A method for equal-length matching optimization of high-speed printed circuit board differential pair transmission path

Abstract

The application discloses a high-speed printed circuit board differential pair transmission path equal-length matching optimization method and belongs to the technical field of signal integrity modeling and simulation. The method comprises the following steps: constructing a multi-conductor transmission line model containing a reference conductor and extracting a per-unit-length parameter matrix; judging structural uniformity and symmetry; introducing a spatially-varying local modal basis and a continuous modal coupling term for describing the modal conversion caused by the gradual change of the structure; calculating the propagation time delay difference of two signal paths and constructing a phase calibration operator; integrating the modal conversion and the time delay mismatch effect into a unified propagation equation for solving; and applying the phase calibration operator to align the voltage and current at the recoupling reference surface, so that the equivalent differential mode impedance is finally calculated. The application integrates the combined effects of modal conversion and time delay mismatch for integrated modeling, and significantly improves the impedance simulation accuracy of the recoupling area of the differential pair serpentine compensation structure.

Description

Technical Field

[0001] This invention relates to the field of signal integrity modeling and simulation technology, specifically to a method for optimizing the length matching of differential pair transmission paths on high-speed printed circuit boards. Background Technology

[0002] In high-speed digital systems, interconnects can no longer be simply viewed as "ideal wires." The book *Signal Integrity and Power Integrity* systematically describes a commonly used engineering criterion: when the electrical propagation time of an interconnect is comparable to the signal rise time, the interconnect should be treated as a transmission line, and issues such as characteristic impedance, reflection, crosstalk, loss, dispersion, and changes in the return path caused by discontinuities in the reference plane must be considered. Consequently, the industry has developed a design centered on "impedance control." With the widespread application of high-speed serial interfaces, inter-chip interconnects, and high-frequency differential signals in electronic systems, differential pairs, as the primary transmission structure for suppressing common-mode noise and improving interference immunity, are extensively used in PCB and package interconnect designs. In high-speed differential interconnect design, to meet signal synchronization and timing requirements, engineering typically requires equal-length matching of the two transmission paths of the differential pair, with serpentine compensation structures being one of the most common methods. Serpentine compensation structures introduce periodic bends on the shorter signal path to maintain the same geometric length as the other path. Impedance control at the decoupling and recoupling points becomes a critical issue in PCB design and also places new demands on the accuracy of impedance simulation.

[0003] The traditional impedance calculation algorithm follows this process: field solution → RLGC(x) extraction → mode decomposition → transmission line cascading → impedance. Specifically, it obtains the RLGC (resistance, inductance, conductance, and capacitance per unit length) of the trace along its length through 2D / 2.5D field solution or parameter extraction. The differential pair is decomposed into odd / even modes (or differential / common modes) at each cross-section, outputting an impedance curve that varies with length (one impedance value at each location), typically Zodd(x) and Zeven(x). The trace is segmented, and cascaded as a "segmented uniform transmission line" to obtain S-parameters and impedance. In current technology, the analysis of differential pair impedance characteristics is usually based on odd / even mode theory, assuming that the differential signal maintains an ideal odd mode state throughout propagation. Based on this, the differential mode impedance is often obtained by simple conversion from the odd mode impedance. This process provides sufficient engineering guidance for many "approximately uniform, symmetrical, and weakly coupled" differential interconnects, and is therefore widely used by traditional impedance simulation and extraction tools.

[0004] However, in actual serpentine compensation structures, the above assumptions often fail due to local geometric asymmetry, continuous parameter variations, and differences in the propagation lengths of the two transmission paths. On the one hand, local structural changes can cause energy conversion (mode transition) between differential and common modes; on the other hand, the propagation delays of the two paths are not perfectly consistent, introducing phase mismatch at the recoupling point. Even if no significant mode transition occurs during propagation, this will manifest as an anomaly in the equivalent impedance during recoupling measurements. Therefore, traditional simulation methods lack sufficient accuracy in the recoupling region after decoupling in the differential serpentine compensation section. Currently, there is a lack of a unified method that can uniformly characterize mode transitions and delay mismatches at the physical propagation level and provide point-by-point impedance calculation results at the recoupling point.

[0005] To address the aforementioned issues, there is an urgent need for a high-speed printed circuit board differential pair transmission path length matching optimization method to solve the problems existing in traditional methods. Summary of the Invention

[0006] The purpose of this invention is to provide a method for optimizing the transmission path length matching of differential pairs on high-speed printed circuit boards. By integrating the combined effects of mode conversion and time delay mismatch for integrated modeling, the impedance simulation accuracy of the recoupling region of the differential pair serpentine compensation structure is significantly improved.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0008] A method for optimizing the transmission path length matching of differential pairs on high-speed printed circuit boards includes:

[0009] Step 1: Obtain the physical and material parameters of the differential pairs on the high-speed printed circuit board and construct a multi-conductor transmission line model;

[0010] Step 2: Establish a parameter matrix per unit length based on the multi-conductor transmission line model, and establish a propagation operator matrix based on the parameter matrix per unit length;

[0011] Step 3: Determine whether the differential pair structure is uniform and strictly commutatively symmetric. Specifically, determining whether the differential pair structure is uniform involves determining whether the local differential pair structure is modally convertible and whether the differential pair structure modes undergo mutual conversion during propagation. Determining whether the differential pair structure is strictly commutatively symmetric involves determining whether the even and odd modes are completely decoupled and each is equivalent to a single-mode transmission line.

