( ) = RI R ( t) V R ( s) = RI R ( s) (1) ( t) = 1 C Q C. ( s) ( ) ( ) = 1 C I C ( t) ( ) ( ) V ( s) ( t) = L d dt I t C

Transcription

1 MTH 352 Real World LC Componens Fall 2008 Prof. Townsend (analysis by Paricia Mellodge) Real world capaciors and inducors include he ideal model along wih resisors and, perhaps, inducors and capaciors. This documen addresses he effec on he volage across he primary resisor in boh parallel and series RLC circuis due o using real world models. There are a number of such models bu we will use he ones ha only inroduce addiional resisances in he circuis o keep he differenial equaions o second order. The capacior and inducor models we are using, shown below, were given in Lessons in Elecric Circuis V. II AC on pages 75 and 97, available from Figure 3.15: Inducor Equivalen circui of a real inducor. Figure 4.15: Real capacior has boh series and parallel resisance. As you migh suspec, inclusion of he addiional resisors resuls in coupled differenial equaions. Thanks o Laplace Transforms, we can significanly simplify he problem by looking in s space. Consider he volage drops across he various elemens. We assume he iniial condiions are zero. There are mehods in exbooks for handling non-zero iniial condiions ha include creaing arificial circui elemens so ha he iniial condiions can sill be aken as zero. Hence aking he derivaive in ime jus leads o muliplicaion by s in he frequency domain. V R V C Time () Frequency (s) Equaion # ( ) = RI R ( ) V R ( s) = RI R ( s) (1) ( ) = 1 C Q C Take he derivaive. d d V C V L ( ) ( ) = 1 C I C ( ) ( ) = L d d I C sv C ( s) = 1 C I C s Divide by s. V C ( s) = 1 sc I C ( ) ( s) (2) ( ) V ( s) = sli ( s) (3) L C Real_RLC_Circuis.doc Page 1 of 6

2 In he able below are he circui diagrams for series and parallel RLC circuis. The diagrams show he circuis, boh wih ideal and he models we are using for capaciors and inducors, in boh he ime domain and he frequency domain. Time Frequency (s) Real_RLC_Circuis.doc Page 2 of 6

3 As indicaed above, ime domain analysis of hese circuis resuls in coupled differenial equaions. We know ha he use of Laplace Transforms akes differenial equaions and urns hem ino algebraic equaions. The analysis of he frequency domain circuis is algebra inensive so I used a symbolic algebra program, LiveMahMaker (nee Theoris), o do he work for me. The resuls are shown in he diagram below. Real_RLC_Circuis.doc Page 3 of 6

4 The words messy equaions in he above are he iles of groups of hidden algebraic manipulaions in LiveMahMaker, viewable upon reques. Consider he resuls for he parallel circui. The final equaion is ( Bs 2 + Ds + A)V ( s) = ( Gs 2 + Fs + E) I s s Recall ha he curren source is represened by I s ( ) (4) ( ) = I 0 sin (!) (5) Is Laplace Transform is "! % I s ( s) = I 0 # $ s 2 +! 2 & ' (6) We can hen solve equaion (4) for he Laplace Transform of he volage ( ) = I 0 Gs 2 + Fs + E V s ( ) ( Bs 2 + Ds + A) "! % # $ s 2 +! 2 & ' (7) Use he TI-89 Expand() command or use a parial fracion expansion o rewrie equaion (7) as a collecion of idenifiable inverse ransforms, hen use he Laplace Transform able o find he volage across all he branches as a funcion of ime. The equaion for he series RLC configuraion follows he same analysis wih he resul being he curren hrough he elemens in series as a funcion of he applied volage, V s ( ) = V 0 sin (!). We can also recover he differenial equaion for he sysem in he ime domain. Sar wih equaion (4). Recall ha muliplicaion by s in he frequency domain is equivalen o a derivaive in he ime domain. Therefore equaion (4) looks like he following in he ime domain. B d 2 d V ( ) + D d 2 d V ( ) + AV ( ) = G d 2 d I 2 s ( ) + F d d I s Since we know he ime dependence of he curren source, I s becomes ( ) + EI s ( ) (8) ( ) = I 0 sin (!), equaion (8) { } (9) B d 2 d V ( ) + D d 2 d V ( ) + AV ( ) = I 0!G" 2 sin (") + F" cos (") + E sin (") The righ hand side of equaion (9) is jus a sine or cosine wave wih a phase shif. B d 2 d 2 V ( ) + D d d V ( ) + AV ( ) = I 0! cos " # $ ( ) (10) where α is a muliplier ha is a funcion of E, F, G, and ω. We find he values of he phase shif, δ, and ampliude facor, α, by using a rig ideniy. Real_RLC_Circuis.doc Page 4 of 6

5 Given! cos (" # $ ) = ( E # G" 2 )sin(" ) + F" cos (") (11) we use he rig ideniy! cos " # $ ( ) =! sin $ { ( )sin(" ) + cos ( $ )cos(")} (12) o arrive a,! sin " ( ) = E # G$ 2 (13) and! cos (" ) = F# (14) Dividing equaions (13) and (14) gives us an (! ) = E " G# 2 F (15) The appropriae quadran for δ is deermined by looking a he signs of equaions (13) and (14). Squaring (13) and (14), adding he resuls, hen aking he square roo of he resul gives ( ) 2 + F 2 (16)! = E " G# 2 We now rewrie equaion (10) so i looks like (almos) an RLC parallel circui wih effecive values of R, L and C.! B d 2 " d V ( ) +! D 2 " d d V ( ) +! A " V ( ) = I 0! cos (! # $ ) (17) Define C eff! " B #, R eff! " # D, L eff! " # A, and! " # 0 (18) Then equaion (17) becomes d 2 C eff d V ( ) R eff d d V ( ) + 1 V L eff ( ) = d d I sin! " 0 ( 0 ) { } (19) which, excep for he shif in ime, is he same as he equaion for a simple parallel RLC circui. C d 2 v d + 1 d 2 R d v + 1 L v = di s (20) d wih i s ( ) = I 0 sin! ( ). A similar analysis can be performed for he series RLC circui case. Real_RLC_Circuis.doc Page 5 of 6

6 Since we assumed zero iniial condiions o arrive a equaion (13) via Laplace Transform mehods, he effecive circui described by equaion (13) will resul in non-zero iniial condiions due o he ime shif in he curren source. The impac of using our models of real world capaciors and inducors leads no only o effecive inducors and capaciors, as provided by he lef hand side of equaion (10), bu i also modifies he curren source in ampliude and phase, as provided by he righ hand side of equaion (10). For he parallel RLC circui case, he volage across he primary resisor is found by solving equaion (7), hen back ransforming or by solving equaion (10) using convenional ime dependen echniques. For he series RLC circui case, we find he curren hrough he resisor hen muliply by he resisance, R, o find he volage across he resisor. I am deeply indebed o Paricia Mellodge for providing he above analysis. She urned my inracable problem ino one ha we could analyze and undersand. Real_RLC_Circuis.doc Page 6 of 6