A Node-to-Node Admittance Functions Implementation of an Improved Frequency Dependent Multiconductor Transmission Line Model

The electromagnetic interference (EMI) performances of the interconnects and cables can be predicted via a standard multi-conductor transmission line (MTL) model, while the latter is not valid for the evaluation of power rail collapse and ground bounce responses. To circumvent the limitations, a more general and feasible improved MTL representation is presented in this paper. It physically incorporates the partial resistance and partial inductance parameters of all signal and reference conductors. To consider the frequency dependent behavior of the per-unit-length (PUL) electrical parameters in time domain simulations, a terminal description for this improved MTL model with any desired length is demonstrated. Subsequently, an equivalent node-to-node admittance functions (NAFs) implementation for this terminal representation is carried out. The correctness and effectiveness of the NAFs circuit model in time domain is then numerically validated by analyzing two dedicated examples.


I. INTRODUCTION
Electromagnetic compatibility (EMC) characteristics involved in electronic and electrical systems have been dramatically aggravated because of the increased operating frequency and decreased rising and falling times [1]- [6]. Generally, for the purpose of EMC predictions, the interconnects and cables in the systems can be analyzed by means of full-wave electromagnetic approaches [7]- [12], an electromagnetic topology principle [13]- [19], a Kron reduction technique [20]- [22], or a standard multi-conductor transmission line (MTL) model [23]- [27].
Specifically, the powerful standard MTL model is widely applied due to the simple implementation. Nevertheless, as can be noted, a subtle characteristic of this commonly used standard MTL model is that the general per-unit-length (PUL) loop electrical parameters, that is, loop resistances and loop inductances are adopted [24]. With this unique feature the wave equations for voltage and current signals can be The associate editor coordinating the review of this manuscript and approving it for publication was Mohd Zainal Abidin Ab Kadir . analytically or numerically solved in time domain [28]- [37] or in frequency domain [38]- [47]. Alternatively, in frequency domain, the MTLs can be modeled via a macro-modeling approach based on this standard MTL representation [48].
As explained in [49]- [51], PUL loop resistances and loop inductances essentially involve the electrical properties both from the signal conductors and the reference one (or named ground). Typically, loop resistances and loop inductances can be placed in either the associated signal conductor or the reference one while they cannot be uniquely assigned to either conductor. Therefore, in the standard MTL model, the reference conductor is implicitly treated as an ideal one so that there is no longitudinal potential difference along it. As a consequence, the application of PUL loop resistance and loop inductance parameters makes the standard MTL model not applicable to predict the EMC phenomena related to the reference conductor. It means that the voltage drops across each conductor, such as the power rail collapse and ground bounce behaviors cannot be computed uniquely [51]. As a matter of fact, they are the primary cause of most EMI and must be evaluated correctly.
To cope with the issues raised above appropriately, the important concepts of partial resistances and partial inductances as opposed to loop resistances and loop inductances which are well discussed in [49]- [51] can be employed. In [35] and [36], an improved MTL model is presented and analyzed by introducing partial resistances and partial inductances. Nonetheless, the voltage across each conductor can not be uniquely computed. This is because in the finite difference time domain solution of the MTL equations the partial resistance and partial inductance parameters are transformed into the corresponding loop ones. As a result, only the loop voltages, that is, the voltages of signal conductors with respect to the reference one can be computed.
As it is well known, EMC predictions in time domain is intuitive, feasible, and highly desired since in this case both linear and nonlinear devices can be easily taken into consideration. Generally, in a wideband frequency range, interconnects and cables reveal to be frequency dependent coupled lossy MTLs. Losses of MTLs due to skin-, proximity-, and dispersive-effects can further adversely degrade the signal and power transmission quality [24]. As a matter of fact, the accurate and efficient representation of the frequency dependent parameters in time domain is very challenge while significant on the MTL modeling. Therefore, in this paper, firstly an improved coupled lossy MTL representation is introduced. The PUL partial parameters, such as self-and mutual-partial resistances and partial inductances for all conductors including reference ones are considered. A terminal description of the improved MTL representation with a desired length is then presented. Next, to involve frequency dependent PUL parameters in time domain, the broadband terminal admittance matrix (TAM) representation for this terminal description model of an MTLs is numerically extracted via a frequency sweep analysis. A passive reduced order model for the TAM is then achieved via a matrix rational approximations (MRAs) technique. Finally, a node-to-node admittance functions (NAFs) implementation for the rational model of the terminal description representation can be applied.
For the numerical validation work in time domain, two dedicated test cases are analyzed in terms of the voltage responses, such as ground bounce, power rail collapse, and crosstalk. The time domain results are verified against a reference Inverse Fast Fourier Transform (IFFT) approach.

