As the data rate of high-speed digital channels is increasing, signal and power integrity mitigation requires accurate calculation and measurement details. S-parameters are widely used to characterize single-ended high-speed data links. As differential transmission lines are predominantly used in high-speed digital channels to achieve faster data transmission at a lower signal loss, the concept of mixedmode S-parameters was put forward in [1] to characterize differential systems physically. A single-ended to mixedmode S-parameter conversion formulation was derived in [2]. The matrix representation of the formulation can be written as , where is the mixedmode S-parameter, is the transformation matrix, and is the single-ended S-parameter. But this formulation is only true for uncoupled transmission lines, thereby making the odd-mode and even-mode impedances equal to the characteristic impedance ( = = ). However, this assumption will not hold for coupled transmission lines. Therefore, the coupling factor between differential transmission lines was introduced in [3] to derive a new formulation. However, the limitation of [3] is that it considers a frequency-independent microstrip model to perform intrapair analysis.
The absence of a multi-pair crosstalk formalization suitable for both tightly and loosely coupled differential transmission lines necessitates further investigation. In this paper, for the first time, we present a generalized mixed-mode interpretation of crosstalk applicable to any number (N) of differential ports, significantly expanding the scope and impact beyond previous research restricted to specific configurations [1]- [3]. The proposed theory was validated through a comparative study encompassing two transmission line structures crucial to highspeed digital channels: intra-pair stripline and microstrip line. Simulations demonstrated significant discrepancies between the novel and old approaches, highlighting the potential advantages of the proposed method. Further, the inter-pair crosstalk for a multi-pair differential stripline is also analysed with the proposed generalized conversion formulation. The results indicate that the old approach may fail to capture and analyse crosstalk in high-speed channels with sufficient accuracy, potentially compromising the design quality and performance of these systems. This paper is divided into five sections. In section a, the newly proposed generalized conversion formulation for an Ndifferential port network, along with the transformation matrices, is discussed. In section c, an intra-pair comparison of the proposed and existing formulations is conducted using stripline and microstrip configurations to evaluate the impact of the new approach. Also, the proposed formulations are validated by considering loosely coupled stripline and microstrip line pairs. The NEXT and FEXT comparison for a multi-pair differential stripline is showcased in section c. Finally, section e summarizes the benefits of the proposed formulation for the generalization of mixed-mode Sparameters for coupled differential high-speed channels.
A generalized N-differential port network, as shown in Fig. 1, was considered. The differential and common-mode normalized waves, considering a low-loss transmission line, will be [5] +
where , , , and are the differential-mode and common-mode forward and reverse power waves at the differential port 8j9, respectively. , , , , , and are the differential-mode and common-mode voltage, current, and impedance at the differential port, respectively. The port number 8j9 varies from 1 to N. The differential and common-mode voltage, current, and impedance are given by [1] 2 ;
(5)
2 ;
(7
where 8i9 is the single-ended port number varying from 1 to 2N and 8j9 is the differential port number varying from 1 to N. A generalized matrix representation of an N-differential port network will become
To convert the single-ended S-parameters into mixed-mode Sparameters, the differential and common-mode power waves should be represented in terms of the single-ended power waves. Since the odd-mode and even-mode impedances of a coupled transmission line are unequal, i.e., b , the coupling factors and need to be introduced [3]. There are a couple of different methods to calculate the mode impedance. One way is to use the closed-form formulations already defined in previous literature [5]. Another way is to use 2D field solvers to generate the odd-and even-mode impedances. The definition of coupling coefficient 8C9 in microwave directional coupler is given by [6] C
The coupling factors and can be calculated from the coupling coefficient, ,
Therefore, the modal impedances for the coupled transmission lines can be calculated from the following equations.
(11)
The generalized differential and common3mode power waves for coupled differential pairs at a differential port 8j9 in terms of the corresponding single-ended power waves can be derived by substituting equations ( 5)-( 7) and ( 11)-( 12) into ( 1)-( 4).
