On the Unique Determination of Modal Multiconductor Transmission-Line Properties
JJOURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 1
On the Unique Determination of ModalMulticonductor Transmission-Line Properties
Stuart Barth,
Graduate Student Member, IEEE, and Ashwin K. Iyer,
Senior Member, IEEE
Abstract —Some modal (or decoupled ) transmission-line prop-erties such as per-unit-length impedance, admittance, or charac-teristic impedance have long been held to be, in general, non-unique. This ambiguity arises from the nature of the similaritytransformations used to relate the terminal and modal domains,for which the voltage transformation matrix has been shown to beonly loosely related to the corresponding current transformationmatrix. Modern methods have attempted to relate the two,but these relations typically rely on arbitrary normalizations,leading to strictly incorrect and/or non-unique results. This workintroduces relations between the two transformations, derivedfrom the physical equivalence of total power and currentsbetween the two domains, by which the transformation matricescan be unambiguously related to each other, and the modalproperties uniquely solved. This technique allows for the correctextraction of the modal transmission-line properties for anyarbitrary system of conductors. Multiple examples are studiedto validate the proposed solution process.
Index Terms —Multiconductor transmission-line theory, di-agonalization, decoupling, similarity transform, characteristicimpedance.
I. I
NTRODUCTION T RANSMISSION-line (TL) theory is a powerful conceptwhich simplifies the fields of transverse electromagnetic(TEM) modes and allows them to be expressed as circuitquantities, making it a critical tool for high-accuracy circuitdesign. Basic TL systems contain only two conductors,whereas generalized multiconductor
TL (MTL) systemsmay contain three or more conductors. The two commonrepresentations (or “domains”) of MTLs are as follows:1) The terminal domain (also referred to as the natural domain), in which the various TL parameters aredefined between each conductor and a pre-selectedreference conductor. These parameters are expressed asmatrices which are (generally) fully populated, whichimplies that the terminals are coupled to each other.2) The modal domain (also referred to as the diagonalized or decoupled domain), in which the TL’s properties aregiven in terms of the TL’s characteristic modes andexpressed as diagonal matrices. The diagonality of thematrices implies that the solutions are isolated from oneanother, and due to this fact, each mode can be expressedby a simple two-conductor TL model. S. Barth and A. K. Iyer are with the Department of Electrical andComputer Engineering, University of Alberta, Edmonton, AB, Canada. e-mail:[email protected]
A process exists by which the terminal-domain parametersmay be transformed into the modal domain, and vice-versa[1]–[3]. This process, known as diagonalization, results in thesolution of two transformation matrices, which are responsiblefor transforming the currents and voltages (and by extension,other parameters) between the two domains. These matricesare solved independently, which is the source of some ambi-guity when they are used together. The specific process of,for example, the diagonalization of the propagation constantrequires that only one of the transformation matrices be usedand solved; however, the process for the diagonalization ofsome properties such as the characteristic impedance involvesboth the transformation matrices. This fact has historicallyresulted in the conclusion that some modal properties such ascharacteristic impedance are generally ambiguous when trans-formed from the terminal domain (although other methodsmay of course be used to obtain the correct values, such asderivation from known modal field quantities [4], [5]).Over roughly the last two decades, solutions have beenproposed in attempts to overcome this ambiguity [6]–[9], andpresent similar processes to what will be introduced in thiswork, but generally either utilize ambiguous normalizationsthat lack physical bases, or produce non-unique results.This work demonstrates that any such ambiguities can beresolved by noting that physical quantities – specifically, totalpower and current – must be equivalent under both formsof representation. These physical facts are used together toconstrain the spectrum of possible mathematical solutions andproduce unique modal results. While attempts have been madeto equate total real power in both domains [6], it will be shownthat the consideration of total power (i.e., including the reactivecomponent) is required in order to produce a unique solution.Additionally, it can be demonstrated that given the equivalenceof these properties in the two domains, the transformationmatrices are required to be real.The layout of this document is as follows: the analyticalprocess of the terminal-to-modal transformation is detailedin Sec. II, along with the mathematical description of theambiguity in the transforms’ relations, a brief overview ofpreviously proposed solutions, and its resolution. Sec. IIIdetails the data obtained for a number of TLs and demon-strates how the correct modal characteristic impedances maybe directly computed using only the terminal domain per-unit-length inductance and capacitance matrices, even in thepresence of extreme loss. a r X i v : . [ phy s i c s . c l a ss - ph ] F e b OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 2
