8. TIME-VARYING ELECTRICAL ELEMENTS, CIRCUITS, & THE DYNAMICS

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1 8. TIMEVARYING ELECTRICAL ELEMENTS, CIRCUITS, & THE DYNAMICS 8..1 Timevarying Resisors Le us firs consider he case of a resisor, a simple saic device. In addiion o he wo variables, volage and curren, which are funcions of ime, we have seen, we can inroduce addiional variables o describe he effec of variaion of some physical propery of he srucure of he device as a funcion of ime. Thus, we may wrie: v[, α] = v i[ ],α for a curren conrolled resisor and as: [ ] (8.1) 8.1 Inroducion o he Chaper In he las chaper, we noed ha some parameers associaed wih physical sysems (ha may or may no appear in he dynamical equaions used o represen he sysems) can vary wih ime (or forced o change) leading o imevarying or nonauonomous sysems. We also found ha, unforunaely, greaer emphasis has been placed on he saemen "vary wih ime " and he firs saemen "parameers associaed wih physical sysems " is mosly ignored in he analyical approach o dealing wih nonauonomous sysems. In his chaper, we will consider how he imevarying characerisics can be inroduced a he (elecrical) elemen level and used o derive physically meaningful dynamics and pracically useful resuls. Of course, he final resuls ha we obain are similar o he ones obained using he analyical approach. However, he elemen or building block approach makes i easier o quanify he ime varying behavior, arrive a meaningful expressions and limis on heir variaions, and o visualize heir effec on he dynamics. Also, he elemen approach makes he disincion beween nonauonomous linear dynamics and nonlinear auonomous sysems very clear. 8. TimeVarying Elecrical Elemens As we did in he case of linear and nonlinear imeinvarian elemens, we can use he basic properies of power consumpion, generaion, sorage and reurn o define various imevarying elemens. We also have o make sure ha he imevarying behavior is properly associaed wih some physical properies or he srucure of he device so ha he basic properies of ha device, for example power consumpion, are mainained. Of course, if our ineres is in more exoic devices, we can relax some of he consrains and sudy he resuling behavior. [ ] = i[ v[ ], β] (8.) i,β for a volage conrolled resisor. Here, α and β are variables used o lump he effecs of he various physical phenomena 1 ha are assumed o change as a funcion of ime. The quesion is, how o consrain he above wo general models so ha hey poin o meaningful devices. We can derive one simple model from (8.1) as: or v[, α] = R[ α] i (8.3a) [ ] = 1 R[ α] v = G α i, α where R[ α] G[ α] [ ]v (8.3b) ( ) represens he ime varying resisance (conducance). The inverse of a ime varying resisance (or conducance) may no exis as in he case of nonlinear resisors. Therefore, we will have o idenify which is he independen variable and which is he dependen variable and use hem accordingly. Some examples under his represenaion ha can be derived from physical consideraions are: R[ α] = R[ T ] (8.) where R[ T ] represens he variaion of he resisance as a funcion of ime. Or, we may have: 1 If we know wha he various phenomena are and associae individual variables o each and every phenomena, we do no have o lump heir effecs ino a single parameer. In such a case, we will have volagecurren relaionships ha include hose parameers as well.

2 3 Moving arm < L min L L max (8.8) 1 Clockwise roaion of he shaf (a) (b) which in urn leads o: and < R min R R max (8.9) Figure 81. a) A poeniomeer wih hree erminals ha can be used as a rheosa (imevarying resisor); b) Symbol of a imevarying resisor. τ R(τ)dτ γ = R min τ (8.1) This in urn implies, for a consan curren, p min p p max (8.11a) R[ α] = R[ L ]= LR u (8.5) where L represens he ap lengh of a rheosa (variable resisor) ha may be varied as a funcion of ime and R u represens he resisance for uni lengh, a posiive quaniy (Fig. 8.1). We can make some imporan observaions from he equaions (8.3a, b). Firs, he model corresponds o a separable model. Second, as indicaed before we need o clearly indicae he inpu and oupu variables and sick o ha assignmen. Finally, he wo examples given are examples of linear imevarying resisors. Considering he power consumed by he resisors described by (8.3), we obain: p[ v, i, α] = R[ α] i = G[ α]v (8.6) Thus, we need o consrain R[ α] = R[ α ] ( G[ α] = G[ α ]) o be posiive for all ime if he resisor has o be passive (passive linear, imevarying resisors) for all he ime. Such consrains can be easily included in he models derived from physical consideraions such as he ones given in (8.) and (8.5). In fac, in hese equaions we may include: and E R (τ) = τ i R R i R R min τ (8.11b) Tha is, power will be consumed coninuously (as long as he independen variable, curren or volage is nonzero) and a minimum amoun of energy will be consumed by he resisor (and ges convered ino hea) during any and every ime inerval τ for a given consan curren or volage. All of hese properies have pracical implicaions as we will see when we consider neworks made of such elemens. An expression ha fis he descripions given above is: R = R min (1 α sin [ ]) ; α > (8.1) where R min is he minimum value of he resisor and he maximum value is given by R max = R min (1 α) (8.13) Time varying expressions ha have been considered by he classical conrol heoriss and cas in he form of imevarying resisances are: and < T min T T max (8.7) The condiion of physical realizabiliy is used o derive he general forms of imevarying expressions in he coninuous domain. These equaions can be used o build analog devices, o model analog sysems, or implemened in he digial domain.

