MODERN systems theory has its roots in electrical. Why RLC realizations of certain impedances need many more energy storage elements than expected

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1 Why RLC realzaton of certan mpedance need many more energy torage element than expected Tmothy H. Hughe arxv:6.0658v [c.sy] Jan 08 btract It a gnfcant and longtandng puzzle that the retor, nductor, capactor (RLC) network obtaned by the etablhed RLC realzaton procedure appear hghly nonmnmal from the perpectve of lnear ytem theory. Specfcally, each of thee network contan gnfcantly more energy torage element than the McMllan degree of t mpedance, and poee a non-mnmal tate-pace repreentaton whoe tate correpond to the nductor current and capactor voltage. Depte th apparent non-mnmalty, there have been no mproved algorthm nce the 950, wth the concurrent dcovery by Reza, Pantell, alkow and Gert of a cla of network (the RPG network), whch are a lght mplfcaton of the ott- Duffn network. Each RPG network contan more than twce a many energy torage element a the McMllan degree of t mpedance, yet t ha never been etablhed f all of thee energy torage element are neceary. In th paper, we preent ome newly dcovered alternatve to the RPG network. We then prove that the RPG network, and thee newly dcovered network, contan the leat poble number of energy torage element for realzng certan potve-real functon. In other word, all RLC network whch realze certan mpedance contan more than twce the expected number (McMllan degree) of energy torage element. Index Term Pave ytem, potve-real, mnmal realzaton, network ynthe, electrc crcut, mechancal control, nerter. I. INTRODUCTION MODERN ytem theory ha t root n electrcal crcut analy and ynthe [, p. 78]. The noton of realzablty, mnmalty, and the relatonhp between the nternal and external properte of ytem, all feature n everal clacal paper on electrcal crcut ynthe, e.g., oter reactance theorem []. The connecton between pavty and potve-real (PR) functon orgnated n the the of Otto rune on electrcal crcut ynthe [3], where t wa etablhed that the mpedance of any pave network necearly PR. Thee concept contnue to play a central role n modern ytem theory. Neverthele, many gnfcant reult n electrcal crcut ynthe contnue to perplex ytem theort. In partcular, there reman everal open queton on the ynthe of PR mpedance wth network comprng Th reearch wa conducted whle the author wa the Henlow reearch fellow at tzwllam College, Unverty of Cambrdge, U.K., upported by the Cambrdge Phloophcal Socety, Tmothy H. Hughe wth the Department of Engneerng, Unverty of Cambrdge, Trumpngton Street, Cambrdge, UK, C PZ, e-mal: thh@cam.ac.uk. c 07 IEEE. Th the accepted veron of the manucrpt: Hughe, T.H.: Why RLC realzaton of certan mpedance need many more energy torage element than expected. IEEE Tran. utom. Control, 6(9), (07). retor, nductor, and capactor (RLC network). Some of the more puzzlng queton concern mnmalty (n term of the number and type of element requred) [4], [5]; controllablty [6], [7]; and obervablty [7], [8]. Notably, an RLC network can contan more energy torage element than the McMllan degree of t mpedance, and poe a nonmnmal tate-pace repreentaton whoe tate correpond to the nductor current and capactor voltage. Indeed, th the cae for the famou ott-duffn network [9], and ther mplfcaton [0] []. The purpoe of th paper to demontrate the necety of th apparent non-mnmalty n the RLC realzaton of certan PR functon. In [3], t wa etablhed that the ott-duffn network contan the leat poble number of energy torage element for realzng certan PR functon (the bquadratc mnmum functon) ung ere-parallel network. However, t poble to realze an arbtrary gven PR functon wth RLC network whch are not ere-parallel and contan fewer energy torage element than the ott-duffn network. Th demontrated by the network dcovered by Reza, Pantell, alkow and Gert [0] [] (hereafter referred to a the RPG network), whch acheve a lght mprovement on the ott- Duffn network. In th paper, we frt preent ome newly dcovered alternatve to the RPG network. We then prove that, among the entre cla of RLC network, the RPG network and our newly dcovered alternatve contan the leat poble number of energy torage element for realzng almot all bquadratc mnmum functon. Th depte the number of energy torage element n thee network beng more than twce the McMllan degree of the correpondng network mpedance. Secondary to the motvaton outlned above, the topc of th paper alo relevant to mechancal control followng the recent nventon of the nerter [4]. Ung the completed electrcal-mechancal analogy (ee ppendx C), any gven RLC network ha a correpondng damper-prng-nerter network whoe tranfer functon from force to velocty equal to the mpedance of the correpondng RLC network. Such damper-prng-nerter network have applcaton n vbraton aborpton ytem [4], e.g., vehcle upenon [5]; tran upenon [6] [8]; motorcycle teerng compenator [9], [0]; and buldng upenon []. The tructure of th paper, and the key contrbuton, are a follow. Secton II dcue tate-pace decrpton of RLC network behavor. In Secton III, we preent the RPG network (g. 4a), and our newly dcovered alternatve (g. 4b). Each network n Secton III contan gnfcantly more energy torage element than the McMllan degree of

2 t mpedance. In Secton IV, we tate our man reult concernng the necety of th apparent non-mnmalty for the realzaton of bquadratc mnmum functon (Theorem 4 6). Secton V nvetgate the realzaton of general (not necearly bquadratc) mnmum functon wth RLC network. The man reult are then proved n Secton VI. Relevant background nformaton ncluded n three appendce. ppendce and contan techncal nformaton on RLC network clafcaton and the graph theoretc analy of RLC network. The reader who whe to follow the proof n Secton V and VI n detal adved to read thee appendce before thoe ecton. nally, n ppendx C, we outlne the electrcal-mechancal analogy and t relevance to th paper. Our notaton a follow. We let R (rep., C) denote the real (rep., complex) number. or z C, we denote the real (rep., magnary) part by R(z) (rep., I(z)), and the complex conjugate of z by z. R[] (rep., R()) denote the polynomal (rep., ratonal functon) n the ndetermnate wth real coeffcent. Wth denotng one of R, C, R[], or R(), then m n and n denote matrce and vector of the repectve dmenon whoe entre are all from. We let dag ( ) M M n denote the block dagonal matrx wth M,..., M n on the man dagonal; and col ( ) M M n := [ ] M T Mn T T. or G R(), we ay G PR f () G analytc n the open rght half plane; and () R(G(λ)) 0 for R(λ) > 0. Equvalently, condton () can be replaced wth () R(G(jω)) 0 for all ω R (except at pole of G), and the pole of G on jr are mple and have real potve redue. G called lole f t PR and R(G(jω)) = 0 for all ω R. When ˆp, ˆq R[] are coprme, and G = ˆp/ˆq, then the McMllan degree of G the maxmum of the degree of ˆp and ˆq, and equal to the number of tate n a mnmal (controllable and obervable) tate-pace realzaton for G. II. STTE-SPCE REPRESENTTIONS O RLC NETWORK EHVIORS The famou ott-duffn network [9], and ther mplfcaton [0] [] (the RPG network), prove that any gven PR functon can be realzed by an RLC network. However, the number of energy torage element n thee network conderably greater than the McMllan degree of ther mpedance. In contrat, there are many RLC network whch poe the ame number of energy torage element a the McMllan degree of ther mpedance. or example, any regular functon of McMllan degree two (bquadratc) can be realzed by an RLC network contanng two energy torage element []. Indeed, n the analy of electrcal network, t not unuual to aume that the behavor of the network ha a mnmal tate-pace realzaton whoe tate correpond to the nductor current and capactor voltage (ee [3, Secton III] for a nce decrpton of th and other commonly held aumpton and ther mplcaton). In th ecton, we provde example of network whch volate th condton. One concluon of th paper