II. T
HEORY
A. Diagonalization Procedure
The analysis of TL systems typically begins with extractingtheir per-unit-length inductance and capacitance matrices ( [ L ] and [ C ] , respectively). These matrices detail the terminal-domain values, but for clarity in the following analysis, thesubscripts T and M will be used to indicate which domain(terminal or modal) is being used in each matrix, e.g., [ L T ] or [ C M ] .The transformation between the two domains makes useof similarity transforms, specified in terms of currents andvoltages, and which are expressed as (nonsingular) matrices[1], [2]. These transformations are defined as (cid:126)V T = [ T V ] (cid:126)V M (1a) (cid:126)I T = [ T I ] (cid:126)I M (1b)such that each column of [ T I ] and [ T V ] corresponds to aparticular mode, and for which the entries describe the relative (i.e., with respect to the modal quantities) currents and voltageson each conductor for that mode. The TL wave equations forpropagation along z in the terminal domain can be expressedas [1] ∂∂z (cid:20) (cid:126)V T (cid:126)I T (cid:21) = (cid:20) [0] − [ Z T ] − [ Y T ] [0] (cid:21) (cid:20) (cid:126)V T (cid:126)I T (cid:21) (2)where [ Z T ] and [ Y T ] are the per-unit-length impedance andadmittance, respectively, in the terminal domain. The modal-domain equations have the same form; that is ∂∂z (cid:20) (cid:126)V M (cid:126)I M (cid:21) = (cid:20) [0] − [ Z M ] − [ Y M ] [0] (cid:21) (cid:20) (cid:126)V M (cid:126)I M (cid:21) (3)Inserting (1a) and (1b) into (2) yields ∂∂z (cid:20) [ T V ] [0][0] [ T I ] (cid:21) (cid:20) (cid:126)V M (cid:126)I M (cid:21) = (cid:20) [0] − [ Z T ] − [ Y T ] [0] (cid:21) (cid:20) [ T V ] [0][0] [ T I ] (cid:21) (cid:20) (cid:126)V M (cid:126)I M (cid:21) (4)which can be re-arranged and compared to (3) to give therelations [ Z M ] = [ T V ] - [ Z T ] [ T I ] (5a) [ Y M ] = [ T I ] - [ Y T ] [ T V ] (5b)Using these definitions, the propagation constants and charac-teristic impedances can be expressed as [ γ M ] = [ Z M ] [ Y M ] = [ T V ] - [ Z T ] [ Y T ] [ T V ] (6a) [ γ M ] = [ Y M ] [ Z M ] = [ T I ] - [ Y T ] [ Z T ] [ T I ] (6b) [ Z cM ] = (cid:0) [ Z M ] [ Y M ] - (cid:1) = (cid:0) [ T V ] - [ Z T ] [ T I ] [ T V ] - [ Y T ] - [ T I ] (cid:1) (7a) [ Z cM ] = (cid:0) [ Y M ] - [ Z M ] (cid:1) = (cid:0) [ T V ] - [ Y T ] - [ T I ] [ T V ] - [ Z T ] [ T I ] (cid:1) (7b) where [ γ ] is the propagation constant matrix, and [ Z c ] is thecharacteristic impedance matrix. Additionally, characteristicimpedances can be defined as (cid:20) (cid:126)V + T (cid:126)V − T (cid:21) = (cid:20) [ Z cT ] [0][0] − [ Z cT ] (cid:21) (cid:20) (cid:126)I + T (cid:126)I − T (cid:21) (8a) (cid:20) (cid:126)V + M (cid:126)V − M (cid:21) = (cid:20) [ Z cM ] [0][0] − [ Z cM ] (cid:21) (cid:20) (cid:126)I + M (cid:126)I − M (cid:21) (8b)Again applying equations (1a) and (1b), and comparing (8a)with (8b) yields the relation [ Z cM ] = [ T V ] - [ Z cT ] [ T I ] (9)Other TL modal properties may be determined in a similarmanner. B. Origins of Ambiguity
Equations (5a) through (9) are well known and unambigu-ously correct. However, ambiguity is introduced in the solutionprocess – specifically, through the use of (6a) and (6b), whichare used to solve for the the transformation matrices [ T V ] and [ T I ] independent of one another [2]. Specifically, since thesetwo equations are of the form [ D ] = [ Q ] - [ M ] [ Q ] , (10)where [ M ] is a matrix to be diagonalized, [ Q ] is the diag-onalization matrix, and [ D ] is a diagonal matrix said to bethe diagonalized form of [ M ] , a regular eigenvalue process istypically used to simultaneously solve for [ Q ] and [ D ] , given [ M ] . Unfortunately, the matrix [ Q ] is not unique. It can beobserved that if [ Q ] is post-multiplied by a diagonal matrix [ g ] , such that [ S ] = [ Q ] [ g ] , (11)then the diagonalization of [ M ] by use of [ S ] is [ D ] = [ S ] - [ M ] [ S ] , (12)and is still valid, since [ D ] = [ g ] - [ Q ] - [ M ] [ Q ] [ g ] . (13)A diagonal [ g ] ensures the diagonality of [ D ] , since any [ Q ] - [ M ] [ Q ] will itself be diagonal (subject to some physicalconstraints – it has been shown that some physical systemscan result in a non-diagonalizable matrix [10], although theseare unlikely to be encountered in practice). Furthermore, itcan be shown that since [ Q ] - [ M ] [ Q ] is diagonal, the product [ g ] - [ Q ] - [ M ] [ Q ] [ g ] reduces simply to [ Q ] - [ M ] [ Q ] , suchthat the value of [ g ] has no bearing on the solution values [ D ] and [ Q ] in (13) [2].This fact allows for the correct diagonalization of thepropagation constants in (6a) and (6b), regardless of the valuesof [ g ] . The problem with this ambiguity arises from attemptingto solve any of (5a), (5b), (9), for which [ T V ] and [ T I ] couldeach be post-multiplied by an arbitrary diagonal matrix –however, in these equations, the value of [ g ] will indeed havea direct impact on the diagonalized results, for example [ Z cM ] = [ g V ] - [ T V ] - [ Z cT ] [ T I ] [ g I ] , (14) OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 3 the solution of which is presently ambiguous. In the followingwork the correction matrices [ g ] will be utilized with thefollowing definitions: [ T V ] new = [ T V ] old [ g V ] (15a) [ T I ] new = [ T I ] old [ g I ] (15b)where [ T I ] old and [ T V ] old are the matrices determined by theoriginal eigenmode solution, and [ T I ] new and [ T V ] new are thecorrected transformation matrices used to produce the correctmodal TL properties. C. Existing Disambiguation Processes
