Transen esponse n Elecrc rcus The elemen equaon for he branch of he fgure when he source s gven by a generc funcon of me, s v () r d r ds = r Mrs d d r (')d' () V The crcu s descrbed by he opology equaons and by he elemen equaons of he form gven by eq.. n m v r r v r = f( r ) = = v r r V r - r r M rs Ths sysem of equaon s negrodfferenal and can be solved by an exsng convenonal mehod of mahemacs. s r r Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Transen esponse n Elecrc rcus Transen ause A change n he crcu operang condons s he cause of a ransen before reachng he seady sae operaon whch can be suded n he me doman by means of he se of he equaons of he crcu analyss. On- off or off- on mode changes of swches or sudden changes of he excaon (volage or curren source modeled as sep funcons) are he cause of ransens followed by a seady sae operaon ha s he response of he crcu o he changed condon. Ø The ransen response s he crcu s emporary response ha wll de ou wh me. Ths s he emporary par of he response. Ø The seady- sae response s he behavor of he crcu a long me afer he sudden change has happened. Ths s he permanen par of he response. omplee response = ransen response seady- sae response Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna
Transen esponse n Elecrc rcus Transen ause The ransen response operaon, whch wll precede he esablshmen of he seady- sae operaon, s due o he me requred by he sorage elemens o bul he new condons under whch hey wll operae a he fnal seady- sae regme. The sorage elemens may change her operaon sae wh fulfllng he energy conservaon prncple. As a consequence of sorage elemens preven nsananeous varaon of energy n he rans from = - o = : ε( - ) = ε( ) An nsananeous varaon of he energy would only be caused by a source of nfne power: - ε( ) ε( ) p( ) lm = = Δ Δ Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Energy onservaon Prncple In order o nhb nsananeous varaons of he sored energy, capacors oppose any sharp varaon of he enson, and nducors oppose any sharp varaon of he curren. Ths can be seen from he expresson of he energy sored n he wo elemens. ε () = magnec energy sored by an nducor Q ε () = v = elecrosac energy sored by a capacor The rans from = - and =, represenng he saus of he crcu operaon mmedaely before he change and mmedaely afer of respecvely, has o fulfll he energy conservaon prncple, whch has he followng consequences: ( - ) = ( ) n branches wh nducors v( - ) = v( ) beween he ermnal of a capacor Q( - ) = Q( ) charge sored n each of he conducng plaes of he capacor Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna
Frs Order rcus: rcu Source Free rcu esponse Naural esponse q A me = he swch of he crcu of he fgure s closed. The capacor before = s charged a Q = Q. A ransen wll sar and wll be exngushed when he regme a he new condons s reached. The volage, due o he elecrosac feld of he capacor acs on he charges sored on a capacor plaes, whch flow hrough he ressor oward he oppose plae. Ths curren ransfers he elecrosac energy sored by he capacor o he ressor where s dsspaed. When all he energy s dsspaed he ransen vanshes. Ø The crcu s beng exced by he energy sored n he capacor. No exernal sources are presen. Ø The am s o deermne he crcu response ha s assumed o be gven by he behavor of he volage v() across he capacor. The naural response of a crcu refers o he behavor (n erms of volages or of currens) of he crcu self, wh no exernal sources of excaon. = - < - < Q = V A Q() v() v() Frs Order rcus: rcu Source Free rcu esponse Naural esponse A me < s Q = Q and v = Q / = V. For he energy conservaon prncple a = s also Q = Q and v = Q / = V. A he < from K a node A s: = as = dv/d and = v/, resuls dv v = () d The soluon of he ransen, whch represens he naural response of he crcu, s gven by he soluon of eq. ha s a frs order homogeneous dfferenal equaon (hs s he movaon of he erm frs order crcu): v() = A e α α = - /() = - /τ where α s he soluon of he characersc equaon assocaed o eq.. The me consan s defne by τ = - /α. = - < - < A Q() Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Q A v() v() 3