II. DESCRIPTION OF THE IMPROVED MTL REPRESENTATION A. IMPROVED MTL REPRESENTATION
Firstly, let us consider a coupled homogeneous (M + 1)conductor (M ≥ 1) MTLs with any desired length. It is sketched in FIGURE 1. As can be noted, the reference conductor (labeled #0) is regarded as a non-ideal one. Observe that as an equivalent, the MTLs with the desired length can be represented by an interface terminal description with in total (2M +2)-terminal (both ends of the (M +1)-conductor). Note that the left and right terminals of the reference conductor are numbered as (2M + 1) and 0, respectively. Therein, terminal 0 is identified as the voltage reference for all the other terminals and its choice can be somehow arbitrary.
In a given frequency range, let us consider an electrically small section ( x) of an MTLs under the quasi transverse electromagnetic (TEM) approximation. This PUL section can be equivalent to an improved distributed parameter MTL representation (FIGURE 2) by incorporating partial resistances and partial inductances [49]- [51]. As illustrated in FIGURE 2, the partial inductances and partial resistances are uniquely ascribed to each conductor; r pii and l pii (i = 0, 1, . . . , M ) represent the PUL self partial resistances and self partial inductances for each conductor, respectively; r pij and l pij (i, j = 0, 1, . . . , M ) are the PUL mutual partial resistances and mutual partial inductances between conductors i and j, respectively. Note that elements g ij and c ij (i, j = 1, 2, . . . , M ) indicate the PUL conductance and capacitance parameters, respectively. With reference to FIGURE 2, the partial voltage drops V pi (x, t) across each signal conductor i (i = 1, 2, . . . , M ) and V p0 (x, t) across the reference one are typically defined as power rail collapse and ground bounce responses, respectively [50], [51]. They can be physically written in a concise matrix form: and are (M + 1)×1 partial voltage and current vectors, respectively; in addition and are (M + 1) × (M + 1) PUL partial resistance and partial inductance matrices, respectively. The quasi-TEM assumption implies the current relations where I p0 (x, t) and I pi (x, t) (i = 1, 2, . . . , M ) are governed respectively by is the voltage across the terminals i and j.
It can be noted that from (1) the partial voltage vector V p (x, t) across each conductor depends on the self-and mutual-partial resistances and inductances and current vector I p (x, t). As shown in (7) and (8) the partial currents are associated with mutual capacitances and conductances and also with the voltage across each two terminals.
A remark is due here. For the case of frequency independent PUL parameters, (2) and (3) in time domain can be numerically obtained by simulating enough cascaded electrically small sections. On the contrary, for the case of frequency dependent PUL parameters, the time domain responses of an MTLs can be obtained by using the IFFT approach [52]. Although it is only applicable for linear terminations and can be time consuming for a wide frequency band and a long MTLs, it provides a reference solution for the validation purposes in this paper. In Section III, an alternative yet more efficient time domain approach is presented for an MTLs with frequency dependent PUL parameters based on the terminal description.