3 1 G 1 = (8.1) (1 ) G = 1 (1 ) (8.15) G 3 = (8.16) Though hese expressions are valid examples of imevarying funcions, he connecion o he physical elemens is no generally made. We can exend he model o include nonlinear imevarian elemens by making he volagecurren relaionship nonlinear when he oher variable α is fixed. An example of a passive nonlinear imevarying resisor relaionship is given by: v[ i, α] = R[ α] i 3 (8.17) v[ i, α] = i{i isin1} (8.) v R for = π, 5π, 9π ec. for = π, 3π, 9π ec. for =, π, π ec. i R for = 5π, 7π, 13π ec. for = 3 π, 7π, 11π ec. where again R[ α] will be made a suiable posiive funcion of he ime o reflec he effec of he changes in physical parameers. A nonpassive nonlinear imevarying resisor will be of he form: Figure 8. The volage Vs curren characerisics (a differen imes) of a imevarying nonlinear passive resisor. v[ i, α] = R[ α] f[ i ] (8.18) where f[ i ] = for i = and he characerisic f[ i ] Vs i can be in any of he four quadrans. We do no have o resric ourselves o separable funcions o describe a imevarying nonlinear resisor. We can choose nonseparable funcion v i, α [ ] of he wo variables iand α which has he required properies. For [ ] will have he following properies: example, for a passive resisor, v i, α and 1) v[ i =, α] = regardless of he value of α (8.19a) ) v[ i, α] > for i> (8.19b) 3) v[ i, α] < for i< (8.19c) For example, he expression: mees he above requiremens. In figure 8., we show he volage Vs curren characerisics (a differen imes) of his imevarying resisor. We can noe ha he waveforms are confined o firs and hirdquadrans confirming ha he imevarying resisor is indeed passive. 8.. TimeVarying Gyraors We can exend he concep of imevarying elecrical elemens o mulipor lossless memoryless devices. Le us consider he case of a wopor imevarying gyraor. The I/O relaionship for such a device will be of he form: i 1 [, α] G[ x, α] [, α = ] G[ x, α] v 1 v (8.1) i where we have assumed he por volages as he independen variables and he por currens as he dependen variables and x is he sae vecor of he nework

4 in which he gyraor has been embedded. We can also use por currens as he independen variables o resul in an I/O model: Elemen Symbol Example equaions & properies v 1 v [,β] R[ x, β] [, β = ] R[ x,β] i 1 i (8.) In eiher case, we find ha: p = v 1 i 1 [, α] v i [,α] (8.3) = Nonlinear, imevarying resisor (NLTV) {omi he black srip if he elemen is imevarying only}. v R i R v R = Ri R : Separable, LTV, > R max R R min > paasive, curren conrolled. v R = Rf[i R ]; f[] = ; Separable, NLTV, curren conrolled. > R max R R min > f[i] > for i > ; passive < for i < i R, α v R [ ] = f[i R ]{i R i R sin 1} indicaing ha a imevarying device described by (8.1) or (8.) is lossless. Again, we can exend he definiion o include mulipor (M>) gyraors, define special classes and add consrains based on physical realizabiliy. For example, for a wopor gyraor, a linear imevarying model will lead o: G[ x,α] = αˆ G [ x] = α G ˆ ; ˆ G > (8.) where α will be defined based on he applicaions need. We can noe ha, unlike he case of he resisors, we can le α become posiive, negaive or zero. In he case of nonlinear imevarying por gyraors, he admiance funcion will ake he form: G[ x,α] = αˆ G [ x] (8.5) Since we do no have any specific resricions on he admiance (or impedance) marix elemens, we can use a general nonseparable model if suiable funcions can be chosen based on he problem being solved TimeVarying Transformers NLTV charge conrolled capacior. NLTV flux conrolled inducor. v L i c Sored energy, E c [ q c =,α ] = for all α; E c > when q c, E c bounded when q c is bounded. (Charge conrolled, wih q= as he relaxaion poin. The effec of imevariance appears as he funcion α ). i L Sored energy, E L [ φ L =,α ] = for all α; E L > when φ L ; bounded when φ L is bounded E L Nonseparable, NLTV, curren conrolled. Elecrical volage = δe c[ q,α] or poenial δq p = de c[ q,α] = δe c[ q,α] dq δq δe c q, α δα curren, i L = δe L δφ L E L = {(φ f 1 ) f }φ k = k{(q f 1 ) f }φ k 1 (φ f 1 )φ k ;f 1 & f bounded. (NLTV, nonseparable model) [ ] dα = p q p α (Power eneering he cap. due o change in charge and due o he imevarying parameer ). E c [ q,α ] = {1 sin }q k Separable, TV, Linear = k{1 sin k 1 }q (k=1), NLr (K>1). E c ={(q f 1 ) f }q k (NLTV = k{(q f 1 ) k 1 nonseparable) f }q (q f 1 )q k ; f 1 & f bounded. We can also exend he definiion for lossless ransformers o nonlinear imevarying ransformers. We omi he deails as he process and he resuls will be very similar o he case of imevarying gyraors. In Table 8.1 we have provided some deails and examples of imevarying ransformers. Table 81. Timevarying elemens, heir symbols, and he defining equaions.

5 Elemen Symbol Example equaions & properies. 1 Movable plae NLTV gyraor i 1 G[x,α] i v 1 v (por gyraor) i 1 [, α ] i [,α ] = G[ x,α ] v 1 G[ x,α ] v Volage conrolled model; x is he sae vecor; α represens he effecs of imevarying elemens. p = v 1 i 1 v i = for all. G[ x,α] =α G ˆ [ x], separable, nonlinear model. G[ x,α ] =α G ˆ [ x] = α G ˆ ; G ˆ >, Linear, ˆ TV. G [x] & G ˆ real. d Ferroelecric Maerial NLTV gyraor 1 : N[x,α] i 1 i v 1` v v i 1 = N[ x,α ] v 1 N[ x,α ] i Po1 volage assumed o be independen & he erminaion a he secondpor mus be volage conrolled. p = v 1 i 1 v i = for all. N[x,α] =α N ˆ, Linear, TV N[x,α] =α N ˆ [x], Nonlinear, TV (a) Figure 83. A concepual realizaion of a imevarying capacior and i s symbol. (b) N ˆ [x]& N ˆ real & posiive. q c = q c [, ε,a,d] (8.6a) Table 8.1 (Cond.) We modeled he effecs of A and d o be linear and he effecs of and ε o be linear or nonlinear o resul in imeinvarian linear and nonlinear models. We can look a some of hese variables as poenial candidaes for obaining imevarying behavior. I is obvious ha we can visualize in paricular varying he dielecric consan ε and or he disance beween he plaes d as a funcion of ime subjec o some consrains (Figure 8.3). Thus we can have: 8.. TimeVarying Capaciors Le us now consider he case of imevarying elemens wih memory capaciy. While considering physically realizable capaciors (which we represened earlier using imeinvarian linear and nonlinear models), we observed ha he charge q c on he plaes of wo parallel plaes subjeced o a volage depends, in addiion o he volage, on he dielecric consan ε of he medium beween he plaes, he area A of he plaes and he disance d beween he plaes. Tha is: < ε min ε ε max (8.6b) < d min d d max (8.6c) For modeling purposes, we can lump he effecs of he change in hese wo parameers ino a single parameer α. For a volage conrolled capacior, we can hen wrie: q c = q c [,α] (8.7)

