Section II: Theoretical Analysis
Design of Tri-Band DGS BSF filter
To demonstrate the proposed
technique, a semi-H shaped DGS with three different defects in the
ground plane with all the marked parameters, Fig. 1, is considered. Its
top plane with marked parameters is depicted in Fig. 2.
Fig.1-> Defected
Ground Plane with 3 different sizes, which represent 3 different
frequencies
Fig.2-> Top plane
with a 50 Ω Microstrip Line(MSL)
Table
I: Parameters for DGS Filter Design
This architecture effectively
translates to three distinct band-stop frequencies for the filter. A
thorough simulation and optimization in CST environment is carried out
and the resulting parameters are given in Table I. It is imperative to
note that the three chosen frequencies in this study are 433 MHz, 700
MHz, and 915 MHz and the resonance in EM simulation is optimized
accordingly. Furthermore, three SMD capacitors need to be loaded in the
respective excitation slots for synthesis of resonances at the chosen
frequencies as well as for improvements of the overall quality factor
and their values are also included in Table I.
π type Equivalent Circuit Model
In order to understand the working of the DGS based filter, firstly a
single band π-type circuit model is developed, using an earlier
approach[7], as shown in Fig. 3. The inductor and capacitor in
parallel represent the resonant frequency of the circuit.In this work,
the resistors are introduced to model dielectric losses. The parallel RC
circuit models the fringing fields at the discontinuity between the
transmission line and DGS. An earlier technique requires all the ABCD
parameters to calculate the resistances, capacitances, and inductance of
this equivalent circuit, whereas we propose a simpler approach in which
a less rigorous computation enables the derivation of all the relevant
parameters of the equivalent circuit. For example, first, the CST
simulation and optimization determines the value of SMD capacitor for
any specified resonant frequency. This can then be used for the
computation of the inductance and resistance of the equivalent circuit
using (1). Furthermore, the calculation of the parallel RC modules can
be done from just two of the ABCD parameters(2)-(3).
Fig.3-> Pi-Type
Circuit model for single band system
R=Real(Z11) ,\(L=\frac{1}{4*\pi^{2}*f^{2}*C_{\text{SMD}}}\) (1)
\(A=\frac{[(1+S_{11})*(1-S_{11})]\ +\ {S_{21}}^{2}}{2*S_{21}}\)(2)
\(B=\frac{[(1+S_{11})*(1+S_{11})\ -{\ S_{21}}^{2}}{2*S_{21}}\)(3)
\(Y_{b}=\frac{A-1}{B}\) ,\(C_{p}=\frac{\text{Imaginary}(Y_{b})}{2*\pi*f}\),\(R_{p}=\frac{1}{\text{Real}(Y_{b})}\) (4)
Fig.4 -> EM and
Circuit Simulation at 915 MHz
Table
II: Parameters for Single Band Circuit Model
The efficacy of the equivalent
circuit is evaluated by simulating it at 915 MHz in the Keysight ADS
environment. A good agreement between the circuit and EM simulation
results, depicted in Fig. 4, is obtained for the optimized equivalent
circuit parameters given in Table II. Here, the parameters for the
single band EM simulation is taken from Table I. In brief, a good
agreement provides the requisite confidence that this circuit model can
be readily used for the development of multi-band equivalent circuit. An
important point to note is that the capacitance affects the resonant
frequency more significantly than inductance or the size of the DGS.
Therefore, keeping that in perspective, the resonant frequency is
controlled with the help of external SMD during design stage while the
inductance plays the role of secondary controller of the resonant
frequency.
Multi band π type
Equivalent Circuit Model
Now, to utilize the single band
concept in the design of multi-band circuit, a flowchart, depicted in
Fig. 5, is developed. In essence, the single-band circuit model is
cascaded multiple times to meet the requirements. For demonstration of
the proposed concept, a tri-band band-stop filter is developed as shown
in Fig. 6 to obtain three distinct band-stop frequency responses at the
frequencies of 433 MHz, 700 MHz, and 915 MHz. The simulations were
carried out at both the structure and circuit levels and the resulting
optimized circuit level parameters are given in Table III. Optimization
of the structure is done to ensure that the three cascaded sections
largely work independently of each other and this situation is observed
in the respective surface current distributions. The apparent excellent
agreement between the EM and circuit simulation results, in Fig. 7,
provide a solid ground for confidence in the developed circuit model.
Fig.5 -> Flow chart
depicting the multi-band stop band filter design process
Fig.6-> Cascaded
Tri-Band Filter using \(\pi\) type circuit model
Fig.7-> Tri band EM
and Circuit Simulation Results.
Table III:Parameters for Sections
2 and 3 of \(\pi\) type model