1. Linear second-order circuits
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1 ear eco-orer crcut Sere R crcut Dyamc crcut cotag two capactor or two uctor or oe uctor a oe capactor are calle the eco orer crcut At frt we coer a pecal cla of the eco-orer crcut, amely a ere coecto of a retor, a uctor a a capactor re by a oltage ource (ee Fg R R (t Fg We ecrbe th crcut ug the capactor oltage oltage a term of R a a arable Therefore we expre R R R, ( Next we wrte KV equato ( + (3 R + a ubttute to th equato relatohp ( a ( R + + After mple rearragemet we obta R For coeece mapulato let u efe two parameter + + (4 R α, ω
2 The parameter α calle the ampg cotat a the parameter ω the agular reoat frequecy Ug thee parameter we rewrte equato (4 the form The tal coto are: α + ω ω + (5 ( V a ( I The eco tal coto ca be expree a follow or ( ( I ( I Equato (5 that ecrbe the crcut of Fg a lear eco orer, ohomogeou fferetal equato wth cotat coeffcet A we ow from mathematc the oluto of th equato cot of two term a f +, (6 f where a geeral oluto of the homogeou equato + α + ω, (7 where f a partcular oluto of the orgal (ohomogeou equato (5 At frt we tae to accout the homogeou equato (7 To ole th equato we wrte the charactertc equato + α + ω The zero of th equato, calle the atural frequece of the crcut, are, (8 α + α ω (9 α α ω The form of the oluto of equato (7 epe o the relate alue of α a ω Therefore we coer the followg cae
3 ae α > ω (oerampe cae O the ba of (8 a (9 we tate that a are real a egate The oluto of the equato (6 a follow t t e + e, ( where a are cotat that epe o the tal coto A typcal plot of epcte Fg (t ( t t Fg ae α ω (crtcally ampe Ug (8 a (9 we obta α, that the charactertc equato ha a ouble zero beg real a egate The oluto of (7 αt ( + t e ( The plot of ( t ha mlar hape a preouly The crtcally ampe cae rather theoretcal oe becaue practcal applcato the equalty α ω caot be atfe ae 3 α < ω (uerampe cae Sce α < ω mplcate 3
4 α ω <, the expreo α ω a magary umber a follow where ω ω α Hece, we hae ( ω α jω α ω, α + jω, α jω Thu, both atural frequece are complex cojugate The oluto of equato (7 ha the form ( ω t + θ αt e co, ( where a θ are cotat that epe upo the tal coto A typcal plot of the waeform t how Fg 3 ( (t t If partcular α, or equaletly R Fg 3, we hae α ω ω ω a equato ( reuce to ( ω +θ co t We ow coer the ohomogeou equato (6 The repoe repoe, epe o the ource oltage f, calle a force ( t It ca be how that t a cotat f a 4
5 D oltage a a uo f a cotat oltage, e ( a uoal oltage We coer oly a tuato whe t V Obere that f V atfe equato (6 V V α + ωv ω + V (3 Thu, obta f V a partcular oluto of the ohomogeou equato a ug (6 we + V, (4 where, a how aboe epe o the relate alue of α a ω We wh to f the oluto ( t o that t atfe equato (6 a the tal coto We wll ere the oluto all the cue aboe cae Oerampe cae ( α > ω Ug ( a (4 we obta t t e + e + V (5 Sce clue two cotat a that mut be eterme, we ee aother equato To form th equato we compute curret The we wrte (5 a (6 at t (6 t t e + e ( V + V +, ( ( I + a ole th et of equato for a We ue the ubttutg metho: Hece, we obta V V, ( V V + I I ( V V, I ( V V V V 5
6 a ubttute to equato (5 I I ( V V ( V V t t ( t V V e + e + V (7 3 rtcally ampe cae ( α ω Ug (6 a ( yel Hece, we obta αt ( t ( + t e V + (8 αt ( α ( + t e et u wrte equato (8 a (9 at (9 t a ubttute ( I, ( V I V + V, ( ( α ( We ole the et of equato ( a ( for a V V, ( I Next we ubttute ( a (3 to (8 4 Uerampe cae ( α < ω Settg ( to (6 we obta Hece, t hol ( V + α V (3 I αt ( t V V + + α ( V V t e + V (4 ( ω + θ V α t e co (5 + αt αt ( αe co( ω t + θ ω e ( ω t + θ (6 6