6 where q c [.] indicaes he funcional relaionship. Similarly, for a charge conrolled capacior, we will have: = [ q c,α] (8.8) We know losslessness and energy sorage are he key properies of a capacior (and an inducor). Thus, we can consider he energy E c [ q c, α] sored in a imevarying capacior and arrive a he following consrains (assuming q c = as he relaxaion poin): E c [ q c =, α] = for all values of α (8.9a) = δe c[ q, α] δq (8.31) An example of linear imevarying capacior is given by he energy expression: E c [ q,α ]= αq = {1 sin }q The corresponding volage expression is: (8.3a) and E c [ q c, α] > when q c (8.9b) = δe q, α c[ ] δq = {1 sin }q (8.3b) E L [q c ] E c [q c, α] E u [q c ] (8.9c) where E L [q c ] and E u [q c ] are wo posiive definie funcions. We arrive a he firs consrain because no charge implies no sored energy regardless of wha values ε and d ake. This is based on he assumpion ha q c = is he relaxaion poin for our imevarying capacior (Of course, we can modify he consrain o ake care of he siuaion where he relaxaion poin is no he origin). The second consrain follows from he definiion for sored energy. The hird equaion simply saes ha he sored energy has o be bounded by some funcions of he charge alone as he device represens a capacior for any fixed value of α. We can ake he differenial of he energy wih respec o ime o obain he power enering he capacior as: p = de q, α c[ ] = p q p α = δe [ q, α ] c dq δq [ ] δe c q,α δα dα (8.3) The power expression has wo componens; he firs one is he power enering or delivered o he capacior by he change in he charge and he second one corresponds o he power delivered by he mechanism ha changes α. We denoe hese wo conribuions as p q and p α. From he expression for p q we ge an expression for he volage across he wo plaes of he capacior as: Tha is, we have a separable model and he volage depends linearly on he charge for a fixed value of α. The energy and volage waveforms corresponding o his model are shown in figure 8.. The corresponding expressions for an example of a nonlinear imevarying capacior are: and is: wih E c [ q,α ]= αq = δe q, α c[ ] δq = {1 sin }q = {1 sin }q 3 (8.33a) (8.33b) An example of a nonseparable model for nonlinear imevarying capaciors E c { } [ q,α ]= q 1q qα 75α 6 = q { q 1.α } 6.6α 6 [ ] = q 3 {q qα5α } (8.3a) (8.3b)

7 8 Energy Energy 8 Time Charge Time Fig. a. Charge Fig. a. 1 Volage 8 Volage 8 Time Charge Time 1 Fig. b. 1 Charge Fig. b. Figure 85. a) Energy & b) volage for a NLTV capacior described by a nonseparable model. Figure 8. a) Energy sored in and b) volage across a imevarying linear capacior as a funcion of ime and charge. where α is a bounded imevarying funcion. The energy and volage waveforms for such a imevarying capacior when α = exp() are shown in figure 8.5.

8 8..5 TimeVarying Inducors A imevarying inducor, being he dual of a capacior, can be defined in a similar manner. We simply summarize he resuls for he imevarying inducors in able TimeVarying Acive and Nonpassive Elemens As in he case of imeinvarian nonlinear elemens, we can consrain he dynamic elemens as lossless elemens and associae nonpassive and or acive behavior o resisive elemens and power sources. Thus, we can have fully passive, fully acive and nonpassive imevarying resisors and use separable and nonseparable, and linear and nonlinear models o describe such elemens. We again omi he deails as he derivaions are sraigh forward. i c v R C=1 (a) G = G min (1α sin []) G min, α > 1 G Ω, p G univol 1 3 (b) Figure 86. a) A firsorder circui wih a LTI capacior and a imevarying resisor. b) The conducance and he power consumed per uni vol as a funcion of ime. 8. Elecrical Circuis wih TimeVarying Elemens & he resuling Dynamics In his secion, we consider some simple elecrical circuis buil using imevarying circuis and sudy he properies of such circuis. We will also make connecions, where ever possible and or necessary, o heory developed using analyical approaches and poin ou he similariies, differences, and he superioriy of he approach advanced in his book. Example 1 Le us consider a firsorder circui (Fig. 8.6) consising of a linear imeinvarian univalued capacior and a linear imevarying resisor wih a imevarying conducance given by: G = G min {1 α sin (β)} (8.35) where G min > and α > 1 for he resisor o be passive. The dynamics of he nework is: v c G = v c G min {1 αsin (β)} = (8.36) We can easily obain a Lyapunov funcion (no jus a candidae) for his dynamics by equaing i o he energy lef in he capacior a any given ime as: LF = 1 v c (8.37) The derivaive of his LF along he sysem rajecory is: = = G min {1 α sin (β)} (8.38) We can noe ha he LF derivaive along he sysem rajecory is nohing bu he negaive of he power going ino he imevarying resisor ha is conneced across he capacior. In his example, he resisor is passive and power is consumed coninuously. The power consumed is bounded as: G min (1 α) p R = G min if α G min v c p R = G min (1α) if α< (8.39) Tha is, minimum power will be consumed by he resisor a any ime unil he capacior volage (and hence he sored energy) becomes zero. Thus, we can conclude ha he dynamics given by equaion (8.36) is absoluely sable. In Fig.