that th aumpton volated for all RLC realzaton of certan PR functon (the bquadratc mnmum functon). In fact, we wll anwer the queton what the mnmum poble number of energy torage element requred v +,v,v N v v = v + v g.. RLC network wth ource. The drvng-pont current and voltage are denoted by and v, repectvely. g.. v + v + Mechancal d dt = k(v+ v ) L k k > 0 v + v prng v + b c v v = b d(v+ v ) dt b > 0 nerter = c(v + v ) c > 0 damper v + v + C R Electrcal v v v Pave electrcal and mechancal element. L d dt = (v+ v ) L > 0 nductor = C d(v+ v ) dt C > 0 capactor R = (v + v ) R > 0 retor for realzng a bquadratc mnmum functon, and we fnd that n mot cae the RPG network actually contan the leat poble number of energy torage element. We conder RLC (one-port) network. Thee network poe a par of drvng-pont termnal acro whch a ource can be attached a n g.. The network compre an nterconnecton of retor, nductor, and capactor, whch have the properte hown n g. (we only allow trctly potve value for R, L, and C). Note that th fgure alo ndcate the mlarte between thee element and three mechancal component: prng, damper, and nerter (th wll be dcued n ppendx C). The drvng-pont current and voltage v are contraned by the network to atfy a lnear dfferental equaton of the form p( d dt ) = q( d dt )v for ome p, q R[]. Provdng q 0, then the mpedance Z of the network defned a Z := p/q. The extence of Z guaranteed f there at leat one path of element between the drvng-pont termnal of the network [4], n whch cae Z PR, and the number of energy torage element n the network greater than or equal to the McMllan degree of Z [5]. emphaed n [6], [8], [4], certan RLC network contan more energy torage element than the McMllan degree of ther mpedance. There are two man way n whch th can happen: () t may not be poble to arbtrarly agn ntal value to the current through the nductor and the voltage acro the capactor; and () the et of nductor current and capactor voltage can be ether uncontrollable or unobervable from the drvng-pont termnal. Cae () llutrated by the network n g. 3a. The three capactor n th network form a crcut, o the um of the voltage acro the capactor mut um to zero by Krchhoff voltage law. It follow that the behavor of th network doe not poe a tate-pace repreentaton dx dt = x +, v = Cx + D whoe tate correpond to the nductor current and capactor voltage. To ee th, note that for any gven (real) x(0) and (locally ntegrable), then x(t) =

3 3 K KW Kφ W ω KφW 3 0 η ζ KW Kφψ φ KφW (a) Network wth lnearly dependent capactor voltage.,v Kφψ K Kζ W 4 5 K Kη φ KW 3 KW φ (b) Network whoe behavor not tablzable. g. 3. Two network realzaton of the functon H() n Lemma 3; φ := W, ψ := + W, η := W, ζ := W φ η, K, > 0 0 < < W ( W )/ W, / < W <. The two network are related through a tar-delta tranformaton nvolvng the three capactor. e t x(0) + t 0 e(t τ) (τ)dτ and v(t) = Cx(t) + D(t) (t 0) a oluton to the tate-pace equaton. If the tate correpond to the nductor current and capactor voltage, then th mple that there are trajectore of the network for whch the ntal voltage acro the capactor do not um to zero: a contradcton. Note, however, that t poble to decrbe the behavor of th network ung a lnear dfferental algebrac equaton [4]. Cae () llutrated by the network n g. 3b. The behavor of th network poee the tate-pace repreentaton: dx dt = x +, v = Cx + D; wth x := [ ] T v 3 v 4 v 5, := ω0φψ ω 0 0 K ω0 K 0 0 ω0w φ ω0φ W K 0 ω0φ W K K K Kζ W ω0k η φ C := [Kφψ KW 0], and D := K,, := ω0φψ ω0w φ K Kζ W 0 and where φ, ψ, η, and ζ are a defned n the capton of g. 3. Now, let x = [ φw φ W K η Kζ K η ] T. Then x = W φ x and C x = 0, o th tatepace [ model not obervable. Smlarly, wth ˆx = 0 0 ηζ W η φζ ] T, then ˆx T = 0 and ˆx T = 0, o th tate-pace model not controllable. In fact, th tatepace model not tablzable owng to an uncontrollable mode at the orgn (a emphaed n [4], th volate an aumpton whch mplct n the a-c teady-tate analy of RLC network adopted n [6]). In th paper, we nvetgate the necety of the apparent non-mnmalty of RLC network uch a thoe n g. 3 for the RLC realzaton of certan PR functon. In fact, we wll how that each network n g. 3 contan the leat poble number of energy torage element for realzng t mpedance. III. RLC NETWORK RELIZTION PROCEDURES The famou ott Duffn procedure [9] provded the frt algorthm for realzng a general potve-real functon a the mpedance of an RLC network. Slght mplfcaton of the ott Duffn network were dcovered by Reza, Pantell, alkow and Gert n the 950 [0] [] (the RPG network). In th ecton, we preent ome newly dcovered alternatve to thee network. It wll follow from the reult n Secton IV that the RPG network, and thee new alternatve, contan the leat poble number of energy torage element for realzng almot all bquadratc mnmum functon (defned below). Defnton. H() R() called a mnmum functon (wth mnmum frequency ) f H PR, not dentcally zero, ha no pole or zero on jr, and atfe R (H(j )) = 0 for at leat one > 0 (whch mple I (H(j )) 0). It called bquadratc f t McMllan degree two. oth the ott-duffn procedure and the RPG mplfcaton are nductve, wth two tage at each nductve tep: ) the problem of realzng an arbtrary gven PR functon G() converted nto the problem of realzng a mnmum functon, derved from G(), whoe McMllan degree no greater than that of G(); ) the problem of realzng an arbtrary gven mnmum functon H() converted nto the problem of realzng two PR functon, derved from H(), whoe McMllan degree are at leat two fewer than that of H(). Stage (decrbed above) acheved by the oter preamble, a dcued n [3, Secton II]. The contrbuton of [0] [] wa the dcovery of the network n g. 4a, whch pertan to tage. In g. 4b, we preent the newly dcovered alternatve to thee network. hown n [7, Secton 3.], the network n g. 4 can be derved from the ott-duffn network by a equence of network tranformaton (ee alo [8] for an alternatve equence of tranformaton relatng the network n g. 4a to the ott-duffn network). a conequence of the followng theorem, the network n g. 4 can be ued n tage of the procedure decrbed above. Theorem. Let H() be a mnmum functon wth H(j ) = Xj and X > 0 (rep., X < 0). Then H() realzed a the mpedance of the network on the top left and bottom rght (rep., top rght and bottom left) of g. 4a and 4b for ome α, µ > 0 (rep., β, ν > 0), and ome PR functon H r () (rep., H r ()) whoe McMllan degree at leat two fewer than that of H(). Proof: Conder frt the cae X > 0. Then, a decrbed n [3, Secton II], there ext a µ > 0 uch that X = H(µ)/µ, and there ext an α > 0 and a PR functon H r () uch that (µh(µ) H())/(µH() H(µ)) = /H r ()+α/( +ω 0). Here, the McMllan degree of H r () at leat two fewer than that of H(). It follow that H() = H(µ) 3 + H r ()(α + µ) + ω0 + H r ()µω0 H r () 3 + µ + H r ()(αµ + ω0 ) +. µω 0 Drect calculaton verfe that th the mpedance of the network on the top left and bottom rght of g. 4a and 4b. If, ntead, X < 0, then there ext a ν > 0 uch that ω 0X = H(ν)ν, and there ext a β > 0 and a PR functon H r () uch that (νh() H(ν))/(νH(ν) H()) = / H r () + β/( + ω 0) [3, Secton II]. In th cae, the

