In various embodiments, the present invention relates to simulation systems for the extraction of transmission-line parameters, in particular extraction from the scattering parameters of a transmission-line system.
Signal integrity analysis of high-performance electronic systems requires knowledge and utilization of transmission-line parameters such as, e.g., per-unit-length resistance, inductance, conductance, and capacitance matrices (collectively “RLGC parameters”) and propagation constants. Transient simulation of transmission lines using their transmission-line parameters tends to be more accurate than direct simulation of network parameters, e.g., scattering (or S-) parameters, of the transmission lines. The scattering parameters typically come from measurements (made, for example, by a vector network analyzer), simulations from numerical three-dimensional electromagnetic solvers, or simulations from circuit solvers or closed-form expressions.
Procedures for the extraction of transmission-line parameters from tabulated network parameters are fairly well known, and typically involve solving the opposite problem—computing network parameters from transmission-line parameters—in reverse (see, e.g., W. R. Eisenstadt and Y. Eo, “S-parameter-based IC interconnect transmission line characterization,” IEEE Trans. on Components, Hybrids, and Manufacturing Technology, Vol. 15, No.4, pp. 483-490 (1992), the entire disclosure of which is incorporated by reference herein). Most extraction algorithms utilize a discontinuity-detection-based phase-unwrapping algorithm, which converts a sequence of cyclic phases to their noncyclic counterparts by adding integer multiples of 2π to each cyclic phase. The multiple is the total number of discontinuities in the sequence of cyclic phases observed between zero and a particular frequency (of a cyclic phase) that is more than π in amplitude. This algorithm is simple and computationally efficient, but it requires that the input cyclic phase constant not have any artificial discontinuities. In theory, the frequency-dependent phase constants of transmission-line systems are smooth functions of frequency (i.e., have no discontinuities). However, in practice, some artificial and unintentional discontinuities may be present in cyclic phase constants due to numerical artifacts in the extraction algorithm.
Further, the use of discontinuity-detection-based phase unwrapping requires that the tabulated network parameters are known for non-arbitrary frequencies. Specifically, the frequency step and starting frequency cannot be more than a particular data-dependent constant. This constant, a positive number, is inversely proportional to the propagation delay in the transmission lines. When these constraints are not met, the discontinuity-detection-based algorithm cannot be applied reliably. In particular, applying this algorithm to data not meeting the constraints results in the noncyclic phase constant computed from the unwrapping being arbitrary by an integer multiple of
where l is the length of the line; this arbitrariness results in incorrect values for per-unit-length inductance and capacitance parameters.
Other phase-unwrapping algorithms have been demonstrated in which, unlike the discontinuity-detection-based algorithm, the unwrapped phase at a particular frequency depends only on the wrapped phase at the same frequency, rather than also on the values of the unwrapped phases prior to the particular frequency (see, e.g., L. F. Knockaert, et al., “Recovering lossy multiconductor transmission-line parameter from impedance or scattering representations,” IEEE Trans. on Advanced Packaging, Vol. 25, No.2, pp. 200-205 (2002), hereafter the “Knockaert reference,” the entire disclosure of which is incorporated by reference herein). While such phase-unwrapping algorithms do not impose the above-described numerical challenges, they tend to be computationally inefficient, as their time complexities are exponential with the number of transmission lines. This time complexity increases with the electrical length of the lines. Further, no existing formulation properly handles singularities, resulting in nonphysical discontinuities in transmission-line parameters. Therefore, these alternative phase-unwrapping algorithms may be unsuitable for real-world problems, and there is a need for simulators and simulation methods for the extraction of transmission-line parameters from tabulated network parameters that utilize discontinuity-detection-based phase unwrapping and that are numerically reliable and computationally efficient.
In various embodiments, the foregoing limitations of existing transmission-line simulations are herein addressed through the use of discontinuity-detection-based phase unwrapping without introducing artificial discontinuities in the cyclic phase constant. Further, prior to extracting the transmission-line parameters, the network-parameter representation of the transmission-line system may be analyzed to determine if discontinuity-detection-based phase unwrapping can be reliably applied. The approach described herein is more accurate and computationally efficient than the prior-art techniques described above.
In an aspect, embodiments of the invention feature a simulator for simulating a transmission-line system comprising at least one transmission line. The simulator includes or consists essentially of an analysis module and a simulator module. The analysis module extracts transmission-line parameters of the transmission-line system from a network-parameter representation thereof using discontinuity-detection-based phase unwrapping without introducing artificial discontinuities. The simulator module simulates the response of the transmission-line system to an input based at least in part on the extracted transmission-line parameters.
The transmission-line system may be lossy or lossless. The simulator may include an input module for receiving the network-parameter representation of the transmission-line system. The simulator may also include a de-embedding module for (a) receiving the network-parameter representation from a measurement system or an electromagnetic field solver, (b) removing measurement artifacts from the network-parameter representation, and (c) sending the network-parameter representation to the input module. The simulated response of the transmission-line system may be a time-dependent response. The simulator may include an output module for supplying the simulated response to a display, a device, and/or another simulator module. Prior to extracting the transmission-line parameters, the analysis module may analyze the network-parameter representation to determine if discontinuity-detection-based phase unwrapping can be reliably applied.