Presently, there exist some strategies for solving the appro-priate values of [ g V ] and [ g I ] . Two of the most common arebased on simple normalizations that are typically arbitrarilychosen and applied; both were detailed extensively in [6]. Thefirst is the normalization of the product [ T V ] T [ T I ] , for whichit was shown that the symmetric nature of [ γ M ] , [ Z T ] , [ Y T ] leads to the conclusion that [ T I ] T new [ T V ] new = [ T V ] T new [ T I ] new = [ D ] , (16)where [ D ] is any diagonal matrix, typically chosen to beidentity for convenience. This is the introduction of oneambiguity, since the choice is entirely arbitrary. The secondmethod involves a form of self-normalization of the matrixdiagonals, such that [ T I ] T new [ T I ] new = [ T V ] T new [ T V ] new = [ I ] (17)This process also introduces some ambiguity, since there is nophysical justification for such a method. However, even thoughthese processes are not rigorously justified or correct, theytypically result in solutions with tolerable error, and thereforehave been widely adopted. Some work has also been done inattempting to normalize via physical quantities such as voltage,current, and power, but this has generally resulted in non-unique solutions [8]. D. Initial Postulates and Terminology
The definition of such a modal-terminal domaintransformation relies on the fact that both domains areequally valid representations of the same physical system.Therefore, various physical properties must be equivalent inboth domains, foremostly, total energy and total charge. Inthe frequency domain, these are directly related to total powerand current. Accordingly, the following are postulated: • That the total power being carried by a TL in the modaldomain is equal to that carried in the terminal domain.This will be elaborated on in Sec. II-E. • That the total co-directed currents in the modal domainare equal to the total of those in the terminal domain.This will be investigated in Sec. II-F.While the total voltages and currents in the terminal domain( (cid:126)V T and (cid:126)I T , respectively) represent a superposition of excited modes, it will be of interest to examine the effects of asingle excited mode in the terminal domain. In this case, theseterminal voltages and currents will be expressed as: (cid:126)V T (cid:12)(cid:12)(cid:12) n = [ T V ] (cid:126)δ n V M n (18a) (cid:126)I T (cid:12)(cid:12)(cid:12) n = [ T I ] (cid:126)δ n I M n (18b)where V M n and I M n are the voltage and current of the excitedmode n , respectively, and the delta vector (cid:126)δ n is defined as (cid:126)δ n = (cid:40) , if i = n , otherwise ∀ indices i. (19)The complex-conjugate transpose of a matrix or vector willbe expressed as [ A ] ∗ = [ A ] T , (20)where the over-bar denotes the element-wise complex conju-gate. The inner product will be used to represent sums overvectors; that is, for vectors (cid:126)a and (cid:126)b (cid:88) k a k b k = (cid:126)a · (cid:126)b = (cid:126)a T (cid:126)b. (21)In this manner, the sum of the elements in a single vector maybe expressed as (cid:88) k a k = (cid:126) T (cid:126)a, (22)where (cid:126) is a vector, the entries of which are each unity. E. Power Equivalence
In accordance with the first postulate, (cid:126)I ∗ M (cid:126)V M = (cid:126)I ∗ T (cid:126)V T (23)Substituting the terminal to modal domain transforms in (1a)and (1b) yields (cid:126)I ∗ M (cid:126)V M = (cid:126)I ∗ M [ T I ] ∗ new [ T V ] new (cid:126)V M (24)to which one of the solutions is the identity matrix [ I ] .Furthermore, equating the carried power by a single mode toits terminal-domain equivalent yields I ∗ M n V M n = (cid:126)I ∗ T (cid:12)(cid:12)(cid:12) n (cid:126)V T (cid:12)(cid:12)(cid:12) n = I ∗ M n (cid:126)δ Tn [ T I ] ∗ new [ T V ] new (cid:126)δ n V M n , (25)from which the scalars can be cancelled to give (cid:126)δ Tn [ T I ] ∗ new [ T V ] new (cid:126)δ n = 1 , (26)which, in conjunction with (24) demonstrates the uniquedefinition of [ T I ] ∗ new [ T V ] new = [ I ] (27)A similar derivation was investigated in [6], however, thiswas only for one component of power (the real part), andsubsequently an equation such as (26) could not be developedthrough the cancellation of complex voltage and current termsto demonstrate the identity matrix as the unique solution to(27). OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 4
F. Current Equivalence
