An Equivalent Circuit Model for a DumbbellShaped DGS Microstrip Line
 Author: Woo DukJae, Lee TaekKyung
 Organization: Department of Electrical Engineering and Computer Science, Seoul National University, Seoul, Korea.; Department of Electrical Engineering and Computer Science, Seoul National University, Seoul, Korea.
 Publish: Journal of electromagnetic engineering and science Volume 14, Issue4, p415~418, Dec 2014

ABSTRACT
This paper presents an equivalent circuit model for a dumbbellshaped defected ground structure (DGS) in a microstrip line. The effects of equivalent circuit elements of a dumbbellshaped DGS and their magnetic coupling to the host transmission line are modeled as a simple lumpedelement circuit. In addition, simple approximate expressions to determine the main circuit parameters for this model are presented. The transfer characteristic calculated by the proposed circuit model is compared with the results of EM simulation and measurement.

KEYWORD
Defected Ground Structure (DGS) , Equivalent Circuit Model , Magnetic Coupling

I. INTRODUCTION
A planar transmission line with a dumbbellshaped defected ground structure (DGS) provides lowpass transmission characteristics with a wide rejection frequency band. Due to the wideband rejection property, DGSs with a dumbbell shape are popularly used in the suppression of undesired harmonics for microwave and millimeter wave circuits and in the design of lowpass filters [1,2]. To date, the analysis of the dumbbellshaped DGS loaded planar transmission lines has almost always involved utilizing commercial electromagnetic fullwave solvers. The simulated
S parameters are converted into the transmission (ABCD) and the impedance (Z) matrices and an equivalent LC resonator circuit model is derived from the matrices [1]. However, in those approaches, there is no direct correlation between the physical dimensions of the structure and the equivalent LC parameters. Furthermore, timeconsuming iterative design procedures are required to accomplish the design goal.In this paper, we propose a new equivalent circuit model for a dumbbellshaped DGS coupled to a transmission line, especially a microstirp line, to provide design rule and physical insight. Also, simple approximate expressions are presented to determine the main circuit parameters for this model.
II. EQUIVALENT CIRCUIT MODEL
Fig. 1 illustrates the geometry of the microstrip line with a dumbbellshaped DGS where two circular etched areas are connected by a narrow etched gap. In the DGS, the electric field is concentrated around the narrow etched gap while the current is confined to the metallic ground plane surrounding the circular etched pattern at resonance.
Hence the DGS in the ground plane can be represented by an LC resonant circuit model as shown in Fig. 2. Since there are two circular etched areas, two
L_{d} are connected parallel to the capacitance of 2C_{d} in the resonant circuit for the DGS.For the dumbbellshaped DGS loaded microstrip line, the equivalent circuit model can be proposed as illustrated in Fig. 3(a). The magnetic flux from the current on the host transmitssion line passes through the wide etched circular area on the ground plane and it causes a magnetic coupling of the DGS to the host line. The
L_{m} represents the mutual inductances coupling the DGS to the host line, and theL_{m_dd} is the mutual inductance between twoL_{d} for etched circular areas. TheL andC are the inductance and capacitance of the microstrip transmission line corresponding to the length occupied by a unit cell of DGS, respectively.To obtain a simplified equivalent circuit of the DGS, the equivalent impedance of the branch between
T andT’ is analyzed by defining mesh currents as shown in Fig. 3(b), in which the mesh currents in the resonant circuit are the same (i_{d1} =i_{d2} ) due to the symmetry. Applying the voltage law around the meshes and solving fori andv , the equivalent impedance betweenT andT’ ,Z_{eq} is given byFrom (1), the complex circuit between
T andT’ can be replaced by a simplified equivalent circuit model as shown in Fig. 3(c). The expressions for the inductance and capacitance of the simplified equivalent circuit are given bywith the resonance angular frequency of At resonance,
L_{d}’ andC_{d}’ are obtained in the simplified formsince
L_{m} =k_{m}(LL_{d}) ^{1/2} andL_{m_dd} =k_{m_dd}L_{d} , wherek_{m} is the magnetic coupling coefficient between the host transmission line and the resonator andk_{m_dd} is the magnetic coupling coefficient between two etched circular areas.III. PARAMETER EXTRACTION FOR EQUIVALENT CIRCUIT
One of the aims in this paper is to find values of parameters of the equivalent circuit model for the dumbbellshaped DGS coupled to the microstrip line. In the patterned ground plane, the current distribution is confined within the metallic periphery of the circular etched pattern, as shown in Fig. 2. Hence the inductance
