Broadband equivalent circuit derivation for multi-port circuits based on eigen-state formulation

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WANE, Sidina. BAJON, Damienne. Broadband equivalent

circuit derivation for multi-port circuits based on eigen-state formulation. In: 2009

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Broadband Equivalent Circuit Derivation for Multi-Port Circuits

Based on Eigen-State Formulation

Sidina Wane

1

and Damienne Bajon

2

1

NXP-Semiconductors, Esplanade Anton Philips 14906, Colombelles BP 2000 CAEN Cedex 9 France

sidina.wane@ieee.org

2

ISAE-Université de Toulouse, 10 avenues Edouard Belin Toulouse-France

damienne.bajon@isae.fr

Abstract — In this paper a passive guaranteed wide-band

equivalent circuit derivation methodology, attempting to bridge physical geometry considerations with equivalent circuits model extractions, is proposed. Electromagnetic (EM) eigen-states formulation is introduced to bridge between physical topology (geometry) and equivalent network architectures. Instead of casting global Z or Y parameters in pole-residue expansions following or T-networks, the proposed methodology considers eigen-states input impedances/admittances as primary goal functions to derive in canonical equivalent circuit models. While classical or T representations are based on global ground assumptions, the proposed methodology refers to local ground references. The validity of the broadband extraction methodology is demonstrated through correlations with RF measurement carried out on CPW transmission lines and coupled RF inductors up to very high frequencies.

Keywords— Bridge/Trellis topology, Mittag-Leffler, Passivity

Preservation, Layout topology, Modal-States.

I.INTRODUCTION

Classical design methodologies built around library-centric concepts and associated tooling flows are generally based on schematic-oriented analysis and simulation. Main limitation of schematic-oriented simulation techniques lies in difficulties to incorporate topology-related parasitic influences requiring full-wave electromagnetic analysis to properly model their frequency dependent attributes and induced potential couplings and interferences.

Topology-oriented and Physics-based Broadband equivalent circuit derivations from EM analysis or measurement results represent real challenges for distributed time-domain and frequency-domain mixed-signal co-simulation and co-design. Although existing full-wave EM solver solutions can provide high accuracy simulations they find some limitations to include in their analysis DC contributions, while quasi-static responses suitable for DC analysis lack high frequency effects (e.g., losses, couplings,

eddy currents, substrate stack influences, etc…).

Available commercial BBS (Broad-Band-SPICE) model extractors are generally based on a black-box approach where the extracted equivalent circuits result from pole-residue numerical expansion techniques including controlled sources. Such numerical techniques remain mathematically abstract with no possibility to link extracted lumped element RLCG parameters to physical layout topologies for optimization or tuning purposes. In addition non-physical elements (e.g.,

negative inductance/resistance/capacitance) can be obtained

which can violate passivity and causality preservations. Foster

Z1 1- Z1 2 Z2 2- Z1 2 Z1 2 E q u i p o t e n t i a l Y1 1+ Y1 2 Y2 2+ Y1 2 -Y1 2 E q u ip o te n tia l (a) (b) S ig n a l-in XO d d XO d d XE v e n XE v e n G ro u n d -in S ig n a l-o u t G ro u n d -o u t S ig n a l-in XO d d XO d d XE v e n XE v e n G ro u n d -in S ig n a l-o u t G ro u n d -o u t (c)

Fig.1 Classical T (a) and (b) equivalent model representations based on equi-potential assumptions. Proposed: Bridge/Trellis architecture (c) based on Y-matrix for two-port representation with floating ground references.

canonical representations associated with pole-residue extractions give straightforward basis for lumped elements equivalent circuit model extractions, with however a limitation to lossless assumptions. When applied to ideal transmission line, Forster canonical representation coincides with Mittag-Leffler expansion [1]. Challenges remain as how to efficiently and systematically select equivalent circuit architecture in order to avoid introduction of “non-physical” lumped components. Innovative methodologies to systematically derive proper equivalent circuit architectures are required. Such topologies should extend classical or T architectures that fail to account for distributed grounding (as

illustrated in Fig.1(a-b) with equi-potential assumptions) and

broadband responses. The proposed extraction methodology circumvents such limitations with appropriate equivalent circuit architecture referenced in this work as Trellis/Bridge architecture [2]. Similar topology can be derived for Z-matrix.

