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