(
(16
where 8i9 is the single-ended port number and 8j9 is the differential port number. Substituting (13)-( 16) into the matrix (8) will give a generalized formula for mixed-mode Sparameters that consider transmission line coupling.
where 8i9 is the single-ended port number ranging from 1 to 82N9. Here, the matrices can be derived into their generalized representation for an N-differential port network, as shown in equations ( 18) and (19). From the generalized equations ( 17), (18), and (19), we will be able to perform a multi-pair analysis of coupled transmission lines. Compared to the stripline, the closed-form formulations for the odd-mode and even-mode impedances of the microstrip line are complicated. Therefore, a 2D-solver, Ansys Q2D, was used to obtain the mode impedances of the microstrip line pair.
The single-ended S-parameter of the stripline and microstrip line was generated from the 3D simulation performed in Ansys HFSS. Two cases were considered-1. Tightly coupled case: The intra-pair spacing (S) was assumed to be 3 mil.
The intra-pair spacing (S) was assumed to be 40 mils. The tightly coupled case was considered to realize the discrepancy between the new and old formulations. Fig. 3 shows the results from the stripline pair. The differential return loss (
) and insertion loss ( ) are plotted in Fig. 3 The following can be summarized from the tightly coupled case:
• For striplines, the new formulation gives lower differential return loss compared to the old formulation. • For microstrip lines, the new formulation gives higher differential return loss compared to the old formulation.
The loosely coupled case was considered to validate the new formulation. An intra-pair spacing of 40 mils (10×W, Wtrace width) was considered for both the stripline and microstrip line. Under the loosely coupled case, the assumption that the odd-mode and even-mode impedance are equal will hold = o . As a result, the coupling factors become unity ( = =1). This will reduce the transformation matrix 2 to zero 2 0 . Therefore,
From the differential return loss and insertion loss plots of the stripline in Fig. 3(c) and 3(d), the old and new conversion results overlap. This validates the explanation given above. The same is true with the microstrip line results in figures 4(c) and 4(d).
The generalized formulation (17) shall be used to perform an inter-pair analysis on transmission line structures. Here, as an example, a 2-differential pair stripline, designed for a 0 1 2 3 4 5 Freq(Hz) 10 10 -80 -60 -40 -20 SDD11 Old Conversion(Zoo=Zoe) New Conversion(Zoo b Zoe) dB (a) (b) (c) (d) Fig. 4. Coupled Microstrip line: (a) Differential Return Loss (tightly coupled), (b) Differential Insertion Loss (tightly coupled), (c) Differential Return Loss (loosely coupled), (d) Differential Insertion Loss (loosely coupled)
dB differential impedance of Zd = 92-ohm is used to compare the NEXT and FEXT between the old and new conversion. The geometry parameters, material properties, and calculated impedances are tabulated in Table 2.
In this work, we proposed a novel and generalizable formulation for converting single-ended to mixed-mode Sparameters. This framework leverages an N-differential port network, representing a significant advancement over existing methods that often lack scalability. Also, the generalization of transformation matrices, and makes it possible to perform multi-pair differential trace analysis. A threedimensional electromagnetic simulation was performed on stripline and microstrip line intra-pairs to compare the differential return loss and differential insertion loss between the old and new conversion formulations. Based on the results shown in Fig 3 0 1 2 3 4 5 Freq(Hz) 10 10 -70 -60 -50 -40 -30 -20 NEXT(SDD21) Old Conversion(Zoo=Zoe) New Conversion(Zoo b Zoe) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Freq(Hz) 10 10 -90 -80 -70 -60 -50 -40 FEXT(SDD41) Old Conversion(Zoo=Zoe) New Conversion(Zoob Zoe)
TABLE II INTER-PAIR STRIPLINE PARAMETERS Dielectric Constant (• 3.1 Dielectric Loss-tangent 0.01 Intra-Pair Spacing (S) 5 mils Inter-pair Spacing (S9) 5 mils Conductor Thickness (t) 1.2 mils Conductor Length (L) 1000 mils (1-inch) Conductivity 5.67e7 S/m Stripline Substrate Thickness (b) 12 mils Line Width (W) 5 mils Stripline Odd-Mode Impedance (Zoo) 45.91 « Stripline Even-Mode Impedance (Zoe) 57.19 «