Consider first a single excited mode, n , which is modelledvia a two-conductor TL. The current magnitude for this modeis expressed, in accordance with the previous definitions, as I M n . While this is a single quantity, there are of course twocurrents present: one on each conductor. Let the magnitude ofthe current on one of the conductors be labelled I M n (+) , andthe magnitude of the current flowing in the opposite direction– that is, on the other conductor – be labelled I M n ( − ) . Theconservation of current enforces the equality of the currents I M n (+) = I M n ( − ) .Consider next a TL with more than two conductors, asdescribed in the terminal domain. Assume for now that thecurrent carried by the reference conductor is included in theset of all currents. It is certain that the conservation of currentspecifies that the sum of the currents on all conductors must bezero. Similar to the modal domain TL, assume that the currentmagnitudes may be divided into two sets (cid:126)I T (+) and (cid:126)I T ( − ) ,with the membership of the sets being determined by the phaseof these currents with respect to the modal currents. Althoughit may seem intuitive for TEM modes, a proof that only twosuch contra-directed phases (corresponding to co- and contra-directed currents) exist is supplied in Appendix A. Accordingto the conservation of current, the sums of each group mustbe equal to each other, that is: (cid:126) T (cid:126)I T (+) = (cid:126) T (cid:126)I T ( − ) .Furthermore, the set of total currents on all conductors is equalto the difference of the two groups, such that (cid:126)I T = (cid:126)I T (+) − (cid:126)I T ( − ) (28)The vector (cid:126)I T (+) will then contain an entry of zero wherever (cid:126)I T ( − ) has a nonzero entry, and vice-versa, such that all vec-tors have the same length (which is the number of conductorsin the TL).It is known from (1b) that the distribution of terminal-domain currents is dictated by the transformation matrix [ T I ] – that is, if a single mode is excited, the currents on eachconductor are uniquely specified by the corresponding columnof [ T I ] . Let the transformation matrix then be separated intotwo new matrices – one, labelled [ T I (+)] , consisting of thecomponents related to the set of currents (cid:126)I T (+) and anotherlabelled [ T I ( − )] , consisting of the components related to theset of currents (cid:126)I T ( − ) . Then, [ T I ] = [ T I (+)] − [ T I ( − )] (29a) (cid:126)I T (+) = [ T I (+)] (cid:126)I M (29b) (cid:126)I T ( − ) = [ T I ( − )] (cid:126)I M (29c)The equality of currents between domains specifies that thetotal co-directed currents must be equal in both the modal andterminal domains. Specifically, for an excited mode n , theremust exist appropriate sets (cid:126)I T (+) new and (cid:126)I T ( − ) new , suchthat I M n (+) = (cid:126) T (cid:126)I T (+) new (cid:12)(cid:12)(cid:12) n (30a) I M n ( − ) = (cid:126) T (cid:126)I T ( − ) new (cid:12)(cid:12)(cid:12) n (30b) Invoking (18b) and (29b) allows the previous expressions tobe expanded to I M n (+) = (cid:126) T [ T I (+)] new (cid:126)δ n I M n (+) (31a) I M n ( − ) = (cid:126) T [ T I ( − )] new (cid:126)δ n I M n ( − ) (31b)The scalars I M n (+) and I M n ( − ) can then be cancelled toyield the expressions: (cid:126) T [ T I (+)] new (cid:126)δ n (32a) (cid:126) T [ T I ( − )] new (cid:126)δ n (32b)which simply imply that the sum of each column in [ T I (+)] new and [ T I ( − )] new must be equal to unity.The unique correction process may then be determinedby substituting (29a) into (15b), and letting the matrices [ T I (+)] old and [ T I ( − )] old be specified to contain the samenon-zero indices as [ T I (+)] new and [ T I ( − )] new , [ T I (+)] new = [ T I (+)] old [ g I ] (33a) [ T I ( − )] new = [ T I ( − )] old [ g I ] (33b)Substituting the previous equations into (32a) and (32b) yields: (cid:126) T [ T I (+)] old [ g I ] (cid:126)δ n (34a) (cid:126) T [ T I ( − )] old [ g I ] (cid:126)δ n (34b)which indicates that the entries along the diagonal of thecorrection matrix [ g I ] are simply the inverse of the sum ofthe corresponding columns in [ T I (+)] old and [ T I ( − )] old .Recall that in practical use, the information related to thereference conductor would not be contained directly in (cid:126)I T new , [ T I ] new , [ T I (+)] new , or [ T I ( − )] new . This is not such a criticalissue, as being a single, scalar value, it would, if accounted for,either appear in the set of currents (cid:126)I T (+) new or (cid:126)I T ( − ) new ,and subsequently be manifested in either of [ T I (+)] old or [ T I ( − )] old . This does mean that, in practice, only one of (34a)or (34b) will be correct – but the issue is simply resolved bynoting that the correct equation will always correspond to thesum with the larger magnitude. Then, g In = max (cid:104)(cid:16) (cid:126) T [ T I (+)] old (cid:126)δ n (cid:17) , (cid:16) (cid:126) T [ T I ( − )] old (cid:126)δ n (cid:17)(cid:105) − (35)and hence the current transformation matrix is uniquely de-fined.It is worth noting that if the transformation matrices (intheir corrected state) are complex, several contradictions mayarise with the previously derived equations. To alleviate suchconcerns, Appendix B offers a proof of the remarkable factthat the transformation matrices must be real – indicating thatthe terminal-domain voltages and currents must always be inphase with their corresponding modal quantities. OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 5 d a a Fig. 1. Setup of the three wire transmission line. The conductors (blackcircles) are similar and form an equilateral triangle.