9 8.7, we show he various responses for wo differen iniial condiions. As expeced, all he variables go o zero as. From his example, we should noe ha even hough he dynamics is imevarying, he Lyapunov funcion need no necessarily be a imevarying funcion. Here, he derivaive of he LF along he sysem rajecory is a imevarying one and he negaive of he derivaive is a decrescen funcion (dominaed by a imeinvarian posiive definie funcion). Example Le us use he same nework of Fig. 8.6 (for his example as well as for examples 3 and ) and replace he conducance funcion by imevarying funcions used in he conrol heory lieraure. Firs, we use: G = G max 1 α ; (8.) wih α a posiive consan. The conducance value which is always posiive becomes smaller as ime progresses and becomes exacly equal o zero as. Thus, is power absorpion capaciy decreases as ime progresses. The dynamics corresponding o his resisance is: v c G = v c G max 1 α = (8.1) Since he reacive elemen is he same, we have he same Lyapunov funcion: LF = 1 v c However, he LF derivaive along he sysem rajecory changes o: (8.) 1, i G = = G max 1 α v c (8.3) i G and is bounded by: 1 1 (a) E c, p G p G 1 (b) G max v c (8.) Again, he derivaive is a decrescen funcion and becomes zero as and when he capacior volage becomes zero and says here. Thus, we can conclude ha he dynamics based on he resisor in equaion (8.) will also be absoluely sable. In Fig. 8.8, we show he various responses for wo differen iniial condiions. Again, as expeced, all he responses go o zero as. Example 3 G = The conducance expression used for his example is given by: G max (1 α) ; (8.5) Figure 87. Response of he dynamics of he imevarying nework of figure 8.6. for some iniial values. a) Volage across he capacior and he curren hrough he imevarying resisor. b) Energy lef in he capacior and he power consumed by he resisor. We can observe ha he conducance value a any imeinsance is posiive for posiive α. Also, he conducance ends o zero much faser as ime progresses as compared o he imevarying conducance of he previous example. In fac, we can show ha he infinie inegral:

10 G(τ)dτ (8.6) τ= is finie for he conducance of his example where as i is infinie for he conducance of he previous example. The dynamics based on his conducance is given by: v c G = v c G max (1 α) v c = (8.7) and he LF derivaive along he sysem rajecory is: or = = G max (1α) v c (8.9) Again, he LF is he same given by he energy lef in he linear capacior: LF = 1 v c 1 E c (8.8) G max v c (8.5) Thus, hough he derivaive is negaive, i can (and does) become zero before he capacior volage becomes exacly equal o zero. Thus, he dynamics resuling from he use of he resisor of equaion (8.5) is sable, bu no absoluely sable. For absolue sabiliy, he inegral of he conducance funcion given by (8.6) has o be infinie. A proof of his requiremen, hough simple o derive, is no really necessary, and hence is omied. In Fig. 8.9, we show he various responses for wo differen iniial condiions. Noe ha hough he curren in he circui becomes zero as, he volage across he LTI capacior and hence he energy lef in he capacior does no become zero. 3 3 (a) 6 i G 3 6 (b) p G Example Finally, we consider he same firs order nework of Fig using a conducance expression of he form: G = α ; α, (8.51) 1 Here we have he diamerically opposie siuaion. The conducance value increases (linearly) as he ime increases, and hence is appeie for power. The dynamics, he LF, and is derivaive along he sysem rajecory using his conducance are: 3 6 (c) 3 6 (d) v c G = v c α = (8.5a) Figure 88. Response of he imevarying nework of figure 8.6. when he imevarying conducance is changed o G = G max (1 α), α a posiive consan. a) Volage across he capacior; b) Energy lef in he capacior; c) Curren hrough he imevarying resisor, and d) Power consumed by he resisor. LF = 1 v c = = α (8.5b) (8.5c)

11 or and β where β > (8.5d) = only when = (3.5e) Thus, we can conclude ha he dynamics will be a leas absoluely sable. From he observaion ha he conducance and hence he power absorpion capaciy increases as ime goes by, we can speculae ha he dynamics will be exponenially sable. In fac, he condiion: τ G(τ)dτ γ > (8.53) ha we encounered before (using he resisance definiion) implies exponenial sabiliy, and is saisfied by he resisor used in his example. In Fig. 8.1, we show he response of he dynamics 8.5 for some iniial condiions. As discussed, he response goes o zero. Also by comparing figures 8.7 o 8.9 wih fig. 8.1, we find ha he response of dynamics of example 1 and go o zero faser, whereas he response of example and 3 are somewha sluggish. In fac, for some iniial condiions, he response of example 3 seles a some nonzero value as indicaed earlier. 1 E c 1 E c 1 6 (a).. (b).. i G p G.5 1 (a) 1 i G.5 1 (b) p G 1. (c)... (d) 1 (c).5 1 (d).5 1 Figure 89. Response of he imevarying nework of figure 8.6. when he imevarying conducance is changed o G = G max (1 α), α a posiive consan. a) Volage and b) Energy lef in he capacior; c) Curren hrough and d) Power consumed by he resisor. Figure 81. Response of he imevarying nework of figure 8.6. when he imevarying conducance is changed o G = α, α a posiive consan. a) Volage and b) Energy lef in he capacior; c) Curren hrough and d) Power consumed by he resisor.

12 i L v L v R L=1 H R=1 i c C=. F Figure 811. A nework wih a LTI capacior and inducor and a imevarying resisor leading o a secondorder imevarying dynamics. Examples 5 & 6 Now, we consider wo examples corresponding o second order dynamics. Boh he examples use a linear imeinvarian capacior, a linear imeinvarian inducor and one or more imevarying linear resisors. The nework corresponding o he firs dynamics is shown in Fig The inducor value is se a one Henry and he capaciance value a. Farad. The imevarying resisance is given by: R =1 (8.5) where he resisance increases linearly wih respec o he ime variable. The dynamics of he nework is: 1 v 5 c i L = 1 1 ( 1 ) i L (8.55) which is linear and imevarian. We find ha [ ] is he only equilibrium poin for his dynamics. We can wrie he LF as he sum of he energy lef in he wo linear reacive elemens as: LF = v c i L (8.56) which is imeinvarian (depends only on he sae vecor and no he ime variable explicily). The derivaive of he LF along he sysem rajecory is: = i L [ ] 1 5 v c i L 1 v = [ i L ] c 1 ( 1 ) i L = ( 1 )i L = for i L = and any value for (8.57) As noed above, he derivaive can become zero as soon as one sae variable, he curren hrough he inducor, becomes zero and hence he dynamics may no reach he equilibrium poin ( ). However, from he condiion i L =, and he nework dynamics we find ha i L and hence will change. The passive resisance presen in he circui which keeps consuming energy as long as he curren hrough is no zero will hen push he sae variables close, and finally o he origin, he equilibrium poin. Hence, we can conclude ha he dynamics is absoluely sable. In Fig. 8.1, we show he various responses for wo iniial condiions. The wo sae variables, he volage across he capacior, and he curren hrough he inducor, are shown in figures 8.1a and 8.1b respecively. The volage across he resisor appears in figure 8.1c. The energy lef in he circui and he power consumed by he resisor are shown in Fig. 8.1d. A Phase plane plo using he sae variables of he nework appears in figure 8.1e. A Phase plane plo using he energy in he capacior and he energy in he inducor is shown in figure 8.1f. We can noe ha he sae goes o he origin, he equilibrium poin, as expeced. Also, we can see from figure 8.1e ha he inducor curren becomes zero making he derivaive of he LF momenarily zero. However, he dynamics moves he sae away from ha poin such ha he oal sored energy ges depleed compleely. In his example, we used a resisor whose power absorpion depends on one sae variable (curren hrough he inducor) which in urn is equal o he derivaive of he second sae variable (volage across he capacior) because of he way he elemens are conneced. Also, as we observed earlier, he resisance value goes o infiniy in a linear fashion. A faser move o infiniy (which implies large power consumpion if he curren is no zero) may make he curren iself zero (zero power consumpion) because of he nework srucure (or equivalenly he dynamics) while he oher sae variable, he volage across he capacior, may no become zero. Tha is, we can end up wih a dynamics whose equilibrium poin is no absoluely sable. We will illusrae his in he nex example.