In another aspect, embodiments of the invention feature a method of simulating a transmission-line system comprising at least one transmission line. Transmission-line parameters of the transmission-line system are extracted using discontinuity-detection-based phase unwrapping without introducing artificial discontinuities, and a response of the transmission-line system to an input based on the extracted transmission-line parameters is simulated.
A network-parameter representation of the transmission-line system may be provided, and the transmission-line parameters may be extracted from the network-parameter representation. Providing the network-parameter representation may include or consist essentially of receiving the network-parameter representation from a circuit solver. Providing the network-parameter representation may include or consist essentially of (a) receiving the network-parameter representation from a measurement system or an electromagnetic field solver, and (b) removing measurement artifacts from the network-parameter representation. Prior to extracting the transmission-line parameters, the analysis module may analyze the network-parameter representation to determine if discontinuity-detection-based phase unwrapping can be reliably applied. The transmission-line system may be lossy or lossless. The simulated response of the transmission-line system may be a time-dependent response. The simulating step may produce simulated output values from the modeled transmission-line system, and the output values may be used to drive a device and/or a second simulation.
In the drawings, like reference characters generally refer to the same parts throughout the different views. Also, the drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the invention. In the following description, various embodiments of the present invention are described with reference to the following drawings, in which:
Simulation System
Referring to
The main memory 110 contains a group of modules that control the operation of CPU 105 and its interaction with other hardware components. An operating system 140 directs the execution of low-level, basic system functions such as memory allocation, file management, and operation of mass storage devices 115. At a higher level, an analysis module 145 and a simulation module 150 direct execution of the primary functions performed by embodiments of the invention, as discussed below, and a user interface module 155 enables straightforward interaction with simulator 100 over display 135.
An input module 160 accepts input data corresponding to a system or network to be simulated from, e.g., a mass storage device 115, keyboard 125, and/or position-sensing device 130, or in some implementations, from an external signal source. The input data may include or consist essentially of digitized information corresponding to the system to be simulated, i.e., one or more transmission lines. For example, the input data may be values representing S-parameters and/or lengths of the transmission lines. Generally, the input data will be a representation of the system in the frequency domain. An optional de-embedding module 165 removes any artifacts (e.g., the effects of measurement probes, connectors, etc.) from the input data. De-embedding may be performed by, e.g., procedures described in W. Kim, et al., “Implementation of broadband transmission line models with accurate low-frequency response for high-speed system simulations,” DesignCon 2006 Technical Paper Proceedings, the entire disclosure of which is incorporated by reference herein. Alternatively, input module 160 may directly accept as input data the set of S-parameters (de-embedded or otherwise), obviating the need for processing by de-embedding module 165.
An output module 170 directs output data from analysis module 145 and/or simulation module 150 to, e.g., a mass storage device 115 for storage, display 135 for presentation, a second simulator 175 for further analysis and/or simulation, and/or an external device 180 to operate as an input thereto (e.g., as a control signal to an electronic device).
Although the modules in main memory 110 have been described separately, this is for clarity of presentation only. As long as simulator 100 performs all necessary functions, it is immaterial how they are distributed therewithin and the programming or hardware architecture of simulator 100. Furthermore, the above-described implementation is exemplary only. Other hardware approaches are possible, e.g., the various modules of embodiments of the invention may be implemented on a general-purpose computer programmed with appropriate software instructions implementing the functions described below, or as hardware circuits (e.g., as an application-specific integrated circuit, or ASIC), or as mixed hardware-software combinations.
Simulation Method
The simulation method according to various embodiments of the invention is described below, beginning with the well-known prior-art formuation for the extraction of transmission-line parameters in reference to
In step 310, the input data is reordered in order of increasing frequency to facilitate the discontinuity-detection-based phase-unwrapping algorithm used to extract the propagation constants (described below in reference to step 320).
For exemplary transmission-line system 200, in step 300 simulator receives input data including or consisting essentially of the 2N-port tabulated network parameters, i.e., S-/Y-/Z-/ABCD-parameters, along with l. The tabulated parameters are known for Nf frequencies {f1, f2, . . . , fN
The two ends of each transmission line (or, the ports) are represented by z=0 and z=l. T(ω)∈C2N×2N denotes the multiport transmission parameters (i.e., the ABCD-parameters) of the lines. Ports on one side of the lines are numbered first followed by the ports on the other side; thus, if i refers to a port of a line at z=0, then the port on the other side is numbered N+i (see
The derivation of R(ω), L(ω), G(ω), and C(ω) from T(ω) begins with formulating the relationship therebetween, as is known in the art. V(z)∈CN×1 represents the vector of voltages across the lines at a distance z and I(z)∈CN×1 is the vector of currents flowing along the lines from z=0 to z=l at a distance z. The transmission line equations in the frequency domain may be written as:
where Z(ω) is the per-unit-length impedance of the lines, defined as Z(ω)=R(ω)+jωL(ω). Similarly, Y(ω)=G(ω)+jωC(ω) is the per-unit-length admittance of the lines. The matrix R(ω) may be decomposed as:
R(ω)=RDC+Rs(ω), (3)
where RDC is the DC resistance matrix and Rs(ω) is the remaining part of the resistance matrix. The matrix R′s(0)=0. The matrix L(ω) may be decomposed as:
L(ω)=Le+Li(ω), (4)
where Le is the external inductance matrix and Li(ω) is the internal inductance matrix. The matrix Li(∞)=0. The matrix G(ω) may be decomposed as:
G(ω)=GDC+Gd(ω), (5)
where GDC is the DC conductance matrix and Gd(ω) is the remaining part of the conductance matrix. The matrix Gd(0)=0.
