Characterization of a CoplanarWaveguide Open Stub-Based Discontinuity for MMICs and Filter Applications

This paper presents the analysis and design of an open stub-based two-port discontinuity in a coplanar waveguide. The discontinuity has a good bandpass response and the required design characteristics were achieved by using the method of moments (MOM) and the finite integration technique (FIT). The analysis is found to be computationally efficient which makes the discontinuity suitable for use in complex structure applications. The analysis of the discontinuity is also formulated using the quasi-static finite-difference technique. The proposed discontinuity structure finds applications in RFICs and MMICs. Finally, the application of the discontinuity is supported by the simulated design of a broadband bandpass filter on an MMIC with a relative bandwidth of 96%. The filter is found to have a miniature size and a fairly good stop band rejection.


Introduction
A coplanar waveguide (CPW), first described by Wen [1], has a uniplanar construction and because of this property, it remarkably simplifies fabrication using on-wafer techniques.Since its advent, it is often the preferred choice in many applications especially in RF circuit design including those in Monolithic Microwave Integrated Circuits (MMICs).A detailed description can be found in [2][3][4][5][6][7][8][9][10].This is owing to the fact that structure dimensions are not dependent on the substrate thickness for CPW and hence can be chosen to be very small.However, the metallization thickness is critical in the case of CPW since, in contrast to microstrip, this thickness is comparable to the overall dimensions of the structure.
A CPW can be on a single substrate or multiple substrates and, moreover, may be with or without a bottom ground plane in addition to the top semi-infinite ground planes.The name conductor-backed CPW (CBCPW) is used when the bottom ground plane is present.The choice of a particular CPW topology depends on the specific requirements of an application and its design.
CPW parameters have a small dispersion, that is, they do not vary significantly with frequency since the field is mainly concentrated in the slot regions between the metallization.This leads to increased frequencies up to which static analysis can be considered to be valid in contrast to microstrip or stripline analysis.Dispersion, however, depends on the width of the signal strip, which means that dispersion is high for narrow strips comparable with the skin depth [11].The CPW supports a quasi-TEM mode of operation but there is no low frequency cutoff unlike the transmission line case.
The CPW short-end and open-end series stubs are two important structures for filter applications.The CPW shortend series stub was modeled as a series inductor and the open-end series stub was modeled as a series capacitor by Houdart [3].These can be attributed to their bandstop and International Journal of Antennas and Propagation bandpass response, respectively.However, both of these stubs are asymmetric and their resonant nature is valid only for small stub lengths (l → 0).Williams and Schwarz [22] extended the model for the open-end stub to a capacitive PImodel and selected a reference plane to remove the element asymmetry.
The open stub-based discontinuity structure presented in this work is symmetric and also has a good bandpass response.This remarkably simplifies the model for use in CAD applications with a significant reduction in computational time and complexity.The fractional bandwidth of the proposed structure is nearly 82% which is a significant value to utilize the structure in broadband filter applications.This aspect is also demonstrated by a design example at the end of this paper.The CPW single open stub also has a bandpass response but has a very low relative bandwidth compared to a cascade combination of the open stubs as proposed in this work.The resulting cell may be reused or cascaded to build filters with ultrabroadband bandwidths.

Reference planes
Semiinfinite ground planes The analysis of the proposed discontinuity is also formulated using the quasi-static finite-difference technique.The proposed analysis is also useful in deriving an equivalent network model for the proposed discontinuity.Different authors have discussed the equivalent models for different scenarios concerning CPW [23][24][25][26][27] and may be useful in this regard.