G. Final Solution Process
The solution process to determine the corrected, uniquetransformation matrices is then as follows:1) Find the original, uncorrected, transformation matrices [ T I ] old and [ T V ] old (for example, via an eigenmodeprocess utilizing (6a) and/or (6b)).2) Apply the equality of currents to obtain [ g I ] from [ T I ] old ,utilizing (35).3) Apply (15b) to obtain the correct current transformationmatrix [ T I ] new from [ g I ] and [ T I ] old .4) Employ the equality of power (27) to obtain the cor-rected voltage transformation matrix [ T V ] new , that is, [ T V ] new = [ T I ] − ∗ new .A proposed algorithm for correcting the transformation matri-ces and associated discussion are given in Appendix C.III. E XAMPLES
This section will examine four examples of TLs in the TEMapproximation, and perform the process used to diagonalizetheir properties, as explained in the previous section. AnsysHFSS was used to extract the per-unit-length inductanceand capacitance matrices in the terminal domain, as wellas determine the propagation constants and characteristicimpedances in the modal domain, all computed at a frequencyof 1 GHz. The convergence criteria for the HFSS simulationswere chosen to be extremely strict, since the produced datawill be the only standard of comparison in this work. Thesecriteria enforced a minimum of 20 converged passes with aconvergence ∆ | S | of 0.0001 and ∆ ∠ S of 0.1 ◦ . The final datamay still have errors of roughly 1 % , due to extensive numericalprocessing. Data which are presented in the form of complexinductance ([ L ]) and capacitance ([ C ]) matrices include lossesfrom the traditional resistance ([ R ]) and conductance ([ G ]).The validity and effectiveness of the proposed process isestablished by the corroboration of the HFSS modal values,given only the known terminal values. A. Three-Wire Line
This TL consists of three cylindrical copper conductors ofdiameter d = 1.00 mm suspended in air and spaced equallyapart by a distance a = 3.46 mm, as shown in Fig. 1. Since the system is symmetric, the choice of reference conductor isarbitrary, and the [ L T ] and [ C T ] matrices are 2 × [ L T ] = (cid:20) . − j . . − j . . − j . . − j . (cid:21) uH/m [ C T ] = (cid:20) . − j . − . j . − . j . . − j . (cid:21) pF/m (36)Employing equations (6a) and (6b) (noting that these twoequations are solved independently of one another) yields themodal propagation constants [ γ M ] , as well as the current andvoltage transformation matrices: [ γ M ] = (cid:20) . j . . j . . j . . j . (cid:21) /m [ T V ] old = (cid:20) . j . − . − j . . j . . − j . (cid:21) [ T I ] old = (cid:20) . j . − . − j . . j . . − j . (cid:21) (37)where it can be noted that the uncorrected transformationmatrices within acceptable error satisfy the equality of totalpower (27). The modal propagation constants agree to within0.2% with the modal solutions given by HFSS, which are [ γ M ] = (cid:20) . j . . j . . j . . j . (cid:21) /m (38)Subsequently, the terminal and (incorrect) modal characteristicimpedances, the latter determined from either (7a) or (7b), are [ Z cT ] = (cid:20) . − j . . − j . . − j . . − j . (cid:21) Ω[ Z cM ] = (cid:20) . − j . . − j . . − j . . − j . (cid:21) Ω (39)the latter of which are significantly different from the HFSSmodal characteristic impedance values of [ Z cM ] = (cid:20) . − j . . − j . . − j . . − j . (cid:21) Ω (40)exhibiting differences of 39.7% and 40.3%, respectively. Thisis expected, as the current and voltage transformation matrices [ T I ] and [ T V ] do not strictly respect current or power equiva-lence between domains, for which they must be first correctedusing the proposed processes.The correction process is then implemented by utilizing(15a), (15b), (27), and (35). Specifically, implementing thelatter gives [ g I ] = (cid:20) . − j . . j . . j . . j . (cid:21) (41)Then, it follows that the corrected transformation matrices are [ T V ] new = (cid:20) . j . − . j . . − j . . − j . (cid:21) [ T I ] new = (cid:20) . − j . − . j . . j . . − j . (cid:21) (42) OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 6
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Fig. 2. Setup of the S-CBCPW transmission line. The conductors (solidblack rectangles) are numbered, with the conductor backing ( which are well within the acceptable error of being entirelyreal. Using these new values in (9) gives [ Z cM ] = (cid:20) . − j . .