Both Z(ω) and Y(ω) are also symmetric, as is known in the art. Also known is the fact that the matrix product Z(ω)Y(ω) can be diagonalized for most transmission-line configurations through the eigenvalue decomposition:
Z(ω)Y(ω)=E(ω)γ(ω)2E(ω)−1, (6)
where E(ω)∈CN×N is the nonsingular matrix of eigenvectors, and
γ(ω)=diag{γ1(ω), γ2(ω), . . . , γN(ω)}, (7)
where γi(ω)∈C is the ith propagation constant. Each γi(ω)=αi(ω)+jβi(ω), where αi(ω)∈R is the ith attenuation constant, βi(ω)∈R is the ith phase constant, and j=√{square root over (−1)}. Also, αi(ω)≧0 and βi(ω)≧0 ∀i and for ω≧0. Further, αi(ω) is decomposed into:
αi(ω)=αi
where αi
Next, matrix Γ(ω)∈CN×N is defined as:
Γ(ω)=E(ω)γ(ω)E(ω)−1. (9)
The characteristic impedance matrix, Zc(ω)∈CN×N, is then computed from Γ(ω) and Z(ω) as:
Zc(ω)=Γ(ω)−1Z(ω), (10)
and is known in the art to be symmetric. The transmission matrix T(ω) is defined as follows:
where
and the matrices A(ω), B(ω), C(ω), and D(ω) are:
A(ω)=cos h(Γ(ω)l),
B(ω)=sin h(Γ(ω)l)Zc(ω),
C(ω)=Zc(ω)−1 sin h(Γ(ω)l), and
D(ω)=Zc(ω)−1 cos h(Γ(ω)l)Zc(ω), (13)
where the quantities
cos h(Γ(ω)l)=E(ω)cos h(γ(ω)l)E(ω)−1, and
sin h(Γ(ω)l)=E(ω)sin h(γ(ω)l)E(ω)−1. (14)
As described above, in various embodiments the input data corresponds to l and the A, B, C, and D matrices on the left-hand side of equation (13) for frequencies f1 . . . fN
A(ω)=E(ω)cos h(γ(ω)l)E(ω)−1. (15)
Since cos h(γ(ω)l) is a diagonal matrix, both γ(ω) and E(ω) may be obtained from the eigenvalue decomposition of A(ω). Denoting ψ(ω) as the diagonal matrix of eigenvalues of A(ω), one might compute γ(ω) as
However, this value of γ(ω) is only the principal value of the propagation constant and therefore may not be the true value. If w=cos h(z), where w and z are complex numbers, then w is also equal to cos h(z+j2πn), where n is an integer. Therefore, the value of cos h−1(w) is arbitrary by j2πn. The value of cos h−1(w) for n=0 is the principal value and may not be true solution. Denoting PV[z] as the principal value of z, it is inferred that
PV[γ(ω)l]=cos h−1(ψ(ω)). (16)
Therefore, the product γ(ω)l is:
γ(ω)l=PV[γ(ω)l]+j2πζ(ω), (17)
where ζ(ω)∈ZN×N is a diagonal matrix with integer entries. From equation (17), it follows that the correct γ(ω)l differs from the PV[γ(ω)l] only by an imaginary number. Then, the real part of γ(ω)l, namely [γ(ω)l], is the same as that of PV[γ(ω)l]. Therefore, the attenuation constant, α(ω), is computed as
From equations (16) and (17), βl is written as:
β(ω)l=PV[β(ω)l]+2πζ(ω). (19)
ζ(ω) in equation (19) is calculated by phase unwrapping, in preferred embodiments performed by a discontinuity-detection-based phase-unwrapping algorithm. Once γ(ω) is known from equations (16)-(19), matrix Γ(ω) is computed from equation (9) using E, known already from equation (15).
In step 330, the characteristic impedance matrix Zc(ω) is extracted from B(ω) and Γ(ω)l using equation (13):
The final step in equation (20), which does not involve any unwrapped quantities, follows from equation (17) and the properties of the hyperbolic sine function. Therefore, the phase constant need not be unwrapped to compute Zc(ω).
Finally, in step 340, the transmission-line parameters are extracted. From equations (20) and (10), the matrix Z(ω) is computed:
And, from equations (21), (6), and (9), the matrix Y(ω) is computed:
Then, from equations (21) and (22), the matrices R(ω), L(ω), G(ω), and C(ω) are computed:
R(ω)=[Z(ω)], (23)
and
The phase constant is unwrapped during computation of Z(ω) and Y(ω) (and thus influences R(ω), L(ω), G(ω), and C(ω), as a result): as γ(ω) also appears outside of the hyperbolic sine function in equations (21) and (22), Z(ω) computed with and without unwrapped phase are typically different. Also, from equation (24), the quantity [Z(ω)] should increase with ω, to prevent L(ω) from becoming zero. However, [Z(ω)] will generally not increase if the phase constant was not unwrapped (the same argument applies for C(ω) in equation (26)).