III. NAFs IMPLEMENTATION FOR THE IMPROVED MTL MODEL WITH FREQUENCY DEPENDENT PARAMETERS A. TAM REPRESENTATION OF THE IMPROVED MTL MODEL WITH A DESIRED LENGTH
Consider an MTLs with the desired length based on an elementary section of improved MTL model in FIGURE 2. Following [53] the (2M + 2)-terminal description (FIGURE 1) of the improved MTL model can be represented by an NAFs circuit network y(s) (s is the Laplace variable) totally including (M + 1) × (2M + 1) equivalent admittance elements. This model is illustrated in FIGURE 3 in detail. It can be noted that, each pair of terminals is associated with an admittance element: element y i,i (s) (i = 1, 2, . . . , 2M + 1) is the self-admittance between terminal i and the reference 0; while element y i,j (s) (i, j = 1, 2, . . . , 2M + 1, and i = j) indicates the mutual-admittance between terminals i and j.
Referring to FIGURE 3, assume that terminal i (i = 1, 2, . . . , 2M + 1) is excited by a sinusoidal voltage source u i with respect to the voltage reference terminal 0. Then the current i i flowing into the terminal i can be given by: where branch currents i i,j (i, j = 1, 2, . . . , 2M + 1) and voltages u i associated with terminal i are linked via admittances y i,j : Substituting (10) into (9) yields: i i = y i, 1 + · · · + y i, i + · · · + y i, M + y i, M +1 + · · · + y i, 2M +1 u i − y i, 1 u 1 + · · · + y i, i−1 u i−1 + y i, i+1 u i+1 + · · · + y i, 2M +1 u 2M +1 (11) or, in a compact matrix form for all terminals, one obtains: (12) where i(s) is terminal current vector and u(s) is the terminal voltage vector with respect to reference 0; and they are: In (12), the symmetric Y(s) represents the TAM of the improved MTL model with any desired length. It reads with matrix elements defined as and for i = j. Substituting (16) into (15) yields so that (16) can be rewritten as for i = j. Remark that in this paper, the entries Y i,j (s) of TAM are defined with uppercase letters while the circuit elements y i,j (s) are defined with lowercase ones.

B. EXTRACTION OF TAM
In this sub-section, two different approaches, that is an indirect method and a direct method are presented in order to numerically extract the TAM Y(s) in (14) for an MTLs. Basically, the improved MTL model in Laplace domain depicted in FIGURE 4 is adopted for the two approaches. Note that enough cascaded sections must be employed to achieve the desired length of an MTLs for the frequency sweep analysis in the prescribed frequency samples.  In FIGURE 4, the frequency dependent PUL partial resistance and inductance parameters can be numerically evaluated by using a quasi-static magnetic field solver, e.g. [54]. Observe that the frequency dependent behaviors of the capacitances and the conductances are neglected. They can be computed via the static electric field solver [54].
Based on the improved MTL model in FIGURE 4, y i,j (s) (i, j = 1, 2, . . . , 2M + 1) of an MTLs with desired length can be computed by using an AC analysis method with the aid of circuit simulators. In this paper, the MATLAB/Simulink platform is preferred since the circuit model in Simulink can be easily invoked through a script in MATLAB.
The evaluation schematic of y i,j (s) for the indirect approach is illustrated in FIGURE 5 [55]. The procedures can be organized as follows: (1) in the frequency range of interest, determine the total sections of an MTLs to achieve the given length. (2) As depicted, apply an AC voltage source U i,j (s) with unit amplitude between terminals i and j. (3) Short all remaining terminals, and then connect this new point of presence G to the negative terminal of the source supply. In this case terminals G and j share the same electric potential. This means that i j (s) = 0. (4) Perform a frequency sweep analysis in the frequency range of interest in order to collect the targeted current i i,j (s). Finally, y i,j (s) can be evaluated as Subsequently, all other admittance elements associated with NAFs y(s) can be obtained by using this well-defined procedure. Therefore, one can easily obtain the desired Y(s) via (15) and (16). However, a remarkable drawback of this indirect method is that (M + 1)×(2M + 1)/2 times simulations must be carried out considering the symmetric property of y(s).