4 4 h µ h H r() ˆN hµ χ αhφ ω 0 χ αhφ χ hh r() hµω 0 χ hν h H r() η h ν h H r() η hω 0 η h βψ βψ hη νω 0 h H r() ˆN h µ αh ω 0 αhφ χ αhµ χ hh r() hµω 0 χ hν h H r() hω 0 β hη βν h H r() hη νω 0 η h βψ hν hω 0 ζ βψ h h νζω0 H r() βψ h νζ h H r() h ζ h µ hγ h ω0 H r() ˆN hh r() hγµ hγµω 0 αφ αφ hγ h H r() hν β h βψ h νζω 0 β hω 0 ζ h H r() h ζ h µ h H r() ˆN h α hγ αω 0 hh r() hγ hγµω 0 αφ (a) The RPG network for realzng a mnmum functon. (b) Newly dcovered alternatve network. g. 4. Illutraton of a ngle tep n the realzaton of a mnmum functon H() wth H(j ) = Xj. Here, h = H(µ), χ = ω0 + αµ, γ = µ + α, φ = χ + µ, h = H(ν), η = ω0 + βν, ζ = ν + β, ψ = η + ν, and H r(), µ, α, Hr(), ν, β are defned n the proof of Theorem. McMllan degree of H r () at leat two fewer than that of H(), and we obtan H() = H(ν) H r () 3 + ν + H r ()(ω0 + βν) + νω0 3 + H r ()(β + ν) + ω0 + H. r ()νω0 y drect calculaton, th the mpedance of the network on the top rght and bottom left of g. 4a and 4b. IV. RLC RELIZTIONS ND MINIMLITY Each network n g. 4 contan many more energy torage element than expected for the realzaton of t mpedance. However, we wll prove that thee network contan the leat poble number of energy torage element (fve) and the leat poble number of retor (two) for realzng almot all bquadratc mnmum functon. Our man reult are tated n Theorem 4 6, whch adopt the parametraton of a bquadratc mnmum functon decrbed n the followng lemma: Lemma 3. Let H() be a bquadratc mnmum functon. Then H() take the form K ω0( W ) + W + ω0w, + ω0( W ) + ω 0 W for ome K, > 0 and for ome W, whch atfy ether () 0 < W < and > 0, or () W > and < 0. Here K,,, and W are unquely determned by H(), wth K = H( ), KW = H(0), and K j = H(j ). Proof: Th mmedate from [6, Theorem 8]. Relatve to the termnology of [6, equaton (4), (8)], we have made the ubttuton R = K, k = W, and X 0 = K (th enable a more conce preentaton of the man reult). Theorem 4. Let N be an RLC network whoe mpedance H() a bquadratc mnmum functon, a n Lemma 3. Then N contan at leat three energy torage element and at leat two retor. If, n addton, N contan exactly three energy torage element, then ether (a) W = and > 0, or (b) W = and < 0. In partcular, H() the mpedance of N (rep., N ) n cae (a) (rep., (b)) (ee g. 5). K K K N K K Kω0 K K N K Kω0 g. 5. Network N and N. The mpedance of both N and N have the form ndcated n Lemma 3; and atfe condton (a) (rep., (b)) of Theorem 4 n network N (rep., N ). The proof of Theorem 4 come at the end of Secton VI, a doe the proof of the followng theorem: Theorem 5. Let N be an RLC network whoe mpedance H() a bquadratc mnmum functon, a n Lemma 3. If N contan four or fewer energy torage element, then ether condton (a) or (b) n Theorem 4 atfed, or one of the followng four condton mut hold: (c) / < W < and = W W /( W ), (d) < W < and = ( W ) W/( W ), (e) < W < and = W 3 ( W )/( W ), (f) / < W < and = W ( W )/ W. In partcular, H() the mpedance of N 3 (rep., N 4, N 5, N 6 ) n cae (c) (rep., (d), (e), (f)) (ee g. 6) We then obtan the followng theorem (alo proved at the end of Secton VI):

5 5 Kφ W Kφ η Kφ γ Kφ KW φψ KW φψ K N 3 K N 5 K KW K Kω0 Kγ φ K KW φ KW K φ K N 4 Kη φ K N 6 Kω0 K Kφψ Kφψ g. 6. Network N 3, N 4, N 5, and N 6. Here, φ = W, ψ = + W, η = W, and γ = W. The mpedance of all four network have the form ndcated n Lemma 3, and and W atfy condton (c) (rep., (d), (e), (f)) of Theorem 5 n network N 3 (rep., N 4, N 5, N 6 ). Theorem 6. Let N be an RLC network whoe mpedance H() a bquadratc mnmum functon, a n Lemma 3. Then the followng condton all hold.. If none of the condton (a) (f) n Theorem 4 and 5 are atfed, then N contan at leat fve energy torage element and at leat two retor.. If > 0 and 0 < W <, then H() the mpedance of the network on the top left and bottom rght of g. 4a and 4b, where X = K/, µ = W /, H(µ) = KW, α = ( + W )( W ) /(W ), H r () = W, and ˆN and are both retor. 3. If < 0 and W >, then H() the mpedance of the network on the top rght and bottom left of g. 4a and 4b, where X = K/, ν = /W, H(ν) = KW, β = ( + W )( W ) /(W ), Hr () = /W, and and are both retor. We alo note that the mpedance of each network n g. 3 ha the form ndcated n Lemma 3, wth K, > 0, 0 < < W ( W )/ W, and < W <. It then follow from Theorem 6 that each network n g. 3 contan the leat poble number of energy torage element and the leat poble number of retor for the realzaton of t mpedance. Th depte the fact that the capactor voltage are lnearly dependent n the network n g. 3a, and the behavor of the network n g. 3b not tablzable. To how Theorem 4 6, we frt determne thoe mnmum functon (not necearly bquadratc) whch are realzed by RLC network contanng four or fewer energy torage element (th reult n Theorem 7 and 8 n Secton V). The proof of Theorem 4 6 (at the end of Secton VI) then amount to determnng whch bquadratc mnmum functon are realzed by the network decrbed n Theorem 7 and 8. We note an mportant dtncton between Theorem 4 6 and the reult n the paper [6]. Specfcally, Theorem 4 6 etablh the mnmum poble number of energy torage element n RLC network realzng certan PR mpedance, wherea [6] conder the mnmum poble number of element. dcued n Secton II, the number of energy torage element more relevant from a lnear ytem theory perpectve. Moreover, Theorem 4 and 5 cover a cla of nfntely many network (a there no retrcton on the number of retor), wherea the reult n [6] only cover a cla of fntely many network (thoe contanng even or fewer element). In fact, there are two notable error n the man reult of [6]. rt, [6, Theorem ] wa hown to be ncorrect by oter, who tated perhap uch a cenu (of bquadratc mnmum functon realzed by RLC network contanng even or fewer element) hould tll be ncluded among thoe problem not completely olved a yet [9]. Second, n [6, p. 349], t clamed that the RPG network are the only general even-element realzaton of the bquadratc mnmum functon. ut th dproved by the network n g. 4b of th paper. More pecfcally, the even-element RPG network (ee g. 4a and Theorem 6) realze the et of all bquadratc mnmum functon,.e., the et of all H() of the form ndcated n Lemma 3. If 0 < W < and > 0, then H() the mpedance of the network on the top left and bottom rght of g. 4a and alo g. 4b (ee condton of Theorem 6); and f W > and < 0, then H() the mpedance of the RPG network on the top rght and bottom left of g. 4a and alo g. 4b (ee condton 3 of Theorem 6). We note that the network n g. 3, 5, and 6 (and other network n [7]) realze mpedance H() of the form ndcated n Lemma 3 for ome but not all poble value of W and, o they are not general realzaton of the bquadratc mnmum functon. The realzaton of bquadratc mnmum functon wth RLC network contanng even or fewer element wa recondered n [7], and the network n g. 3 and 4b were dcovered. There, t hown that the network n g. 4 are the only even-element realzaton of certan bquadratc mnmum functon, e.g., ( )/( ), whch ha the form ndcated n Lemma 3 wth K = = = and W = 3. V. RLC RELIZTIONS O MINIMUM UNCTIONS In th ecton, we nvetgate the realzaton of general (not necearly bquadratc) mnmum functon wth RLC network contanng lmted number of energy torage element. We tate the man reult n Theorem 7 and 8, whch adopt the network clafcaton cheme decrbed n ppendx. Theorem 7. Let N be an RLC network whoe mpedance H() a mnmum functon. Then N contan at leat three energy torage element. Moreover, f N contan exactly three energy torage element, then H() the mpedance of a network from Q 7 (ee g. 7). Theorem 8. Let N be an RLC network whoe mpedance H() a mnmum functon. If N contan four or fewer energy torage element, then H() the mpedance of a network from one of the clae defned n g. 7 to

