# Coupled Slow-wave CoPlanar Waveguide (CS-CPW)

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## Classical coupled microstrip line

The symmetrical coupled microstrip lines consist of a two signal strips integrated on a substrate with a backside ground plane, as shown in Figure 2-1 (a). Being a three-conductor device, two modes will propagate in such structure. The symmetry in the configuration allows us to use the even- and odd-mode approach; which consist in divide the structure into two independent cases (even and odd) and at the end superpose the results of each one of them .
The even-mode, plotted in Figure 2-1 (b), is excited when a voltage of the same magnitude and phase is applied to each one of the signal strips. While the odd-mode plotted in Figure 2-1 (c), is excited when a voltage of the same magnitude and opposite phase is applied to each one of the signal strips.
The section of coupled lines can behave as a 4-port coupler which its design has to be optimized in order to get the desired coupling, matching, and isolation. The couplers’ design is specified in chapter 3. To begin with, it is necessary to detail the equivalent lumped circuit model from which the coupling coefficients can be calculated.

Classical coupled microstrip line

The equivalent circuit is derived from the electrical model given in Figure 2-2 (a), composed of a self and a mutual inductance 𝐿0−𝜇 and 𝐿𝑚−𝜇, along with a self and a mutual capacitance, 𝐶0−𝜇 and 𝐶𝑚−𝜇. The μ subscript stands for microstrip, the subscript m for mutual and 0 for self. As losses are not taken into account, the resistances and conductances will not appear in the presented model. An even- and odd-mode analysis (Figure 2-2 (b) and (c)) is made to derive the coupling coefficients: magnetic coupling 𝑘𝐿−𝜇 and electric coupling 𝑘𝐶−𝜇.
For the even-mode, there is an even symmetry about the center of the structure which means that no current flows between the two strips; thus the symmetry plane acts as a magnetic-wall (open circuit). In Figure 2-2 (b), the equivalent capacitance is determined by the capacitance of either line with the magnetic wall (H-plane). 𝐿μ
𝑒𝑣𝑒𝑛 and 𝐶μ
𝑒𝑣𝑒𝑛 are the even-mode
effective inductance and capacitance respectively, they are expressed in equations (2-1) and (2-2).
Similarly for the odd-mode, there is an odd symmetry about the center of the structure which means that a voltage between the two strips is null, thus the symmetry plane acts as an electric-wall (short circuit). In this case Figure 2-2 (c), the equivalent capacitance is determined by the capacitance of either line with the electric-wall (E-plane). 𝐿μ
𝑜𝑑𝑑 and 𝐶μ
𝑜𝑑𝑑 are the evenmode
inductance and capacitance respectively, they are expressed in equations (2-3) and (2-4).
𝐿0−μ, 𝐿𝑚−μ, 𝐶0−μ and 𝐶𝑚−μ are imposed by the permittivity of the dielectric substrate and the dimensional parameters, i.e. the separation 𝑆 between the coupled strips, the height ℎ𝜇 between strips and ground plane, and the strips widths W. The variation of the electrical parameters versus the dimensions are presented in ,, and .
This expression is valid for coupled microstrip case as well as for any type of structure where 𝑘𝐿 is the corresponding magnetic coupling, 𝑘𝐶 is the electrical coupling and 𝑘 the total coupling level.
It can be shown that infinite directivity of a coupled line structure is reached if phase velocities are identical. See condition (2-8):
𝑣𝜙𝑒𝑣𝑒𝑛=𝑣𝜙𝑜𝑑𝑑
It is important to notice that condition (2-8) is equivalent to have: 𝑘=|𝑘𝐿|=|𝑘𝐶| (2-9)
Equation (2-9) presents the ideal case to provide theoretically infinite directivity. This condition is most of the time not achieved because of the inhomogeneity presented in several technologies. For this reason, in this thesis an approach to achieve (2-9) with any kind of integrated technology is proposed. This approach is not only compatible with any CMOS/BiCMOS technology but also it is straightforward to implement because of its simplicity. In this way, high-performances couplers at mm-wave could be now designed.