[0012] Step 4: If the mode is uniform and symmetrical, perform modal decomposition using a fixed modal basis and calculate the differential mode impedance.

[0013] Step 5: If the structure is non-uniform or asymmetric, then introduce a spatially varying local modal basis transformation matrix at each location;

[0014] Step 6: Calculate the continuous modal coupling terms caused by the variation of the basis space based on the local modal basis transformation matrix;

[0015] Step 7: Calculate the propagation delay difference when the two signal paths reach the recoupled reference plane, and construct the diagonal phase shift matrix for phase calibration;

[0016] Step 8: Incorporate the continuous mode coupling terms and the diagonal phase shift matrix into the multi-conductor transmission line model to construct a unified propagation equation that includes mode transitions;

[0017] Step 9: Calculate the equivalent differential-mode impedance at the recoupling point based on the unified propagation equation.

[0018] Furthermore, the multi-conductor transmission line model is expressed in the frequency domain as follows:

[0019] ;

[0020] ;

[0021] In the formula, The spatial coordinates along the propagation direction of the transmission line, This refers to the angular frequency in frequency domain analysis.

[0022] The unit length parameter matrix is ​​as follows:

[0023] ;

[0024] ;

[0025] In the formula, ∈C （N×1） Let each conductor be a voltage vector relative to the reference conductor. ∈C （N×1） Let each conductor have its current vector. This is the series impedance matrix per unit length, used to describe conductor losses, inductive effects, and inter-conductor coupling. R is a parallel admittance matrix per unit length, used to describe dielectric leakage conductance, capacitance effect, and inter-conductor coupling. L G C ∈R （N×N） These are the resistance, inductance, conductance, and capacitance matrices per unit length, respectively.

[0026] Furthermore, in step 2, the propagation operator matrix is ​​established based on the unit-length parameter matrix, specifically as follows:

[0027] The multi-conductor transmission line model is rewritten in first-order state-space form, and the state vector is:

[0028] ;

[0029] Define the propagation operator matrix as follows:

[0030] ;

[0031] Then we have:

[0032] .

[0033] Furthermore, in step 4, if the mode is uniform and symmetrical, modal decomposition is performed using a fixed modal basis to calculate the differential-mode impedance, specifically as follows:

[0034] Will Abbreviated as When the structure is uniform, let V, I∝ Substituting into the multi-conductor transmission line model, we get:

[0035] ;

[0036] ;

[0037] In the formula, The propagation constant is undifferentiated by mode.

[0038] Propagation constant and corresponding eigenvector The eigenvalue problem arising from matrix ZY is as follows:

[0039] ;

[0040] For each mode m, the relationship between the voltage mode and the current mode is obtained as follows:

[0041] ;

[0042] In the formula, Let be the voltage eigenvector corresponding to the m-th propagation mode. Let be the current eigenvector corresponding to the m-th propagation mode. Let be the propagation constant of the m-th propagation mode;

[0043] The modal characteristic impedance is defined as the proportional relationship between voltage and current in the m-th mode, as follows:

[0044] ;

[0045] ;

[0046] In the formula, This is the series impedance matrix corresponding to the m-th propagation mode. Let m be the parallel admittance matrix corresponding to the m-th propagation mode. Let L be the modal characteristic impedance corresponding to the m-th propagation mode. Under strict geometric and environmental exchange symmetry, L and C have typical structures. The symmetric structure of the even / odd transformation matrix is ​​simultaneously diagonalized by the same constant matrix. The even / odd modes are completely decoupled and are each equivalent to a single-mode transmission line, resulting in:

[0047] ;

[0048] ;

[0049] Z diff Z is the differential-mode impedance of the differential pair. odd Z is the odd-mode impedance of the differential pair. cm Zeven is the common-mode impedance, and Zeven is the even-mode impedance of the differential pair.

[0050] Furthermore, in step 5, a spatially varying local modal basis transformation matrix is ​​introduced at each location, specifically as follows:

[0051] At each position x, a local modal transformation is performed, introducing a spatially varying local modal basis transformation matrix T(x), which is:

[0052] ;

[0053] .

[0054] In the formula, , In position respectively The modal coordinate vectors relative to the local modal basis transformation matrix T(x) are the expansion coefficients of the actual conductor domain voltage and current in the local modal domain, respectively.

[0055] Further, in step 6, the continuous modal coupling term caused by the variation of the basis space is calculated based on the local modal basis transformation matrix, specifically as follows:

[0056] Differentiating V after local mode transformation, we get:

[0057] ;

[0058] Substitute it into the multi-conductor transmission line model and multiply it by T on the left. -1 ,get:

[0059]

[0060] Similarly, the same local mode transformation is applied to I as to V, that is, let And then regarding its Differentiate and substitute into the multi-conductor transmission line equation. Finally, multiply both sides of the equation by the left side. The expression for the current in the local modal domain is obtained as follows:

[0061]

[0062] The final continuous modal coupling term is:

[0063] ;

[0064] In the formula, This is a simplified representation of the local modal basis transformation matrix T(x), where, Let be the equivalent series impedance matrix of the m-th mode in the local modal domain. Let be the equivalent parallel admittance matrix of the m-th mode in the local modal domain.