The original wideband, physics-based, equivalent circuit models synthesis methodology proposed in this paper considers, for bridging between physical topology

(layout-driven) and equivalent network representations (lumped elements circuit driven), electromagnetic eigen-states as

introduced in [3] for filter analysis. The proposed equivalent circuit architectures shown in Fig.1(c) allows eigen-states responses to be mapped in equivalent circuit derivation and extend to distributed local ground references where input and output ports are referenced to different ground nodes. Link

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between physical layout and equivalent circuit representations is suggested for optimization/response-tuning purposes. The proposed methodology is presented for symmetrical two ports and multiports with N=2n ports and is validated by comparison with high frequency measurement.

II.THEORY AND PROPOSED METHODOLOGY

A. Problem Statement, Previous Work & Originality of the Proposed Contribution

Most of published broadband synthesis approaches, for extracting RLCG lumped elements equivalent circuits, are generally based on pole-residue expansions applied to cast Y or Z parameters in a or T network with global ground

assumption. Although application of pole-residue based

techniques can lead to efficient synthesis of the diagonal terms of Z or Y matrix elements, it faces challenging issues, especially for wideband extractions, with the off-diagonal matrix elements constituting the coupling branches in or T architectures. The problems with the off-diagonal Y and Z elements concern passivity and stability preservation, at

branch level, as for symmetrical structures they involve

subtractions of even and odd contributions. To overcome such problems, this contribution proposes synthesizing alternative state variables in place of Y and Z parameters. For symmetrical structures, such alternative independent state variables are the even and odd modal contributions. Furthermore, in the context of netlist oriented extractions, classical lumped elements equivalent circuit representations generally assume global grounding references which fail to reflect full-wave attributes. Major attributes of Trellis architectures in Fig.1(c) include possibility, by construction, to account for floating ground references not possible to deal with using classical and T representations in Fig.1(a-b). The associated Y or Z matrix are linear combinations of the

contributions of these two eigen-states (designated with the subscript Even/Odd for symmetrical two-port systems) following equation (1):

>

@

»

¼

º

«

¬

ª

»

¼

º

«

¬

ª

»

¼

º

«

¬

ª

»

¼

º

«

¬

ª

1

2

1

2

1

11 12 12 11 Odd Even Odd Even Odd Even Odd Even Odd Even port Two

X

(1)

where the even and odd impedance or admittance, respectively

X

_Even and

X

_Odd, are scalar complex functions. The notation X stands for impedance or admittance parameters. Challenges with passivity preservations when Z21

or Y21 is considered as goal function to cast in pole-residue

form result from the difficulty to ensure passivity of the difference

G

_OddEven between even and odd mode contributions:

Even Odd Odd

Even

X

G

(2) It should be underlined that, in some special cases, negative elements do not necessarily mean passivity violation, as pole-residue contributions can be combined/clustered in such a way

to preserve passivity for the global system. However, when all branches are composed of passive elements the passivity of the global system is always guaranteed. For example, in reference to conventional -model representation, the conductance given by the real part of

G

_OddEven, in order to be

positive valued, involves opposite sign for real part of

G

_OddEven. Such constraints seem difficult to fulfill in a wide frequency band. Similar reasoning can be considered for T model representation. It is essential to underline that equation (1) has the same structure for Z and Y matrix representations.

B. Case of Transmission Line Structures: Challenges and Derivation of Bridge/Trellis Architecture

A better understanding of the aforementioned synthesis difficulties can be obtained when considering transmission line structures defined with the following impedance/admittance matrix:

>

@

»

¼

º

«

¬

ª

c c c c c port Two

Z

X

T

H

T

H

T

H

coth

sinh

coth

1 1 (3)

X

Two-port stands both for impedance or admittance matrix (for

compact representation), with H=+1 for the impedance matrix

and H=-1 for the admittance matrix. In reference to the CPW transmission line in Fig.2, it can be demonstrated that the impedance parameters are function of even and odd modal input impedances respectively corresponding to the electrical states where a magnetic and electric wall is placed on the symmetry plane (Fig.2). These states can be obtained by appropriate excitations from electromagnetic analysis. For example the even state can be obtained by imposing equal electric fields and current densities at both access ports 1 and 2. Thus even and odd states can be associated, in simulations tools, to excitation states [+1,+1] and [+1,-1], respectively.

