Rational Function Fitting of Electromagnetic Transfer ... · domain response data, provides for accurate and broadband rational function fitting of the DUT. The resulting rational

Rational Function Fitting of Electromagnetic Transfer Functions from Frequency-Domain and Time-Domain Data

Se-Jung Moon, and Andreas C. Cangellaris

University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

Abstract — This paper presents a new methodology for syn-

thesis of broadband equivalent circuits for multi-port high speed interconnect systems from numerically obtained and/or meas-ured frequency-domain and time-domain response data. The equivalent circuit synthesis is based on the rational function fit-ting of admittance matrix, which combines the frequency-domain vector fitting process VECTFIT with its time-domain analog TDVF to yield a robust and versatile fitting algorithm. The generated rational fit is directly converted into a SPICE-compatible circuit after passivity enforcement. The accuracy of the resulting algorithm is demonstrated through its application to the fitting of the admittance matrix of a power/ground plane structure.

Index Terms — Rational function fitting, VECTFIT, TDVF

I. INTRODUCTION

Rational function fitting of transfer functions of passive electromagnetic devices and interconnect structures provides for their convenient incorporation in nonlinear circuit simula-tors used for subsystem and system-level simulation. Numer-ous methodologies have been proposed for the development of such rational function approximations of electromagnetic transfer functions. Most of the proposed methods to date util-ize either frequency-domain data or transient data, obtained either through numerical simulation or through measurement, as input to the fitting algorithm. From these two classes of methods we mention as representative examples the fre-quency-domain vector fitting process VECTFIT [1], [2] and its time-domain counterpart TDVF [3], since they are most closely related to this paper.

Transient responses tend to be rich in high-frequency in-formation, while frequency-domain responses are most suit-able for capturing the low and moderate frequency attributes of the device under test (DUT). This observation has prompted researchers to examine the possibility of combining data from both transient and frequency-domain responses to facilitate the broadband approximation of the electromagnetic transfer function of the DUT [4].

The methodology proposed in this paper falls in this latter category of transfer function approximation methodologies. The basic idea is to hybridize VECTFIT and TDVF into a single algorithm which, through the enrichment of the TDVF-extrapolated (rich in high frequencies) early-time transient response of the DUT with low and moderate frequency-domain response data, provides for accurate and broadband rational function fitting of the DUT. The resulting rational

fitting algorithm will be referred to as CRAFT (Combined Rational Approximation from Frequency-domain and Time-domain data).

The paper is organized as follows. In Section II, the motiva-tion and rationale for the CRAFT process are elaborated fur-ther. This is followed by the mathematical formulation of the proposed fitting methodology in Section III. In Section IV the passivity enforcement and the equivalent circuit generation method are discussed briefly. In Section V, the implemented CRAFT algorithm is validated and its accuracy examined through its application to the fitting of the two-port admit-tance matrix of a generic power/ground plane pair geometry. The paper concludes with a summary and an overview of on-going further development.

II. HYBRID TIME AND FREQUENCY DOMAIN FITTING

For high-speed nonlinear circuit analysis, and in particular for applications involving high-speed digital circuitry with switching speeds in the order of a few tens of picoseconds, passive component electromagnetic response data in the order of several tens of GHz is required. Data from the upper end of such a broadband response can be extracted from the early-time response the DUT. Whether obtained through numerical simulation or through measurement, early-time response cal-culation is most expedient. On the other hand, extracting mul-tiple-tens-of-GHz-frequency response data using frequency-domain numerical methods or network analysis is more cum-bersome and expensive. Frequency domain analysis tends to be most effective and useful for lower and moderate frequen-cies for which the electrical length of the characteristic di-mensions of the DUT is smaller than or in the order of the wavelength. In contrast, time-domain analysis, especially when done numerically, must be run for a sufficiently long interval of time in order to capture the low-frequency attrib-utes of the DUT that govern its steady-state response.

The aforementioned discussion hints to the opportunity provided by utilizing both early-time response data with fre-quency-domain data for the more robust and accurate fitting of the DUT response over a broad frequency bandwidth. One additional attribute of such a hybrid process is that the avail-ability and utilization of frequency-domain data can help re-duce spurious high-frequency noise associated with the pre-mature truncation of the recorder time-domain response. This noise is evident when the TDVF process of [3] is applied for

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the fitting of the DUT transfer function from prematurely truncated portions of its transient response.