[0065] Further, in step 7, the propagation delay difference between the two signal paths when they reach the recoupled reference plane is calculated, and a diagonal phase shift matrix for phase calibration is constructed, specifically as follows:

[0066] Calculate the propagation delay difference when the two signal paths reach the recoupled reference plane. If the recoupling section is considered as the reference surface for DM / CM decomposition and reconstruction, then the diagonal phase shift matrix for single-ended variables is used. This indicates phase alignment, where the diagonal phase shift matrix... for:

[0067] .

[0068] Furthermore, by incorporating the continuous mode coupling term and the diagonal phase shift matrix into the multi-conductor transmission line model, a unified propagation equation including mode transitions is constructed, specifically:

[0069] Incorporating the continuous mode coupling term into the multi-conductor transmission line model, we get:

[0070]

[0071] In the formula, For in position The equivalent series impedance matrix corresponding to the m-th mode at position m, For in position The equivalent parallel admittance matrix corresponding to the m-th mode at position m, For continuous modal coupling terms Incorporating parameters that affect frequency, For in position Place, No. The propagation operator matrix corresponding to each mode;

[0072] The defined differential-mode / common-mode voltage and current can be written in matrix form as follows:

[0073] ;

[0074] ;

[0075] ;

[0076] ;

[0077] In the formula, , These are the differential mode voltage and differential mode current, respectively. , These are the common-mode voltage and common-mode current, respectively. and These represent the single-ended voltages of the two signal lines relative to the reference conductor. and These represent the single-ended current of the two signal lines, respectively. This is the transformation matrix that maps single-ended voltage to differential / common-mode voltage. This is the transformation matrix that maps single-ended current to differential / common-mode current;

[0078] Let the coupling reference plane be defined. Adding the diagonal phase shift matrix to the single-ended voltage and current, we get:

[0079]

[0080] In the above formula, and These represent the single-ended voltage vector and single-ended current after phase calibration, respectively. To recouple the reference plane Location, frequency The voltage vector at that time;

[0081] The unified propagation equation that includes mode transition is:

[0082] ;

[0083] In the formula, For x= Place, The propagation operator matrix corresponding to the mode, To apply phase alignment at the final reference plane, For in position Place, No. Voltage vectors corresponding to each mode For in position Place, No. The current vector corresponding to each mode and Positions Place, No. The modal voltage vector and modal current vector corresponding to each mode.

[0084] Further, in step 9, the equivalent differential-mode impedance at the recoupling point is calculated based on the unified propagation equation, specifically as follows:

[0085] The voltage and current vectors obtained from the unified propagation equations that include mode transitions are used to calculate the equivalent differential-mode impedance as follows:

[0086]

[0087] In the formula, This represents the differential-mode voltage and current after phase calibration at the recoupled reference plane. To recouple the reference plane Location, frequency The equivalent differential mode impedance at that time, and These represent the single-ended voltage vector and single-ended current after phase calibration, respectively.

[0088] In summary, the present invention has at least one of the following beneficial technical effects:

[0089] 1. This invention introduces a local modal basis and its spatial derivative to continuously model the mode conversion effect caused by structural gradual changes or local asymmetries into the propagation process, overcoming the limitation of traditional methods that can only handle mode conversion at discrete boundaries. Traditional impedance calculation algorithms assume that the differential signal remains in an ideal odd-mode state throughout the propagation process. However, this invention introduces a local modal basis transformation matrix T(x) at each position x and calculates the continuous modal coupling term K(x) caused by its spatial changes. This accurately characterizes the energy exchange process between differential and common modes caused by changes in line spacing, turning structures, and inconsistencies in the reference plane in the serpentine compensation structure. This provides a more accurate physical model for impedance calculation in the recoupling region and significantly improves simulation accuracy.

[0090] 2. This invention explicitly incorporates the propagation delay mismatch effect of two signal paths into the impedance calculation process by introducing a phase calibration operator. Addressing the phase inconsistency problem caused by the difference in geometric lengths of the two paths in a serpentine compensation structure, this invention constructs a diagonal phase shift matrix P(ω) to perform unified phase reference processing on the voltage and current variables at the recoupling point, accurately quantifying the differential-mode to common-mode conversion caused by phase inconsistency and its impact on the equivalent differential-mode impedance. Theoretical derivation shows that when the phase difference Δφ≠0, even if the input signal is ideally differential-mode, a non-zero common-mode component will appear at the recoupling cross-section, and this component exhibits frequency-dependent peak-valley characteristics. This invention is the first to incorporate this effect into impedance calculation in an explicit mathematical form, solving the problem that traditional methods cannot explain impedance anomalies caused by time delay differences.