Frs Order rcus: rcu Source Free rcu Naural esponse The consan A s derved from he nal condons a = : herefore: v( ) = V ( ) = Q( ) v() = V e /τ where τ = s he me consan. The me consan τ of a crcu s he me requred o he response o decay o a facor /e or 36.8 % of s nal value. V.368 V Every me nerval of τ he volage s reduced by.368 % of s prevous value: v(τ) = v()/e =.368 v(). The dervave of v() a = s - /τ. Thus τ s he nal rae of decay or he me for v o decay from V O o zero, assumng a consan rae of decay. I akes 5τ o he crcu o reach s fnal sae (seady sae). < V τ () v() Q() e / τ V()/V τ.36788 τ.3534 3 τ.4979 4 τ.83 5 τ.674 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Frs Order rcus: rcu Source Free rcu Naural esponse The curren n he crcu s: () = v() = V e /() The power dsspaed by he ressor s p() = v() () = V e /() and he energy ransfered from he capacor and absorbed by he ressor up o me s ε () = p() d = d = = τ V e /() V e /() = V ( e ) /() and for, ε( ) (V ) /. v() < () v() Q() τ = τ = τ =.5 The crcu can be an equvalen crcu (Thévenn/Noron crcu). The key quanes are:. The nal capacor volge he v( - ) = v( ) = V. The me consan τ = Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna 4
Sep esponse of an rcu The ransen s caused by a sep of he volage. Ths can be done by a source volage whch s suddenly appled by closng a swch. In hs case s: - < - : Q=Q, v = Q /=V A < from he KT s V (') d' -V = () Frs Order rcus: rcu From he me dervave of eq. s obaned: d d = () The soluon of he ransen soluon of eq. (a frs order homogeneous dfferenal equaon) : () = A e α (3) where agan α = -/τ = -/(). The value of he consan A s deermned hrough he analyss of he nal condons. V V - - = Q = V - < - < Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna v() Q() Frs Order rcus: rcu Sep esponse of an rcu Analyss of he nal daa a = - : v ( - ) = V ; Q( - ) = V (nal daa) ( - ) = Analyss of he nal condons a = : - Across he capacor: v( ) = v( - )=V and from he KT: - V V = ( ) = (V -V )/ - From eq. 3 [ ()=Ae - / ], s: ( ) = A A = (V -V )/ Hence he expresson of eq. 3 s () = V V - e () - V - V = = V = Q / V = Q Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna 5
Frs Order rcus: rcu Sep esponse of an rcu The response of he crcu (also for a Thévenn equvalen crcu) o a sudden volage source excaon n erms of he curren () s : V V - () () = e The response of he crcu n erms of he volage v() across he capacor s derved from he elemen equaon of he capacor: v() = V (') d' from he soluon of he negral resuls () ( V - V ) v() = V e.368 V V () V e / (V -V ) The analyss of he regme a = gves he seady- sae response (forced response) [( ) =, v( ) = V ] ha added o he ransen response (naural response) gves he complee response of he crcu suddenly exced. () V -V V -V v() V -V () e / Frs Order rcus: rcu Sep esponse of an rcu When a volage source s suddenly appled by swchng on he crcu s: - < - : =, A < from he KT s d d - V = () The soluon of he ransen s gven by he soluon of he homogeneous dfferenal equaon (naural componen of he response): () = A e α where α s gven by he characersc equaon: α = -/ = -/τ. The complee response of he crcu exced by a volage source s V - = - < - () = A e f f s a parcular soluon of eq.. For he seadysae-soluon a = (forced'response) s aken: f = ( ). The consan A s gven by he nal condon a = : ( ) = ( - ) =. V - < 6
Frs Order rcus: rcu Sep esponse of an rcu Analyss of he seady sae a = : - V = f = = V / - V () = A e Analyss of he nal daa a = - : ( - ) = Analyss of he nal condons a = : ( ) = ( - ) = A = - V / Hence he complee response of he crcu when suddenly exced by a volage source, s () = V - e- ( ) V - V /.63 V / < τ The complee response of he crcu s he seady- sae response (forced response) [( ) = ] added o he ransen response (naural response). Frs Order rcus esponse of Frs Order rcus Ø A frs order crcu s a crcu whch can be a Thévenn conanng a ressor and one memory elemen. The soluon of he response of he crcu o a sudden excaon s gven by he soluon of a frs order dfferenal equaon. Ø The response of he crcu, when exced by an exernal source (complee response) s gven by he response of he crcu when no exernal sources are presen (ransen response or naural response) supermposed o he seady sae response ha s he poron of he complee response (seady sae response forced response) whch remans acve when he ransen response has ded ou. omplee response = ransen response seady- sae response naural response forced response (emporary par ) (permanen par) The naural response, ha gves he ransen par of he response, s he soluon of he homogeneous me dfferenal equaon The forced seady- sae response of he crcu s he parcular soluon of he non- homogeneous me dfferenal equaon a =. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna 7