One way to avoid the occurrence of such spurious high-frequency noise is by complementing the truncated transient response data with any available high-frequency data, ob-tained via frequency-domain modeling and/or measurement. Hence, we conclude that frequency-domain data, both at the low and the moderate-to-high end of bandwidth of interest, are useful in order to improve the accuracy and bandwidth generated rational fit of the transfer function. Use of a uni-form sampling of the frequency data over the bandwidth of interest provides good enough accuracy in the rational func-tion fitting.

III. THE CRAFT PROCESS

It is assumed that the DUT is a passive P-port system repre-sented by its matrix transfer function, (s)H . Let tM denote the recorded time-domain samples of the input and output signals and fM the number of frequencies at which the trans-fer function matrix is computed in the frequency domain. The rational function fit of the (ij)th element of (s)H is given by

{ }M N N

ij n,ijn nn n ij ijij

j nn 0 n 0 n 1

Y (s) rB(s)(s) b s a s d seX (s) A(s) s q= = =

= ≈ = = + +−∑ ∑ ∑Η , (1)

where n,ij{r } and n{q } denote, respectively, the residues and poles, both of which are complex, in general, while the terms

ij{d } and ij{e } are real constants. CRAFT is established to calculate all coefficients in (1) using both frequency-domain and the time-domain samples of the transfer function, which is realized by combining the VECTFIT and TDVF processes of [1] and [3]. The theory behind VECTFIT is described in [1] and will not be repeated here. We start our discussion with the equation used by VECTFIT for the iterative calculation of the rational fit. Let n{q }% be an initial guess for the poles, with

n{r }% the unknown residues. Then the following equation, with i,j = 1, 2,…, P, is solved in a least squares sense

N N

n,ij nij ij ij ij

n 1 n 1n n

r rd se H (s) H (s).s q s q= =

′ ′ ′+ + − ⋅ = − − ∑ ∑ %

% % (2)

With T

1,ij N,ij ij ij 1 Nr r d e r r′ ′ ′ ′ = u % %L L denoting the vector of unknowns, the system is written in compact form as

f f⋅ =A u b , (3)

where it is

f f

f

f f f f

f

ij 1 ij 11

1 1 1 N 1 1 1 N

f ,ij

ij M ij MM

M 1 M N M 1 M N

T

f ,ij ij 1 ij M

H (s ) H (s )1 1 1 ss q s q s q s q

,H (s ) H (s )1 1 1 s

s q s q s q s q

H (s ) H (s ) , i, j 1,2, ,P.

− −

− − − − =

− − − − − −

= =

A

b

L L% % % %

M M M M M M M M

L L% % % %

L K

(4)

Multiplication of (2) on the right by the Laplace transform of the input jX (s) followed by an inverse Laplace transform yields

N N

ij n,ij n, j ij j n n,ijn 1 n 1

y (t) r x (t) d x (t) r y (t)= =

′ ′≈ ⋅ + ⋅ − ⋅∑ ∑ % (5)

where, with ijY (s) denoting the output quantity resulting from the operation of the right-hand side of (2) on jX (s) the fol-lowing time-dependent quantities have been introduced

n

tj q ( t )1

n, jn 0

X (s)x (t) L e x( )d

s q−τ−

= = τ τ − ∫ %

%, (6)

n

tij q ( t )1

n,ij ijn 0

Y (s)y (t) L e y ( )d

s q−τ−

= = τ τ − ∫ %

%. (7)

Because jx (t) and ijy (t) are not continuous but sampled with a time step, t∆ , a linear interpolation method is utilized to calculate the convolution integrals of (6)-(7). This leads to the following equations

n, j m 1 n n, j m n,1 j m 1 n,2 j mx (t ) x (t ) x (t ) x (t )+ += µ + ν + ν (8)

n,ij m 1 n n,ij m n,1 ij m 1 n,2 ij my (t ) y (t ) y (t ) y (t )+ += µ + ν + ν (9)

nn

n

q tq t n

n n,1 2n

q tn

n, 2 2n

1 q t ee , ,q t

1 (q t 1)e .q t

∆∆

∆

− − ∆ +µ = ν =∆

+ ∆ −ν =∆

%%

%

%

%

%

%

(10)

Hence, in matrix form we write

t t⋅ =A u b , (11)

t t t t

t

1 1 N 1 1 1 N 1

t

1 M N M 1 M N M

T

t 1 M

x (t ) x (t ) 1 0 y (t ) y (t ),

x (t ) x (t ) 1 0 y (t ) y (t )

y(t ) y(t ) .