13 1 5 i L R = α 1 α e β (8.58) wih α 1 and α > (resisor becomes passive) and β > makes he resisance grow exponenially. The corresponding dynamics, Lyapunov funcion, and is derivaive along he sysem rajecory are: (a) (b) 8 v R 5*E, p R v c i L = 1 1 ( α 1 α e β ) i L (8.59a) LF = 1 ( v c i L ) (8.59b) 1 (c) (d) i L 8 8 (e) 5 E c 1 (f) Example 6 We use he same nework of Fig for his example, bu change he inducor value o one and he resisor characerisic o: 1 8 p R E L Figure 81. Response of he imevarying dynamics of nework of figure for some iniial condiions. a) The volage across he capacior; b) Curren hrough he inducor; c) The energy lef in he circui and d) he power consumed by he resisor; e) Phase plane plo; f) Phase plane plo using he energy in he capacior and he inducor. = [ i L ] v c i L 1 = [ i L ] 1 ( α 1 α e β ) i L = ( α 1 α e β )i L = for i L = and any value for (8.59c) Again, we noe ha he derivaive of he Lyapunov funcion can become zero when one sae variable becomes zero regardless of he value of he oher sae variable. The fac ha he resisance value moves o infiniy much faser (in an exponenial manner) makes he siuaion much worse. From a nework perspecive, his implies ha he resisor ges removed (or becomes an open circui) before all he energy in he capacior is dissipaed. Therefore, he sae will no reach he equilibrium poin, and hence he equilibrium poin is no sable. In fac, for any () and i L () = ( α 1 ( α 1 α )) (), we can obain a closed form soluion for he dynamics as: = α v () 1 c α e α 1 α 1 α α 1 i L = α 1 () α 1 α e α 1 ; (8.6) implying ha α () ( α 1 α ) and i L as. In Fig we show he resuls of simulaion for some iniial condiions ha are conneced as discussed above. We find ha he volage across he capaciance seles a some nonzero value as expeced.

14 Referring back o he resisor expression, we find ha in effec we have wo resisors in series, one LTI and passive wih a value of α 1 ohms and he oher imevarying and passive wih a value given by α e β. In ordinary circumsance, he presence of he LTI passive resisor alone would have made he equilibrium poin absoluely sable. However, he series inerconnecion and he fac ha he second resisor becomes a open circui as ime progresses makes he presence of he oher LTI resisor irrelevan. The building block approach and he nework perspecive can help overcome such problems. In his paricular case, he sabiliy of he equilibrium poin can be achieved by compleely removing ha resisor (he nework becomes LTI) or by making he imevarying resisor value bounded. Example 7 In his example we consider a slighly more complex nework consising of wo linear imeinvarian capaciors and a imevarying wopor nework as shown in Fig. 8.1a. The wopor nework can be represened by a imevarying wopor gyraor and a number of imevarying passive resisors as shown leading o he archiecure shown in Fig. 8.1b. The nework herefore is passive. The dynamics of his nework is: v c1 1 e v c = (8.61) The dynamics has only one equilibrium poin, he origin. Since he sorage elemens are imeinvarian, we can sill wrie he Lyapunov funcion as he energy lef in he sorage elemens: 1 i L 1 (a) 8 (b) 8 v R E, p R (c) 8 (d) 8 i L ( ) (8.6) LF = 1 v c1 v c and he derivaive of he LF along he sysem rajecory is: 6 (e) = [ 1 ] v c1 v c 1 e v = [ 1 ] c1 1 1 = ( v c1 v c ) 1 1 e = 1 α ( ) ( ) ( ) 1 α where α = ( 1 e ) 1 (8.63) Figure 813. Response of he imevarying nework of figure wih he imevarying resisance as R = α 1 α e β where α 1, α,& βare posiive consans. a) The volage across he capacior; b) Curren hrough he inducor; c) The volage across he resisor; d) Energy lef in he circui and he power consumed by he resisor; e) Phase plane plo.

15 1 ohm 1 ( ) i p1 i C1 v p1 C=1 A imevarying wopor nework Y = e 1 i p v p i C C=1 1 mho (a) (b) (a) v ( ) c1.5(1 e ) υ i p1 i C1 C=1 v p1.5(1 e ) υ A imevarying wopor gyraor Y G = y 1 y 1 i p v p i C C=1.5(1 e ) υ 1 (c) 1 e y 1 = 16 E c (b) Figure 81. a) A nework wih wo LTI capaciors and a imevarying wopor nework; b) An alernae realizaion of he same dynamics using imevarying resisors and a imevarying gyraor. 8 (d) 8 1 E c1 which remains negaive as long as any of he sae variable is nonzero and is a decrescen funcion. Thus, we can conclude ha he dynamics in (8.61) is absoluely sable. In figure 8.15, we show he various responses for some iniial condiions. We can noe ha he sae value reaches he origin as expeced. Figure 815. The response of he nework of figure 8.1: a) & b) The volage across he wo LTI capaciors; c) The phase plane rajecory; d) The phase plane rajecory in erms of he energy lef in he wo capaciors.

