of|Γin(f)| corresponding
to the dry soil, d= 7 cm and TE polarization.Figure 10
The magnitude of|Γin(f)| corresponding
to the dry soil, d= 7 cm and TM polarization.


## 5. Conclusion

Inhomogeneous planar layers (IPLs) were utilized as
impedance matchers between two mediums and a method was introduced to optimally
synthesize them in a desired frequency band. First, the electric permittivity
function of the IPL is expanded in a truncated Fourier series. Then, the
optimum values of the coefficients of the series are obtained through an
optimization approach. The usefulness of the proposed method is verified using
some examples.  It is observed that the solutions yield a good impedance
matching and thicker matching layer may have higher efficiency and easier
fabrication. The proposed method can be extended for IPLs, whose magnetic
permeability is inhomogeneous solely or along with their electric permittivity.
Also, we can consider IPLs for spherical
wave fronts instead of planar ones in the future.



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*Source: 869720-2008-04-07.xml*