Still referring to
Eigenvalue Position
In various implementations, the simulation method described above is not robust. For example, the discontinuity-detection-based phase-unwrapping algorithm utilized in step 320 may introduce artificial discontinuities. Specifically, during operation of simulator 100 the algorithm computes the unwrapped phase of ith phase constant, βi,i, at frequency fk, namely βi,i(ωk)l, from its wrapped counterpart, PV[βi,i(ωk)l], as:
βi,i(ωk)=PV[βi,i(ωk)l]+2πdi,i(ωk), (27)
where di,i(ωk) is the number of discontinuities from ω1 through ωk of more than π in magnitude among the differences between adjacent values of PV[βi,i(ωk)l]. Thus, the discontinuity-detection-based phase-unwrapping algorithm is generally accurate, simple to implement, and computationally efficient (i.e., memory and runtime scale linearly with Nf). However, in various embodiments, the algorithm operates incorectly, leading to gross inaccuracies in the values of the extracted transmission-line parameters.
In a first exemplary embodiment, the eigenvalues of A(ω) do not retain their respective positions in ψ(ω) with frequency. That is, if r is an eigenvalue of A whose position as a function of frequency is being sought, and if r(ωk−1) is at the ith diagonal element of ψ(ωk−1), then r(ωk) need not be at the ith diagonal element of ψ(ωk). Instead, r(ωk) may be at one of the other N−1 diagonal locations in ψ(ωk). In such an embodiment, the phases PV[βi,i(ωk)l] and PV[βi,i(ωk−1)l] computed from equation (16) may not be phases of the same phase constant, and the unwrapped phase computed from equation (27) may not correspond to one particular phase constant. Because βi(ωk)≠βj(ωk) for i≠j, discontinuities result in PV [βi,i(ωk)l]. Accordingly, discontinuities also result in the extraction of γi,i(ωk), R(ωk), L(ωk), G(ωk), and C(ωk). And, in some embodiments, di,i(ωk) is extracted incorrectly. These discontinuities lead to inaccuracies when the transmission-line parameters are extracted. Because there must be at least two eigenvalues of A(ω) for them to swap positions in ψ(ω), the above-described discontinuities generally result only for transmission-line systems having two or more lines.
Embodiments of the present invention prevent artificial discontinuitites during step 320 by preserving the position of each eigenvalue of A(ω) with frequency, i.e. (as described above), the position of r in ψ is held constant for varying ωk. Because propagation constants of transmission-line systems tend to be close in value, it may be difficult to hold each eigenvalue's position constant by tracking its value; but the respective positions of the eigenvalues may be held constant relatively easily by instead tracking the position of the eigenvectors. The eigenvectors corresponding to the eigenvalues are orthogonal to each other at a particular frequency, and the eigenvectors corresponding to an eigenvalue at two nearby frequencies tend to be oriented in nearly the same direction. That is, if e(ωk−1)∈CN×1 represent the eigenvector corresponding to r(ωk−1), then e(ωk−1) and e(ωk) are oriented in approximately the same direction, or equivalently e(ωk−1)He(ωk)≈1. This property is used to find the new position, say j, of each eigenvalue ψi,i(ωk−1). If j≠i, then ψj,j(ωk) is moved to ψi,i(ωk), and the corresponding eigenvector is also relocated to the ith column
Example
In an exemplary embodiment, the transmission-line parameters of a six-line transmission-line system are extracted from its S-parameters. The S-parameters are known from 10 MHz to 10 GHz with a uniform spacing of 10 MHz. The length, l, is 0.508 m. In this case, the transmission-line parameters were already known and were used to compute the S-parameters, in order to demonstrate discontinuity avoidance in accordance with embodiments of the invention. The transmission-line parameters were computed from the S-parameters with and without preserving the relative positions of eigenvalues.
Phase Constant Sign
In various embodiments, the discontinuity-detection-based phase-unwrapping algorithm computes di,i(ωk) incorrectly, primarily for lossless transmission-line systems, due to random (i.e., unpredictable) changes in the sign of PV[βi,i(ωk)l] with ωk (equivalent to a discontinuity of more than π in magnitude in PV[βi,i(ωk)l]. In theory, for lossless lines, values of γi,i(ωk) are imaginary, which implies that values of ψi,i(ωk) are real. However, in practice, e.g., when S-parameters are obtained from microwave circuit simulators, –i,i(ωk) has a nonzero but approximately negligible imaginary part. The quantity βi,i(ωk) has the same sign as [ψi,i(ωk)]. Because |[ψi,i(ωk)]| is approximately negligible, the sign of [ψi,i(ωk)] is typically random, leading to a corresponding randomness in the sign of βi,i(ωk). For lossy lines, this scenario typically does not occur, as |[ψi,i(ωk)]| (or alternatively αi,i(ω)) is not negligible.
In various embodiments, correcting for random fluctuations of the sign of PV[βi,i(ωk)l] enables the accurate extraction of transmission-line parameters. In such embodiments, merely ignoring the sign of PV[βi,i(ωk)l] by making it non-negative is insufficient, as PV[βi,i(ωk)l] is typically negative for some k. Thus, in preferred embodiments, PV[βi,i(ωk)l] is reconstructed from its magnitude, |PV[βi,i(ωk)l]|. Then, the correct sign of PV[βi,i(ωk)l] is determined from the slope of |PV[βi,i(ωk)l]| at fk (in general, the sign of PV[βi,i(ωk)l] is same as the sign of the slope of |PV[βi,i(ω)k)l]|). With this reconstruction, the aforementioned harmful effects are avoided.