Theory and Simulation
The layout of the discontinuity structure unit-cell (DS-UC) is shown in Figure 2. The structure is symmetric which provides an advantage as the computational size of the problem is halved for such a structure.l 1 and l 2 are the lengths of two uniform coplanar waveguides feeding the DS-UC reference planes.Notice the gap dimension Δ which has a critical influence in achieving the passband response of the DS-UC.XX and YY are the reference planes where the field disturbances caused by the discontinuity have decayed down to zero [28].Hence, l is the effective length of the DS-UC.
The scattering parameters for the discontinuity were obtained by the method of moments, [29], using an electromagnetic (EM) simulator [30].Only a fundamental even mode has been considered here for simulation (called CPW mode for CPW), whereas the odd mode (called slot mode for CPW) has been suppressed in the simulation.Practically, this is done by using air bridges between the two semiinfinite ground planes and hence keeping them at same potential (i.e., E-field is then always oriented from center strip conductor to the semi-infinite ground planes, that is the case for even mode or CPW mode).
The substrate is RT/duroid 6010LM with a dielectric constant of 10.2 and height of 625 μm with no bottom ground plane.A magnetic wall was assumed at the input and output ports of the discontinuity where the strip conductor A perfect magnetic conductor (PMC) was assumed on the side walls where the input and output ports intersect the feeding CPW (marked by blue ground symbols); perfect electrically conducting (PEC) walls were assumed on the top and bottom faces (marked by green ground symbols).and the shielding would intersect, whereas electric walls were assumed for the remaining boundary conditions.The MOM setup included 257 cells and the computation was completed in a CPU time of 130 seconds only which is an efficient computational overhead for using the proposed discontinuity in complex structure applications.The scattering parameters for the DS-UC are shown in Figure 3.The 3-dB bandwidth of the first passband covers approximately 4.95 GHz to 11.78 GHz.
The scattering parameters were verified by the finite integration technique (FIT), using an electromagnetic simulator (CST Microwave Studio) [31].The boundary conditions used for the numerical setup are shown in Figure 4 and the scattering parameters are shown in Figure 5.The scattering parameters show a good agreement for both the methods which authenticates the validity of the response for the proposed DS-UC.The effect of the dimensions g 1 , g 2 , and Δ on the response of the DS-UC was studied by performing a parameterization sweep and results are shown in Figure 6.The gap dimension Δ mainly controls the upper cutoff of the passband, whereas the dimensions s 1 = s 2 and g 1 = g 2 mainly control the fractional bandwidth of passband as is evident from Figures 6(b) and 6(c), respectively.
The characteristic impedance and effective dielectric constant for CPW can be computed by quasi-static conformal mapping techniques and given by where C a is the capacitance of CPW with air on both sides (top and bottom) without any intervening dielectric and ε eff is the effective dielectric constant.
For the case of CPW on a dielectric substrate of finite thickness, C a and ε eff are, respectively, given by , ( Applying in (1) yields where K(k 1 ) and K(k 1 ) represent the complete integral of first kind and its complement and k 1 and k 1 given by where The ratio between K (k) and K(k) in ( 3) is an important one in CPW terminology and given by [38] as follows The characteristic impedance as given by ( 1) is plotted for the case of ε r = 10.2 against the normalized w 1 /s ratio on a logarithmic scale and is shown in Figure 7.The characteristic impedance of the structure is found to decrease by increasing the ratio w 1 /s 1 .
The electric field distribution due to the metallization area of the discontinuity gives rise to the capacitance which can be computed by quasi-static finite difference method [28].The setup for applying the analysis for E-field is shown in Figure 8. Magnetic walls were assumed at both the input and output ports for computation of E-field (where the strip conductor and shielding intersect).XX and YY are the reference planes where the E-field caused by the metallization of the discontinuity has decayed to absolute zero and hence they define the geometrical size of the discontinuity.l 1 and l 2 are lengths of two uniform coplanar waveguides feeding the discontinuity.
The electric field can be determined from the static electric potential using where ϕ is the static electric potential.The charges Q 1 and Q 2 on the metalized areas with potentials ϕ 1 and ϕ 2 , respectively, are given by ( 7) and (8) as follows: where D is the electric flux density around the metalized area and E n is the normal component of the electric field.The integration is to be done over the total area A of the metalized region with potentials ϕ 1 and ϕ 2 , respectively, as shown in Figure 8.
Similarly the charges Q 3 and Q 4 deposited on the surface of metalized strips with potentials ϕ 3 and ϕ 4 , respectively, are given by where the integration is to be done over the area as shown in Figure 8.The area of the strips can be determined since the length l and the gap dimensions g 1 and g 2 are known (lengths of sections with potentials ϕ 3 and ϕ 4 would be l − g 1 − g 2 ).
Similarly, the charges on the semi-infinite ground plane sections with potentials ϕ G and l 1 and l 2 (lengths beyond the discontinuity region) are given by where the integration is done along a contour in the magnetic walls as shown in the inset of Figure 8.
The net charge Q total on the discontinuity region (length l) is given by the sum of ( 7)- (10), that is, Finally, the charge stored on the metallization of the discontinuity region is given by the difference of total charge Q total and the charges on two uniform CPWs, that is, Q G1 and Q G2 as given by (11).
Assuming the potential difference between the CPW structure representing the overall discontinuity region and the ground planes is given by Then the capacitance of the discontinuity region would be given by The E-field distribution plotted using CST Microwave Studio is shown in Figure 9 with respect to x, y, and z axes.Port 1 was excited from the left under a quasi-TEM mode to investigate the electric field distribution inside the slotted region of discontinuity structure.Most important is the component E y since the structure has been placed along xz-plane.E y shows a strong electric field distribution near the gap discontinuity which decays down as we move towards the ends of structure from the center.
The magnetic field distribution of the discontinuity gives rise to the inductance so the computation of magnetic field intensity will give the equivalent inductance.The setup for computing the static magnetic field is shown in Figure 10.
The magnetic field can be computed from the static magnetic potential using The magnetic field distribution inside the discontinuity is computed using the quasi-static finite difference method as given in [28].
The magnetic flux through the discontinuity region is given by where the integration is to be done along the slot region over the area A as shown in Figure 10.The assumption made was that the metallization thickness is negligible (t → 0) and the substrate material has no effect on magnetic field due to the metallization, that is, μ r = 1.Likewise, the magnetic flux due to the two uniform coplanar waveguides of lengths l 1 and l 2 is given by The magnetic flux due to the discontinuity metallization is given by the difference of the total magnetic flux and the

