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

Abstract

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.

1. 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 [2–10]. 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.

CPW discontinuities are an integral part of practical CPW circuits [12]. A number of researchers have worked on CPW discontinuities [13–21]. The CPW open-end, short-end, series gap, step change in the width of center conductor, interdigitated capacitance, and the bend are some of such discontinuities. Some of these discontinuities are shown in Figure 1.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 1 

CPW discontinuities (a), open-end (b), short-end (c), gap (d), step (e), right-angled bend (f), open-end series stub (g) and short-end series stub.

The CPW short-end and open-end series stubs are two important structures for filter applications. The CPW short-end 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 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 PI-model 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.

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 [23–27] and may be useful in this regard.

2. 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. 𝑙₁ and 𝑙₂ 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.

Figure 2 

The layout of the discontinuity structure unit-cell (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 semi-infinite 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 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.

Figure 3 

Scattering parameters for the DS-UC by MOM. 𝑠1=𝑠2=0.22 mm; 𝑔1=𝑔2=0.1 mm, Δ=0.1 mm, 𝑤1=0.9 mm, 𝑤2=1.34 mm; 𝑙1=𝑙2=0.45 mm, 𝑙=7.24 mm, ℎ=0.635 mm, 𝜀𝑟=10.2.

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.

Figure 4 

Boundary conditions enforced for application of FIT. 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).

Figure 5 

Scattering parameters for the DS-UC by FIT. 𝑠1=𝑠2=0.22 mm; 𝑔1=𝑔2=0.1 mm, Δ=0.1 mm, 𝑤1=0.9 mm, 𝑤2=1.34 mm; 𝑙1=𝑙2=0.45 mm, 𝑙=7.24 mm, ℎ=0.635 mm, 𝜀𝑟=10.2.

The effect of the dimensions 𝑔₁, 𝑔₂, 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 𝑠₁=𝑠₂ and 𝑔₁=𝑔₂ mainly control the fractional bandwidth of passband as is evident from Figures 6(b) and 6(c), respectively.

(a)

(b)

(c)

Figure 6 

Parameterization of dimensions (a) Δ, (b) 𝑠1, 𝑠2, and (c) 𝑔1, 𝑔2.

3. Quasi-Static Finite-Difference Analysis

The characteristic impedance of a CPW is found through quasi-static analysis by conformal mapping [32–38].

The characteristic impedance and effective dielectric constant for CPW can be computed by quasi-static conformal mapping techniques and given by 𝑍₀=1𝑐𝐶_𝑎√𝜀_eff, (1) 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 𝐶_𝑎=4𝜀₀𝐾𝑘₁𝐾𝑘^′₁,𝜀_eff𝜀=1+^𝑟2. (2) Applying in (1) yields 𝑍=30𝜋√𝜀_eff𝐾𝑘^′₁𝐾𝑘₁, (3) where 𝐾(𝑘^′₁) and 𝐾(𝑘₁) represent the complete integral of first kind and its complement and 𝑘₁ and 𝑘^′₁ given by 𝑘₁=𝑤₁𝑤₁,𝑘+2𝑠_′1=1−𝑘²₁, (4) where 𝑠₁=𝑠₂=𝑠.

The ratio between 𝐾^′(𝑘) and 𝐾(𝑘) in (3) is an important one in CPW terminology and given by [38] as follows 𝐾(𝑘)𝐾^′(𝑘)=𝜋log_𝑒21+𝑘′/1−𝑘′_{0≤𝑘≤0.707},𝐾(𝑘)𝐾^′(𝑘)=log_𝑒21+√𝑘/1−√𝑘𝜋_{0.707≤𝑘≤1}. (5) The characteristic impedance as given by (1) is plotted for the case of 𝜀_𝑟=10.2 against the normalized 𝑤₁/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 𝑤₁/𝑠₁.

Figure 7 

Characteristic Impedance of the CPW for the case of 𝜀𝑟=10.2 versus normalized 𝑤1/𝑠1 ratio on logarithmic scale.

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. 𝑙₁ and 𝑙₂ are lengths of two uniform coplanar waveguides feeding the discontinuity.

Figure 8 

Setup for E-Field computation, 𝑙1=𝑙2=0.45 mm, 𝑔1=𝑔2=0.1 mm, and 𝑙=7.24 mm.

The electric field can be determined from the static electric potential using ^→𝐸=−^→∇𝜑, (6) where 𝜑 is the static electric potential.

The charges 𝑄₁ and 𝑄₂ on the metalized areas with potentials 𝜑₁ and 𝜑₂, respectively, are given by (7) and (8) as follows: 𝑄₁=𝐷_𝐴^→⋅^→𝑛𝑑𝐴=𝜀_𝑜𝜀_𝑟𝐴𝐸_𝑛𝑄𝑑𝐴,₂=𝐷_𝐴^→⋅^→𝑛𝑑𝐴=𝜀_𝑜𝜀_𝑟𝐴𝐸_𝑛𝑑𝐴, (7)(8) 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 𝜑₁ and 𝜑₂, respectively, as shown in Figure 8.