Example
In an exemplary embodiment, the S-parameters of a lossless transmission line are obtained from a commercial microwave simulator. The length of the line is 75 mm, and its characteristic impedance is 25 Ω. Air is the dielectric medium. The S-parameters are obtained for frequencies from 1 MHz to 5 GHz with a uniform spacing of 1 MHz. For comparison in this example, analytical values for the transmission line parameters may be computed. The attenuation constant, α, is zero for a lossless line. The phase constant, β(f), is
where c is the velocity of light in air. The matrices R(ω) and G(ω) are zero for a lossless line, and L(ω) and C(ω) may be analytically computed as 83.33 nH/m, and 0.133 nF/m, respectively.
Reliable Application of Discontinuity-Detection-Based Phase Unwrapping
In various embodiments, prior to extraction of the transmission-line parameters, simulator 100 first determines if the discontinuity-detection-based phase-unwrapping algorithm may be reliably applied to the input data. The unwrapped phase computed from the discontinuity-detection-based algorithm may be arbitrary by an additive factor of 2nπ, where n is an integer (see equation (27)). To illustrate this, the wrapped phase PV[β(ω)l] is assumed to have a constant periodicity of fc, implying that there is one discontinuity of more than π in magnitude among the values obtained by subtracting the adjacent values of the wrapped phases in a span of fc frequency. Thus, the unwrapped phases computed from considering the first k cycles (or periods) of PV[β(ω)l] are same as those computed from considering the next k cycles (i.e., from k+1 to 2k cycles), as both cases will have the same number of discontinuities (the factor d in equation (27)). In general, the same unwrapped phases are computed considering any k consecutive cycles. However, the same unwrapped phase is computed even if the k cycles are not consecutive—a spurious result. Although all of the results with k cycles are the same, only one result is the correct one—the one obtained from the first k cycles. The other results differ from this correct result by multiples of 2π. Therefore, it is necessary to make sure no cycles of PV[β(ω)l] are omitted.
Unfortunately, when the input data includes or consists essentially of tabulated data, some cycles of PV[β(ω)l] may be omitted, as when the data are tabulated, the S-parameters are known generally only at discrete frequencies. Therefore, in some embodiments or applications, the S-parameters (and therefore PV[β(ω)l]) are not known for some cycles of PV[β(ω)l].
To ensure that no cycles are omitted, during step 300 simulator 100 ensures that the input data meets two conditions for discontinuity-detection-based phase unwrapping to be applied. First, the first cycle of PV[β(ω)l] (the one that starts at ω=0) should clearly not be omitted, thus forcing starting frequency, f1, to be within the first cycle. In other words, f1 is less than the periodicity of the first cycle. Second, the maximum frequency step, max{fk+1−fk}, should be less than the periodicity of the smallest cycle. If there are multiple propagation constants, then there are N PV[β(ω)l] waveforms. Therefore, f1 should be smaller than the smallest periodicity among the N first cycles, and the maximum frequency step should be smaller than the smallest periodicity among all cycles.
Computing the periodicity in PV[βi(ω)l], though possible, is often difficult, especially when the periodicity changes with frequency. Thus, simulator 100 utilizes the below approximate equivalent sufficient conditions that do not require computing the periodicities in PV[βi(ω)l]:
f1<fc
and
fk+1−fk<fc
where fc
where the quantity εr
If fc
Example
As an exemplary demonstration of the necessity for the determination of sufficient conditions, a lossless transmission line of length 0.3 m with known S-parameters for 0-10 GHz (in steps of 10 MHz) is considered. Then, S-parameters at some frequencies are selectively removed. The unwrapped propagation constant and other transmission-line parameters are computed for three cases: (1) when all data is present, (2) when the data from zero frequency to 1 GHz is removed, and (3) when the data from 4.01 GHz to 6.15 GHz is removed. Ideally, the unwrapped phases for each case should match for at least the portion of the data set containing data present in all three cases, i.e., from 6.16 GHz to 10 GHz. However,
Singularities Arising from the Inverse Hyperbolic Sine Function
As described above, the characteristic impedance matrix Zc(ω) is generally computed from equation (20). This computation is prone to numerical problems because of a singularity of the quantity (sin h(γ(ω)l))−1 in equation (20)—a condition where |sin h(γi,i(ωk)l)|≈0. The frequency at which singularity occurs is referred to as a singular frequency, fs. Herein, the set of all fs is denoted as fs. At each fs, there is a discontinuity in Zc(ω), which is a nonphysical artifact of this mathematical operation. Therefore, in preferred embodiments, at fs, Zc(ω) is computed differently from the equation (20) formualation.
This singularity issue arises mostly in lossless lines. In all lines (lossy or lossless), the quantity βi(ωk)l may be close to nπ, where n=0,1,2, . . . , at some ωk. In addition, in lossless lines, γi(ωk)l is imaginary, making |sin h(γi(ωk)l)|≈0 at these frequencies. For lossy lines, however, this situation generally does not arise, because the attenuation constant, αi(ω), is nonzero for f>0 and for all i, and |sin h(γ(ωk)il)|≠0 for f>0. However, for lossy lines, a singularity may still arise at f=0, described below in the next section.