[0091] 3. This invention constructs an impedance calculation framework that integrates mode conversion and time delay mismatch, enabling point-by-point impedance distribution results in the recoupling region in physical space. By incorporating the continuous coupling term K(x) into the unified propagation equation and combining it with phase alignment processing at the recoupling point, this invention achieves a logical closed loop from the "propagation evolution stage" to the "measurement definition stage." Compared with traditional methods, the method of this invention significantly improves the simulation accuracy at the AB coupling point and can intuitively display impedance changes as scalars in physical space, greatly enhancing the accuracy and interpretability of impedance simulation for differential serpentine compensation structures, and providing a more reliable signal integrity analysis tool for high-speed PCB design. Attached Figure Description

[0092] Figure 1 This is a schematic diagram of the process of the present invention;

[0093] Figure 2 This is a schematic diagram showing the impedance simulation comparison of the AB section at the recoupling point of a high-speed printed circuit differential pair transmission path equal length matching optimization method provided in an embodiment of the present invention. Detailed Implementation

[0094] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0095] like Figure 1 As shown, this invention provides a method for optimizing the transmission path length matching of differential pairs on high-speed printed circuit boards, comprising:

[0096] Step 1: Obtain the physical and material parameters of the differential pairs on the high-speed printed circuit board and construct a multi-conductor transmission line model;

[0097] Step 2: Establish a parameter matrix per unit length based on the multi-conductor transmission line model, and establish a propagation operator matrix based on the parameter matrix per unit length;

[0098] Step 3: Determine whether the differential pair structure is uniform and strictly commutatively symmetric. Specifically, determining whether the differential pair structure is uniform involves determining whether the local differential pair structure is modally convertible and whether the differential pair structure modes undergo mutual conversion during propagation. Determining whether the differential pair structure is strictly commutatively symmetric involves determining whether the even and odd modes are completely decoupled and each is equivalent to a single-mode transmission line.

[0099] Step 4: If the mode is uniform and symmetrical, perform modal decomposition using a fixed modal basis and calculate the differential mode impedance.

[0100] Step 5: If the structure is non-uniform or asymmetric, then introduce a spatially varying local modal basis transformation matrix at each location;

[0101] Step 6: Calculate the continuous modal coupling terms caused by the variation of the basis space based on the local modal basis transformation matrix;

[0102] Step 7: Calculate the propagation delay difference when the two signal paths reach the recoupled reference plane, and construct the diagonal phase shift matrix for phase calibration;

[0103] Step 8: Incorporate the continuous mode coupling terms and the diagonal phase shift matrix into the multi-conductor transmission line model to construct a unified propagation equation that includes mode transitions;

[0104] Step 9: Calculate the equivalent differential-mode impedance at the recoupling point based on the unified propagation equation.

[0105] In the serpentine compensation section and its transition region, due to factors such as changes in line spacing, turning structures, and incomplete consistency of the reference plane, the modal basis will change with spatial position, resulting in continuous coupling between differential and common modes. To describe this phenomenon, this invention introduces a position-dependent modal basis transformation matrix and explicitly incorporates its spatial derivative into the propagation equation. Thus, mode transition is no longer considered a discrete boundary perturbation, but rather modeled as a continuous physical process along the propagation direction. This modeling approach allows for an accurate description of differential / common mode energy exchange caused by structural variations or local asymmetries at the propagation level.

[0106] In step 1, considering N signal conductors including a reference conductor (return surface / ground), under quasi-TEM conditions, the multi-conductor transmission line model can be expressed in the frequency domain using a parameter matrix per unit length as follows:

[0107] ;

[0108] ;

[0109] In the formula, The spatial coordinates along the propagation direction of the transmission line. Here, ω represents the angular frequency in frequency domain analysis. The above is the frequency domain transmission line equation, and all parameter matrices change with frequency.

[0110] The unit length parameter matrix is ​​as follows:

[0111] ;

[0112] ;

[0113] In the formula, ∈C （N×1） Let each conductor be a voltage vector relative to the reference conductor. ∈C （N×1） Let each conductor have its current vector. This is the series impedance matrix per unit length, used to describe conductor losses, inductive effects, and inter-conductor coupling. R is a parallel admittance matrix per unit length, used to describe dielectric leakage conductance, capacitance effect, and inter-conductor coupling. L G C ∈R （N×N） These represent the resistance, inductance, conductance, and capacitance matrices per unit length. In engineering, "impedance calculation / extraction" essentially involves obtaining these matrix quantities that vary with frequency and, possibly, with location.

[0114] In step 2, the propagation operator matrix is ​​established based on the unit-length parameter matrix, specifically as follows:

[0115] To facilitate subsequent derivation, the multi-conductor transmission line model is rewritten in first-order state-space form, and the state vector is... for:

[0116] ;

[0117] Define the propagation operator matrix for:

[0118] ;

[0119] Then we have:

[0120] .

[0121] In step 4, if the mode is uniform and symmetrical, modal decomposition is performed using a fixed modal basis to calculate the differential-mode impedance, specifically:

[0122] Will Abbreviated as When the structure is uniform, let V, I∝ Substituting into the multi-conductor transmission line model, we get:

[0123] ;

[0124] ;

[0125] In the formula, The propagation constant is undifferentiated by mode.