L_{d} for the left half of the dumbbellshaped DGS can be calculated from the metallic split ring model seen in Fig. 4. As the etched gap is very narrow, the inductance of the split ring may be obtained from the formula for a ring as [3]where
R_{m} =r + d / 2 ,t is the conductor thickness, andμ_{0} is the freespace permeability.In Fig. 2, the capacitance of the DGS on the ground plane consists of the gap capacitance and the surface capacitance. The gap capacitance is calculated by the expression for the parallelplate capacitor of the gap with the correction due to the fringing fields. The surface capacitance is calculated by the analytical expression for the surface capacitance of a metallic split ring [4].
By replacing the air and substrate regions of the microstrip by a homogeneous medium with an effective permittivity
ε_{e} , the capacitance of the split ring for the narrow gap isIn (5) the first two terms are the expressions for the gap capacitance that accounts for the fringing effect [5], the third term is an analytical expression for the surface capacitance, and
ε_{e} is the effective permittivity of the homogeneous medium withε_{e} =ε _{0} (ε_{r} +1) / 2 , whereε_{r} is the dielectric constant of the substrate.In the DGS cell, since the inductance of the signal line is strongly coupled with the inductance of the split ring surrounding the etched area, the coupling coefficient
k_{m} is assumed to be a unity. It is also assumed that thek_{m_dd} is zero because the magnetic coupling between two split rings on the ground plane is negligible.IV. SIMULATION AND EXPERIMENTAL RESULTS
Having extracted circuit parameters using the analytic expressions in the previous section,
L_{d}’ andC_{d}’ of the simplified equivalent circuit are calculated from (3). The circuit parameters are calculated for various radii of the etched circular pattern and the results 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 transmission line was designed to be 50 Ω (w = 1.2 mm) and the gap width of DGS wasg = 0.4 mm. The width of the ring (d ) is chosen here to be 0.6 mm. The width of 0.6 mm was chosen as this is the width that corresponds to the maximum concentration of the current distribution [6]. This optimum width changes slightly with an abrupt increase or decrease of the etched circular pattern radii, but 0.6 mm is a good approximation for which the computed results match with the simulated or measured ones.The numerical calculations are performed for the simplified equivalent circuit model (Fig. 3(c)) with the extracted parameters in Table 1 by using the ADS, and the results are compared with the results from the electromagnetic (EM) simulation using the CST Microwave Studio.
In Fig. 5, the resonance frequencies and 3dB cutoff frequencies calculated by the simplified circuit simulation and the EM simulation are plotted as functions of the radius of the etched circular pattern in the dumbbellshaped DGS. The comparison between the results from the circuit modeling and the EM simulation shows that the proposed circuit model predicts the resonance frequencies and 3 dB cutoff frequencies with reasonable accuracy for a wide variation of radii from 2 to 10.
Fig. 6 illustrates the comparative transfer characteristics from the simulations and the measurement for a fabricated DGS with
r = 4 mm. The resonance frequency and the stop bandwidth obtained from the measurements agree well with those from the circuit simulation and the EM simulation.V. CONCLUSION
This paper has presented a new equivalent circuit model of a dumbbellshaped DGS and simple approximate expressions to extract equivalent circuit parameters. The proposed equivalent circuit model provides an improved design method and physical insight into the electromagnetic behavior of the dumbbellshaped DGS.

[Fig. 1.] Configuration of microstrip line with a dumbbellshaped defected ground structure.

[Fig. 2.] Sketch of the current flows and electric field lines for the dumbbellshaped defected ground structure in the ground plane of the microstrip line and its equivalent circuit model.

[Fig. 3.] (a) Lumped element equivalent circuit model for a microstrip line with a dumbbellshaped defected ground structure. (b) Defining the mesh currents for the circuits in the branch between T and T’. (c) Simplified equivalent circuit of the branch between T and T’ obtained from the expression for equivalent impedance.

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[Fig. 4.] Topology of the metallic split ring to extract the inductance and the surface capacitance of the defected ground structure on the ground plane.

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[Table 1.] Extracted circuit parameters for various radii of the etched circular pattern (km = 1, km_dd = 0)

[Fig. 5.] Calculated results from simplified equivalent circuit model and EM simulation. (a) Resonance frequency. (b) 3 dB cutoff frequency.

[Fig. 6.] Sparameters by measurement, simplified circuit simulation, and EM simulation.

[Fig. 7.] Bottom view of the fabricated defected ground structure.