6 6 C Cω0 C C C ω0 CEω0 C C ω0 E ω0 N 7 N 7 g. 7. Quartet Q 7 :,, C > 0. The proof of Theorem 7 and 8 come at the end of th ecton, and rely on Lemma 9 3. The tructure of the argument a follow. We let N be a network whoe mpedance atfe the properte requred of a mnmum functon, and we conder a nuodal trajectory of N at the frequency (.e., the drvng-pont and nternal element current and voltage all vary nuodally at th frequency). rom equaton (), the energy uppled to N over a ngle perod of the nuodal trajectory equal to the energy dpated n N over the ame nterval. ut the properte of a mnmum functon mply that the energy uppled to N over a ngle perod zero, o there no energy dpated n N. In partcular, there are no current flowng through the retor n N. More generally, there are ubnetwork wthn N whch are blocked,.e., they have no current flowng through them, and ther vertce are all at the ame potental. lo, the only unblocked element are energy torage element. We then fnd that f N contan four or fewer energy torage element, then N mut compre the one-port ˆN to ˆN 5 connected a n g. E C C E D N 8 D N 8 D D C E E D N d 8 C D N d 8 g. 8. Quartet Q 8 :,, C, D > 0; E := CD C+D. g. 9. E C N 9 DE E C N 9 Quartet Q 9 :,, C, D > 0; E := + +D. D D DE N0 g. 0. Quartet Q 0 :,, C, ( D)(C D) > 0, C; E := D C D. E E E C D N E E E C E C ω D 0 D E E C D N E N N E E Retor (replaced wth a hort crcut when = 0). Retor (replaced wth an open crcut when C = 0). g.. Network clae N, N, N, and N :,, C, D, E > 0; N a, Na, N a, and Na : = 0,, C, D, E > 0; N b, Nb, N b, and Nb : C = 0,,, D, E > 0 (e.g., N n g. ha th form, where ˆN the oneport correpondng to the parallel connecton of a retor and a capactor n ere wth another retor). urther condton (manly relatng to the abence of pole and zero at the orgn and nfnty) then reult n Theorem 7 and 8. To formale th argument, we ue the herarchcal graph theory baed approach to the analy of RLC network outlned n ppendx (ee alo [4], [7]). The reader who whe to follow the detaled proof n th ecton adved to read that appendx now. It contan numbered note (,, etc) whch wll be referred to n the proof. ormal defnton for the termnology n th ecton can alo be found n that appendx (e.g., ubnetwork, one-port, nuodal trajectory, drvng-pont trajectory, phaor current and voltage). We emphae here the dtncton between a ubnetwork and a one-port (whch a pecal type of ubnetwork). or example, the two capactor n network N n g. form a ubnetwork of N, but th not a one-port. lo, wthout lo of generalty, we wll only conder network whch are bconnected (ee 5). Lemma 9. Let N be an RLC network wth mpedance H(),

7 7 and let N compre the one-port ˆN,..., ˆN m. Conder a nuodal trajectory of N at an arbtrary but fxed frequency ω R. Denote the mpedance of the one-port ˆNk by Z k (), t phaor current by ĩ k, and t phaor voltage by ṽ k (k =,..., m). Then the followng hold:. If Z k () ha a pole at = jω, then ĩ k = 0, otherwe ṽ k = Z k (jω)ĩ k.. Suppoe that ether () H() ha a pole at = jω, or () R (H(jω)) = 0. If Z k () doe not have a pole at = jω and R(Z k (jω)) 0, then ĩ k = ṽ k = 0. Proof: To how condton, we let k (t) = R(ĩ k e jωt ) and v k (t) = R(ṽ k e jωt ) for all t R. Then col ( ) k v k a nuodal drvng-pont trajectory (at frequency ω) for the one-port ˆNk, o condton follow from note. or condton, we denote the phaor current and voltage for the ource by ĩ and ṽ, repectvely. rom 8, we obtan ṽ ĩ + ĩ ṽ = m ṽkĩk + ĩ kṽ k. () k= rom note, f H() ha a pole at = jω (rep., H(jω) = 0), then ĩ = 0 (rep., ṽ = 0), and o ṽ ĩ = ĩ ṽ = 0. Otherwe, ṽ ĩ+ĩ ṽ = R (H(jω)) ĩ. Hence, the left-hand de of () zero f ether H() ha a pole at = jω, or R (H(jω)) = 0. urthermore, f Z k () ha a pole at = jω, then ĩ k = 0 by condton, and o ṽ kĩk = ĩ kṽk = 0. Otherwe, ṽ kĩk+ĩ kṽk = R (Z k (jω)) ĩk, whch non-negatve nce Zk () PR. Snce all term n the ummaton n () are non-negatve then they mut all be zero n order that ther um zero. Hence, f Z k () doe not have a pole at = jω and R (Z k (jω)) 0, then ĩ k = 0, o ṽ k = Z k (jω)ĩ k = 0 by condton. rom the above lemma, f N an RLC network whch realze a mnmum functon, then for any nuodal trajectory of N at the mnmum frequency there can be no current through or voltage acro the retor n N. We call the retor blocked, n accordance wth the followng defnton. Defnton 0. Let N be an RLC network and let ˆn be a (not necearly one-port) ubnetwork of N. or an arbtrary gven trajectory of N, we call ˆn blocked f both the current through and the voltage acro all of the element n ˆn are dentcally zero, and unblocked otherwe. We call ˆn a maxmal-blocked ubnetwork of N f t blocked and t not contaned wthn any larger blocked ubnetwork of N. We now decrbe the tructure of any RLC network whoe mpedance ha a mnmum frequency n term of t maxmal-blocked ubnetwork and unblocked element (wth repect to a nuodal trajectory at frequency ). In the followng lemma, we let > 0 be fxed but arbtrary (and content wth the n ppendx ). Lemma. Let N be an RLC network wth mpedance H() whch not lole, doe not have a pole at = j, and atfe R (H(j )) = 0 and I (H(j )) 0. There ext a nuodal trajectory of N at frequency for whch the correpondng drvng-pont trajectory non-zero. Moreover, for any uch trajectory, the followng hold:. Nether the drvng-pont current nor the drvng-pont voltage are dentcally zero.. ll retor n N are blocked. 3. If the ource ncdent wth a vertex n a maxmalblocked ubnetwork of N, then an unblocked element alo ncdent wth th vertex. 4. If an unblocked element ncdent wth a vertex n a maxmal-blocked ubnetwork of N, then ether () the ource, or () a econd unblocked element, alo ncdent wth th vertex. 5. Nether () the ource, nor () an unblocked element, can be ncdent wth two vertce of the ame maxmal-blocked ubnetwork of N. If, n addton, N contan four or fewer energy torage element, then the followng hold: 6. There are ether three or four unblocked element n N, and each of thee an energy torage element. 7. There are ether one or two maxmal-blocked ubnetwork of N, and each of thee ubnetwork a one-port. 8. Let ˆn a be one of the maxmal-blocked one-port n N. Then by ether hortng or openng ˆn a n N we obtan an RLC network N a whoe mpedance H a () atfe H a (j ) = H(j ). lo, when applcable, let ˆn b be the econd maxmal-blocked one-port n N, and uppoe ˆn b alo a one-port n N a. Then by ether hortng or openng ˆn b n N a we obtan an RLC network N b whoe mpedance H b () atfe H b (j ) = H(j ). Proof: rom, there ext a nuodal trajectory of N at frequency wth a non-zero drvng-pont trajectory. We conder any uch trajectory; we denote the mpedance of ˆN k by Z k (), t phaor current by ĩ k, and t phaor voltage by ṽ k (k =,..., m); and we denote the phaor current and voltage of the ource by ĩ and ṽ, repectvely. Proof of Snce H() doe not have a pole at = j, H(j ) 0, and col ( ĩ ṽ ) 0, then t follow from note that ĩ 0 and ṽ 0. Proof of If the element ˆNk a retor, then Z k () doe not have a pole at = j and R (Z k (j )) 0, hence ĩ k = ṽ k = 0 by Lemma 9. Proof of 3 Let ˆn be a maxmal-blocked ubnetwork n N, and let x a be a vertex n ˆn. If the ource ncdent wth x a, but no unblocked element are ncdent wth x a, then ĩ = 0 by Krchhoff current law. Th contradct condton. Proof of 4 Let ˆn and x a be a n the proof of 3. If only one unblocked element ˆNk ncdent wth x a, and the ource not ncdent wth x a, then ĩ k = 0. ˆN k a retor, nductor, or capactor, then Z k () doe not have a pole at = j, o ṽ k = Z k (j )ĩ k = 0 by Lemma 9. Thu, ˆn not a maxmal-blocked ubnetwork: a contradcton. Proof of 5 Let ˆn be a maxmal-blocked ubnetwork n N, and uppoe ether () the ource, or () a ngle unblocked element ˆNk, ncdent wth two vertce x a and x b n ˆn. Snce ˆn connected, then there a path between x a and x b n ˆn. In cae (), by conderng Krchhoff voltage law for the crcut compred of the unon of th path and the ource, t evdent that ṽ = 0, whch contradct condton. Smlarly, n cae (), we fnd that ṽ k = 0. Smlar to the