S G G S G G LC P W S G G S G G S G G S G G S G G S G G LC P W S y m m e t r y p l a n e ( LC P W) / 2 A c c e s s p o r t 1 A c c e s s p o r t 2 S G G S G G LC P W S G G S G G S G G S G G S G G S G G LC P W S y m m e t r y p l a n e ( LC P W) / 2 A c c e s s p o r t 1 A c c e s s p o r t 2

Fig.2. Illustration of CPW-Line with symmetry plane with local ground distribution.

Since the symmetry plane is at the halves the CPW line length (LCPW/2) ZEven and ZOdd are given by:

¸ ¹ · ¨ © § ¸ ¹ · ¨ © § 2 coth 2 coth 1 12 11 12 11 c C Odd c C Even Z Z Z Z Z Z Z Z T T

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leading to the following expressions for the impedance parameters:

>

@

>

Even Odd

@

c c c c c Odd Even c c c c c Z Z Z Z Z Z Z Z Z Z » ¼ º « ¬ ª ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § » ¼ º « ¬ ª ¸ ¹ · ¨ © § ¸ ¹ · ¨ © § 2 1 2 coth 2 coth 2 sinh 2 1 2 coth 2 coth 2 coth 1 1 12 1 11 T T T T T T (5) 306

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Applying Mittag-Leffler expansion to the admittance parameters gives, in the lossless case, without loss of generality:

¸¸_¹ · ¨ ¨ © §

¦

f 1 2 2 1 1 1 2 1 cot Im k C C ii k Z Z Y S T T T T (6)

_¸¸ ¹ · ¨ ¨ © §

¦

f 1 2 2 1 1 1 ₂ 1 sin 1 Im k k C C ij k Z Z Y S T T T T (7)

where for lossless case

T

Im

T

_c

E

L

_CPW the electrical length of the line, Z_C being the characteristic impedance. The following Mittag-Leffler expansion of cot-1[T]:

> @

_¦

f » » ¼ º « « ¬ ª ¸ ¹ · ¨ © § 1 1 2 2 2 1 2 1 2 cot k k S T T T (8) combined with the identity sin-1[T]=(cot[T/2]+cot-1[T/2])/2,

gives a straightforward pole-residue expansions for the modal states impedance/admittance representation. While synthesis of Yii parameter is straightforward, extraction of lumped

element models for Yij brings some difficulties as how to

traduce the alternate contribution (-1)k in terms of passive circuit elements in (7). To solve these difficulties the modal state responses are mapped into a Bridge/Trellis architecture. The Odd and Even modal states contributions in relation with the Mittag-Leffler expansions can be represented with lumped equivalent circuit model elements synthesis giving:

¦

f » ¼ º « ¬ ª 1 1 1 1 k Sk Sk Sk LF LF Odd C L j R L j R Y Z Z Z (9)

¦

f

»

¼

º

«

¬

ª

1 1 0 0 0

1

k k k k Even

C

L

j

R

Y

Z

(10) where the different equivalent circuit parameters are given by the closed-form expressions completely determined by the effective permittivity and characteristic impedance easily deduced from EM analysis or measurement results :

1 1 2 1 1 0 1 1 2 1 1 0 2 , 2 1 2 2 , 2 , ¸¹ · ¨© § ¸ ¹ · ¨ © § r eff CPW c C k C Sk LF C k Sk L v k Z C Z k C L L Z L L L H Z SZ SZ SZ

wherev_cis the light velocity in free space, r

eff

H being the real part of the effective permittivity. LSk and L0k are taken

different from branch to branch in order to capture frequency dependence of Zc and effective permittivity.