### New Coupled Slow-wave CoPlanar Waveguide (CS-CPW) concept

The topology of the CS-CPW is presented in Figure 2-3. The structure is composed of two central signal strips, with coplanar lateral ground strips. Thin floating ribbons (also called floating shielding) of width 𝑆𝐿, separated by a gap 𝑆𝑆, are placed below as for classical S-CPWs, in order to create the CS-CPW structure. As explained in Chapter 1, S-CPW lead to high miniaturization and high quality factor Q, compared to microstrip or CPW transmission lines implemented in silicon technologies, thanks to their slow-wave behavior .
In this structure, as for S-CPWs, the electric field will be confined between the main strips and the floating shielding; therefore specific capacitances, not present in microstrip or CPW models, will appear in the model. Meanwhile, the magnetic field will be almost not undisturbed by the floating ribbons due to their extremely thin thickness and length, hence the model will present the same inductance as in a classical CPW. This brings a separation of the magnetic and electric fields, which brings the slow-wave effect. This particularity offers a new degree of freedom in couplers’ design by letting the magnetic and electric coupling coefficients vary independently. Consequently, magnetic and electric couplings can be controlled separately, depending on the floating shield design.
Based on this statement, the interesting idea developed in the next sections consists in modifying the coupling coefficients 𝑘𝐶 and 𝑘𝐿, by cutting the shielding ribbons, either between the two coupled strips (cut in the center, CC-ribbons), or between the coupled strips and the ground strips (cut on the sides, CS-ribbons) . This will be explained in details in sections 2.6 and 2.7.

#### Electrical equivalent model for CS-CPW

Equivalent electrical model

For coupled microstrip lines, the electrical model has been used. The developed model for CS-CPW differs due to the presence of the floating shield ribbons. The resulting model can take three forms according to the configuration of the floating shield: uncut, center cut, or side cut. The first one will be detailed below and the last two will be explained in sections 2.6.2 and 2.6.3.
For better understanding, Figure 2-4 presents the steps for the development of the CS-CPW model. The model can be considered as “electrical” since it is based on the electromagnetic behavior. Conductive losses are not considered. First, according to the electric field distribution in Figure 2-4 (a), the equivalent capacitances are shown in Figure 2-4 (b). Instead of going directly from the central strips to the ground (as in CPW), the electric field is taking a shortcut through the floating ribbons, which adds new capacitances in the model (𝐶𝑠 and 𝐶𝑔). Then in Figure 2-4 (c) the magnetic flux of each conductor in the structure is presented, and its equivalent parameter 𝐿0 is drawn in Figure 2-4 (d). The last interaction to review is the one of the magnetic field between the conductors (Figure 2-4 (e)), represented by a mutual inductance 𝐿𝑚 in Figure 2-4 (f). As mentioned before, the magnetic field is almost not disturbed by the presence of the floating shield underneath the coupled CPW, therefore the self and mutual inductances, 𝐿0 and 𝐿𝑚, are almost similar to the ones of a coupled CPW.