16 Example 8 In his example, we consider a firsorder nework wih a charge conrolled linear imevarying capacior and a linear imeinvarian resisor of value G mhos as shown in figure The capaciance is described by he sored energy expression: E c [q,] = 1 {1α sin (β)}q (8.6) wih α > 1, and he volage expression: = δe q, c[ ] δq i c = q c = ={1α sin (β)}q (1 sin )q c G mhos Figure 816. A firsorder nework wih a imevarying capacior and a linear resisor. (8.65) The dynamics of he nework is given by: q G. = q G{1α sin (β)}q = (8.66) which is same as he dynamics of a nework consising of a linear capacior and a imevarying resisor as given by example 1. However, he rue Lyapunov funcion, he sored energy in he reacive elemen for his nework, is differen and is given by: LF = E c = 1 {1 α sin (β)}q (8.67) which is a decrescen funcion. The derivaive of his funcion along he sysem rajecory is: = de c { } = q αβ sin(β)cos(β){1 α sin (β)}q q = 1 q αβ sin(β) G{1 α sin (β)} q = q { G{1 αsin (β)}.5 αβ sin[β] }.5αβ if G ( 1 α) when 1< α <.5αβ when α (8.68) which is also differen from he one seen in example 1. From he above expression, we find a sufficiency condiion for absolue sabiliy is ha he conducance value mus be greaer han some minimum value. This will ensure ha he exra energy which is being added by he imevarying mechanism will also be consumed in a imely manner so as o bring he sae o he equilibrium value. Noe ha his condiion is differen from wha we obained in example 1 which simply saes ha he conducance has o be posiive. We show in Fig simulaion resuls for boh examples for wo values of G. We show no only he sae rajecory (which is same for boh) bu also oher variables such as energy lef ec. (which are differen). In figures 8.17 a o c, we show he resuls when α = 3, β = π and G =. From he above expression we find ha hese values lead o a derivaive of he Lyapunov funcion ha is always negaive implying ha he dynamics is sable and he simulaion resuls confirm he same. In figures 8.17 d o f, we show he resuls of simulaion wih G =.5. For his value of G, he derivaive of he Lyapunov funcion becomes boh posiive and negaive (as can be seen from he above expression as well as figure f) implying ha we canno say if he equilibrium is sable based on his Lyapunov funcion. However, he simulaion resuls (as well he equivalence o example 1 and he resuling condiion ha he conducance only needs o be posiive) indicae ha he dynamics is sable for his small value of G as well. This example illusraes he key aspec of sabiliy analysis using Lyapunov mehod. I is only a sufficiency condiion and no a necessary one. From a nework perspecive, Lyapunoondiion amouns o saying ha a dynamics is sable if he ne power going ino he energy sorage devices is always negaive. And as his example demonsraes, we can have siuaions where his need no necessarily be rue, bu he dynamics can sill be sable. In design, where we invariably use conservaive figures, we may use elemen values ha obey he Lyapunoondiion. This is especially rue for higher order sysems where we may no be able o arrive a alernae Lyapunov funcions (as we did for his firs order example) which may poin o differen elemen values. We will illusrae his in example 1.

17 q, q, q..8.1 (a) E c &.5q 1 E c..8.1 (b) 8 p c & p R 6 p c. (c) (d) E c &.5q E c 1 (e) p c & p R p c (f) Figure 817. The response of he firsorder imevarying nework of figure 8.16 wih α = 3, β = π and G =. a) The charge and he volage across he imevarying capacior; b) The energy lef in he imevarying capacior; c) The negaive of he derivaive of he Lyapunov funcion (= p c, ne power leaving he capacior). We also show he response from a ne wih a LTI uni valued capacior & a imevarying resisor ha leads o he same dynamics (example 1). In figure b, we have he energy (=.5q ), and in figure c, we show he power consumed by he imevarying resisor ( p R ). We can see ha hough he sae is he same for boh dynamics, he res of he responses are no. Fig Cond. The response of he firsorder imevarying nework of figure 8.11 wih α = 3, β = π and G =.5. d) The charge and he volage across he imevarying capacior; e) The energy lef in he imevarying capacior; f) The negaive of he derivaive of he Lyapunov funcion (=p c, ne power leaving he capacior). In his case, we can observe ha he ne power leaving he capacior becomes negaive implying ha a imes he power consumed by he LTI resisor is less han he power injeced by he imevarying phenomena. However, evenually, he sae variable goes o zero indicaing ha he dynamics is sill absoluely sable.

18 Example 9 In his example we consider he dynamics obained from a nework consising of a charge conrolled nonlinear imevarying capacior and a linear resisor of conducance G mhos (Fig. 8.18a). The capaciance is described by he expression for sored energy: = de c = 1 { q β 1 β sin(β )cos(β ) {1 sin (β )}q 3 q } E c [ q,α ]= αq = 1 {1β sin 1 (β )}q wih β 1 > 1 for he energy o be posiive and he volage expression: (8.69) = 1 q β 1 β sin(β ) G(1 sin (β )) q 6 = q G(1 sin (β )) q 1 β β sin(β ) 1 { } = q G α 1 (q,) α (8.7a) = δe q, α c[ ] δq = {1β 1 sin (β )}q 3 (8.7) where α 1 (q,) = (1β 1 sin (β )) q (8.7b) The dynamics of he nework is given by: and q G. = q G{1β 1 sin (β )}q 3 = (8.71) α = 1 β 1β sin(β ) (8.7c) Again, we can use he energy lef in he capacior (equaion 8.69) as he Lyapunov funcion. The derivaive of he LF along he sysem rajecory is: i c = q (a) G mhos = { 1 β 1 sin (β ) }q 3 i c = q c = 1 F i R = G{ 1 β 1 sin (β ) }v 3 R = q Figure 818. a) A firsorder nework wih a nonlinear, imevarying capacior and a LTI resisor; b) Anoher nework wih a LTI capacior and a nonlinear, imevarying resisor wih he same dynamics. (b) Thus, when he magniude of q is large, he derivaive will be negaive. However, when he magniude of q becomes smaller han some value ha is dependen on β 1, β and G, he derivaive will sar oggling from posiive o negaive and vice versa. Therefore, we canno say anyhing conclusively abou he absolue sabiliy of he equilibrium poin using his LF which happens o be he rue energy funcion. However, being a firs order dynamics, we can equae his dynamics o ha of a nework consising of a linear uni valued capacior and a nonlinear ime varying capacior as shown in Fig. 8.18b. Since he resisance in he new circui is always posiive when G >, we can conclude ha he dynamics will be sable as long as G >. This equivalence also poins o anoher imporan aspec of his dynamics: since he curren hrough (and hence he power absorpion) he nonlinear ime varying resisor becomes very small as q < 1, he dynamics will no be exponenially sable. We may ask how we can avoid such problems. I is obvious ha he problem arises due o he use of a higher power (only) of q for he volage across he capacior. Insead, We can include a square erm for he energy ( resuling in a linear erm for he volage expression) as: E c [ q,α ]= αq = 1 {1 sin }(kq q ) ; k > (8.73a) and

