An Equivalent Circuit Model for a Dumbbell-Shaped DGS Microstrip Line

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  • ABSTRACT

    This paper presents an equivalent circuit model for a dumbbell-shaped defected ground structure (DGS) in a microstrip line. The effects of equivalent circuit elements of a dumbbell-shaped DGS and their magnetic coupling to the host transmission line are modeled as a simple lumped-element 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 dumbbell-shaped defected ground structure (DGS) provides low-pass 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 low-pass filters [1,2]. To date, the analysis of the dumbbellshaped DGS loaded planar transmission lines has almost always involved utilizing commercial electromagnetic full-wave 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, time-consuming iterative design procedures are required to accomplish the design goal.

    In this paper, we propose a new equivalent circuit model for a dumbbell-shaped 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 dumbbell-shaped 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 Ld are connected parallel to the capacitance of 2Cd in the resonant circuit for the DGS.

    For the dumbbell-shaped 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 Lm represents the mutual inductances coupling the DGS to the host line, and the Lm_dd is the mutual inductance between two Ld for etched circular areas. The L and  C 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 and T’ is analyzed by defining mesh currents as shown in Fig. 3(b), in which the mesh currents in the resonant circuit are the same (id1 = id2) due to the symmetry. Applying the voltage law around the meshes and solving for i and v, the equivalent impedance between T and T’, Zeq is given by

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    From (1), the complex circuit between T and T’ 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 by

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    with the resonance angular frequency of At resonance, Ld and Cd are obtained in the simplified form

    image

    since Lm=km(LLd)1/2 and Lm_dd=km_ddLd, where km is the magnetic coupling coefficient between the host transmission line and the resonator and km_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 dumbbell-shaped 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 Ld for the left half of the dumbbell-shaped 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]

    image

    where Rm = r + d / 2 , t is the conductor thickness, and μ0 is the free-space 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 εe, the capacitance of the split ring for the narrow gap is

    image

    In (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 εe is the effective permittivity of the homogeneous medium with εe = ε0 (εr +1) / 2 , where εr 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 km is assumed  to be a unity. It is also assumed that the km_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, Ld and Cd 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 was g = 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 cut-off 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 dumbbell-shaped 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 cut-off 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 dumbbell-shaped 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.

  • 1. Chang I. S., Lee B. S. 2002 "Design of defected ground structures for harmonics control for active microstrip antenna," [in Proceedings of IEEE Antennas Propagation Society International Symposium] P.852-855 google
  • 2. Lim J. S., Kim C. S., Ahn D., Jeong Y. C., Nam S. W. 2005 "Design of low-pass filters using defected ground structure," [IEEE Transactions on Microwave Theory and Techniques] Vol.53 P.2539-3545 google doi
  • 3. Grover F. W. 2004 Inductance Calculations: Working Formulas and Table. google
  • 4. Sydoruk O., Tatartschuk E., Shamonina E., Solymar L. 2009 "Analytical formulation for the resonant frequency of split rings," [Journal of Applied Physics] Vol.105 google doi
  • 5. Froncisz W., Hyde J. S. 1982 "The loop-gap resonator: a new microwave lumped circuit ESR sample structure," [Journal of Magnetic Resonance] Vol.47 P.515-521 google
  • 6. Karmakar N. C., Roy S. M., Balbin I. 2006 "Quasi-static modeling of defected ground structure,” [IEEE Transactions on Microwave Theory and Techniques] Vol.54 P.2160-2168 google doi
  • [Fig. 1.] Configuration of microstrip line with a dumbbell-shaped defected ground structure.
    Configuration of microstrip line with a dumbbell-shaped defected ground structure.
  • [Fig. 2.] Sketch of the current flows and electric field lines for the dumbbell-shaped defected ground structure in the ground plane of the microstrip line and its equivalent circuit model.
    Sketch of the current flows and electric field lines for the dumbbell-shaped 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 dumbbell-shaped 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.
    (a) Lumped element equivalent circuit model for a microstrip line with a dumbbell-shaped 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.
    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)
    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 cut-off frequency.
    Calculated results from simplified equivalent circuit model and EM simulation. (a) Resonance frequency. (b) 3 dB cut-off frequency.
  • [Fig. 6.] S-parameters by measurement, simplified circuit simulation, and EM simulation.
    S-parameters by measurement, simplified circuit simulation, and EM simulation.
  • [Fig. 7.] Bottom view of the fabricated defected ground structure.
    Bottom view of the fabricated defected ground structure.