− − = − −

=

A

b

L LM M M M M M M M

L L

L

(12)

The combination of (3) and (11) yields

, .

⋅ =

= =

f f

t t

A u bA b

A bA b

(13)

This is the system solved in the least squares sense for the calculation of the poles and residues of the rational fit. The iterative process described in [1] is followed for the calcula-tion of the calculation of the poles first and then the extraction of the residues n,ij{r } and other coefficients ij{d } and ij{e } directly from the linearized equation (1) using the given tM samples of jx (t) and ijy (t) and fM of (s)H .

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IV. PASSIVITY ENFORCEMENT AND CIRCUIT GENERATION

Even though the generated rational fit contains only stable poles by construction, its passivity cannot be guaranteed. This is due to the well-known fact that while a passive system is also stable, the reverse is not necessarily true. The passivity requirement for the rational fit of the multi-port transfer func-tion of a passive system is particularly important when the synthesized equivalent circuit is to be utilized in a general-purpose, network analysis-oriented, nonlinear simulator.

Mathematically, passivity can be defined through the re-quirement that the admittance matrix of the system is positive real. Once the CRAFT process is completed, the passivity of the fitted admittance matrix is examined and, if violations are found, one of several methodologies (for example, [5]) can be used to rectify the situation. Subsequently, the process of [2] is used for the development of an equivalent circuit represen-tation of the rational fit.

V. NUMERICAL VALIDATION

The implemented CRAFT algorithm was used for the fit-ting of the transfer function the 2-port power-ground plane pair of Fig. 1. The depicted geometry is a generic example of a portion of the power distribution network used in the inte-grating substrate (PCB or package) of high-speed digital and mixed digital/RF systems. SPICE-compatible multiport mac-romodels of these structures are used for the predictive analy-sis of noise generation and coupling during high-speed switching. The plane pair is composed of two, parallel, 8cm×4cm conducting plates, separated by an insulating slab of relative permittivity of 4 and thickness 1mm. The two power pins form (with the adjacent circular edge of the ground plane serving as the second terminal) the two ports of the system. The position of two pins is 0.5 mm away from the two corners.

Fig. 1. A generic power-ground plane pair structure. Top plate is

ground, bottom plate is power. First, the time-domain responses associated with the extrac-

tion of the admittance parameters of the two-port structure are obtained using the quasi-three-dimensional FDTD modeling approach of [6]. A Gaussian pulse of 20dB bandwidth of 4 GHz is used as a voltage source at one port while the other port is short-circuited. The transient port currents are recorded over the interval 0 t 16 ns with a time step of 5 ps. As far as the frequency-domain responses are concerned, the model order reduction (MOR) methodology described in [7] is used in conjunction with the aforementioned quasi-3D-FDTD

model for the computation of the frequency-dependent admit-tance parameters over the bandwidth 4 MHz - 4 GHz. Over this bandwidth the admittance parameters are recorded at a total of 1000 frequency points using linear sampling. The recorded time-domain and frequency-domain data is subse-quently used both for the selection of the subset of data points used for the fitting and as the reference solution for testing the accuracy of the generated rational fit.

With the order N for the rational fit set at 100, the number fM of frequency-domain data was taken to be 51. Time-

domain data over the time interval 0 t 4 ns was used, which amounted to tM = 800. These are depicted in Figs. 2 and 3 for the current responses at Port 1 and Port 2, respectively, when Port 1 is driven. The selected samples from the fre-quency-domain responses that were used to complement the time-domain data in the application of the CRAFT process, are depicted with the dots in Figs. 4 and 5.