Similarly the charges 𝑄₃ and 𝑄₄ deposited on the surface of metalized strips with potentials 𝜑₃ and 𝜑₄, respectively, are given by 𝑄₃=𝐷_𝐴^→⋅^→𝑛𝑑𝐴=𝜀_𝑜𝜀_𝑟𝐴𝐸_𝑛𝑄𝑑𝐴,₄=𝐷_𝐴^→⋅^→𝑛𝑑𝐴=𝜀_𝑜𝜀_𝑟𝐴𝐸_𝑛𝑑𝐴, (9)(10) 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 𝑔₁ and 𝑔₂ are known (lengths of sections with potentials 𝜑₃ and 𝜑₄ would be 𝑙−𝑔₁−𝑔₂).

Similarly, the charges on the semi-infinite ground plane sections with potentials 𝜑_𝐺 and 𝑙₁ and 𝑙₂ (lengths beyond the discontinuity region) are given by 𝑄_𝐺𝑠=_𝑐𝐷_𝑛𝑑𝑠=𝜀_𝑜𝜀_𝑟𝑐𝐸_𝑛𝑑𝑠𝑠=1,2, (11) 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, 𝑄_total=𝑄₁+𝑄₂+𝑄₃+𝑄₄. (12)

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 𝑉=𝜑_𝐷−𝜑_𝐺. (13) Then the capacitance of the discontinuity region would be given by 𝐶_𝐷=𝑄_𝐷𝑉=𝑄_total−𝑄_𝐺1−𝑄_𝐺2𝑉. (14)

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.

(a)  

(b)

(c)

Figure 9 

Electric field (V/m) distribution inside the discontinuity structure at 8 GHz under quasi-TEM fundamental mode (a) 𝐸𝑥, (b) 𝐸𝑦, and (c) 𝐸𝑧.

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.

Figure 10 

Setup for magnetic field computation, 𝑙1=𝑙2=0.45 mm, 𝑔1=𝑔2=0.1 mm, and 𝑙=7.24 mm.

The magnetic field can be computed from the static magnetic potential using 𝐻^→=−^→∇𝜓. (15)

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 𝜙_𝑇=_𝐴^→𝐵⋅^→𝑛𝑑𝐴=𝜇_𝑜𝐴𝐻_𝑛𝑑𝐴, (16) 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 𝑙₁ and 𝑙₂ is given by 𝜙_𝐺𝑡=_𝑐𝐵𝑑𝑠=𝜇_𝑜𝑐𝐻𝑑𝑠𝑡=1,2. (17)

The magnetic flux due to the discontinuity metallization is given by the difference of the total magnetic flux and the magnetic flux due to the uniform CPWs (lengths 𝑙₁ and 𝑙₂), that is, 𝜙_𝐷=𝜙_𝑇−𝜙_𝐺1𝑙₁−𝜙_𝐺2𝑙₂. (18)

From the net magnetic flux, the current in the centre conducting plane can be determined as follows: 𝐼=𝐻_𝑐^→⋅𝑑^→𝑠=𝜓_𝑇, (19) 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: 𝐿_𝐷=𝜙_𝐷𝐼. (20)

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.

(a)  

(b)

(c)

Figure 11 

Magnetic field (A/m) distribution inside the discontinuity structure at 8 GHz under quasi-TEM fundamental mode (a) 𝐻𝑥, (b) 𝐻𝑦, and (c) 𝐻𝑧.

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/m³) are observed using CST Microwave Studio and are shown in Figure 12 where the peak electric energy density = 0.0872 J/m³ and peak magnetic energy density = 0.0459 J/m³_.

(a)

(b)

Figure 12 

Energy density (J/m3) under fundamental mode (a) electric energy density and (b) magnetic energy density.

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 mm²).

Figure 13 

Top view of bandpass structure achieved by cascading three cells of DS-UC.

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.

Figure 14 

Scattering parameters for the bandpass resonator constructed by cascading three cells of DS-UC by MOM.

Figure 15 

Scattering parameters for the bandpass resonator constructed by cascading three cells of DS-UC by FIT.

The bandwidth of the filter can be controlled by adjusting the dimensions Δ, 𝑠₁, 𝑠₂, 𝑔₁, and 𝑔₂ 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, 𝑠₁, 𝑠₂, 𝑔₁, and 𝑔₂. This can be achieved by performing a parametric study of the dimensions Δ, 𝑠₁, 𝑠₂, 𝑔₁ and 𝑔₂ 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 𝑠₁=𝑠₂ and 𝑔₁=𝑔₂ 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.

4. 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 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 quasi-static 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.