To prevent artificial discontinuities in Zc(ω), simulator 100 proceeds as follows. First, the input data is characterized as lossless or lossy. If the data is lossy, simulator 100 calculates the characteristic impedance matrix as described above. However, if the data is lossless, simulator 100 identifies singular frequencies in the data before Zc(ω) is computed. The singular frequencies usually correspond to non-negative integer multiples of half wavelengths. This identification may be performed by verifying whether or not |sin h(jβi(ωk)l)| is numerically close to zero at each k. However, generally, for a spike to occur in Zc, the quantity |sin h(jβi(ωk)l)| need not even be close to zero—it must only be “small,” i.e., below a threshold that may change from data set to data set. Thus, simulator 100 defines, for each input data set, a threshold, t, the smallest value of |sin h(jΔβi(ωk)l)|. ∀k, ∀i, where Δβi(ωk)=βi(ωk+1)−βi(ωk). As may readily be shown, |sin h(jβi(ωk)l)|≧t ∀fk∉fs. In some embodiments, simulator 100 may utilize a threshold t even smaller than that defined above. Thus, simulator 100 identifies singular frequencies as whose for which the above inequality is not true.
At each singular frequency fs, the matrix Zc(ω) is computed by simulator 100 as the average of Zc(ω) at the two frequencies on either side of fs:
In this manner, the characteristic impedance matrix is extracted without introducing artificial discontinuities, and thus so are the transmission-line parameters of the transmission-line system.
Transmission-Line Parameter Extraction at Zero Frequency
At ω=0, computing Zc(ω), R(ω), L(ω), G(ω), and C(ω) for a lossy transmission-line system may result in artificial discontinuities in some embodiments or applications. In the extraction method described above, extracting L(0) and C(0) (see equations (24) and (26)) may result in undefined quantities due to, e.g., division by zero. And, it is generally not possible to compute L(0) and C(0) from only the input-data network parameters at ω=0, as these parameters at ω=0 do not contain any information about L and C (instead they contain information only about R and G). Thus, to accurately compute L(0) and C(0) without introducing artificial discontinuities, the network parameters, or equivalently, the L and C values, at neighboring frequencies may be utilized to extrapolate L(0) and C(0).
In other embodiments, a different procedure is proposed to compute RDC and GDC from S(0). For Y(ω)∈C2N×2N,the open-circuit impedance parameters, and Z(ω)∈C2N×2N, the short-circuit impedance parameters, then, for small values of αi
and
Equations (33) and (34) are good approximations because αi
In this manner, simulator 100 extracts accurrate transmission-line parameters at zero frequency without introducing artificial discontinuities.
Example
As an exemplary demonstration of the solution described above, S-parameters for a lossless transmission line having a length of 75 mm were obtained from a commercial microwave simulator. DC losses were then introduced into the data, and the values of DC resistance, conductance, and impedance were calculated both with and without the above-described correction.
Improved Simulation
Next, similar to step 320 described in relation to
In step 1160, analysis module 145 identifies all singular frequencies in the input data, as described above. Then, in step 1170, the characteristic impedance matrix is extracted. For all non-singular frequencies, the characteristic impedance is calculated utilizing equation (20) above, and, if the transmission-line system is lossless, then the characteristic impedance is calculated at the singular frequencies utilizing equation (32) above. In step 1180, the transmission-line parameters are extracted. Specifically, the RLGC parameters are calculated utilizing equations (23)-(26) above for all non-singular frequencies. And, as in step 1170, if the transmission-line system is lossless, the RLGC parameters are calculated at the singular frequencies utilizing equation (32) above. Further, if the first frequency in the input data is equal to zero and is a singular freqency, L(0) and C(0) are calculated using extrapolation, as describe above, and RDC, GDC, and Zc(0) are calculated using equations (33), (34), and (35), respectively.
Finally, as in step 350 of
In this manner, the full set of transmission-line parameters is extracted without introducing artificial discontinuities. Further, the memory complexity of the simulation method of
The terms and expressions employed herein are used as terms and expressions of description and not of limitation, and there is no intention, in the use of such terms and expressions, of excluding any equivalents of the features shown and described or portions thereof. In addition, having described certain embodiments of the invention, it will be apparent to those of ordinary skill in the art that other embodiments incorporating the concepts disclosed herein may be used without departing from the spirit and scope of the invention. Accordingly, the described embodiments are to be considered in all respects as only illustrative and not restrictive.