2) DIRECT METHOD
Alternatively, a direct approach is introduced in this subsection. It is a preferred and more efficient numerical method compared with the indirect one. In this approach, the numerical simulation setup for evaluating TAM Y(s) for an MTLs with the desired length can be represented in FIGURE 6 (a). Therein, terminal i (i = 1, 2, . . . , 2M + 1) is activated by an AC voltage source U i (s) while the others are short-circuited to the reference terminal 0.
The equivalent circuit of FIGURE 6 (a) can be detailly represented in FIGURE 6 (b). Due to the short-circuit connections, only circuit elements y i,j (s) (j = 1, 2, . . . , 2M + 1) remain. According to the Kirchhoff's current law and in terms of terminal i, one obtains: In FIGURE 6 (b), the relationship between I i,j (s) and voltage source U i (s) is given by Substituting (15) and (21) into (20) yields: Substituting (16) into (21) yields: Combining (22) and (23), the i-th row entry Y i,j (s) (j = 1, 2, . . . , 2M + 1) can be obtained by evaluating I i,j (s), so that Remark that with this direct approach, one can easily achieve a row of Y(s) with only once frequency sweep computation. As a result, (M + 1) circuit simulations in total are required for an (M + 1)-conductor MTLs. Clearly the direct approach for evaluating Y(s) is more computationally efficient compared to the indirect one.
Obviously, from (17) and (18), if Y i,j (s) can be rationally approximated, as a consequence y i,j (s) will be automatically achieved also in a rational representation. Note that the two rational models share the same set of poles. The following sub-section introduces a methodology that represents Y(s) with a stable and passive rational approximation. Then a circuit synthesis method of rational model of y(s) is given.

C. MRAs FOR TAM
With a straightforward rational approximation by using the MRAs approach [56]- [58] and a passivity enforcement method [59]- [60], the obtained TAM Y(s) for an improved MTL model with the desired length can be expressed by a passive partial fraction expansion: which satisfies the following passivity requirements where N is the degree of the approximation; a and C are the vector of the common poles and the matrix of the residues, respectively; D and H are the matrices of constant and proportional terms, respectively. Specifically, ''eig'' denotes evaluation of eigenvalues. The numerical implementation of (26) can be achieved by perturbation of the eigenvalues of these matrices with minimal changes [59]- [60]. From (25), using (17)-(18) one obtains k n s − a n + p + sq (27) with and for i = j and in which To this extent, the rational approximation representations of NAFs y(s) are obtained based on the admittances defined by (27) and (29).
Then following the circuit synthesis approach in [61] the frequency dependent rational representations in (27) and (29), that is, constant terms and s-proportional terms, real pole terms, and complex pole terms can be synthesized as equivalent circuits that only with constant circuit elements. VOLUME 9, 2021 Eventually, the NAFs circuit model for the improved MTL representation with the desired length can be realized.

IV. NUMERICAL VALIDATIONS
Practical applications of the presented NAFs circuit model based on the improved MTL representation are illustrated in this section. Two numerical examples for the analysis of power rail collapse, ground bounce, and crosstalk voltage responses are carried out. The well-known IFFT approach [52] is performed for the numerical validation purpose.