Use of these time-domain and frequency-domain data in CRAFT required 10 iterations for it to converge to a rational fit. Once the rational fits have been generated and checked for passivity, a SPICE netlist was used to: a) extrapolate the trun-cated time-domain data up to 16 ns; b) interpolate the fre-quency-domain data over the 4 GHz bandwidth. Figures 2-5 compare the generated responses with those from FDTD and MOR. Excellent agreement is observed.

V. CONCLUSION

In summary, a new methodology has been introduced for the rational function fitting of numerically obtained and/or measured frequency-domain and time-domain electromag-netic responses. Once such rational fits have been generated for the admittance or impedance matrix parameters of elec-tromagnetic multiports, SPICE-compatible equivalent circuits can be synthesized. The presented fitting methodology, called CRAFT, can be thought of as the hybridization of the fre-quency-domain rational fitting process VECTFIT and its time-domain counterpart TDVF. The utilization of both time-domain and frequency-domain data is shown to improve the accuracy and numerical robustness of the rational function fitting of electromagnetic responses over broad frequency bandwidths. On-going work examines the impact of time-response duration and frequency-domain data bandwidth on the accuracy of the generated fit.

ACKNOWLEDGEMENT

This work was supported in part by the Semiconductor Re-search Corporation and by the Air Force Office for Scientific Research.

Port 1 Port 2

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REFERENCES

[1] B. Gustavsen, and A. Semlyen, “Rational approximation of frequency responses by vector fitting,” IEEE Trans. Power De-livery, vol. 14, no. 3, pp. 1052-1061, July 1999.

[2] B. Gustavsen, “Computer code for rational approximation of frequency dependent admittance matrices,” IEEE Trans. Power Delivery, vol. 17, no. 4, pp. 1093-1098, Oct. 2002.

[3] S. Grivet-Talocia, F. G. Canavero, I. S. Stievano, and I. A. Maio, “Circuit extraction via time-domain vector fitting,” Proc. Int. Symposium on EMC, pp. 1005-1010, August 2004.

[4] M. Yuan, T.K. Sarkar, B.H. Jung, Z. Ji, and M. Salazar-Palma, “Use of discrete Laguerre sequences to exprapolate wide-band response from early-time and low-frequency data,” IEEE Trans. Power Systems, vol. 16, no. 1, pp. 97-104, Feb. 2001.

[5] B. Gustavsen, and A. Semlyen, “Enforcing passivity for admit-tance matrices approximated by rational functions,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 7, pp. 1740-1750, Jul. 2004.

[6] M. J. Choi, and A. C. Cangellaris, “A quasi three-dimensional distributed electromagnetic model for complex power distribu-tion networks,” Proc. 51st Electron. Comp. and Technol. Conf., pp. 1111-1116, May 2001.

[7] M. J. Choi, K. -P. Hwang and A. C. Cangellaris, “Direct gen-eration of SPICE-compatible passive reduced-order models of ground/power planes,” Proc. 50th Electron. Comp. and Tech-nol. Conf., pp. 775-780, May 2000.

Fig. 2. Transient response of Y11. Solid line: FDTD result; Dotted line: Response obtained from CRAFT-generated equivalent circuit.

Fig. 3. Transient response of Y21. Solid line: FDTD result; Dotted line: Response obtained from CRAFT-generated equivalent circuit. Fig. 4. Plot of |Y11||, |Y22| in dB. Solid line: MOR result; Dots: Frequency domain data used for fit; Dashed line: CRAFT rational fit. Fig. 5. Plot of |Y21|, |Y12|. Solid line: MOR result; Dots: Fre-quency domain data used for fit; Dashed line: CRAFT rational fit.

extrapolated data

Cur

rent

at p

ort 1

(A

)

−0.02

0.02

0

0.04

0.06

0.08

4 8 12 16 0 Time (ns)

0 2 4 1 3 Frequency (GHz)

Mag

nitu

de o

f Y11

, Y

22 (

dB)

-30

-20

-10

0

10

-50

-40

0 2 4 1 3 Frequency (GHz)

-30

-20

-10

0

10

-40

Mag

nitu

de o

f Y11

, Y

22 (

dB)

Cur

rent

at p

ort 2

(A

)

extrapolated data

4 8 12 16 0 Time (ns)

−0.04

0

0.04

0.08

0.12

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