000 + j . − . − j . . − j . (cid:21) Ω (43)which are in much closer agreement (a percent difference of0.27% for both values on the diagonal) to the modal valuesgiven by HFSS (40) than the uncorrected values in (39). Foradditional verification, it can be shown that, within the givenerror, [ T V ] Tnew [ T I ] ∗ new = [ I ] , and that [ γ M ] remains consistentwhether solved with (6a) or (6b).The current and voltage transformation matrices containinformation regarding the relative voltage and current distribu-tion of each mode in each column. Observing the first column(mode), it is noted in [ T I ] that the relative currents have equalmagnitudes and are in the same direction. By deduction, itmay be inferred that the reference conductor then carries arelative current with double the magnitude of either of theother two conductors, and from [ T V ] it is observed that thetwo non-reference conductors have the same relative voltage.This mode then corresponds to a “common” mode, where thetwo-non reference conductors have the same relative currentand voltage characteristics. Observing the second mode, itcan be seen that in the corresponding second column of [ T I ] ,there are opposite and equal-magnitude relative currents on thenon-reference conductors, leading to the conclusion that therelative current on the reference conductor is zero. Since thesecond column entries in [ T V ] are also equal in magnitude, butoppositely directed, it can be inferred that this is a “balanced”(or “differential”) mode. B. Shielded, Conductor-Backed, Coplanar Waveguide
The shielded, conductor-backed coplanar waveguide (S-CBCPW) TL used is shown in Fig. 2. The lower dielectricis RO-3010, and the upper dielectric is air. The boundaryconditions used on the vertical sides are (fictitious) perfectmagnetic conductors (PMCs), indicated by the dashed lines.The numbers indicate the conductors, with number 0 (theconductor backing) being taken as the reference. Conductors 1 and 3 are the CPW grounds, while conductor 2 is the CPWstrip line, and conductor 4 is the shield. The various parametersused were h l = 1.524mm, h u = 100mm, s = 1.00mm, g =1.00mm, w = 10mm, (cid:15) l ≈ (cid:15) u ≈ [ L T ] = . . . . . . . . . . . . . . . . µH/m [ C T ] = . − . − . − . − . . − . − . − . − . . − . − . − . − . . pF/m (44)These yield modal propagation constants and (uncorrected)transformation matrices [ γ M ] = j . . . . . . . . . . . . . . . . /m (45)and transformation matrices [ T I ] old = . . − . − . . . . − . . − . − . − . . . . . [ T V ] old = . − . − . . . . . . . . − . . . . . . (46)The modal propagation constants are once again very close(within 0.2%) to the HFSS-given values, which have imaginarycomponents of [ γ M ] = j . . . . . . . . . . . . . . . . /m (47)The (real components of the incorrect) modal characteristicimpedances derived using the above transformation matricesare [ Z cM ] = . . . . . . . . . . . . . . . . Ω (48)which are not very similar to those given by HFSS: [ Z cM ] = . . . . . . . . . . . . . . . . Ω (49) OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 7 with percentage differences of 8.6%, 56.7%, 36.5%, and17.0% respectively. The correction process yields the matrices [ g I ] = . . . . . . . . . . . . . . . . [ T I ] new = . . − . − . . . . − . . − . − . − . . . . . [ T V ] new = . . − . . . . . . . − . − . . . . − . . (50)which, in turn, allow for the computation of the correctedcharacteristic impedance matrix, the real values of which are [ Z cM ] = . . . . . . . . . . . . . . . . Ω (51)the values of which are very close to those given by HFSSin (49): percentage differences of 0.17%, 0.06%, 0.63%, and0.04% respectively.The nature of the various modes can be determined fromobservation of the transformation matrices. The first mode(column) is characterized by a relative current value of ap-proximately 0.4 on each of the CPW ground conductors(numbers 1 and 3), and approximately 0.2 on the CPW stripline (conductor 2). There is a negligibly small relative currentcomponent on the shield (conductor 4), and the same relativevoltage is present on conductors 1 through 4, indicating thatthere is a negligibly small electric field in the upper region (theregion between the plane of conductors 1, 2 and 3, and theplane of conductor 4). Since the same relative voltage exists onthe three CPW conductors (1 through 3), it can be concludedthat this mode is a parallel-plate-waveguide (PPW)-type modein the lower dielectric region. The relative current magnitudesare a result of a nearly equally distributed (relative) currentdensity over unequally sized conductors.The second mode can be analyzed simply, as only two ofthe conductors – the two CPW ground planes (conductors 1and 3) – support non-zero relative currents and voltages. Thiscorresponds to the coupled-slotline (CSL) mode, for whichthere are indeed currents induced on the remaining conductors,but the net currents on each conductor are zero.The third mode is characterized by a positive relative currenton the CPW strip line (conductor 2), and negative relativecurrents of nearly half that magnitude on each of the CPWgrounds (conductors 1 and 3). This is what would be expectedof a CPW mode, although it can be observed that the sum ofthe negative relative currents implies that there is some smallrelative current component on the reference conductor backingas well. This would be expected from a conductor-backedCPW mode, and also it can be observed from the relativevoltages that there is some coupling with the shield as well. Fig. 3. Setup of the two-layer PPW transmission line. The conductors (solidblack rectangles) are numbered, with the conductor backing (conductor 0)taken as the reference.
