Transient Analysis: Series RLC Circuit

Transcription

1 Transent Analyss: Seres L rcut SW L V - urrent n an L crcut lke shown Is governed by the equaton V d 1 L dt dt We wll analyse the stuatons wth and wthout The source (V). The stored energy n or L wll force the current V V Once the swtch (SW) s closed, after some oscllatory perod, current And voltage wll settle. In steady state, apactor voltage (V ) wll approach V t Sajjad Hadar

2 LT SPIE Smulaton: Addng components After addng the component and components values, adhe SPIE DIETIVE onsderng there s no stored energy n the nductor (L) or apactor, At tme, t, I At tme t, V Sajjad Hadar

3 un: Smulaton 1. Smulaton> Edt Smulaton. UN Sajjad Hadar

4 un: Smulaton V V(n3) unnng the smulaton and placng the voltage probe at Node, n3 and clckng we fnd capactor voltage and clckng the current probe ether on or L, we fnd current urrent apactor voltage, V Sajjad Hadar

5 Stored Energy n apactor (): No Power Source SW L When the capactor s charged and connected as shown, energy wll be exchanged back and forth nbetween the nductor and capactor. However the resstor wll start dsspatng the energy. The resultng current s governed by the equaton d 1 L dt dt L d d o t Sajjad Hadar

6 Smulaton: Stored apactor n the apactor (): No Power Source Put ntal condtons usng spce drectve..i V(N)1 means ntal capactor voltage s 1 V Sajjad Hadar

7 un Smulaton UN unnng the smulaton and placng the current Probe on ether the resstor or the nductor, we fnhe oscllatory current as shown urrent Placng the voltage probe at node:n and lckng we fnhe capactor voltage waveform apactor voltage Sajjad Hadar

8 LTSpce to fnd Power and energy Press down ALT and put the cursor on 1 (You wll See a thermometer con) and clck You wll get the power data as shown Press down TL and place the cursor on V(N1)*1 as shown and clck Power dsspated n resstor You wll get the wndow lke ths Sajjad Hadar

9 Stored Energy n Inductor (L): No Power Source L When the swtch n poston 1, maxmum current V/ L reaches n steady state. Now the swtch s placed n poston, the stored energy n the nductor wll cause the current to oscllate n the L crcut. V - L 1 L In practce: Whenever the swtch s about to release from poston 1, there wll be abrupt change n current, causes a hgh voltage to develop governed by the equaton: V L Ld/dt. The stored energy n the nductor (1/LI ) wll be lost at the swtch juncton (1) (hgh voltage> Ionzaton>archng (heat)). However by electronc devces t s possble to release nductor-energy nto an L d crcut. I hope to dscuss t later L d However Let us consder an dealsed stuaton that when the swtch s n poston the energy (1/LI ) s released n the L crcut Nature of current can be expressed by the equaton Sajjad Hadar

10 LT SPIE Smulaton: Addng components otate resstor 1 twce, whch wll gve you the current n postve drecton Now ntal nductor current s 1 Amp. anhe capactor voltage s V Set the spce drectve Sajjad Hadar

11 UN: Smulaton un the smulaton We get current and capactor voltage as shown: urrent apactor voltage Sajjad Hadar

12 LTSpce to fnd Power and energy Press down ALT and put the cursor on 1 (You wll See a thermometer con) and clck You wll get the power data as shown Press down TL and place the cursor on V(N1)*1 as shown and clck You wll get the wndow lke ths Sajjad Hadar

13 Under, over and rtcally damped oscllaton n L crcut SW L d L d 1 Let us consder: st Ae Puttng equaton n 1: L S Ae St S Ae St Ae st LS S 1 1 > L L Overdamped Ths s the characterstc equaton Whch determnes the crcut behavour oots of ths equaton: 1 < L L Underdamped S 1 L L 1 L 1 L L rtcally damped S 1 L L 1 L Sajjad Hadar

14 Under, over and rtcally damped oscllaton n L crcut d L d The solutons of the dfferental equaton for these three condtons: Overdamped 1 > L L Let us consder, α L And ω 1 L t α > ω ( t) s t s t Ae 1 Where, Be S 1, S α ± α ω Underdamped α < ω ( Acos ω t Bsn ω t ) α t ( t) e Where ω ω o α t rtcally damped α ω ( t) α t ( A B t ) e t Sajjad Hadar

15 Smulaton - Undedamped: LTSpce SW 1 Ω L 1 mh L 5x1 6 1 µf 1 L 1x1 9 As, 1 < L L It s a case of underdamped oscllaton as we found before ( Acosω t Bsn ω t ) α t ( t) e t e α ω ω o α ad/sec π Tme perod, T ω 1. µs T Sajjad Hadar

16 Smulaton - Overdamped: LTSpce Let us consder, Ω L 1 L 1x1 9 1 x > L L s t st ( t) Ae 1 Be unnng LTSpce smulaton the same way as Beforewe fnhe overdamped behavour as shown Sajjad Hadar

17 Smulaton - rtcally Damped: LTSpce d L d α ω Or 1 L L In our case wth, L 1mH, 1µF: Ω ( t) α t ( A B t ) e To fnhe tme at whch the current reaches the peak, we should dfferentate (t) and equate to : ( t ) d d 1 L t c α Puttng Ω t c 31.6 µ s Sajjad Hadar

18 Smulaton - rtcally Damped: LTSpce unnng the smulaton gves us the current response, (t)as shown The mportance of crtcally damped crcut s, currently quckly reaches wthout oscllatng tc t c 31.6 µ s Sajjad Hadar