8 8 proof of 4, nce ˆN k ether a retor, capactor, or nductor, and ṽ k = 0, we conclude that ĩ k = 0, o ˆn not a maxmalblocked ubnetwork: a contradcton. Proof of 6 Snce H() not lole, then N mut contan at leat one retor [3, Secton III]. It then follow from condton that there at leat one maxmal-blocked ubnetwork n N. Now, let ˆn be a maxmal-blocked ubnetwork, and uppoe ˆn contan exactly p vertce at whch ether () the ource, or () an unblocked element, ncdent. It follow from condton 5 that there mut be at leat p unblocked element n N, and all of thee are energy torage element. Snce N bconnected (ee note 5), then p. Moreover, f there are four or fewer energy torage element n N, then we requre p =, mplyng that there mut be ether three or four unblocked element n N. Proof of 7 Suppoe there are exactly q maxmal-blocked ubnetwork ˆn,..., ˆn q n N. rom the proof of 6, there are exactly two vertce of ˆn k at whch unblocked element are ncdent (k =,..., q). Th mple that each maxmalblocked ubnetwork of N a one-port. Snce, n addton, any vertex n N n at mot one maxmal-blocked ubnetwork, then there are exactly q vertce n the maxmal-blocked ubnetwork of N at whch unblocked element are ncdent. urthermore, from condton 3 and 4, then each of thee vertce ncdent wth ether unblocked element or unblocked element and the ource. each element (and the ource) ncdent wth exactly two vertce, then there mut be at leat q unblocked element n N. Thu, f there are four or fewer energy torage element n N, then q. Proof of 8 rom 3, the network N a (and N b when applcable) ha a nuodal trajectory at frequency whoe drvng-pont trajectory the ame a the drvng-pont trajectory of N condered n th proof. Snce th drvng-pont trajectory non-zero, then condton 8 follow from. In [3], the realzaton of mnmum functon ung ereparallel network wa condered. We now generalze [3, Theorem 5]. Lemma. Let N be an RLC network whch contan four or fewer energy torage element, and let the mpedance of N be a mnmum functon. Then N cannot be a ere connecton, nor a parallel connecton, of two RLC network. Proof: Suppoe that N a ere connecton of the RLC network ˆN and. Then H() = Z () + Z () where H(), Z (), and Z () are the mpedance of N, ˆN, and, repectvely (ee note 0). Now, let be a mnmum frequency. Then nether Z () nor Z () have any pole on jr, and R (Z (j )) = R (Z (j )) = 0 [3, Lemma ]. In partcular, nether Z () nor Z () lole, o both ˆN and contan at leat two energy torage element [3, Lemma ]. Snce I (H(j )) 0 then ether I (Z (j )) 0 or I (Z (j )) 0, o ether ˆN or contan at leat three energy torage element by Lemma. Hence, N contan at leat fve energy torage element: a contradcton. The cae where N a parallel connecton of two RLC network mlar, and complete the proof. We now decrbe thoe RLC network contanng four or fewer energy torage element whch realze a mnmum functon. Lemma 3. Let N be an RLC network contanng four or fewer energy torage element, and let the mpedance of N be a mnmum functon (wth mnmum frequency ). Then N compre the one-port ˆN to ˆN 5 connected a n g., and ˆN to ˆN 5 atfy at leat one of the followng condton.. ˆN contan only retor, ˆN and are both capactor (rep., nductor), ˆN4 and ˆN 5 are both nductor (rep., capactor), and Z (j )(Z 3 (j ) + Z 4 (j )) + Z 4 (j )(Z 3 (j ) + Z 5 (j )) = 0.. ˆN and are both capactor, ˆN3 contan only retor, ˆN4 and ˆN5 are both nductor, and Z (j )Z (j ) = Z 4 (j )Z 5 (j ) where Z (j ) Z 4 (j ) and Z (j ) Z 5 (j ). 3. ˆN contan retor together wth at mot one energy torage element, ˆN contan only retor, ˆN3 contan only capactor (rep., nductor), ˆN4 and ˆN 5 contan only nductor (rep., capactor), and Z 3 (j ) = Z 4 (j ) = Z 5 (j ). 4. ˆN and each contan only retor, ˆN3 a capactor (rep., nductor), ˆN4 compre a ere or parallel connecton of an nductor and capactor, ˆN5 an nductor (rep., capactor), and Z 3 (j )= Z 4 (j )= Z 5 (j ). Proof: Let H() denote the mpedance of N. Snce H() a mnmum functon, then t atfe the condton n Lemma. We wll frt how that N compre the one-port ˆN,..., ˆN 5 connected a n g., and one of the followng two condton hold: (a) Exactly one of the one-port ˆN,..., ˆN 5 correpond to a maxmal-blocked ubnetwork, and the remanng oneport are compred of energy torage element. (b) Exactly two of the one-port ˆN,..., ˆN 5 correpond to maxmal-blocked ubnetwork, thee two one-port are not ncdent wth the ame vertex, and the remanng oneport are compred of energy torage element. To how th, we note ntally that N compre ether one or two maxmal-blocked ubnetwork whch are one-port, and ether three or four unblocked element whch are all energy torage element, by Lemma. Now, let G denote the aocated graph of N (we recall that G nclude an edge correpondng to the ource). Then G contan at mot even edge, bconnected, and, by Lemma, t mut not correpond to a ere or parallel connecton of two RLC network. y [30, p. 36], t mut be ether the complete graph on four vertce (graph G a n g. 3), or the graph obtaned by replacng any ngle edge n th graph by ether two edge n ere or two n parallel (graph G b and G c n g. 3). y Lemma, f G take the form of G b (rep., G c ), then nether of the edge connected n ere (rep., parallel) can correpond to the ource. Thu, rrepectve of whether G take the form of G a, G b or G c, we fnd that N compre the one-port ˆN,..., ˆN 5 connected a n g.. lo, from Lemma, then ether (a) or (b) mut hold. To complete the proof, we wll how that condton or (rep., 3 or 4) mut hold n cae (a) (rep., (b)). To how th, we note ntally that H() ha no pole or zero on jr,

9 9 o N mut not contan a drvng-pont L-cut-et, C-cut-et, L-path or C-path (ee 4). Cae (a) Wthout lo of generalty, we can let the blocked ubnetwork be ether ˆN or (th follow from 6). urthermore, the remanng ubnetwork mut each compre a ngle energy torage element. Suppoe ntally that ˆN correpond to the blocked oneport. Then, wth Ȟ() and Ȟ() a n g. 4, we requre Ȟ (j ) = Ȟ(j ) by condton 8 of Lemma. Snce, n addton, N mut not contan a drvng-pont L-cut-et, C-cutet, L-path or C-path, we fnd that condton mut hold. Next, uppoe that ˆN3 correpond to the blocked oneport. Then, wth Ȟ3() and Ȟ4() a n g. 5, we requre Ȟ 3 (j ) = Ȟ4(j ). N mut not contan a drvng-pont L-cut-et, C-cut-et, L-path or C-path, we fnd that condton mut hold. Cae (b) Wthout lo of generalty, we can let the blocked ubnetwork be ˆN and (th agan follow from 6). lo, nce there mut be at leat three energy torage element whch are not n thee one-port, and N contan four or fewer energy torage element, then at leat one of thee two one-port mut contan only retor. Hence, wthout lo of generalty, we can let ˆN contan only retor. Now, wth H (), H (), H 3 (), and H 4 () a n g. 6, we requre H (j ) = H (j ) = H 3 (j ) = H 4 (j ) by Lemma. Th mple Z 3 (j ) = Z 4 (j ) = Z 5 (j ). () There are now three cae to conder: () ˆN3, ˆN4 and ˆN 5 each compre one energy torage element, and ˆN contan retor and at mot one energy torage element; () ˆN3 compre two energy torage element, ˆN4 and ˆN 5 each compre one energy torage element, and ˆN contan only retor; and () compre two energy torage element, and ˆN 5 each compre one energy torage element, and ˆN contan only retor. In cae (), equaton () mple that ˆN4 and ˆN 5 mut compre energy torage element of the ame type, and of oppote type to, and o condton 3 hold. In cae (), equaton () mple that ˆN4 and ˆN 5 compre energy torage element of the ame type. If compre an nductor and a capactor, then contan a drvng-pont L- cut-et, C-cut-et, L-path or C-path, o mut contan only one type of energy torage element. rom equaton (), th energy torage element of oppote type to thoe n and ˆN 5. Thu, condton 3 alo hold n th cae. In cae (), equaton () mple that ˆN3 and ˆN 5 compre energy torage element of oppote type, and the energy torage element n cannot all be of the ame type a. It then clear that ether condton 3 or 4 hold. Note that each of the one-port n Lemma 3 contan at mot two type of element. Ung etablhed reult concernng uch network, we now prove Theorem 7 and 8. Proof of Theorem 7: The mpedance of any RLC network contanng only one type of element equvalent to the mpedance of a ngle element of that type. Theorem 7 then follow from 9 and Lemma 3, notng that condton 3 n that Z 4() Z () ˆN Z 3() Z () g.. Network decrbed n Lemma 3. g. 3. Z 5() ˆN 5 G a G b G c Three bconnected graph wth even or fewer edge. lemma mut hold f N contan exactly three energy torage element. Proof of Theorem 8: It well known that the mpedance of any RLC network whch contan only two type of element equvalent to the mpedance of one of the Cauer canoncal network (ee, e.g., [3]). Theorem 8 then follow from 9 and Lemma 3. VI. RLC RELIZTIONS O IQUDRTIC MINIMUM UNCTIONS In th ecton, we prove Theorem 4 and 5. Th nvolve determnng whch of the network n Lemma 3 realze bquadratc mnmum functon. The network n g. 9 and 0 are elmnated by the followng lemma. Lemma 4. Let N be an RLC network whoe mpedance a bquadratc mnmum functon. Then the retor n N cannot all be contaned n a ngle one-port ubnetwork of N compred of retor alone. Proof: Let H() denote the mpedance of N, Then H() ha nether a pole nor a zero at = 0 or =. lo, H() take the form ndcated n Lemma 3, whence W > 0 and W, and o H(0) H( ). Now, uppoe all the retor n N are contaned n a ngle one-port ˆNk compred of retor alone. Then the mpedance of ˆN k equal to ome potve contant R, and the mpedance of N the ame a the network N a obtaned by replacng the one-port ˆNk n N wth a ngle retor ˆN of retance R (ee note 9). y applyng the reult n 4 nductvely to open all the capactor and hort all the nductor n N a, we obtan a network N b whch ether () the retor ˆN, () an open crcut, or () a hort crcut. gan from 4, nce H() doe not have a pole or zero at = 0, then N b mut be the retor ˆN, and H(0) = R. Smlarly, by conderng the network obtaned by hortng all the nductor and openng all the capactor n N a, we conclude that H( ) = R. Th mple H(0) = H( ): a contradcton. Ung an algebrac argument, we now prove Theorem 4. In th proof, we ue the notaton R k (p(), q()) to denote the Sylveter determnant of two polynomal p() = p m m +

