### Introduction

Coupled microstrip lines exhibit strong far-end crosstalk (FEXT) due to unequal phase velocities of the even and odd propagation modes. The diagram in *Figure 1** *shows that, in the case of the even-mode, a larger portion of E-field is travelling in the substrate material and therefore the effective dielectric constantis greater compared to the odd-mode dielectric constant . The characteristic impedance of the even-modeis larger than the one of the odd-modeas well. This is because the characteristic impedance is defined asin the lossless case series inductance per unit length,shunt capacitance per unit length). Reduction of microstrip FEXT has been addressed in high-speed digital design by the use of stubbed transmission lines^{1,2} or dielectric overlays^{3}. This phenomenon also plays an important role in microwave engineering, as microstrip coupled line backward couplers suffer from poor directivity when not compensated. Techniques to improve directivity are similar to those for high-speed digital design and include dielectric overlays^{4}, the use of wiggly lines^{5,6} and capacitive compensation^{7,8,9}. Reference 8 proposes equations for the calculation of optimum capacitance values for compensation. However, it requires the modification of the dimensions of the coupling structure itself. In this article, a more general approach, which does not require a modification of the dimensions of the coupled line structure, is presented, and closed form solutions for optimum capacitor values are provided

### Analysis of Compensated Coupled Microstrip Lines

** Figure 1 **shows two microstrip lines, which are coupled through the lengthThe feedlines at all four ports are considered to be uncoupled and having the system’s characteristic impedance. At both ends of the coupled line section, the lines are connected via two capacitors having the capacitanceEach of those can be split up into two capacitors ofconnected in series. The circuit is considered lossless and perfectly two-fold symmetric with the symmetry planes A and B. It can be easily observed that the capacitors are, with respect to symmetry plane A, only effective for the odd-mode. Otherwise, one of the capacitor's terminals is connected to the virtual open circuit in even mode. The relation between even-mode impedanceodd-mode impedanceand the system’s impedanceis defined as:

(1)

The characteristic impedances of the coupled transmission lines can be normalized as follows:

(2)

The normalized characteristic admittances of the even- and odd-mode can be calculated by inversion of the impedances:

(3)

The two-fold symmetrical four-port can be decomposed into four one-ports^{10,11}, as depicted in *Figure 2***.** These four one-ports are terminated by different combinations of o.c.and s.c. within the symmetry planes. Alternatively, o.c. and s.c. are often called magnetic and electric walls, respectively. The corresponding normalized eigenadmittances of the network can be calculated by transmission line theory:

(4-7)

The first and second index denote the mode with respect to symmetry plane A and B, respectively. The electrical length of the even and odd-mode is referred to asand The eigenreflections, then, can be calculated according to:

(8)

Scattering parameters can be easily calculated as they are linear combinations of the eigenreflections:

(9, 10, 11, 12)

A typical constellation of eigenreflections for two coupled microstrip lines with is shown in** Figure 3**. The uncompensated eigenreflections are represented by the dashed arrows, while the compensated are represented by the solid arrows. The effect of the capacitors can be regarded as an extension of the electrical lengthThus, the eigenreflectionsand are rotated clockwise, which is indicated by the red arrows. Note that

**represents the principle of compensation, but a perfect compensation by two capacitors as shown is not possible forandsimultaneously when. This will become clear by the equations presented below.**

*Figure 3*As the FEXT shall be minimized, the desired scattering parameteris zero. From Equation 11 and the assumed constellation of eigenreflections in ** Figure 3**, two criteria for= 0 can be derived by what is essentially complex vector addition:

(13)

These two criteria must be fulfilled at the same time in order to forceto be zero. As a 180-degree phase shift of a reflection coefficient is equal to the inversion of the corresponding impedance or admittance, the two criteria can be written as:

(14)

By inserting the Equations 4-7 into Equation 14 and solving for the capacitance value, the two competing results can be obtained:

(15, 16)

With the help of common trigonometric addition formulas, Equations 15 and 16 can be written as:

(17, 18)

It can be seen thatwhen= 0. Therefore, the argument of the cosine function has to be:

(19)

The electrical lengthandcan be expressed as

(20, 21)

with the angular frequencyand the speed of light. By inserting Equations 20 and 21 into Equation 19, the “free-space,” electrical length can be calculated as:

(22)

Therefore, perfect isolationis possible for thesingle frequency points:

(23)

When assuming thatthe Equations 17 and 18 become identical and Equations 20-22 can be inserted. Therefore, the choice of the following capacitances values leads to perfect isolation:

(24)

ForEquations 23 and 24 simplify to:

(25, 26)

Equation (25) presents the relation between the length of the coupled line structure and the frequency where perfect isolation can be achieved when compensation capacitances as calculated in (26) are used. For small arguments of the tangent-function, i.e. small differences betweenand, the equation (26) can be approximated by:

(27)

This approximation may be sufficient in many cases, however, in the following the exact solution according to equation (25) and (26) is used. The example in the next section shows that improvement of isolation is not only obtained for the frequency but for a wide bandwidth around that frequency.