[0126] Propagation constant and corresponding eigenvector The eigenvalue problem arising from matrix ZY is as follows:

[0127] ;

[0128] For each mode m, the relationship between the voltage mode and the current mode is obtained as follows:

[0129] ;

[0130] In the formula, Let be the voltage eigenvector corresponding to the m-th propagation mode. Let be the current eigenvector corresponding to the m-th propagation mode. Let be the propagation constant of the m-th propagation mode;

[0131] The modal characteristic impedance is defined as the ratio between voltage and current in that mode. (Strictly speaking, this is in the sense of a vector; in engineering, it is often further taken as an equivalent scalar), which is:

[0132] ;

[0133] ;

[0134] In the formula, This is the series impedance matrix corresponding to the m-th propagation mode. Let m be the parallel admittance matrix corresponding to the m-th propagation mode. The modal characteristic impedance is the impedance corresponding to the m-th propagation mode.

[0135] It is evident that the "characteristic impedance" of a coupled system naturally corresponds to modes rather than a single scalar; only when the specific conditions for modal decoupling are met will it degenerate into a familiar scalar form and directly output a curve.

[0136] When strictly geometrically and environmentally interchangeable symmetric (the two lines interchange invariant), L and C have typical structures. The symmetric structure of the even / odd transformation matrix is ​​simultaneously diagonalized by the same constant matrix. The even / odd two modes are completely decoupled, each equivalent to a single-mode transmission line, resulting in:

[0137] ;

[0138] ;

[0139] In the formula, Z diff Z is the differential-mode impedance of the differential pair. odd Z is the odd-mode impedance of the differential pair. cm Zeven is the common-mode impedance, and Zeven is the even-mode impedance of the differential pair; in a single-mode transmission line, the differential-mode impedance Z is the common-mode impedance. diff Equal to odd-mode impedance Z odd Twice that the common-mode impedance Z cm Equal to even-mode impedance Z even Half of. That is, from arrive Then .

[0140] In step 5, a spatially varying local modal basis transformation matrix is ​​introduced at each location, specifically as follows:

[0141] At each position x, a local modal transformation is performed, introducing a spatially varying local modal basis transformation matrix T(x), which is:

[0142] ;

[0143] .

[0144] In the formula, , In position respectively The modal coordinate vectors relative to the local modal basis transformation matrix T(x) are the expansion coefficients of the actual conductor domain voltage and current in the local modal domain, respectively.

[0145] In step 6, the continuous modal coupling terms caused by the variation of the basis space are calculated based on the local modal basis transformation matrix, specifically as follows:

[0146] Differentiating V after local mode transformation, we get:

[0147] ;

[0148] Substitute it into the multi-conductor transmission line model and multiply it by T on the left. -1 ,get:

[0149]

[0150] Similarly, the same local mode transformation is applied to I as to V, that is, let And then regarding its Differentiate and substitute into the multi-conductor transmission line equation. Finally, multiply both sides of the equation by the left side. The expression for the current in the local modal domain is obtained as follows:

[0151]

[0152] The final continuous modal coupling term is:

[0153] .

[0154] This is a simplified representation of the local modal basis transformation matrix T(x), where, Let be the equivalent series impedance matrix of the m-th mode in the local modal domain. Let be the equivalent parallel admittance matrix of the m-th mode in the local modal domain;

[0155] That is, the structure maintains strict exchange symmetry throughout the propagation direction, T is a constant matrix and K(x) ≡ 0, and the modes will not couple due to "coordinate system rotation"; when the structure cannot maintain strict exchange symmetry, T(x) must be used, and the off-diagonal elements of K(x) will lead to energy exchange between modes. ,Right now ).

[0156] Its effect on impedance is as follows: along the base rotation (asymmetry / parameter change) → continuous mode coupling → when reaching the recoupling surface, DM / CM is impure, so the port sees "equivalent impedance" that deviates from the pure odd mode impedance, accompanied by reflection and additional loss.

[0157] The continuous coupling described above refers to the mode conversion caused by the rotation of the path base. In serpentine / recoupled structures, another common mechanism exists: when the two lines reach a certain "recoupling / merging" reference section, there is a phase inconsistency, resulting in common-mode components even with differential-mode excitation. When the two paths merge again to form a recoupled structure, this propagation delay difference manifests as frequency-dependent phase mismatch. At this point, even if the input signal maintains an ideal differential-mode ratio during propagation, common-mode components will still be introduced due to the reference phase inconsistency during recoupling measurements or equivalent impedance calculations, leading to a shift in the differential-mode impedance calculation results. This invention introduces an equivalent phase calibration operator to provide a unified reference for the voltage and current variables at the recoupling point. This method allows the time delay mismatch effect caused by propagation path differences to be incorporated into the impedance calculation process in a clear mathematical form.