The last mode is similar to the first, in that all of theconductors except for one are equipotential. This, along withthe relative currents on all conductors, readily identifies thismode as the PPW-like mode in the upper region.
C. Lossy Two-Layer Parallel-Plate Waveguide
To confirm the validity of the proposed solution, this exam-ple will deal with artificially inflated losses. The TL consistsof three rectangular conductors, as shown in Fig. 3, which areassumed to be perfectly conducting. It is also assumed that theupper and lower faces of the middle conductor (conductor 1)possess the same voltage and current values, such that thereis coupling between the upper and lower regions. The verticalboundary conditions are again modelled as PMCs. Let w =10 mm, h l = 1.524 mm, h u = 10 mm, (cid:15) l = 3.0 − j (cid:15) u = 1.0 − j [ L T ] = (cid:20) . − j . . − j . . − j . . − j . (cid:21) µH/m [ C T ] = (cid:20) . − j . − . j . − . j . . − j . (cid:21) pF/m (52)These data give modal propagation-constant and (uncorrected)modal characteristic-impedance and transformation matrices of [ γ M ] = (cid:20) . j . . j . . j . . j . (cid:21) /m [ T I ] old = (cid:20) − . j . . j . . j . . j . (cid:21) [ T V ] old = (cid:20) . j . − . j . . − j . − . j . (cid:21) [ Z cM ] = (cid:20) . j .
151 0 . − j . . − j . . j . (cid:21) Ω (53)from which a number of observations may be made. Firstly,the modal propagation constants are in excellent agreement OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 8 (within 0.03% and 0.4%) of HFSS, which gives [ γ M ] = (cid:20) . j . . j . . j . . j . (cid:21) /m (54)Secondly, the transformation matrices do not satisfy the currentequalities, since the sum of the either of the positive ornegative currents is not unity, and moreover, one entry of [ T V ] has a significant imaginary component. Thirdly, the modalcharacteristic impedances differ greatly (44.8% and 17.2%,repsectively) from those given by HFSS: [ Z cM ] = (cid:20) . j . . j . . j . . j . (cid:21) /m (55)The correction process yields the matrices [ g I ] = (cid:20) . j . . j . . j . . j . (cid:21) [ T V ] new = (cid:20) . j . . j . . j . . j . (cid:21) [ T I ] new = (cid:20) − . j . . j . . j . . j . (cid:21) [ Z cM ] = (cid:20) . j . . − j . . − j . . j . (cid:21) Ω (56)from which it can be seen that the transformation matrices arereal, and the characteristic impedance values are very closeto those given by HFSS (0.03% and smaller than 0.01%),indicating that the modal transformation process proposed inthis work is valid even in the presence of extreme loss.IV. C ONCLUSION
It has been shown that a unique relationship between voltageand current transformation matrices for TL modes can beextracted from the physical equivalence of power and currentbetween domains, which represents an important improvementover previously proposed processes. Various examples furtherdemonstrated the use and accuracy of the proposed correctionprocess, through comparison with HFSS simulations. Theunique determination of the properties of TL modes mayfind meaningful applications in fields such as electromagneticcompatibility, signal and power integrity in printed-circuit-board environments, and the analysis of coupling betweenclosely spaced power system circuits.A
CKNOWLEDGMENT
The authors would like to thank Mr. David Sawyer forinspirational discussion and proof-reading of this work.A
PPENDIX AP ROOF OF C ONTRA -D IRECTED P HASES
It is typically assumed that the forms of the transforma-tion matrices [ T V ] and [ T I ] are not limited. However, thisassumption implies that for a given excited mode, currents maybe excited which possess any arbitrary phase, with respect tothe excitation. Here, this possibility is formally investigated, assuming only the equivalence of power between the twodomains.Comparing (6a) to the transpose of (6b) in the case ofcorrected transformation matrices, and noting that the matrices [ Z T ] and [ Y T ] are symmetric, it can be concluded that [ T V ] new = [ T I ] - Tnew [ D ] (57)where [ D ] is an arbitrary matrix, which is required to bediagonal. Inserting [ T V ] new into the proposed equality ofpower (27) yields [ D ] = [ T I ] - new [ T I ] new (58)The right-hand side of the preceding equation is not generallydiagonal for any complex [ T I ] new , whereas [ D ] is requiredto be diagonal. This apparent contradiction is resolved byrearranging (58) to give [ T I ] new = [ T I ] new [ D ] (59)Observe that this can also be expressed as [ T I ] new = [ T I ] new [ D ] [ D ] (60)Which can be further simplified to [ T I ] new = [ T I ] new [ | D | ] (61)This interesting result yields a pair of useful conclusions.Firstly, observing each column n of the transformation matrix,where (cid:126)T I n = (cid:126)T I n | D n | , it is noted that if | D n | (cid:54) = 1 , then (cid:126)T I n must equal (cid:126) , a case that can be physically discounted, sinceit represents a mode that excites no terminal-domain currents.Secondly, it is then noted that D n must be of the form e jθ n ,where θ n is any real number. Inserting this observation