Number | Name | Date | Kind |
---|---|---|---|
4303892 | Weller et al. | Dec 1981 | A |
5313398 | Rohrer et al. | May 1994 | A |
5553097 | Dagher | Sep 1996 | A |
5946482 | Barford et al. | Aug 1999 | A |
6349272 | Phillips | Feb 2002 | B1 |
6675137 | Toprac et al. | Jan 2004 | B1 |
6785625 | Fan et al. | Aug 2004 | B1 |
6832170 | Martens | Dec 2004 | B2 |
6961669 | Brunsman | Nov 2005 | B2 |
7034548 | Anderson | Apr 2006 | B2 |
7127363 | Loyer | Oct 2006 | B2 |
7149666 | Tsang et al. | Dec 2006 | B2 |
7389191 | Furuya et al. | Jun 2008 | B2 |
7539961 | Dengi et al. | May 2009 | B2 |
7627028 | Frei et al. | Dec 2009 | B1 |
7865319 | Jacobs et al. | Jan 2011 | B1 |
8063713 | Cheng et al. | Nov 2011 | B2 |
8245165 | Tiwary et al. | Aug 2012 | B1 |
8386216 | Al-Hawari et al. | Feb 2013 | B1 |
20030208327 | Martens | Nov 2003 | A1 |
20070038428 | Chen | Feb 2007 | A1 |
20070073499 | Sawyer et al. | Mar 2007 | A1 |
20080120083 | Dengi et al. | May 2008 | A1 |
20080120084 | Dengi et al. | May 2008 | A1 |
20090184879 | Derneryd et al. | Jul 2009 | A1 |
20090284431 | Meharry et al. | Nov 2009 | A1 |
20090314051 | Khutko et al. | Dec 2009 | A1 |
20100318833 | Reichel et al. | Dec 2010 | A1 |
20110010410 | DeLaquil et al. | Jan 2011 | A1 |
20110218789 | Van Beurden | Sep 2011 | A1 |
20110286506 | Libby et al. | Nov 2011 | A1 |
20120326737 | Wen | Dec 2012 | A1 |
Entry |
---|
Vijai K. Tripathi et al., “A SPICE model for multiple coupled microstrips and other transmission lines,” 1985, IEEE Transactions on Microwave Theory and Techniques, vol. MTT-33, No. 12, pp. 1513-1518. |
Woopoung Kim et al., “Implementation of Broadband Transmission Line Models with Accurate Low-Frequency Response for High-Speed System Simulations,” 2006, DesignCon 2006, 25 pages. |
Jose M. Tribolet, “A new phase unwrapping algorithm,” 1977, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-25, No. 2, pp. 170-177. |
Hamid Al-Nashi, “Phase unwrapping of digital signals,” 1989, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, No. 11, pp. 1693-1702. |
Zahi N. Karam et al., “Computation of the One-Dimensional Unwrapped Phase,” 2007, Proceedings of the 2007 15th International Conference on Digital Signal Processing, pp. 304-307. |
G. Hebermehl et al., “Improved numerical methods for the simulation of microwave circuits,” 1997, Weierstrass Institute for Applied Analysis and Stochastics, pp. 1-14. |
Antoulast A.C., On the Scalar Rational Interpolation Problem, IMA Jrl. of Mathematical Control and Information, 3:61-88 (1986). |
Benner et al., Partial Realization of Descriptor Systems, System and Control Letters, 55(11):929-936 (Jun. 13, 2006 preprint). |
Blackburn, Fast Rational Interpolation, Reed-Solomon Decoding, and the Linear Complexity Profiles of Sequences, IEEE Transactions on Information Theory, 43(2): 537-548 (Mar. 1997). |
Bracken et al, S-Domain Methods for Simultaneous Time and Frequency Characterization of Electromagnetic Devices, IEEE Transactions on Microwave Theory And Techniques, 46(9):1277-1290 (1988). |
Cai et al, Displacement Structure of Weighted Pseudoinverses, Applied Mathematics and Computation, -153(2):317-335 (Jun. 4, 2004). |
Fitzpatrick, On the Scalar Rational Interpolation Problems, Math, Control Signal Systems, 9:352-369 (1996). |
Gallivan et al. Model Reduction MIMO Systems Via Tangential Interpolation, SIAM J Matris Anal. Appl., 26(2):328-349 (2004). |
Gallivan et al, Model Reduction via tangental interpolation, MTNS2002 (15th Symp. on the Mathematical Theory of Networks and Systems) (2002) 6 pages. |
Gallivan et al, Model Reduction via Transaction: An Interpolation Point of View, Linear Algebra and its Applications, 375:115-134 (2003). |
Hiptmair, Symmetric Coupling for Eddy Current Problems, SIAM J. Numer. Anal. 40(1):41-65 (2002). |
Lee et al. A Non-Overlapping Domain Decomposition Method with Non-Matching Grids for Modeling Large Finite Antenna Arrays, J. Comput. Phys., 203:1-21 (Feb. 2005). |
Lefteriu et al, Modeling Multi-Port Systems from Frquency Response Data via Tangential Interpolation, IEEE, 4 pages (2009). |
Li et al, Model Order Reduction of Linear Networks with Massive Ports via Frequency-Dependent Port Packing, 2006 43rd ACM/IEEE Design Automation Conference, pp. 267-272 (2006). |
Schrama, Approximate Identification and Control Design with Applicaiton to a Mechanical System. Delft University of Technology, Thesis, 294 pages (1992). |
Vandendorpe, Model Reduction of Linear Systems, and Interpolation Point of View, Univ. Catholique de Louvain, Center for Systems Engineering and Applied Mechanics, 162 pages (Dec. 1, 2004). |
Woracek, Multiple Point Interpolaton in Nevanlinna Classes, Integral Equations and Operator Theory, 28(1):97-109, (Mar. 1997). |