7 Note that the ame curret traere uctor a capactor oequetly, I et u wrte equato (5 a (6 at t ( ( ( V coθ V +, (7 ( I ( αcoθ + ω θ (8 We ole th et of equato for a θ a follow Equato (7 lea to coθ V (9 V Settg (9 to (8 we obta, after mple mapulato, O the ba of (9 a (3 we f I α θ (3 ω ω ( V V a ext I α ω ω taθ V V ( V V I α ( V V - ω ω θ ta V V Hag eterme θ a ue (9 to f V V coθ ( t Now we ubttute θ a to (5 a obta the oluto equato (5 ecrbg the crcut a the tal coto Example et u coer the crcut how Fg where ( V F, ( V 3V, I 9 3 At frt we compute α a ω that atfe both t V, R Ω, ( A We wh to eterme ( t R α 5, ω 3 [ for t, H, 7
8 Sce α > ω we hae the oerampe cae The atural frequece are α + α ω, α α ω 9, a the oluto ( t ge by (5 repeate below t t e + e + V We et to th equato, 9, V e + e t 9t + (3 To f a o that the tal coto are atfe we create aother equato 9 t 9t ( e 9 e (3 Next we wrte equato (3 a (3 at t a apply the relatohp I ( ( ( 3 +, ( ( We ole the aboe et of equato, ug ubttutg metho, fg The we ubttut a to (3 a (3 3, The waeform of ( t ( ( 3 t e e 9 t + t, 6 t 9t ( t e + e a t are plotte Fg 4 a 5 8
9 (t t Fg 4 (t / t Fg 5 Example oer the ere R crcut wth tal oltage acro the capactor t the crcut hort crcute (ee Fg 6 F ( t ( V At t R 5μF, mh, R Ω Fg 6 We calculate the cotat α a ω : R α 5, ω 44 Sce α < ω we hae the uerampe cae, hece, the atural frequece are complex cojugate: α + jω, 9
10 α jω, where α 5 a ω ω α 33 Ug (7 yel To eterme a θ we f the curret ( t +θ t e 5 co 33 (33 5 e 6 5t 5t ( 5e co( 33t + θ 33 ( 33t + θ 5t ( 5co( 33t + θ + 66 ( 33t + θ e (34 We wrte the equato (33 a (34 at Hece, we f or a t a ubttute ( V, ( coθ, ( 5coθ 66θ + ta θ 379, θ 74, 6 9 co (- 74 Subttutg a θ to (34 yel 6 9e 5t ( 5co( 33t ( 33t 7 After rearragemet we obta 5 State equato 5t ( t 7 48e 33t et u coer the equato (3 repeate below R + +
11 a ubttute R R a, where ca be coere a a curret flowg through the uctor After mple mapulato we obta R + (35 Dfferetal equato (35 cota two arable a Aother equato clug thee arable, whch ca be rewrtte the form (36 The equato (35 a (36 form a et of tate equato ecrbg the crcut of Fg The curret flowg through the uctor a the oltage acro the capactor are calle the tate arable The tal coto (tal tate are ( ( I, ( V Summarzg we coclue that the eco orer yamc crcut how Fg ca be ecrbe by the tate equato ( ( V The oluto of the tate equato pecfe by ( t wth the tal coto I, ( V R +, (37, (38 a ( t for all t At t, ( I ca be coere a the coorate of a pot o plae A t creae from to the pot ( ( t,( t trace a recte cure tartg at ( V, I Th cure geerate by the tal coto ( t, ( t calle a trajectory To llutrate th ea we tae to accout the crcut coere Example The oluto of th crcut pecfe by t, t plotte Fg 4 a 5 O the ba of thee ( ( waeform we create the trajectory o the plae a how Fg 7
12 /3 98 t t 4 Fg 7 6 Parallel R crcut et u coer a parallel coecto of a retor, a uctor a a capactor (ee Fg 8 R (t R Fg 8 We apply K at the top oe + (39 + R a ubttute to equato (39 the followg relatohp R G G,
13 Hece, we obta the eco orer fferetal equato + G + We ere both e of th equato G a efe the parameter α a ω a follow + + (4 G α, ω The parameter α calle the ampg cotat a the parameter ω calle the agular reoat frequecy Ug α a ω we rewrte equato (4 a follow + (4 α + ω ω The eceary tal coto are ( I a ( V be expree term of the erate of The eco tal coto ca V ( ( ( or ( V From mathematcal pot of ew equato (4 ha the ame form a equato (5 Hece, we coclue that the oluto of equato (4 +, f where the oluto of the homogeou equato + + ω α a f where a partcular oluto of the orgal (ohomogeou equato I a pecal cae a D curret ource, e t I, t hol ( f I 3
14 The oluto of the homogeou equato epe o the relate alue of α a ω mlarly a the cae of the ere R crcut The form of the oerampe, crtcally ampe a uerampe cae the ame a Secto Thu, orer to f ere Secto replacg by The approach leag to the oluto ( the ere R crcut coere we etfy the cae a ue the correpog equato of the parallel R crcut the ame a the cae of the ere R crcut a the proceure ecrbe Secto ca be repeate tep by tep replacg by, V by by I a well a by a ce era I V 4
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