intoeach row of (59) yields (cid:126)T I n = (cid:126)T I n e jθ n (62)Generally, expressing each entry in (cid:126)T I n in polar form givesthe following: | T I mn | e jφ mn = | T I mn | e − jφ mn e jθ n (63)where φ mn is the phase angle of the complex entry T I mn .Note that this equation implies e jφ mn = e j ( θ n − φ mn ) (64)and thus, φ mn = θ n iπ, (65)where i is any integer. This result relates the phase of anyelement in a given column of [ T I ] new to the value θ n , whichonly depends on the column n . That is to say, the elementsof each column n can only possess one of two possiblephases, which are 180 ◦ out of phase with each other. A similarconclusion can be demonstrated for the phases of [ T V ] new . OURNAL OF L A TEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 9 A PPENDIX BP ROOF OF R EAL T RANSFORMATION M ATRICES
Having established that in the terminal domain, there existonly two phases of currents which are 180 ◦ apart, let the set [ T I (+)] new be defined as consisting of the terms of [ T I ] new which possess a phase of e jθ n , and [ T I ( − )] new as consistingof the terms of [ T I ] new which possess a phase of e j ( θ n + π ) (removing the π phase shift): [ T I (+)] new = [ | T I (+) | ] new (cid:2) e jθ (cid:3) (66a) [ T I ( − )] new = [ | T I ( − ) | ] new (cid:2) e jθ (cid:3) (66b)where [ | T I (+) | ] new and [ | T I ( − ) | ] new are matrices for whicheach entry is the magnitude of the corresponding entry in [ T I (+)] new and [ T I ( − )] new , respectively, and (cid:2) e jθ (cid:3) is a di-agonal matrix, the entries of which are e jθ n . Inserting theserelations into (32a) and (32b), respectively, gives: (cid:126) T [ | T I (+) | ] new (cid:2) e jθ (cid:3) (cid:126)δ n (67a) (cid:126) T [ | T I ( − ) | ] new (cid:2) e jθ (cid:3) (cid:126)δ n (67b)which establish that θ n = 2 iπ . This is a remarkable resultsince, in conjunction with (27), it clearly indicates that thecorrected transformation matrices are real.A PPENDIX CP ROPOSED C ORRECTION A LGORITHM
Obtaining the sum of each of the (non-reference) compo-nents of the columns of [ T I ] will yield two quantities: if thesevalues are equal, then it can be stated conclusively that thecurrent component on the reference conductor is zero, and thatthe value of g I is simply the inverse of either the positive ornegative sum. If the two sums do not have the same magnitude,then the larger sum is the correct value (with the differencebeing the current component on the reference conductor), theinverse of which is g I . With this knowledge, the pseudocodeAlg. 1 is proposed, where all scalar or matrix values areassumed to be of a complex, floating-point type, unless theyare indices.This algorithm has an outermost loop which iterates overeach column. For each column, a guard loop ensures that thecolumn is sufficiently corrected before proceeding to the nextcolumn – if it is not, then the inner loop is repeated. Theinner loop has four main processes: in order of operation, thepositive and negative sums are computed, and the larger sum isselected and evaluated. If the sum is sufficiently close to 1, theguard loop is notified that the process is complete, otherwisethe column’s g I is multiplied by the inverse of the sum andthe loop is repeated. Lastly, once the complete [ g I ] matrixhas been calculated for all columns, the corrected [ T I ] can becalculated, and utilizing the power equality, [ T V ] is calculateddirectly from this. Both corrected matrices are then returned. Algorithm 1 [ T V ] and [ T I ] Correction Procedure procedure C ORRECT T V T I ( [ T V ] , [ T I ] ) [ g I ] ← [ I ] (cid:46) Initialize [ g I ] with identity for Each Column n do Balanced ← f alse (cid:46) True if sum is close to 1 while Balanced = f alse do N egSum ← P osSum ← LargerSum ← for Each Row k do a ← T I ( k, n ) × g I ( n, n ) if (cid:60) ( a ) ≤ then N egSum ← N egSum − a else P osSum ← P osSum + a end if end for if (cid:60) ( N egSum ) ≥ (cid:60) ( P osSum ) then LargerSum ← N egSum else
LargerSum ← P osSum end if if |(cid:60) ( LargerSum ) − | ≤ then Balanced ← true else g I ( n, n ) ← g I ( n, n ) ÷ LargerSum end if end while end for [ T I ] ← [ T I ] × [ g I ] [ T V ] ← [ T I ] − T return ([ T V ] , [ T I ]) end procedure R EFERENCES[1] C. R. Paul,
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Stuart Barth (S’07-GSM’11) received the B.Sc. andM.Sc. degrees in electrical engineering in 2012 and2015, respectively, from the University of Alberta,Edmonton, AB, Canada where he is currently work-ing towards the Ph.D. degree.His current research interests include the study ofmulticonductor transmission-line RF/microwave cir-cuits, dispersion engineering of periodic structures,fundamental electromagnetic theory, and antennaradiation-pattern shaping.Mr. Barth received the IEEE AP-S Pre-DoctoralResearch Award in 2014, and the IEEE AP-S Doctoral Research Grant in 2016for his ongoing research into electromagnetic bandgap structures for antennaand waveguide applications. He serves as an Officer of the IEEE NorthernCanada Section MTT-S/AP-S joint chapter.