A Simple Model for a DGS Microstrip Line with Stepped Impedance SlotLines
 Author: Woo DukJae, Lee TaekKyung, Nam Sangwook
 Publish: Journal of electromagnetic engineering and science Volume 15, Issue1, p26~30, 00 Jan 2015

ABSTRACT
In this paper, a simple equivalent circuit model for a defected ground structure (DGS) microstrip line with stepped impedance slotlines in the ground plane is presented. In addition, an analytic expression for the resonance frequency of the proposed structure is derived. In equivalent circuit modeling, the capacitance and the inductance of the resonance circuit are evaluated from the dimensions of the etched pattern in the ground plane. The resonance frequencies calculated from the proposed method are compared with those obtained with an electromagnetic (EM) simulation.

KEYWORD
DGS , Equivalent Circuit Model , Stepped Impedance SlotLine

I. INTRODUCTION
The planar transmission line with a spiralshaped defect in the ground plane is one of the most popular slotshaped defected ground structures (DGSs) [1]. Based on this structure, several modified slotshaped DGSs have been proposed to comply with required performances [28]. In terms of circuit modeling, many other researchers have presented equivalent circuits for various slotshaped DGSs. Simple lumped elements circuit models [8,9] and geometric models based on transmission lines were proposed in [1013]. These efforts have provided improved physical insight into the operation principle of the DGS.
The stepped impedance resonator (SIR) is used in the filter design in order to push the spurious passband to a higher frequency range [14,15] and to reduce the circuit size [16]. In addition, this resonator has become very popular in the design of dualband filters since the dual passband behavior can easily control the second passband [17].
In this paper, we describe the dualband property of the microstrip line with a stepped impedance slotline DGS in the ground plane. This paper also proposes an equivalent circuit model that provides insight into the coupling mechanism between the microstrip line and the stepped impedance slotline, as well as a technique for obtaining analytic expression of the resonance frequencies.
II. CIRCUIT MODEL AND RESONANCE PROPERTIES
The configurations of the proposed stepped impedance slotline DGS on the ground plane of the microstrip line are shown in Fig. 1, where two shortcircuited stepped impedance slotlines with different characteristic impedances Z_{1} and Z_{2} and electrical lengths
θ _{1}(=β _{1}l _{1}) andθ _{2}(β _{2}l _{2}) are connected by a narrow etched gap. In the DGS, the narrow etched gap can be modeled as a quasistatic capacitance [18]. The transmission line model for the etched pattern on the ground plane is shown in Fig. 2(a), in which two stepped impedance slotlines are shortended and are connected in parallel with a gap capacitance of 2C_{s} . In order for the etched pattern in the ground plane to operate as a resonant circuit, the input impedance Z_{in}(=jX_{in} ) of the stepped impedance slotline must have an inductive reactance asThe transmission line model of the stepped impedance slotlines is replaced by an equivalent circuit as depicted Fig. 2(b), where the equivalent inductance is
Here,
c is the speed of light, andε_{e1} andε_{e2} represent the effective permittivity of the first and the second slotlines, respectively.Based on the equivalent resonance model of the etched pattern on the ground plane, the equivalent circuit model of the microstrip line with stepped impedance slotline DGS is shown in Fig. 2(c) [13]. The
L and theC are the inductance and the capacitance of the microstrip line corresponding to the length occupied by the DGS. The DGS on the ground plane is modeled as a parallel resonant circuit with inductanceL_{s}(ω) and capacitanceC_{s} that is coupled to the microstrip line through mutual inductance,L_{m} .In the design of a DGS, it is important to find an analytic expression for the resonance frequency, which can be directly derived from the dimensions of the etched pattern. From Fig. 2(c), the resonance angular frequency is
By substituting Eq. (2) for