[0158] In step 7, the propagation delay difference between the two signal paths when they reach the recoupled reference plane is calculated, and a diagonal phase shift matrix for phase calibration is constructed, specifically as follows:

[0159] Let the single-ended voltage at the reference section be... , Differential-mode and common-mode voltages are defined as follows:

[0160] ;

[0161] ;

[0162] If the ideal differential-mode excitation introduces a phase difference after propagation ,make:

[0163] ;

[0164] ;

[0165] To reference the cross-section and determine the differential-mode voltage amplitude corresponding to the differential-mode excitation, the single-end excitation of the two conductors is taken as follows: Substituting into the above formula, we get:

[0166] ;

[0167] ;

[0168] The final result is:

[0169] ;

[0170] From the above formula, we can conclude that as long as... ≠0, even if the excitation is essentially differential mode, a non-zero common mode will appear at this cross section; and when (ω) As the frequency changes, the common-mode component will exhibit frequency-dependent peak-valley characteristics;

[0171] If the recoupling section is considered as the reference surface for DM / CM decomposition and reconstruction, then the diagonal phase shift matrix for single-ended variables is used. This represents phase alignment (or equivalent delay compensation), where the diagonal phase shift matrix... for:

[0172] .

[0173] First, it should be noted that mode transition and... Phase mismatch For two-mode coupling, the transition amplitude from odd excitation to even excitation under the first-order approximation can be written as:

[0174] ;

[0175] The term represents the complex amplitude of the even mode at x=E (the end before recoupling), the term κ(x,ω) represents the coupling strength between odd and even, and Δ The (ξ, ω) term represents the phase mismatch, and the formula is the cumulative mode transition of its integral accumulation. B represents the total length of the coupling region. This is represented by the position variable along the propagation direction, and ω represents the angular frequency. That is, coupling strength, propagation constant, phase accumulation, etc., all change with frequency. The imaginary unit represents the phase factor, which is the rotation of the complex amplitude. Is the even mode in position? ,frequency The local phase evolution rate describes how much phase accumulates per unit length during propagation of an even mode. For positional variables, representing values ​​from 0 to... The phase mismatch at every point along this path is accumulated. Describes the odd mode in position ,frequency The local phase evolution rate under the given conditions. It can be seen that DM→CM (odd→even) is usually affected by two points: one is... Secondly, the coherent accumulation of phase mismatch.

[0176] The overall process is as follows: At the propagation level, a transmission operator including mode conversion terms is used to uniformly solve the propagation process of the differential pair from any position to the terminal or recoupling surface, thereby obtaining the voltage and current distribution including the cumulative effect of mode conversion; at the recoupling measurement level, a phase alignment operator is introduced to correct the propagation delay mismatch, ensuring that the differential-mode / common-mode decomposition is based on a unified phase reference; after completing the above two steps, the equivalent differential-mode impedance at the recoupling point or any position is calculated based on the uniformly defined differential-mode voltage and differential-mode current. The following is a detailed explanation of the steps:

[0177] Incorporating the continuous mode coupling term into the multi-conductor transmission line model, we get:

[0178]

[0179] In the formula, For in position The equivalent series impedance matrix corresponding to the m-th mode at position m, For in position The equivalent parallel admittance matrix corresponding to the m-th mode at position m, For continuous modal coupling terms Incorporating parameters that affect frequency, For in position Place, No. The propagation operator matrix corresponding to each mode;

[0180] The defined differential-mode / common-mode voltage and current can be written in matrix form as follows:

[0181] ;

[0182] ;

[0183] ;

[0184] ;

[0185] In the formula, , These are the differential mode voltage and differential mode current, respectively. , These are the common-mode voltage and common-mode current, respectively. and These represent the single-ended voltages of the two signal lines relative to the reference conductor. and These represent the single-ended current of the two signal lines, respectively. This is the transformation matrix that maps single-ended voltage to differential / common-mode voltage. This is the transformation matrix that maps single-ended current to differential / common-mode current;

[0186] Let the coupling reference plane be defined. Adding the diagonal phase shift matrix to the single-ended voltage and current, we get:

[0187]

[0188] In the above formula, and These represent the single-ended voltage vector and single-ended current after phase calibration, respectively. To recouple the reference plane Location, frequency The voltage vector at that time;

[0189] The final unified propagation equation, which includes mode transitions, is:

[0190] ;

[0191] In the formula, For x= Place, The complex amplitude of the mode, To apply phase alignment at the final reference plane, For in position Place, No. Voltage vectors corresponding to each mode For in position Place, No. The current vector corresponding to each mode and Positions Place, No. The modal voltage vector and modal current vector corresponding to each mode.

[0192] The equivalent differential-mode impedance at the recoupling point is calculated based on the unified propagation equation, specifically as follows:

[0193] The equivalent differential-mode impedance is:

[0194]

[0195] In the formula, This represents the differential-mode voltage and current after phase calibration at the recoupled reference plane. To recouple the reference plane Location, frequency The equivalent differential mode impedance at that time, and These represent the single-ended voltage vector and single-ended current after phase calibration, respectively.