In (9) and (10) the losses are extracted by identification applying pole-residue expansion on the modal state input admittances, including clustering of pole-residue terms.

XOdd XOdd XEven XEven Signal-in Ground-in Signal-out Ground-out XOdd XOdd XEven XEven Signal-in Ground-in Signal-out Ground-out

XOdd XOdd XEven XEven Signal-in Ground-in Signal-out Ground-out XOdd XOdd XEven XEven Signal-in Ground-in Signal-out Ground-out (a) (b)

Fig.3. Schematic representation of a Trellis architecture in bridge-analogy (a), and twisted representation (b).

C. Generalization to Symmetrical 2n-port Systems

Let’s consider extending the Bridge architecture to 4-port symmetric systems, following the port numbering in Fig.4. The even and odd admittance matrix are complex matrix blocks and relations similar to those of equations (1), can be derived following a two-port block-partitioning:

> @

>

_>

@

_@

>

_>

@

_@

» ¼ º « ¬ ª 2 2 11 2 2 12 2 2 12 2 2 11 4 4 x x x x x YB YB YB YB Y (11)

Following the same expansion technique as in (1), it is seen convenient to split multi-port matrix form in (11) such as:

> @ > @ > @ ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª » ¼ º « ¬ ª ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª » ¼ º « ¬ ª U U U U YB U U U U YB YB YB YB YB YB YB YB YB YB YB YB YB YB YB YB YB Y x O x E x 2 2 2 2 12 11 12 11 12 11 12 11 12 11 12 11 12 11 12 11 4 4 2 1 2 1 (10) where U is the two-by-two identity matrix. The two-by-two symmetrical matrix

> @

YB_E and

> @

YB_O may, at their turn, be split following (1) such as :

> @

_» ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 OO OE x O EO EE x E YB YB YB YB YB YB (12)

leading to the nested trellis architecture in Fig.4.

Z o Z _o Z e Z e Z o Z _o Z e Z e

Fig.4 Nested Trellis architecture applied to symmetrical four-port with Z0

and Ze having the architecture given in Fig. 3(b).

For general symmetrical multiports such as N=2n, the same procedure applies with canonical equivalent circuits for odd and even states readily deduced from the direct pole-residue extraction performed on open and short circuit input impedances of the defined elementary two-ports.

III.DISCUSSION OF MAIN RESULTS,CORRELATION ANALYSIS

AND VALIDATIONS

The proposed extraction methodology is applied to CPW transmission lines and coupled RF-inductors, both structures

requiring local ground distribution (ring). The even and odd

modal-state contributions are extracted from the S-parameters measurement. In Fig.5, correlations between extracted equivalent circuit model and measurement for a CPW transmission line are shown. The broadband behaviour of the Y21 parameter is properly captured. The proposed extraction

methodology is applied to the analysis of coupling between inductors shown in Fig.6. The structures use a ground-seal ring with and without a barrier (shielding). The derived wideband equivalent model of the coupled RF-inductors is shown in Fig.7. Both configurations have been extracted and compared to measurements in Fig.8(a-b). The analysis of extracted even and odd modes circuit elements demonstrates that the influence of the shielding can be efficiently captured.

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Fig.5. Correlations of results obtained applying the proposed extraction approach with measurement for CPW transmission lines (Log scale).

(a) G G S G G S G r o u n d S e a l - r i n g G G S G G S G G S G G S G r o u n d S e a l - r i n g (b)

Fig. 6. Photograph of designed and measured coupled inductors on Silicon (a) without barrier (Shield), (b) with barrier (Shield).

S ig n a l- in S ig n a l- o u t G r o u n d - in G r o u n d - o u t ZL e f t S ig n a l- in S ig n a l- o u t G r o u n d - in G r o u n d - o u t ZL e f t S ig n a l- in S ig n a l- o u t G r o u n d - in G r o u n d - o u t ZL e f t S ig n a l- in S ig n a l- o u t G r o u n d - in G r o u n d - o u t ZL e f t

Fig.7 Wideband bridge-architecture for the coupled RF-inductors in Fig.6.