A. A TWO-CONDUCTOR DIGITAL MTLs SYSTEM
As a first example, consider a typical two-conductor digital MTLs. It is interfaced with transmitter and receiver drivers, as illustrated in FIGURE 7 (a). The interconnects include 2 coupled parallel circular power pins (S1 and S2) and a shared ground pin (G). The pin pattern and its geometrical dimensions are described in FIGURE 7 (b). For simplicity, suppose the excitation voltage source of this MTLs is characterized by a trapezoidal pulse (FIGURE 8 (a)) with A = 5, t on = 50 ns, and T = 100 ns. In this test, two values for rising and falling times t r = t f = 100 ps and t r = t f = 200 ps are considered to investigate the influence on transient responses. Simplifying the MTLs system with resistive source loads at the near end (NE) and frequency dependent capacitive loads at the far end (FE) leads to an equivalent circuit representation as in FIGURE 8 (b).
The bandwidth concerned herein is from 10 MHz to 10 GHz with a uniform step of 10 MHz. To extract the TAM accurately and efficiently, 10 cascaded PUL sections and the direct method are employed. Then, a 12th-order rational model (6 real poles and 3 complex conjugate pole pairs) in (25) is applied to achieve a good approximation for TAM. In this paper, a PC with an Inter(R) Core(TM) i7 CPU at 3.07 GHz with 12 GB of memory is employed. For this test, the computational times of the NAFs model and the IFFT approach with the same time step of 1 ps are reported in TABLE 1. This confirms that the NAFs model is more computationally efficient than the reference IFFT solution. The ground bounce voltage responses (V GB ) obtained by using different rising times and excitation source patterns are presented in FIGURE 9. Therein, ''S1'' indicates that only signal conductor 1 is excited, while ''S1S2'' implies that conductors 1 and 2 are activated simultaneously.
The power rail collapse voltage responses V 13 and V 24 are reproduced in FIGURE 10 (a) and (b), respectively. In addition, the NE and FE crosstalk responses V NE and V FE are presented in FIGURE 11 (a) and (b), respectively. Note that only signal conductor 1 is activated (''S1'') in this case.
From FIGURE 9, FIGURE 10, and FIGURE 11, a general conclusion can be drawn that using a smaller rising time (100 ps) can further degrade the EMC performances, since the ground bounce, power rail collapse, and crosstalk responses are increased accordingly. Additionally, from FIGURE 9, the same conclusion holds when more signal conductors are activated (''S1S2'') simultaneously. The oscillations in the voltage responses implies that the MTLs cannot transmit the signals with sufficient fidelity.
To characterize the accuracy of the proposed NAFs model, a root-mean-square (RMS) error is adopted. It can be

B. A FOUR-CONDUCTOR SHIELDED POWER CABLE
In this test, a four-conductor shielded power cable for low voltage applications is considered. Its geometric description and dimensions are illustrated in FIGURE 12 and TABLE 3. The three-phase conductors U, V, and W, the ground conductor G, and the shield S are considered for the cable modeling. The frequency range from 100 kHz to 100 MHz with 1000 linearly spaced frequency samples is concerned. A cable sample of 1 m is investigated. 10 electrically small sections are cascaded to acquire the TAM. Then an 18th-order   rational model in (25) is applied in order to achieve a good approximation for the TAM of the power cable.
The time domain simulation setup for the validation of the proposed NAFs model is illustrated in FIGURE 13. Again, a trapezoidal voltage source in FIGURE 8 (a) is used, where A = 1, T = 10 µs, t on = 900 ns, and t r = t f = 10 ns.       The voltage responses V 19 , V 50 , V 15 , and V GB defined in FIGURE 13 are shown in FIGURE 14 (a) and (b). The corresponding RMS errors are reported in TABLE 5. It can be noted that very accurate results are obtained compared with the reference IFFT solution, confirming the effectiveness and accuracy of the presented model.

V. CONCLUSION
In this paper, an improved MTL representation based on the introduction of the PUL partial resistance and partial inductance parameters is presented. It circumvents the limitations of the standard MTL representation and enables the unique computation of power rail collapse and ground bounce responses. Then the terminal description of the improved MTL representation with a desired length is demonstrated. In time domain, to consider the frequency dependent PUL parameters, an NAFs circuit model is implemented for the terminal description of improved MTL model with the aid of matrix rational approximations and circuit synthesis technique. Time domain voltage responses obtained from the NAFs model indicate a high computational accuracy against to the reference IFFT solution and confirm a computational effort reduction. The implemented model can be extended to nonuniform and more complex MTLs with any desired length and geometry. By utilizing this model, EMC issues, especially the responses of ground bounce and power rail collapse, raised by interconnects and cables can be addressed and further minimized efficiently.