[0196] This invention provides an embodiment in which the simulated differential pair impedance diagram is as follows: Figure 2 As shown, Figure 2 a) Figure 2 b) in the diagram represents two scenarios in traditional impedance simulation. The different colored plots on the left represent the simulated equivalent differential-mode impedance (50-120 ohms). Figure 2 In the example, a) indicates that the equivalent differential-mode impedance increases to 120 ohms due to the increased spacing caused by the differential trace serpentine compensation. Figure 2 b) shows the simulation result of a blue 50 ohm blue line appearing in the single-ended trace due to decoupling caused by increased spacing in the differential trace serpentine compensation. This is compared to traditional impedance calculation methods, such as... Figure 2 As shown in c), AB is the cross-section of the serpentine line perpendicular to the conductor at the coupling point. The impedance optimization algorithm significantly improves the simulation accuracy at the AB coupling point. It also incorporates the effects of mode conversion and propagation delay, and can be intuitively displayed as scalar quantities in physical space. Through this integrated method, mode conversion and delay mismatch are no longer treated separately, but rather each performs its function in the propagation evolution stage and the measurement definition stage, forming a logical closed loop. This method not only explains the formation mechanism of impedance anomalies in the recoupling region but also provides spatial impedance distribution results, intuitively reflecting the impact of the serpentine compensation structure on signal integrity.

[0197] Embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0198] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0199] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0200] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0201] Contents not described in detail in this specification are prior art known to those skilled in the art. It is hereby indicated that the above description is intended to help those skilled in the art understand this invention, but does not limit the scope of protection of this invention. Any equivalent substitutions, modifications, improvements, or simplifications of the above descriptions that do not depart from the essential content of this invention fall within the scope of protection of this invention.

Claims

1. A method for optimizing the transmission path length matching of differential pairs on high-speed printed circuit boards, characterized in that, include: Step 1: Obtain the physical and material parameters of the differential pairs on the high-speed printed circuit board and construct a multi-conductor transmission line model; Step 2: Establish a parameter matrix per unit length based on the multi-conductor transmission line model, and establish a propagation operator matrix based on the parameter matrix per unit length; Step 3: Determine whether the differential pair structure is uniform and strictly commutatively symmetric. Specifically, determining whether the differential pair structure is uniform involves determining whether the local differential pair structure is modally convertible and whether the differential pair structure modes undergo mutual conversion during propagation. Determining whether the differential pair structure is strictly commutatively symmetric involves determining whether the even and odd modes are completely decoupled and each is equivalent to a single-mode transmission line. Step 4: If the mode is uniform and symmetrical, perform modal decomposition using a fixed modal basis and calculate the differential mode impedance. Step 5: If the structure is non-uniform or asymmetric, then introduce a spatially varying local modal basis transformation matrix at each location; Step 6: Calculate the continuous modal coupling terms caused by the variation of the basis space based on the local modal basis transformation matrix; Step 7: Calculate the propagation delay difference when the two signal paths reach the recoupled reference plane, and construct the diagonal phase shift matrix for phase calibration; Step 8: Incorporate the continuous mode coupling terms and the diagonal phase shift matrix into the multi-conductor transmission line model to construct a unified propagation equation that includes mode transitions; Step 9: Calculate the equivalent differential-mode impedance at the recoupling point based on the unified propagation equation.

2. The method for optimizing the transmission path length matching of differential pairs on a high-speed printed circuit board according to claim 1, characterized in that, The multi-conductor transmission line model is expressed in the frequency domain as follows: ； ； In the formula, The spatial coordinates along the propagation direction of the transmission line, This refers to the angular frequency in frequency domain analysis. The unit length parameter matrix is ​​as follows: ； ； In the formula, ∈C （N×1） Let each conductor be a voltage vector relative to the reference conductor. ∈C （N×1） Let each conductor have its current vector. This is the series impedance matrix per unit length, used to describe conductor losses, inductive effects, and inter-conductor coupling. R is a parallel admittance matrix per unit length, used to describe dielectric leakage conductance, capacitance effect, and inter-conductor coupling. L G C ∈R （N×N） These are the resistance, inductance, conductance, and capacitance matrices per unit length, respectively.

3. The method for optimizing the transmission path length matching of differential pairs on a high-speed printed circuit board according to claim 2, characterized in that, In step 2, the propagation operator matrix is ​​established based on the unit-length parameter matrix, specifically as follows: The multi-conductor transmission line model is rewritten in first-order state-space form, and the state vector is: ； Define the propagation operator matrix as follows: ； Then we have: 。 4. The method for optimizing the transmission path length matching of differential pairs on a high-speed printed circuit board according to claim 3, characterized in that, In step 4, if the mode is uniform and symmetrical, modal decomposition is performed using a fixed modal basis to calculate the differential-mode impedance, specifically: Will Abbreviated as When the structure is uniform, let V, I∝ Substituting into the multi-conductor transmission line model, we get: ； ； In the formula, The propagation constant is undifferentiated by mode. Propagation constant and corresponding eigenvector The eigenvalue problem arising from matrix ZY is as follows: ； For each mode m, the relationship between the voltage mode and the current mode is obtained as follows: ； In the formula, Let be the voltage eigenvector corresponding to the m-th propagation mode. Let be the current eigenvector corresponding to the m-th propagation mode. Let be the propagation constant of the m-th propagation mode; The modal characteristic impedance is defined as the proportional relationship between voltage and current in the m-th mode, as follows: ； ； In the formula, This is the series impedance matrix corresponding to the m-th propagation mode. Let m be the parallel admittance matrix corresponding to the m-th propagation mode. Let L be the modal characteristic impedance corresponding to the m-th propagation mode. Under strict geometric and environmental exchange symmetry, L and C have typical structures. The symmetric structure of the even / odd transformation matrix is ​​simultaneously diagonalized by the same constant matrix. The even / odd modes are completely decoupled and are each equivalent to a single-mode transmission line, resulting in: ； ； Z diff Z is the differential-mode impedance of the differential pair. odd Z is the odd-mode impedance of the differential pair. cm Zeven is the common-mode impedance, and Zeven is the even-mode impedance of the differential pair.