19 = δe q, α c[ ] δq The new dynamics is: = 1 {1 sin }{kq q 3 } (8.73b) q 1 A wopor imevarying gyraor G 1 υ y 1 [q] G υ 1 [q 1 ] y = v y 1 [q] c [q ] q q G = q 1 G{1 sin }{kq q 3 } = (8.7) The derivaive of he energy along he sysem rajecory is: Figure 819. A secondorder nework wih wo linear imevarying capaciors, a wopor nonlinear gyraor, and wo LTI resisors. = de c { } sys.r = 1 (kq q )sin()(1 sin )(kq q 3 ) q = 1 {(kq q )sin() G{(1 sin )(kq q 3 )} } (8.75) wih he LF given by he sum of he wo sored energy funcions. The nework dynamics is given by: ( ) = 1 q (k q ) G (1 sin )(k q ) (k q ) sin() q 1 q = G 1 y 1 [q] 1 [q 1 ] y 1 [q] G [q ] (8.77) from which we can noe ha a finie value of G (for any given posiive value of k) can be chosen so as o ensure ha he derivaive is negaive. Thus, any amoun of energy injeced ino he dynamics by he imevarying mechanism will be absorbed by he resisor forcing q o reach he equilibrium poin. The dynamics is herefore absoluely sable. Also, he presence of he linear erm in he volage expression makes he dynamics exponenially sable as we discussed before. Example 1 In his example, we will look a he various equaions using a second order nework wih wo linear ime varying capaciors as shown in figure The wo capaciors are described by he wo expressions for sored energy as: E c1 [ q 1,α]= 1 {1 α 1 e α }q 1 ; α 1, α > E c [ q,α]= 1 {1 β sin 1 (β )}q ; β 1, β > (8.76) where 1 [q 1 ] = {1 α 1 e α }q 1 [q ] = {1β 1 sin (β )}q (8.78) and y 1 [q] is some nonlinear funcion of he sae vecor q. The derivaive of he LF along he sysem rajecory is given by: = d(e c1 E c ) = q 1 G 1 (1 α 1 e α ).5α 1 α e α { } (8.79) q { G {1β 1 sin (β )}.5β 1 β sin(β ) } which again is he difference beween he power injeced ino he nework by he ime varying mechanism and he power consumed by he wo resisors. We find ha we can selec proper values for he conducances G 1 and G ha will ensure ha he dynamics is sable. We should also noe ha i is no really

20 possible o come up wih an equivalen nework where he capaciors are linear and he resisors are ime varying. Example 11 In he above example, we concluded ha proper values for he conducances can be seleced such ha ne power is aken ou of he circui. This is no always he case as we will see in his example. We consider he same circui of las example wih G 1 =, G = G, y 1 q energy expressions as: E c1 [ q 1,α]= 1 {1 sin [ ]}q 1 E c [ q,α]= 1 {1 cos [ ]}q [ ] = 1, and he capacior (8.8) Tha is, he second capacior is nonlinear and imevarying. The nework dynamics is given by: where q 1 q = 1 1 [q 1 ] 1 G [q ] (8.81) is only a sufficien condiion and no a necessary one. In our case, we have wo energy sorage devices and he effec of he imevarying phenomena on he energy sored is limied. Tha is: E c1_min E c_min [ q 1 ]= 1 q 1 E c1 [ q 1, α ] E c1_max q 1 [ ] =q 1 [ q ]= 1 q E c [ q, α ] E c_min q [ ] = 1 q (8.8) which implies ha he capaciors change from one LTI (or NLTI in he case of he second capacior) elemen o anoher. For a given value of q 1 and q, once he imevarying mechanism pushes he energy envelope o he maximum level, here is nohing much i can do. I can only ake back some of he energy i gave o he elemens. Therefore, for all pracical purposes, he circui behaves like a nonlinear circui. The linear damper (as opposed o a nonlinear or imevarying damper) will consume power uniformly making he dynamics asympoically sable. This example, demonsraes how we can make complex imevarying dynamics and a he same ime preserve sabiliy. We can make he damper imevarying as well leading o exoic behavior. I is fair o say ha we have jus scrached he surface and much work needs o be done. 1 [q 1 ] = {1 sin [ ]}q 1 (8.8) [q ] = {1 cos 3 [ ]}q are he capacior volages. The LF for he dynamics is he sum of he capacior energy funcions. Is derivaive along he sysem rajecory is: LF = d(e E ) c1 c = G{1 sin [ ]} q.5 q ( 1.5q )sin[ ] (8.83) I is no clear from he above expression if a finie value for G can be found such ha he derivaive is always negaive. Noe ha here we are using he rue energy funcion and no an arbirarily chosen LF candidae and sill he derivaive is no negaivedefinie. Upon some reflecion, we can conclude ha here is no reason o be surprised. Afer all, we have a imevarying mechanism here and a imes i can injec more power han is being consumed making he energy lef increase (or equivalenly, he derivaive becoming posiive). The key is ha any peak value (or local maxima) ha he energy akes a any ime is less han he previous peak as shown in Fig. 8.. Thus, he derivaive of he LF has o be negaivedefinie Figure 8. A Lyapunov funcion which is no monoonically decreasing (derivaive no negaivedefinie) bu goes o zero as.