Introduction
Chapter 1 Directional Couplers in Millimeter-Waves: Presentation
1.1. Classical directional couplers theory
1.1.1. General definitions for a 4-port symmetrical, reciprocal directional coupler
1.1.2. Coupled lines directional coupler
1.2. General types of directional couplers and applications
1.2.1. Branch-line coupler
1.2.2. Coupled lines coupler
1.2.3. Lange coupler
1.2.4. Discussion
1.3. State-of-the-art for integrated coupled lines directional couplers
1.3.2. Integrated technologies available at mm-wave: Discussion
1.4. Solution: coupled lines based on slow-wave effect
1.4.1. Characteristic parameters of the transmission lines
1.4.2. Slow-wave coplanar waveguide S-CPW
1.4.3. Discussion
1.5. Conclusion
1.6. References
Chapter 2 Coupled Slow-wave CoPlanar Waveguide (CS-CPW)
2.1. Introduction
2.2. Classical coupled microstrip line
2.3. New Coupled Slow-wave CoPlanar Waveguide (CS-CPW) concept
2.4. Electrical equivalent model for CS-CPW
2.4.1. Equivalent electrical model
2.4.2. Even- and odd-mode Analysis
2.5.1. Validation of the method with the microstrip case
2.5.2. Simulation method: even- and odd-mode
2.5.3. Electrical parameters extraction from even- and odd-mode simulation
2.5.4. Simulation method on the CS-CPW
2.5.5. Simulation results: electrical performance of CS-CPW vs C-μstrip
2.6. Cutting the floating shielding
2.6.1. Concept
2.6.2. Cut in the Center (CC)
2.6.3. Cut on the sides (CS)
2.7. Coupling coefficient vs dimensions variation
2.7.1. Variation of coupling coefficients with 𝑾 and 𝑺
2.7.2. Variation of magnetic coupling with G
2.7.3. Cutting the floating ribbons for 𝒌𝑪
2.7.4. Effect of the cut width
2.7.5. Effect of strip spacing SS and strip length SL
2.8. Analytical model and abacus
2.8.1. Analytical model
2.8.2. Charts
2.9. Technological issues
2.9.1. Effect of dummies in CS-CPW
2.9.2. Dimensions limitations
2.10. Conclusion
2.11. References
Chapter 3 Millimeter-Waves CS-CPW High-Directivity Directional Couplers
3.1. Introduction
3.2. Design method with CS-CPW
3.2.1. Principle
3.2.2. Example
3.3. Practical couplers design
3.3.1. 3 dB coupler at 50 GHz
3.3.2. 18 dB coupling at 150 GHz
3.4. Comparison of CS-CPW with C-μstrip directional couplers
3.5. Multimode TRL de-embedding method
3.6. Measurement results
3.6.1. 18-dB coupling at 150 GHz
3.7. Discussion
3.8. References
Chapter 4 Millimeter-Waves Parallel-Coupled Line Filters with CS-CPW
4.1. Introduction
4.2. Classical parallel coupled line filter theory
4.3. Odd and even-modes characteristic impedances
4.3.1. Characteristic impedances when cutting the floating ribbons
4.4. Design
4.4.1. Parallel-coupled lines resonator
4.4.2. Parallel-coupled lines third-order filter
4.4.3. Practical issues
4.5. Comparison of CS-CPW with state-of-the-art
4.5.1. Simulation with HFSS of the CS-CPW based resonator
4.5.2. Comparison with the state-of-the-art
4.6. Conclusion
4.7. References
Chapter 5 Applications of the CS-CPW Directional Couplers
5.1. Introduction
5.2. Reflection-Type Phase Shifter (RTPS) with CS-CPW
5.2.1. State-of-the-art of mm-wave phase shifters
5.2.2. Principle of the RTPS with CS-CPW
5.2.3. Results of the CS-CPW directional coupler for the RTPS
5.2.4. Results of the RTPS using the 3-dB CS-CPW directional coupler
5.3. Isolator
5.3.1. State-of-the-art for mm-wave isolators
5.3.2. Principle of the isolator with CS-CPW
5.3.3. Results of the CS-CPW directional coupler for the isolator
5.3.4. Results of the isolator using the 7.5-dB, CS-CPW directional coupler
5.4. Baluns
5.4.1. State-of-the-art for mm-wave baluns
5.4.2. Principle of the balun with CS-CPW
5.4.3. Results of the CS-CPW directional coupler for the balun
5.4.4. Results of the balun using the 80 GHz, 3-dB, 50 Ω, CS-CPW directional coupler
5.5. Conclusion
5.6. References
General Conclusions

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