10 0 Z 4() Z 3() Z () Z 5() ˆN 5 Z 3() Z 4() Z 5() ˆN 5 Ň Ň Z () g. 4. Network Ň and Ň decrbed n the proof of Lemma 3. Ň (rep., Ň ) obtaned by openng (rep., hortng) ˆN n the network N n g.. Here, Ȟ = ((Z 3 + Z 5 )(Z + Z 4 ) + Z Z 4 )/(Z + Z 3 + Z 5 ), and Ȟ = Z 5 (Z Z 3 + Z Z 4 + Z 3 Z 4 )/((Z 3 + Z 4 )(Z + Z 5 ) + Z 3 Z 4 ), where Ȟ and Ȟ denote the mpedance of Ň and Ň, repectvely. Z 4() Z () ˆN Z () Z 5() ˆN 5 Z 4() Z () ˆN Ň 3 Ň 4 Z () Z 5() g. 5. Network Ň3 and Ň4 decrbed n the proof of Lemma 3. Ň 3 (rep., Ň 4 ) obtaned by openng (rep., hortng) n the network N n g.. Here, Ȟ 3 = (Z + Z 5 )(Z + Z 4 )/(Z + Z + Z 4 + Z 5 ), and Ȟ 4 = (Z Z 4 (Z +Z 5 )+Z Z 5 (Z +Z 4 ))/((Z +Z 4 )(Z +Z 5 )), where Ȟ 3 and Ȟ4 denote the mpedance of Ň3 and Ň4, repectvely. Z 4() Z 4() Z 3() N Z 5() ˆN 5 ˆN 5 Z 5() ˆN 5 Z 4() Z 3() Z 5() ˆN5 N N3 N4 g. 6. Network N, N, N3, and N 4 decrbed n the proof of Lemma 3. N (rep., N ) obtaned by openng (rep., hortng) n the network Ň n g. 4; and N 3, (rep., N4 ) obtaned by openng (rep., hortng) n the network Ň n g. 4. Here, H = Z 3 +Z 4 +Z 5, H = Z 4, H3 = Z 5, and H 4 = Z 3 Z 4 Z 5 /(Z 3 Z 4 + Z 3 Z 5 + Z 4 Z 5 ), where H, H, H 3, and H 4 denote the mpedance of N, N, N 3, and N 4, repectvely. p m m + and q() = q n n + q n n +, where p m 0 and q n 0: } n k row R k (p(), q()) := }, m k row p m p m 0 p m p m... q n q n 0 q n q n... for k = 0,,..., (mn {m, n} ). rom [3, Theorem 5], p() and q() have at leat r root n common (countng accordng to multplcty) f and only f R 0 (p(), q()) = = R r (p(), q()) = 0. Proof of Theorem 4: That N contan at leat three energy torage element and at leat two retor follow from Theorem 7 and Lemma 4. Snce H() a bquadratc mnmum functon, then H() take the form ndcated n Lemma 3. lo, by drect calculaton, the mpedance of N and N atfy condton (a) and (b), repectvely. Now, let N contan exactly three energy torage element. It then follow from Theorem 7 and Lemma 4 that H() the mpedance of a network from Q 7 (ee g. 7). To complete the proof, we wll how that the functon decrbed n (a) and (b) are the only bquadratc mnmum functon whch can be realzed a the mpedance of a network from Q 7. We frt conder thoe crcumtance n whch the mpedance H 7 () of network N 7 n g. 7 take the form ndcated n Lemma 3. We thu requre H 7 (j ) = K j, H 7 (0) = KW, and H 7 ( ) = K. or H 7 (j ) = K j, we requre C = K, and hence > 0 whch mple 0 < W <. Let g := K/ and g := K/. Then, we requre g, g > 0 and = H 7(0) g + g K = W, and g + g = H 7( ) =. (3) g g K Now, let p() and q() be the polynomal of degree three n uch that p(0) = ω0 3 and p()/q() = H 7 ()/(K ). It ealy verfed that the term n p() and q() of degree three n cannot be zero, o for H 7 () to be bquadratc we requre R 0 (p(), q()) = ω0(g 9 g ) ( + g g ) 4 = 0. Together wth (3), and the condton,, g, g > 0, th mple g = g = and W = /. Thu, f the mpedance of a network from N 7 a bquadratc mnmum functon, then condton (a) hold. y a dualty argument (ee ppendx ) t ealy hown that f the mpedance of a network from N7 d a bquadratc mnmum functon, then condton (b) hold. Th complete the proof. Proof of Theorem 5: Snce H() a bquadratc mnmum functon, then t take the form ndcated n Lemma 3. Drect calculaton verfe that the mpedance of network N 3, N 4, N 5, and N 6 atfy condton (c), (d), (e), and (f), repectvely. Now, let N contan at mot four energy torage element. It then follow from Theorem 7 and Lemma 4 that H() the mpedance of a network from one of the network clae n g. 7, 8, and. We wll how that the functon decrbed n condton (a) (f) of Theorem 4 and 5 are the only bquadratc mnmum functon realzed a the mpedance of a network from one of thee clae. The cae of N belongng to Q 7 wa condered n the proof of Theorem 4, o there are three remanng cae to conder: () N belong to Q 8 (ee g. 8). () N belong to one of the clae N, N, N a, Na, N b or Nb (ee g. ). () N belong to one of the clae N, N, N a, Na, N b or Nb (ee g. ). Cae () Let the mpedance H 8 () of network N 8 n g. 8 take the form ndcated n Lemma 3. or H 8 (j ) = K j we requre D = > 0, whch mple 0 < W <. Let g := K/, g := K/, and c := K/C, and o g, g, c > 0. Then we requre = H 8(0) g + g K = W, and = H 8( ) =. (4) g K

11 Next, let p() and q() be the polynomal of degree four n uch that p(0) = ( + c )ω0 4 and p()/q() = H 8 ()/(K ). We fnd that the term n p() and q() of degree four n cannot be zero, o for H 8 () to be bquadratc we requre R 0 (p(), q()) = R (p(), q()) = 0. Here, R 0 (p(), q()) = c ω0 6 ( + c )( + g g ) 4 f and R (p(), q()) = c ω0( 9 + g g ) f where f and f are both polynomal n c, g, g and. We thu requre f = f = 0, o, n partcular, R 0 (f ( ), f ( )) = c 6 g 0 (+ c ) (c g + c (g g ) + g 3g ) = 0. Taken together wth (4) and the condton c, g, g > 0 and 0 < W <, th mple g = ( W )/W, g =, c = (W )/( W ), and W > /. Then R 0 (p(), q()) = 0 and > 0 mply = W W /( W ), o condton (c) hold. It then ealy hown from dualty and frequency nveron argument (ee ppendx ) that, n cae (), one of the condton (c) (f) mut hold. Cae () We let the mpedance H () of N n g. take the form ndcated n Lemma 3. or H (j ) = K j we requre E =, whch mple > 0 and 0 < W <. Let r := /K, g 3 := K/, g := KC, and c := K/D, o g 3, c > 0 and r, g 0. Then, by conderng H (0) and H ( ), we obtan + r g g ( + r g 3 ) + g 3 = W, and + r g 3 g 3 =. (5) Let p() and q() be the polynomal of degree four n uch that p(0) = ( + r g )ω0 4 and p()/q() = H ()/(K ). We fnd that the term n p() and q() of degree four n cannot be zero, o for H () to be bquadratc we requre R 0 (p(), q()) = R (p(), q()) = 0. In th cae, R 0 (p(), q()) = 4 ω0 6 c (c (r + g 3 ) + ( + r g + g g 3 ) ) f and R (p(), q()) = c ω0f 9 where f and f are both polynomal n c, r, g, g 3, and. We thu requre R 0 (f (c ), f (c )) = 0 g3(r 5 g 3 )((g ( r g 3 )(r + g 3 )+ g3 +g 3 r ) +g3 ) (g 3 (3 r g 3 + r g ( r g 3 )) g ) = 0, and f = c ( r g 3 )+ g 3 (g 3 g ( r g 3 )) = 0. Taken together wth (5), and the condton r, g 0, g 3, c > 0, th mple r = (g 3 )/g 3, g = g 3 (4 g 3 )/(g 3 ), c = g3/( g 3 ), W = /, and g 3 4, g 3, o condton (a) of Theorem 4 hold. Thu, we conclude by dualty and frequency nveron argument that, n cae (), one of the condton (a) or (b) n Theorem 4 mut hold. Cae () Let the mpedance H () of network N n g. take the form ndcated n Lemma 3. In th cae, H (j ) = K j mple E =, and o > 0 and 0 < W <. Now, let r := /K, g := KC, g 3 := K/, and x := K/D, o g 3, x > 0 and r, g 0. Then let p() and q() be the polynomal of degree 4 n uch that p(0) = r x ω0 4 and p()/q() = H ()/(K ). We fnd that the term n p() and q() of degree four n cannot be zero, and n th cae we fnd R 0 (p(), q()) = 6 ω0 6 x (( g 3 + r ) x + ( + r g + g g 3 ) ) f and R (p(), q()) = 6 ω 9 0f where f and f are both polynomal n x, r, g, g 3, and. We thu requre R 0 (f (x ), f (x )) = g 5 3((g ( r g 3 )(r + g 3 ) g 3 r ) +g3 ) (g 3 g ( r g 3 ))(g ( r g 3 ) + g 3 ( + r g 3 )) = 0, together wth f = g 3 ( r g 3 )x + g 3 g ( r g 3 ) = 0. It may be verfed that thee equaton have no oluton