21 Example 11 We now consider he hree secondorder dynamics used in chaper o illusrae differen limi cycle behaviors and explain heir behavior using heir nework equivalence. The dynamics are: and G = G = 1 x 1 x (8.86b) x 1 x = 1 x 1 1 x x {x 1 1 x 1} x {x 1 x 1} x 1 x = 1 x 1 1 x x {1 x 1 1 x } x {1 x 1 x } (8.85a) (8.85b) G = G 3 = 1 x 1 x (8.86c) leads respecively o he hree dynamics given in (8.8). The hree conducances become zero when x 1 x =1 (8.87) and x 1 x = 1 x 1 1 x x 11 x 1 x x 1 x 1 x (8.85c) leading o a lossless LC nework. The parameers of his lossless nework are such ha he resuling oscillaion is given by: x 1 x = r (8.88) I is easy o see ha a linear ime varying nework archiecure of he form shown in Fig. 8.1 wih wo univalued LTI capaciors, a wopor LTI lossless gyraor wih an admiance marix as shown in he figure and wo ime varying resisors wih he same conducance values a boh pors: G = G 1 = x 1 x 1 x 1 A wopor LTI gyraor x x 1 1 x y = G υ c=1 F 1 c=1 F G υ (8.86a) Figure 81. A secondorder nework wih wo imevarying resisors, wo LTI capaciors and a LTI gyraor. By choosing he imevarying conducances properly we can obain dynamics ha exhibi sable, unsable, and semisable limi cycle oscillaions. where r is a consan ha will depend on he iniial charge on he capaciors. This expression includes he expression (8.87) as a special case. Hence, he limi cycle oscillaion. In he nework corresponding o he firs dynamics, he conducances become passive when x 1 x >1 forcing E =.5(x 1 x ), he remaining energy in he circui, o decrease. When x 1 x <1, he conducances become negaive forcing E =.5(x 1 x ), he remaining energy in he circui, o increase. Therefore, he circle of radius one in he phase plane becomes he aracing limi cycle for he firs dynamics. I should be noed ha he behavior of he imevarying resisors, changing from a passive one o an acive one and back, has more influence on he circui dynamics han he iniial energy sored in he capaciors and herefore a limi cycle oscillaion ha is independen of he iniial condiions. In he nework corresponding o he second dynamics, he conducances become negaive when x 1 x >1 forcing E =.5(x 1 x ), he remaining energy in he circui, o increase. When x 1 x <1, he conducances become posiive forcing E =.5(x 1 x ), he remaining energy in he circui, o decrease. Therefore, he limi cycle becomes an unsable one. The origin and infiniy are he wo sable equilibrium poins for his dynamics. In he nework corresponding o he hird dynamics, he conducances become passive when x 1 x >1 as well as when x 1 x <1 forcing E =.5(x 1 x ), he remaining energy in he circui, o increase. Therefore, he circle of radius in he phase plane becomes a semisable limi cycle.

22 As we noed, In all he hree dynamics in (8.85), he sae rajecory for which he imevarying resisance becomes a open circui also happens o be he one soluion for he remaining lossless LTI circui. We may ask if his is really necessary for limi cycle oscillaion o exis. The answer is no as we show using a new example. The chosen dynamics is: x 1 x = 1 x 1 x x {x 1 1 x 1} x {x 1 x 1} (8.89a) which differs from expression (8.85a) corresponding o sable limi cycle dynamics by jus one parameer in he consan marix (,1 elemen). The dynamics can be rewrien as: x 1.5 x = 1 x 1 1 x x 1 {x 1 x 1}.5 x {x 1 x 1} (8.89b) which again poins o a nework wih wo capaciors (one of uni value and he oher.5 F), a lossless wopor LTI gyraor and wo imevarying conducances. The wo conducances sill become zero (open circui) on he uni circle in he phase plane (rajecory # 1 in figure 8.). On he oher hand, he soluion of he res of he LTI lossless circui is given by: x 1.5 x = r (8.9) which doesn include he uni circle. We show wo of hese rajecories (# and # 3) when r =1 and.5 in he figure. We also show he response of he dynamics as rajecory #. The limi cycle oscillaion is neiher he uni cycle nor any of he ellipsoids described by (8.9). Insead, we ge a response ha goes inside of he uni circle for some ime (A o B and C o D in he figure) when he conducances are negaive and ouside of he uni circle for res of he cycle (B o C and D o A in he figure) when he conducances are posiive. Tha is, he imevarying resisors characerisics dominae he dynamics. As here are no oher passive resisors in he circui, he capaciors ge charged for par of he cycle and discharge for he res of he cycle. The various parameers and he erms in he nonlinear dynamics deermine he exac shape of he response. The above example demonsraes how we can use nework conceps o explain easily he behavior of complex dynamics and or o obain dynamics wih he required characerisics. In a laer chaper, we will use similar echniques o explain furher he conceps of chaos and fracals. 1 1 Trajecory # 1 on which he conducances become zero. Trajecory # 3. Anoher sae rajecory for he lossless LTI ne Summary D A x 1 Trajecory #. One possible sae rajecory for he lossless LTI ne. x Trajecory #. Response of he nonlinear dynamics Figure 8. The phaseplane response of he nonlinear dynamics of equaion (8.8) which exhibis limi cycle behavior (Sae variable x is along he x axis and sae variable x 1 is along he y axis). The response is influenced by he rajecory # 1 which separaes he regions in which he imevarying conducances become passive and acive, and no on he response of he LTI lossless nework ha resuls when he conducances become zero. In his chaper we learn ha he phenomena of imevariance can be inroduced properly a he elemen level using he conceps of power and energy. We firs sudied imevarying resisors ha can be linear or nonlinear, fully passive or fully acive (negaive) or nonpassive. We saw he definiions for imevarying gyraors and ransformers, mulipor devices ha are also lossless. We hen looked a imevarying capaciors and inducors, dynamic devices ha are also lossless. Proper inerconnecion of hese building blocks lead o complex imevarying circuis. Using KCL and KVL one can wrie he dynamics corresponding o such circuis. From he nework elemens, we can wrie he rue energy funcion as he Lyapunov funcion and he derivaive of he LF along he C B

23 sysem rajecory easily by inspecion. A number of firs and secondorder circuis were provided o illusrae he power of he new approach. This approach allows us o correcly differeniae nonlinear dynamics from imevarying dynamics. From he examples provided, i became clear ha he LF need no be a imevarying funcion even if he dynamics is imevarying. Also, we can have imevarying dynamics in which he rue LF is no monoonically decreasing (he derivaive along he sysem rajecory no negaivedefinie or negaivesemidefinie) bu sill he dynamics is absoluely sable. Though he examples provided are limied o secondorder dynamics, i should be noed ha he approach is applicable o any higherorder sysems. Furher, he building block approach can be of remendous help in a number of applicaions as we will see in par II of his book.