Eisenstadt et al, “S-Parameter-Based IC Interconnect Transmission Line Characaterization,” IEEE Transactions on Components, Hybrids, and Manufacturing Technology, vol. 15, No. 4, Aug. 1992, pp. 483-490. |
Leung, et al, “Characterization and Attenuation Mechanism of CMOS-Compatible Micromachined Edge-Suspended Coplanar Waveguides on Low-Resistivity Silicon Substrate,” IEEE Transactions on Advanced Packaging, vol. 29, No. 3, Aug. 2006, pp. 496-503. |
Narita et al, “An Accurate Experimental Method for Characterizing Transmission Lines Embedded in Multilayer Printed Circuit Boards,” IEEE Transactions on Advanced Packaging, vol. 29, No. 1, Feb. 2006, pp. 114-121. |
Chen, et al, “Per-Unit-Length RLGC Extraction Using a Lumped Port De-Embedding Method for Application on Periodically Loaded Transmission Lines,” 2006 Electronic Components and Technology Conference, pp. 1770-1775. |
Knockaert, et al, “Recovering Lossy Multiconductor Transmission Line Parameters From Impedance or Scattering Representations,” Proceedings of the 10th Topical Meeting on Electrical Performance of Electronic Packaging, Cambridge, MA, Oct. 2001, pp. 35-38. |
Knockaert, et al, “Recovering Lossy Multiconductor Transmission Line Parameters From Ipedance or Scattering Representations,” IEEE Transactions on Advanced Packaging, vol. 25, No. 2, May 2002, pp. 200-205. |
Kim, et al “Implementation of Boradband Transmission Line Models with Accurate Low Frequency response for High-Speed System Simulations,” DesignCon 2006, 25 pages, 2006. |
Han, et al, “Frequency-Dependent RLGC Extraction for a Pair of Coupled Transmission Lines Using Measured Four-Port S-Parameters,” 63rd ARTFG Conference Digest, Jun. 2004, pp. 211-219. |
Degerstrom, et al “Accurate Resistance, Inductance, Capacitance, and Conductance (RLCG) From Uniform Transmission Line Measurements,” Proceedings of the 18th Topical Meeting on Electrical Performance of Electronic Packaging, Oct. 2008, pp. 77-80. |
Oh et al, “Improved Method for Characterizing Transmission Lines Using Frequency-Domain Measurements,” Proceedings of the 13th Topical Meeting on Electrical Perofrmance of Electronic Packaging, Jul. 2004, pp. 127-130. |
Gruodis, et al, “Coupled Lossy Transmission Line Characterization and Simulation,” IBM J. Res. Develop., vol. 25, No. 1, Jan. 1981, pp. 25-41. |
Kiziloglu, et al, “Experimental Analysis of Transmission Line Parameters in High-Speed GaAs Digital Circuit Interconnects,” IEEE Transactions on Microwave Theory and Techniques, vol. 39, No. 8, Aug. 1991, pp. 1361-1367. |
Sampath, “On Addressing the Practical Issues in the Extraction of RLGC Parameters for Lossy Multiconductor Transmission Lines Using S-Parameter Models,” Proceedings of the 18th Topical Meeting on the Electrical Performance of Electronic Packaging, Oct. 2008, pp. 259-262. |
Lalgudi, et al, “Accurate Transient Simulation of Interconnects Characterization by Band-Limited Data With Propagation Delay Enforcement in a Modified Nodal Analysis Framework,” IEEE Transactions on Electromagnetic Compatability, vol. 50, No. 3, Aug. 2008, pp. 715-729. |
Hill, et al, “Crosstalk Between Microstrip Transmission Lines,” IEEE Transactions on Electromagnetic Compatibility, vol. 36, No. 4, Nov. 1994, pp. 314-321. |
Karam, “Computation of the One-Dimensional Unwrapped Phase,” Massachusetts Institute of Technology Thesis, Jan. 2006, 101 pages. |
Antoulas, A new result on passivity preserving model reduction, Systems & amp: Control Letters, 54(4): 361-374, Apr. 2005. |
Astolfi, A new look at model reduction by moment matching for linear systems, Decision and Control, 2007 46th IEEE Conference, pp. 4361-4366, Dec. 12-14, 2007. |
Peng et al, Non-conformal domain decompostion method with second-order transmission conditions for timeharmonic electromagnetics, Journal of Computational Physics 229, Apr. 10, 2010, pp. 5615-5629. |
Zhao et al, A Domain Decomposition Method With Nonconformal Meshes for Finite Periodic and Semi-Periodic Structures IEEE Transactions on Antennas and Propagation, vol. 55, No. 9, Sep. 2007. |
Wolfe et al, A Parallel Finite-Element Tearing and Interconnecting Algorithm for Solution of the Vector Wave Equation with PML Absorbing Medium IEEE Transaction on Antennas and Propagation, vol. 48, No. 2, Feb. 2000. |
Gutknecht, Block Krylov Space Methods for Linear Systems with Multiple Right-hand sides: An introduction, pp. 1-22, 2006. |
Badics et al, A Newton-Raphson Algorithm With Adaptive Accuracy Control Based on a Block—Preconditioned Conjugate Gradient Technique, pp. 1652-1655, 2005. |
Mayo et al., A Framework For The Solution of the Generalized Realization Problem, Linear algebra and its applications 425.2 (2007):634-662. |