It is observed that the main influence of the shield barrier (located on the symmetry plane) is lowering the magnetic couplings through the values of the extracted inductances. Extracted equivalent circuit parameters of Fig.7 in the inset of Fig.8(a-b), illustrating such effects, show ability of the proposed methodology to capture geometry related effects in network circuit representation : only the odd modal state is

impacted by the shielding. Hence possible link between

eigen-states EM field spatial distributions (layout related) and equivalent network elements (circuit related) is demonstrated. Such link renders possible mapping global EM attributes e.g., Q-factor, power-distribution into canonical modal-states contributions, for physics-based optimization and tuning towards desired performances.

RO d d= 2 5 0 m: R sO d d= 5 5 0 m: LO d d= 0 . 7 5 8 n H CO d d= 9 8 f F GO d d= 2 . 5 m S RE v e n= 5 0 0 m: R sE v e n= 2 5 0 m: LE v e n= 0 . 7 4 3 n H CE v e n= 1 1 3 . 6 f F GE v e n= 0 . 9 m S RO d d= 2 5 0 m: R sO d d= 5 5 0 m: LO d d= 0 . 7 5 8 n H CO d d= 9 8 f F GO d d= 2 . 5 m S RE v e n= 5 0 0 m: R sE v e n= 2 5 0 m: LE v e n= 0 . 7 4 3 n H CE v e n= 1 1 3 . 6 f F GE v e n= 0 . 9 m S (a) RO d d= 2 5 0 m: R sO d d= 5 5 0 m: LO d d= 0 . 7 7 9 n H C_{O d d}= 1 0 5 f F GO d d= 2 . 3 m S RE v e n= 5 0 0 m: R sE v e n= 2 5 0 m: LE v e n= 0 . 7 4 3 n H C_{E v e n}= 1 1 3 . 6 f F GE v e n= 0 . 9 m S RO d d= 2 5 0 m: R sO d d= 5 5 0 m: LO d d= 0 . 7 7 9 n H C_{O d d}= 1 0 5 f F GO d d= 2 . 3 m S RE v e n= 5 0 0 m: R sE v e n= 2 5 0 m: LE v e n= 0 . 7 4 3 n H C_{E v e n}= 1 1 3 . 6 f F GE v e n= 0 . 9 m S (b)

Fig.8 Correlations of results obtained applying the proposed wideband extraction approach for S21 parameters without (a) and with shielding (b).

IV.CONCLUSION

An original methodology attempting to bridge physical topology considerations with equivalent circuits model extractions has been proposed. Instead of casting Z or Y parameters in pole-residue expansions, the proposed approach considers modal state impedances/admittances as primary goal functions. These goal functions allow the derivation of lumped element equivalent circuits. The validity of the proposed extraction methodology is demonstrated by comparison with measurement. It is demonstrated that inductive couplings can be accurately represented in a wide frequency range without use of mutual term components (mutual elements should not be considered as circuit

components but as connection elements). Application of the

methodology to non-symmetrical passive structures is under investigation, with extension of the notion of even and odd modal states. The used Bridge/Trellis architecture can be understood as a potential way to restore passivity conditions when conventional /T model representations fail (with the

twisting properties in reference to central/plane symmetry considerations). The proposed approach opens the door for

filter synthesis analysis and metamaterial structures modeling.

ACKNOWLEDGEMENT

The authors would like to thank Prof Russer for stimulating discussions. The authors also thank Prof. Kuznetsov and his team.

REFERENCES

[1] P. Russer, “Electromagnetics Microwave circuit and Antenna Design for Communications Engineering“ , 2nd_{edition 2006 Artech House.}

[2] B. Boite and J. Neirynck, “Théorie des réseaux de Kirshhoff”, Traité d’Electricité, d’Electronique et d’Electrotechnique, Dunod 1983. [3] T.Caillet et al, “The Compound Resonator Approach : Parity Control

and Selectivity Enhancement in N-Resonator Planar Systems”, Microwave Symposium Digest, 2008 IEEE MTT-S pp:1015-1018.