for r, g 0 and g 3, x, > 0. It follow from dualty and frequency nveron argument that H() cannot be bquadratc n cae (): a contradcton. Proof of Theorem 6: Condton follow from Theorem 4 and 5. To ee condton, frt note from Theorem and Lemma 3 that H(j ) = Xj = K j, o X = K/. Thu,,, K > 0 mply X > 0, o H() realzed a the mpedance of the network on the top left and bottom rght of g. 4a and 4b by Theorem, where µ, H(µ), α and H r () are a defned n that theorem. It t then ealy verfed that when H() take the form ndcated n Lemma 3, then µ, H(µ), α and H r () are a n condton. Snce H(µ)/H r () and H(µ)H r () are both potve contant, then both ˆN and can be replaced wth a retor. Condton 3 may then be hown mlarly. VII. CONCLUSIONS The network dcovered n the 950 [0] [], whch mplfy the famou ott-duffn network [9], contan a urprngly large number of energy torage element, and have non-mnmal tate-pace realzaton wth tate a nductor current and capactor voltage. In th paper, we howed that thee network actually contan the leat poble number of energy torage element for realzng certan mpedance (almot all bquadratc mnmum functon). In partcular, we proved x theorem on the realzaton of mnmum functon wth RLC network. The man argument wa ummared after Theorem 8. It baed on the obervaton that, for an RLC network N realzng a mnmum functon, and a nuodal trajectory of N at the mnmum frequency, there no energy dpated n N over a ngle perod, o only the energy torage element can tranmt current at th frequency. CKNOWLEDGMENTS The author would lke to thank R..V. Robon and M.C. Smth for many helpful dcuon, and the anonymou revewer for ther valuable comment. PPENDIX NETWORK CLSSIICTION Here, we provde ome network clafcaton termnology, whch enable a conce preentaton of our man reult. n [3], we wll defne network ung dagram wthn fgure (e.g., N n g. 7), n whch we ndcate the drvngpont termnal wth dot, we wrte the mpedance of an element above the element, and we lt contrant on the element mpedance n the fgure capton. Each dagram alo defne a network cla correpondng to the et of all network of the type ndcated whoe element mpedance atfy the contrant lted n the correpondng capton. The concept of dualty and frequency nveron n RLC network analy were exploted n [3], []. n [3], we let > 0 be arbtrary but fxed, and we conder frequency nveron wth repect to. If H() a mnmum functon wth a mnmum frequency, then o too are /H() and H(ω 0/) [3]. In partcular, conder the parametraton of

12 a mnmum functon decrbed n Lemma 3 a a functon of, K,, W, and,.e., H(, K,, W, ) = K ω0( W ) + W + ω0w. + ω0( W ) + ω 0 W Then we note the relatonhp: H( ω 0, K,, W, ) = H(, KW,, W, W ), and H(,K,,W, ) = H(, K,, W, ). Now, let N be an RLC network wth mpedance H(). Then N ha a frequency nverted network N whoe mpedance H(ω 0/) [3]. If, n addton, N planar (and H() 0), then N ha a dual network N d whoe mpedance /H(). Hence, any gven network cla N nduce a econd (pobly dentcal) cla N, contanng the frequency nverted network of every ngle network from N. If, n addton, the network n N are planar, then N nduce two further clae: () N d, contanng the dual network of every ngle network from N ; and () N d := (N d ). Thu, N nduce the network quartet Q, whch the unon of N, N, N d, and N d f the network n N are planar; and the unon of N and N otherwe. PPENDIX HIERRCHICL NLYSIS O RLC NETWORKS In [4], [7], a framework for the analy of RLC network preented. Th framework nfluenced by the behavoral approach to dynamcal ytem [], and graph theory reult from [3] (we note alo reference [33], whch apple graph theory reult from [3] to the analy of port-hamltonan ytem). In th ppendx, we ummare relevant reult from [7, Part ], and we refer to [7] for detaled proof. graph a par (V, E) where V a et {x,..., x n } whoe element are called vertce and E a et of unordered par of vertce called edge,.e., E = {y,..., y q } where y k = (x k, x k ) for k =,..., q and for ome k, k,... n. equence of edge n a graph between two vertce x a and x b called a path from x a to x b. crcut a path from a vertex x a to telf n whch all of the edge are dtnct and no vertce other than x a are repeated. graph called connected f, for any gven par of vertce x a and x b, there a path from x a to x b. cut n a connected graph a et of edge whoe removal partton the vertce nto two djont et. It called a cut-et f, n addton, t contan no ubet whch alo a cut. graph called orented when each edge ha one of t two vertce agned a a head vertex and the other a a tal vertex (we ay the edge orented toward the head vertex). ollowng [4, Secton 3], we aocate any gven RLC network N wth a connected orented graph G whch ha two degnated drvng-pont vertce and contan edge y,..., y m correpondng to the element ˆN,..., ˆN m n the network. The edge y k ha a current k, voltage v k, and a relatonhp p k ( d dt ) k = q k ( d dt )v k correpondng to the properte of element ˆNk (k =,..., m). There one addtonal edge y 0 n G whch ncdent wth the two drvng-pont vertce, and ha a current and voltage v. Th correpond to a ource beng connected to N a n g. (ee Secton II). 3 or the RLC network N n, we let := col ( m ), and v := col ( v v v m ). (6) If () p k ( d dt) k = q k ( d dt) vk (k =,..., m), and () (t) atfe Krchhoff current law and v(t) atfe Krchhoff voltage law for all t R, then we call col ( v m v v m ) a trajectory of N, wth col ( v ) the correpondng drvng-pont trajectory. The behavor (rep., drvng-pont behavor) of N defned a the et of all network trajectore (rep., drvng-pont trajectore). 4 Krchhoff law are related to the cut-et and crcut pace of a graph (ee [4]). Thee two pace correpond to fundamental ubpace of the graph ncdence matrx. Specfcally, let N and G be a n. The ncdence matrx M of G an n (m+) matrx whoe jth entry (rep., +) f edge y j ncdent wth vertex x and orented toward (rep., away from) x, and 0 otherwe. It can then be hown that the R-vector pace panned by the row of M the cutet pace of G (and ha dmenon n a G connected); the orthogonal R-vector pace {z R m+ Mz = 0} the crcut pace of G; and Krchhoff current (rep., voltage) law mple that (t) (rep., v(t)) n the crcut (rep., cut-et) pace of G for all t R [4], [7]. 5 vertex n a connected graph whoe deleton render the graph dconnected called a cut vertex. graph (or an RLC network) called bconnected f t connected and t ha no cut vertce. bconnected component of a graph G a bconnected ubgraph of G whch not a ubgraph of any larger bconnected ubgraph of G. Note, f G contan an edge whch ncdent wth a ngle vertex (a loop), then th edge a bconnected component. It can be hown that the drvng-pont behavor of an RLC network N unchanged by removng element whch are not n the bconnected component of N contanng the ource [7, proof of Lemma..3]. Conequently, we retrct our attenton n th paper to bconnected network. 6 We conder two network to be dentcal f there an orderng and orentaton of the edge n ther repectve graph uch that () the two graph have the ame crcut pace and cut-et pace; and () the relatonhp aocated wth the repectve edge are dentcal. Wth th orderng and orentaton, the two network have the ame behavor. 7 Let N be an RLC network and G the correpondng graph (a n ). non-empty ubet ˆn of the element n N called a ubnetwork f the correpondng edge form a connected ubgraph of G (we do not allow a ubnetwork to contan the ource). Let ˆN be a ubnetwork of N whch contan exactly two vertce x a and x b where the ource and/or element n N but not n ˆN are ncdent. Then ˆN an RLC network wth drvng-pont termnal x a and x b, and we call ˆN a one-port (ubnetwork) n N. or any gven trajectory of N, we defne the current î and voltage ˆv n ˆN a follow. We let î := î + î, where î + (rep., î ) the um of the current through the element n ˆN whch are ncdent wth x a and orented away from (rep., toward) x a. To defne ˆv, we pck an arbtrary path n ˆN from x a to x b, and we let ˆv := ˆv + ˆv, where ˆv + (rep., ˆv ) the um of the voltage acro the element n the path whch are orented wth (rep.,