L_{s} (ω _{0}) in Eq. (3), we can finally obtain the following expressionThe closedform expression for the effective permittivity and the characteristic impedance of the slotline were reported in [19]. The narrow etched gap on the ground plane can be modeled as a microstrip gap [18], and a closedform expression for microstrip gap capacitance 2
C_{s} can be obtained from [11]. With the help of MATLAB, we can calculate the fundamental resonance frequency and the spurious resonance frequencies from Eq. (4).For the proposed structure, it would be of interest to see how the fundamental resonance frequency (
f_{r} ) and the first spurious resonance frequency (f_{s} _{1}) change as the slot width ratio (ω _{2}/ω _{1}) and the slot length ratio (l _{2}/l _{1}) are modified. To simplify the proposed structure, we chose the slotline lengthsl _{1} =l _{2} = 12 mm.For the dimensions
l _{1} =l _{2} =12 mm and g = 0.3 mm, the fundamental resonance frequencies, the first spurious resonance frequencies, and the normalized first spurious resonance frequencies (η =f_{s} _{1}/f_{r} ) of the proposed structure are calculated for the changes in the slotline width, and are summarized in Table 1. In the design, a circuit board RO3010 with a dielectric constant of 10.2, copper thickness of 0.016 mm, and substrate thickness of 1.27 mm is used. The characteristic impedance of the microstrip line is designed to be 50 Ω (ω = 1.2 mm). When the width of the first slotline is fixed asω _{1} = 0.3 mm, the normalized first spurious resonance frequencyη increases as the width of the second slotline (ω _{2}) grows. In addition, for the fixed width of the second slotline withω _{2} = 0.3 mm, it is confirmed that the calculated normalized first spurious resonance frequencyη decreases as the width of the first slotline grows. From the results in Table 1, it becomes evident that the normalized first spurious resonance frequencies can be controlled by the width ratio (slot impedance ratio), and this is the special feature of the SIR [17].We compared the predicted results of the proposed model with the EM simulated results. The simulation was performed using EM software HFSS version 11 (ANSYS Inc., Canonsburg, PA, USA). The resonance frequencies and the first spurious resonance frequencies from the proposed model and those from EM simulation are plotted in Fig. 3 as the slot width ratio changes. In the proposed circuit model, we ignored the influence of the step and short discontinuities in the slotline, since no analytical results are available for various slotline discontinuities. The discrepancies may be attributed to these factors. However, the predicted results of the proposed model agree with the EM simulated results.
Fig. 4 illustrates the comparative
S parameters from the EM simulation (HFSS) and the measurement of a fabricated DGS with the dimensionsl _{1} =l _{2} = 9 mm,ω _{1} = 0.9 mm,ω _{2} = 0.3 mm, andg = 0.3 mm (Fig. 5). The characteristic impedance of the microstrip line is designed to be 50 Ω (ω = 1.2 mm). The normalized first spurious resonance frequencies obtained from the EM simulation and the measurement are 2.32 and 2.23, respectively.III. CONCLUSION
This paper has presented an analytical expression for the resonance frequencies and the equivalent circuit model of a DGS with stepped impedance slotlines in the ground plane of the microstrip line. The theoretical prediction was in reasonable quantitative agreement with the EM simulated resonance property.

[Fig. 1.] Configurations of the microstrip line with a stepped impedance slotline defected ground structure (DGS) in the ground plane. (a) ω1 < ω2. (b) ω1 > ω2.

[Fig. 2.] (a) Shortended slotline model and (b) equivalent inductance model for the etched pattern in the ground plane. (c) Lumped element equivalent circuit model for a microstrip line with stepped impedance slotlines.

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[Table 1.] Calculated fundamental resonance frequencies, first spurious resonance frequencies, and normalized first spurious resonance frequencies for various slot widths (l1 = l2 = 12 mm)

[Fig. 3.] Comparative resonance frequencies between EM simulation and calculation. (a) ω1 is fixed at 0.3 mm. (b) ω2 is fixed at 0.3 mm.

[Fig. 4.] Sparameters from the EM simulation and the measurement.

[Fig. 5.] View of the bottom of the fabricated defected ground structure.