5. The method for optimizing the transmission path length matching of differential pairs on a high-speed printed circuit board according to claim 4, characterized in that, In step 5, a spatially varying local modal basis transformation matrix is ​​introduced at each location, specifically as follows: At each position x, a local modal transformation is performed, introducing a spatially varying local modal basis transformation matrix T(x), which is: ； ； In the formula, , In position respectively The modal coordinate vectors relative to the local modal basis transformation matrix T(x) are the expansion coefficients of the actual conductor domain voltage and current in the local modal domain, respectively.

6. The method for optimizing the transmission path length matching of differential pairs on a high-speed printed circuit board according to claim 5, characterized in that, In step 6, the continuous modal coupling terms caused by the variation of the basis space are calculated based on the local modal basis transformation matrix, specifically as follows: Differentiating V after local mode transformation, we get: ； Substitute it into the multi-conductor transmission line model and multiply it by T on the left. -1 ,get: Similarly, the same local mode transformation is applied to I as to V, that is, let And then regarding its Differentiate and substitute into the multi-conductor transmission line equation. Finally, multiply both sides of the equation by the left side. The expression for the current in the local modal domain is obtained as follows: The final continuous modal coupling term is: ； In the formula, This is a simplified representation of the local modal basis transformation matrix T(x), where, Let be the equivalent series impedance matrix of the m-th mode in the local modal domain. Let be the equivalent parallel admittance matrix of the m-th mode in the local modal domain.

7. The method for optimizing the transmission path length matching of high-speed printed circuit board differential pairs according to claim 6, characterized in that, In step 7, the propagation delay difference between the two signal paths when they reach the recoupled reference plane is calculated, and a diagonal phase shift matrix for phase calibration is constructed, specifically as follows: Calculate the propagation delay difference when the two signal paths reach the recoupled reference plane. If the recoupling section is considered as the reference surface for DM / CM decomposition and reconstruction, then the diagonal phase shift matrix for single-ended variables is used. This indicates phase alignment, where the diagonal phase shift matrix... for: 。 8. The method for optimizing the transmission path length matching of differential pairs on a high-speed printed circuit board according to claim 7, characterized in that, In step 8, the continuous mode coupling terms and the diagonal phase shift matrix are incorporated into the multi-conductor transmission line model to construct a unified propagation equation that includes mode transitions, specifically: Incorporating the continuous mode coupling term into the multi-conductor transmission line model, we get: In the formula, For in position The equivalent series impedance matrix corresponding to the m-th mode at position m, For in position The equivalent parallel admittance matrix corresponding to the m-th mode at position m, For continuous modal coupling terms Incorporating parameters that affect frequency, For in position Place, No. The propagation operator matrix corresponding to each mode; The defined differential-mode / common-mode voltage and current can be written in matrix form as follows: ； ； ； ； In the formula, , These are the differential mode voltage and differential mode current, respectively. , These are the common-mode voltage and common-mode current, respectively. and These represent the single-ended voltages of the two signal lines relative to the reference conductor. and These represent the single-ended current of the two signal lines, respectively. This is the transformation matrix that maps single-ended voltage to differential / common-mode voltage. This is the transformation matrix that maps single-ended current to differential / common-mode current; Let the coupling reference plane be defined. Adding the diagonal phase shift matrix to the single-ended voltage and current, we get: In the above formula, and These represent the single-ended voltage vector and single-ended current after phase calibration, respectively. To recouple the reference plane Location, frequency The voltage vector at that time; The unified propagation equation that includes mode transition is: ； In the formula, For x= Place, The propagation operator matrix corresponding to the mode, To apply phase alignment at the final reference plane, For in position Place, No. Voltage vectors corresponding to each mode For in position Place, No. The current vector corresponding to each mode and Positions Place, No. The modal voltage vector and modal current vector corresponding to each mode. B This represents the total length of the coupling region.

9. The method for optimizing the transmission path length matching of differential pairs on a high-speed printed circuit board according to claim 8, characterized in that, In step 9, the equivalent differential-mode impedance at the recoupling point is calculated based on the unified propagation equation, specifically as follows: The voltage and current vectors obtained from the unified propagation equations that include mode transitions are used to calculate the equivalent differential-mode impedance as follows: In the formula, This represents the differential-mode voltage and current after phase calibration at the recoupled reference plane. To recouple the reference plane Location, frequency The equivalent differential mode impedance at that time, and These represent the single-ended voltage vector and single-ended current after phase calibration, respectively.

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