Second Order rcus: rcu A second order crcu s characerzed by a second order dfferenal equaon. I consss of ressors and he equvalen of wo energy sorage elemens. The analyss of a second order crcu response s smlar o ha used for frs order. The naural response s orgnaed by he excaon/de- excaon wh ransfer of he energy sored n he sorage elemens. = - : The nal daa are essenal o defne he excaons of he ransen. They are he magnec energy sored n nducors and he elecrosac energy n capacors. Therefore hs s deermned by he currens of he nducors and he volages (or he charges) of he capacors a = -. These nal daa are gven by he soluon of he crcu a he seady- sae condons a = -. = : The nal condons are derved from he nal daa consderng ha durng he ranson from = - o = he energy connuy prncple mus be fulflled: ( ) = ( - ) = I n nducors and v( ) = v( - ) = V n capacors. The naural response s calculaed by solvng he homogeneous dfferenal equaons. The characersc mes n he exponen are he soluons of he characersc equaon, he consan of negraon are derved from he nal condons. = : The seady sae soluon a = gves he parcular soluon of he non- homogeneous dfferenal equaons n he forced response case when ndependen sources are presen. Second Order rcus: rcu esponse of Second Order rcus Ø The response of he crcu, when exced by an exernal source (complee response) s gven by he response of he crcu when no exernal sources are presen (ransen response - naural response) supermposed o he seady sae response ha s he poron of he complee response (seady sae response forced response) whch remans when he ransen response has ded ou. omplee response = ransen response seady- sae response naural response forced response (emporary par ) (permanen par) As a general case of second order crcus, n he followng of hs chaper a seres crcu (,, and n seres) exced by an ndependen volage source s consdered. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna 8
j Second Order rcu: Seres rcu A - < - : Q = Q, v = V = Q /, = A = : he swch s closed. A < : from KT resuls: * Q() (')(d' *V ((( ( d( d = ((((((((((((() ( ( (')(d' *V ((( ( d( d = v () (')(d' *V ( d( d = (((() from(he(me(dervaon((resuls(: d ( d d( d ( = ((((((((((((((((((((((((((((3) The(response,(n(erms(of(((),(s(gven(by( he(soluon(of(eq.(3,(ha(s(an(ordnary( dfferenal(equaon(of(he(second'order. V V - - = - < - < Q Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Second Order rcus: Seres rcu Analyss of he nal daa a = - : v ( - ) = V ; Q( - ) = V ( - ) = Analyss of he nal cond. a = : - Across he capacor: v ( ) = v ( - ) = V - Through he nducor: ( ) = ( - ) = d( ) V - V = d d( ) = (V - V )/ d Analyss of he seady sae a = : ( ) = ; v ( ) = V V - A = : apacors exce he crcuas ndependa volage sources: v ( ) = v ( - ) = Q( - )/; Inducors exce he crcuas ndependa curren sources: ( ) = ( - ). V - = ( - ) = - v ( - ) =V V Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna 9
Second Order rcus: Seres rcu Transen analyss d d = (3) d d The characersc equaon of eq. 3 s: x x = x = - ± x = - α ± α ω x, x α = /() ω = naural frequences dampng facor he resonance frequency s also he underdumped naural frequency V - = < ase A:"""α " > ""ω """ """"" > """ " """""""""""""""" x ""and"" x ""are"real"and"dsc" """""""""""""""""(negave):!!!overdamped!case ase B:"""α " = ""ω """ """"" = """ " """"""""""""""""" x ""and"" x "real"double"soluon """"""""""""""(negave):!!crcally!damped!case ase :"""α " < ""ω """ """"" < """ " """"""""""""""""" x ""and"" x ""complex"conjugae" """""(negave"real"par)"underdamped)case""" Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Second Order rcu: Seres rcu ase A - overdamped case : α > ω > ; x, x negave, real and dnsc The soluon of eq. 3 s: () = A e A e From he nal condons a = : A A = ( ) = x x d ( ) xa xa = = ( V - V) d where x = - ± A - A V - V Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna = =
Second Order rcu: Seres rcu Transen soluon n case A Overdamped case : α > ω, > () = V - V e e e - V - = < Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Second Order rcu: Seres rcu ase B - crcally damped case : α = ω = ; x, x negave double soluon The soluon of eq. 4 s: () = A e A e From he nal condons a = : A = ( ) = α α where α = d ( ) A = = ( V - V ) d Transen soluon n ase B crcally damped case: () = V - V e - Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna
Second Order rcu: Seres rcu ase - underdamped case : α < ω < ; x, x complex conjugae (negave real par) The soluon of eq. 3 s: From he nal condons a = : A = () = x = -α jβ x = -α jβ α ( β β ) () = A cos A sn e where where d ( ) αa βa = = ( V - V) d α = β = A A (dampng fac.) = = V - V Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Second Order rcu: Seres rcu Transen soluon n case Underdamped case: < V - V α () = e sn V - = T T = π T < Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna
Transen Analyss: General Mehod Analyss of he ransen n he me doman. Wre he se of equaons (for example by means of he volage subsuon mehod). Defnon of a ordnary dfferenal equaon n one unknown by subsuon and dervaon: n n- d d d n n n-... n- a a a a = f() (4) d d d he soluon of whch s: () = () () where T S n xk T() Ake ransen response k= S = ( lm T() = ) ( ) () seady sae response parcular negral a = of eq. 4 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Analyss of he ransen n he me doman 3. Dervaon of he parcular negral of eq. 4 a = by means of he analyss of he seady sae a =. 4. Dervaon of he naural frequences of T (),x, x,, x 3,.. x n,, whch are he soluons of he characersc equaon of he homogeneous dfferenal equaon assocaed o eq. 5. 5. Dervaon of he negraon consans A, A, A 3,.A n by means of he nal condons. 6. Dervaon of he oher unknowns. Hence, n order o derve he complee response of he crcu, s necessary o fnd he crcu operaon a: = o deermne he seady sae soluon, ha s he parcular negral of eq. 4 a = ; = - o deermne he nal daa; = o deermne he nal condons. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna 3
Analyss of he ransen n he me doman A = he swch s closed and he crcu response wh a ransen s naed.. Se of smulaneous equaons descrbng he crcu a < < : V - = Q d - V = d Q (') d' - V = - - =. Successve dervaon and subsuon o oban an ordnary dfferenal equaon n one unknown: d d V = (5) d d Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Analyss of he ransen n he me doman Analyss of he nal daa a = - : ( - ) = ( - ) = ( - ) = v ( - ) = as ( - ) = ( - ) = ; Q ( - ) = Analyss of he nal cond. a = : - Across he capacor: v ( ) = v ( - ) = - Through he nducor: ( ) = ( - ) = V - ( ) - ( ) = ( ) - ( ) = ( ) = ( ) = V /( ) ( d ) - ( ) = d d ( ) = V /[ ( )] d V V - = Q ( - ) - = - = v (-) = 4
Analyss of he ransen n he me doman Analys. of he seady sae a = : ( ) = ( ) = V /; ( ) = ; v ( ) =. Soluon of eq. 5:!!!!!()!! =!!A e x!!a e x V where! x!and!! x!are!he!soluon!of he!characersc!equaon!of!eq.!5: % x x = %% The!negraon!consans!A!and!A! are!deermned!by!he!nzal!condzons:!!!! ( )! =!A!!A! =!!!!! d! ( ) = x d A!! x A! =!!!!!!!!!!!!!!!!!!!!!!!!! =! V! V - = Q = - = = V /( ) V / V /( ) V / V d /d Q v Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna Termnology Englsh hnese complee response 完全响应 source free response 零输入相应 frs order crcu 一阶电路 seady- sae 稳态 forced response 强迫相应 seady- sae response 稳态响应 naural response 自由响应 sep response 阶跃响应 response 响应 me consan 时间常数 second order crcu 二阶电路 ransen 暂态 naural response 自由响应 ransen response 暂态响应 Deparmen of Elecrcal Engneerng Unversy of Bologna 3 5
Termnology complee response rsposa complea source free response rsposa lbera frs order crcu crcuo del prmo ordne seady- sae regme sazonaro forced response rsposa forzaa seady- sae response rsposa d regme naural response rsposa naurale sep response rsposa al gradno response rsposa me consan cosane d empo second order crcu naural response crcuo del secondo ordne rsposa naurale ransen ransen response ransoro rsposa ransora 3 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unversy of Bologna 6