13 3 agant) the path. It follow from [7, proof of Theorem.9.6] that ˆv doe not depend on the choce of path, and col ( î ˆv ) a drvng-pont trajectory for ˆN. 8 We ay an RLC network N compre the one-port ˆN,..., ˆN m f each element n N belong to one and only one of thee m one-port. There an aocated graph G whch obtaned from the graph G decrbed n by replacng the edge n G correpondng to the element n ˆN k by a ngle edge y k between the drvng-pont termnal of ˆN k (k =,..., m). Snce N bconnected (ee 5), t ealy hown that G too. Now, conder a trajectory of N, let k denote the current and v k the voltage n ˆN k (k =,..., m), let denote the current and v the voltage n the ource, and let and v be a n (6). Then t can be hown that (t) (rep., v(t)) n the crcut (rep., cut-et) pace of G for all t R [7, proof of Theorem.9.6]. lo, f () (t) n the crcut pace and v(t) n the cut-et pace of G for all t R, and () col ( ) k v k a drvng-pont trajectory of ˆNk (k =,..., m), then col ( v ) a drvng-pont trajectory of N. In partcular, nce the cut-et and crcut pace of a graph are orthogonal (ee 4), then T (t)v(t) = 0 for all t R. 9 rom 8, t ealy hown that the drvng-pont behavor (rep., mpedance) of an RLC network N unchanged f we replace a one-port ˆN n N wth a network whch ha the ame drvng-pont behavor (rep., mpedance) a ˆN. 0 Let N be an RLC network (followng 5, N bconnected). We ay that N a ere (rep., parallel) connecton of two RLC network ˆN and f () ˆN and are both one-port n N; () all the element n N are ether n ˆN or ; and () there exactly one vertex (rep., two vertce) where element from both ˆN and are ncdent. It then ealy hown from 9 that H() = Z () + Z () (rep., /H() = /Z ()+/Z ()), where H(), Z (), and Z () denote the mpedance of N, ˆN, and, repectvely. trajectory n whch the current and voltage n the network are all varyng nuodally at a fxed but arbtrary frequency ω R called a nuodal trajectory (at frequency ω). The correpondng drvng-pont trajectory called a nuodal drvng-pont trajectory. The extence of a nonzero nuodal drvng-pont trajectory at frequency ω for any gven RLC network N and ω R guaranteed by [4, Theorem 5]. Conder a nuodal trajectory of N and a one-port ˆNk n N. It ealy hown that the current and voltage n ˆN k are alo varyng nuodally (and correpond to a nuodal drvng-pont trajectory of ˆN k ). In other word, there ext ĩ k, ṽ k C uch that the current k and voltage v k n ˆN k atfy k (t) = R(ĩ k e jωt ) and v k (t) = R(ṽ k e jωt ) for all t R. We call ĩ k the phaor current, and ṽ k the phaor voltage, of ˆNk (correpondng to th pecfc nuodal trajectory). lo, the drvng-pont current and voltage v take the form (t) = R(ĩe jωt ) and v(t) = R(ṽe jωt ), repectvely, for all t R and for ome ĩ, ṽ C, and we call ĩ the phaor current, and ṽ the phaor voltage, of the ource. nally, denotng the mpedance of the network by H(), then ĩ = 0 f H() ha a pole at = jω, wth ṽ = H(jω)ĩ otherwe [4, Theorem 5]. Let N be an RLC network, and let ˆN be a one-port n N wth drvng-pont vertce x a and x b. y openng (rep., hortng) ˆN n N, we mean the operaton of removng all of the element n ˆN from N (rep., connectng the two vertce x a and x b n N), and then removng all element whch are not n the ame bconnected component a the ource. We note that the reultng network N a could contan no element, wth the ource ncdent wth two dtnct vertce (rep., two concdent vertce), n whch cae N a repreent an open crcut (rep., hort crcut), and N a doe not poe an mpedance (rep., the mpedance of N a dentcally zero). 3 Let N be an RLC network comprng the element ˆN,..., ˆN m ; let Ñ be a one-port n N; let N a be obtaned by openng (rep., hortng) the one-port Ñ n N, and (wthout lo of generalty) let N a compre the element ˆN,... ˆN r ; and let b := col ( v m v v m ) be a nuodal trajectory of N at frequency ω. If the phaor current (rep., voltage) n Ñ zero, then b a := col ( ) v r v v r a nuodal trajectory of N a [7, proof of Lemma 3.5.0]. In partcular, f N a oneport n N and alo n N a, then t ha the ame phaor current and voltage n the two nuodal trajectore b and b a, o we may repeat th proce wth the one-port N f t phaor current or voltage alo zero. 4 Let N be an RLC network wth mpedance H(). We ay that N ha a drvng-pont C-cut-et (rep., L-cut-et) f removal of all the capactor (rep., nductor) n the network leave the drvng-pont termnal dconnected, and a drvngpont C-path (rep., L-path) f there a path between the drvng-pont termnal compred olely of capactor (rep., nductor). It well known that () H() ha a pole at = 0 (rep., = ) f N ha a drvng-pont C-cut-et (rep., L-cutet); and () H( ) = 0 (rep., H(0) = 0) f N ha a drvngpont C-path (rep., L-path) [34, Theorem 8.3]. Now, uppoe H() doe not have a pole at = 0 (rep., = ), and let N a be the network obtaned by ether openng (rep., hortng) a capactor n N, or hortng (rep., openng) an nductor n N. Then N a ha mpedance H a () whch atfe H a (0) = H(0) (rep., H a ( ) = H( )) [7, Corollary 3.5.]. PPENDIX C SYNTHESIS O PSSIVE MECHNICL CONTROLLERS gure (ee Secton II) ndcate the properte of the two-termnal mechancal component: damper, prng, and nerter [4]. Ung the force-current analogy, there a one to one correpondence between thee element and the electrcal element retor, nductor, and capactor (ee g. ). Th analogy extend to the nterconnecton law: the net um of all current/force at any vertex zero; and the net um of all voltage/velocte around any crcut zero. Note that there are retrcton to the analogy between a damper-prng-ma network and an RLC network [4]. Specfcally, the force appled to an nerter proportonal to the relatve acceleraton of t two termnal, wherea for the ma t proportonal to t acceleraton relatve to ground (formally, the fxed pont n the nertal reference frame). Conequently, a ma analogou to a grounded capactor, o damper-prng-ma network are analogou to RLC network n whch all capactor are grounded. On the other hand,

14 4 every ngle RLC network ha an equvalent damper-prngnerter network. The tranfer functon from the force appled to the damper-prng-nerter network to the relatve velocty of the network termnal equvalent to the mpedance of the correpondng RLC network. or reaon of cot, complexty, relablty, regulaton, and power requrement, t often derable n mechancal applcaton to ue a pave controller uch a a damperprng-nerter network. pplcaton of damper-prng-nerter network to vehcle upenon, tran upenon, motorcycle teerng compenator, and buldng upenon are decrbed n [4] []. The preent paper condered the realzaton of a PR mpedance ung the mnmum poble number of element, and therefore relevant to the degn of pave mechancal controller. REERENCES [] J. C. Wllem, The behavoral approach to open and nterconnected ytem, Control Sytem Magazne, vol. 7, pp , 007. [] R. M. oter, reactance theorem, ell Syt. Techn. Journ., vol. 3, p. 59, 94. 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Tellegen, Geometrcal confguraton and dualty of electrcal network, Phlp Techncal Revew, vol. 5, pp , 940. [3] T. H. Hughe, On connecton between the Cauchy ndex, the Sylveter matrx, contnued fracton expanon, and crcut ynthe, Proc. of the th Internatonal Sympoum on Mathematcal Theory of Network and Sytem, Gronngen, Netherland, July 04. [3]. ollobá, Modern graph theory. New York : Sprnger, 998. [33]. J. van der Schaft and. M. Machke, Port-Hamltonan ytem on graph, SIM Journal on Control Optm., vol. 5, no., pp , 03. [34] S. Sehu and M.. Reed, Lnear Graph and Electrcal Network. ddon-weley, 96. Tmothy H. Hughe receved the M.Eng. degree n mechancal engneerng, and the Ph.D degree n control engneerng, from the Unverty of Cambrdge, U.K., n 007 and 04, repectvely. rom 007 to 00 he wa employed a a mechancal engneer at The Technology Partnerhp, Hertfordhre, U.K. He currently the Henlow Reearch ellow at tzwllam College, Unverty of Cambrdge. He ha a general nteret n ytem and control theory, and a pecfc nteret n pave mechancal and electrcal control and network ynthe.