International Journal of Antennas and Propagation
Semiinfinite ground planes Reference planes Magnetic walls (PMC) Figure 8: Setup for E-Field computation, l 1 = l 2 = 0.45 mm, g 1 = g 2 = 0.1 mm, and l = 7.24 mm.magnetic flux due to the uniform CPWs (lengths l 1 and l 2 ), that is, From the net magnetic flux, the current in the centre conducting plane can be determined as follows: where the line integral is to be taken along the electric wall shown in Figure 10.Now the equivalent inductance due to the discontinuity can be evaluated as follows: The magnetic field distribution inside the substrate under the quasi-TEM mode assumption was plotted using CST Microwave Studio and is shown in Figure 11, which shows that the magnetic field is stronger near the ends of discontinuity in contrast to the electric field distribution, which is concentrated at the centre of the gap discontinuity.
To support the fact that electric field is concentrated at the middle of gap discontinuity and magnetic field near to the ends of discontinuity, the electric and magnetic field energies (J/m 3 ) are observed using CST Microwave Studio and are shown in Figure 12 where the peak electric energy density = 0.0872 J/m 3 and peak magnetic energy density = 0.0459 J/m 3 .
To support the usefulness of the discontinuity structure for filter applications, a bandpass response with a good rolloff was achieved by cascading three cells of DS-UC as shown in Figure 13.The structure had a miniature size (a sectional area of 1.34 × 24 mm 2 ).
The scattering parameters for the bandpass structure were verified by both the method of moments, using Momentum [30], and FIT using another simulator, CST Microwave Studio [31]; these are shown in Figures 14 and 15, respectively.The relative bandwidth of the bandpass structure was computed to be 96%.MOM setup for the computation of bandpass filter utilized a processor overhead of 71.42 MB and a CPU time of 22 minutes and 6 seconds for computation of the final results.
The bandwidth of the filter can be controlled by adjusting the dimensions Δ, s 1 , s 2 , g 1 , and g 2 of the unit cell.This can be done by first designing a unit cell for the desired bandwidth and then cascading multiple cells for desired roll-off.The design of individual cell can be done using perturbation of dimensions Δ, s 1 , s 2 , g 1 , and g 2 .This can be achieved by performing a parametric study of the dimensions Δ, s 1 , s 2 , g 1 and g 2 using the simulator program.A rule of thumb as mentioned already is that the gap dimension Δ mainly controls the upper cutoff of the passband, whereas the dimensions s 1 = s 2 and g 1 = g 2 mainly control the fractional bandwidth of passband as illustrated in Figure 6 above.
Once a unit cell is designed, a filter with desired bandwidth can be designed by cascading multiple sections for desired rolloff.As in our case, cascade of two cells provided an acceptable rolloff.

Conclusions
An open stub-based two-port discontinuity in a coplanar waveguide is proposed with a much better return loss and relative bandwidth than the single open stub section.The International Journal of Antennas and Propagation    agreement between the results obtained by the classical methods of computational electromagnetics, that is, MOM and FIT supports the validity of the structure's usefulness.The discontinuity has a good bandpass response and an efficient computational overhead which makes it suitable for embedding in complex structure applications.The analysis of the discontinuity was also formulated using the quasistatic finite-difference technique.The proposed approach can be used for deriving a lumped element equivalent network for the discontinuity.The application of the discontinuity was supported by the simulated design of a miniature-sized broadband bandpass filter on MMIC with a significant fractional bandwidth (96%) and a fairly good stop band rejection.

Figure 2 :
Figure 2: The layout of the discontinuity structure unit-cell (DS-UC).

Figure 7 :
Figure 7: Characteristic Impedance of the CPW for the case of ε r = 10.2 versus normalized w 1 /s 1 ratio on logarithmic scale.

Figure 9 :Figure 10 :
Figure 9: Electric field (V/m) distribution inside the discontinuity structure at 8 GHz under quasi-TEM fundamental mode (a) E x , (b) E y , and (c) E z .

Figure 11 :
Figure 11: Magnetic field (A/m) distribution inside the discontinuity structure at 8 GHz under quasi-TEM fundamental mode (a) H x , (b) H y , and (c) H z .

Figure 12 :
Figure 12: Energy density (J/m 3 ) under fundamental mode (a) electric energy density and (b) magnetic energy density.

Figure 13 : 21 Figure 14 : 21 Figure 15 :
Figure 13: Top view of bandpass structure achieved by cascading three cells of DS-UC.