### Example

Let’s assume an example with two microstrip lines of width 1.73 mm with 1 mm space between them on a Rogers RO4003C high frequency laminate, having a dielectric permittivity of 3.55 and a substrate height of 32 mils (which equals 0.813 mm). The length of the coupled section shall be 50 mm. The even and odd parameters are approximatelyandWith Equations 25 and 26, perfect isolation is achieved at a frequencyfor a capacitor of*Figure 4** *shows calculated scattering parameters for the uncompensated and compensated coupled microstrip lines. It can be observed that for the compensated case, the isolation improves considerably over the whole frequency range whilst the matching worsens and the coupling to port 4 increases with frequency. In order to achieve a better performance at higher frequencies, the 50 mm line can be split up into segments, for example, each 10 mm long. Every 10 mm section is compensated perfectly atusing a capacitance ofaccording to Equations 25 and 26.

**shows the circuit topology. Simulation results are depicted in**

*Figure 5***. It can be observed that the isolation performance has improved over a wide frequency range while proper matching and low backwards coupling (equivalent to near end cross-talk, NEXT) are maintained. It should be noted that the approximation in equation (27) leads to essentially the same results, as the capacitance values differ only around 0.12% with respect to the ideal results of Equations 25 and 26. Time domain simulations have been carried out using Keysight ADS;, results are shown in**

*Figure 6***. A step signal having a rise time of 100 ps excites the circuits while all ports are terminated with the system’s impedance ofThe input signalat port 1 is clearly affected by the capacitances. In the case of the two large capacitors, the rise time is slowed down a bit; after around twice the delay time of the transmission line, the reflection from the end of the line can be clearly observed at around 600 ps. Multiple capacitors lead to multiple smaller discontinuities, of which smaller reflections occur. The output signalat port 2 exhibits an additional delay of around 11 ps for both compensated networks. The FEXT at port 3 is considerably reduced from more than 100 mV maximum amplitude to around 31 mV for the compensation with two capacitors and 14 mV for the compensation with multiple capacitors. The NEXT at port 4 increases from a maximum amplitude of 54 mV to 107 mV and 75 mV.**

*Figure 7*

**
**### Conclusion

A method for the determination of capacitor values for compensation of unequal phase velocities in coupled microstrip lines has been presented. Simulation results show that broadband performance can be achieved with multiple capacitors along the coupled lines. In practice, the low capacitance values (a couple of femto-Farads) can be realized by stubs, interdigital capacitors, or the modulation of the space between adjacent microstrip lines. The presented equations help to find the maximum allowed distance between the capacitive elements and their required optimum capacitance values. In reality, there are, of course, often more than two transmission lines. Losses have to be considered, as perfect symmetry is not a given and other non-idealities have to be taken into account.

### References

1. R. K. Kunze, Y. Chu, Z. Yu, S. K. Chhay, M. Lai, and Y. Zhu, “Crosstalk mitigation and impedance management using tabbed lines,” Intel White Paper, 2015.

2. Wu, C.H., Shen, L., Zhang, H. et al. Equivalent circuit parameters of planar transmission lines with spoof surface plasmon polaritons and its application in high density circuits. Sci Rep 9, 18853 (2019).

3. Q. -M. Cai, Y. Zhu, R. Zhang, Y. Ren, X. Ye and J. Fan, "A Study of Coverlay Coated Microstrip Lines for Crosstalk Reduction in DDR5," 2020 IEEE International Symposium on Electromagnetic Compatibility & Signal/Power Integrity (EMCSI), Reno, NV, USA, 2020, pp. 581-585.

4. A. Podell, "A high-directivity microstrip coupler technique," 1970 IEEE MTT-S Int. Microwave Symp. Dig., pp. 33-36, 1970.

5. G. Haupt and H. Delfs, “High-directivity microstrip directional couplers,” Electron. Lett., vol. 10, pp. 142–143, May 1974.

6. Broadband directional couplers using microstrip with dielectric overlays," IEEE Trans. Microwave Theory and Tech., vol. MTT-22, pp. 1216-1220, Dec. 1974.

7. G. Schaller, “Optimization of microstrip directional couplers with lumped capacitors,” Int. J. Electron. Commun. (AEÜ), vol. 31, pp. 301–307, 1977.

8. M. Dydyk, “Accurate design of microstrip directional couplers with capacitive compensation,“ IEEE MTT-S Symp. Dig., pp.581-584, 1990.

9. C. T. Nghe, F. Rautschke and G. Boeck, "Performance optimization of capacitively compensated directional couplers," 2017 47th European Microwave Conference (EuMC), Nuremberg, Germany, 2017, pp. 408-411.

10. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits. New York, NY, USA: McGraw-Hill, 1948.

11. H. Mextorf and R. Knöchel, "The Intrinsic Impedance and Its Application to Backward and Forward Coupled-Line Couplers," in IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 2, pp. 224-233, Feb. 2014.