EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8
EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio > Polyomil expreio dy () t () t y Y ( ) Y ( ) The LT i trformig DE from the time domi (domi i et of vlue tht decribe fuctio, i tht ce the vrible i the time) to other complex domi (i.e. the vrible h rel d imgiry prt). { f ( t) } t F( ) L e f ( t), σ ± j Ue formul tble > Eier Propertie:. Differetitio. b. df L d L ( ) f ( ) F () f () F () f () df ( ) Chpter /8
f lim F ( ) EEE 87. Fil vlue theorem, where f i the vlue of f(t) fter ifiite time. Trfer fuctio The rtio of the Lplce trform of the put over the Lplce trform of the iput. Exmple: f m f Sprig d x m X X X X d ( xi x ) m ( x x ) i i x d x m () X () m X () () X ()( m ) () () m i i LT IC Chpter 3/8
EEE 87 To fid the chrcteritic equtio of ODE: d x i d x x m xi m x x d x The homogeeou ytem i: m x d i i.e. exctly the deomitor of TF. mr therefore the CE Exmple: f i A > f i A x di A L ( v i) I( ) ( V ( ) I( ) ) i x B x m x A L I( ) X ( ) BX ( ) m X ( ) LI ( ) I( ) V ( ) A I( ) X ( ) BX ( ) m X ( ) Chpter 4/8
EEE 87 ) ( ) ( ) ( ) ( ) ( ) ( X m BX X L V L V I A ( ) ) ( ) ( X B m L V A ( )( ) B m L V X A ) ( ) ( Exmple: θ ϕ ϕθ θ ϕ θ ϕ θ θ di L i v di L i v B i J i T T T B T J T m T T T m m Chpter 5/8
EEE 87 () () () () () () () () ( ) () () ( ) () () () ( ) ( ) () () () () ( ) ( ) ( )( ), B J L B J L V B J L V I L V B J I I L I V B I J di L i v B i J LT T T T T Θ Θ Θ Θ Θ Θ Θ Θ ϕ ϕ θ ϕ θ ϕ θ Chpter 6/8
EEE 87 Bloc digrm Bloc Digrm Algebr. To um (ubtrct) two igl, we ue ummig poit:. To ditribute igl, we ue brch poit: 3. Serie coectio: Chpter 7/8
EEE 87 4. Prllel coectio Pole loctio / -ple I previou exmple ( x x ) i d x m LT IC X X i ( ) () m The order of the ODE i order of the deomitor order of the ytem. Chpter 8/8
EEE 87 Exmple Z Y () > order 4. () ( m )( m ) G ) () () N D ( : root of the umertor re clled zero, while the root of the deomitor re clled pole. ( )( 3) oe zero t - d two pole t - d 3: j σ G ( ) Chpter 9/8
EEE 87 j σ Time domi chrcteritic Typicl iput igl. The Dirc fuctio. The tep or the pule fuctio (!!!) 3. The mp fuctio 4. Prbolic fuctio Chpter /8
EEE 87 r () t δ () t () () A r ( t) A r () t At A () ( t) At r A () 3 Firt order ytem emember: HE: t x Ce x ' x b ; HE: ' x x, CE: m olutio of hece tble olutio if < or the pole i t the LHS. The me t -ple: () I V () L Chpter /8
EEE 87 j L L σ I V (), / d τl/. > I() V () () τ Step repoe: I() V τ V V V lim τ or t if e t I V lim V τ τ V i t e t > () V τ (V/) Step epoe.9(v/).8(v/).7(v/).6(v/) Amplitude.5(V/).4(V/).3(V/).(V/).(V/) T T 3T 4T 5T 6T Time (ec) Chpter /8
EEE 87 (V/) Step epoe.9(v/).63 (V/).8(V/).7(V/).6(V/) Amplitude.5(V/).4(V/).3(V/).(V/).(V/) T T 3T 4T 5T 6T Time (ec) Secod order ytem x i (t) x (t) Frictio ( x x ) i dx B d x m LT IC X X i i () X () BX ( ) m X ( ) () ( B m ) X ( ) X X i () () m B X X i () () m B m m Chpter 3/8
EEE 87 C () (), B, m m oot of CE: ± Ce : > The the ytem h two egtive rel root d i clled overdmped: j σ t t () e e c t >> > c() t t e > Overdmped ytem will be lie very low repoe of firt order ytem Chpter 4/8
EEE 87 Step epoe Amplitude.9.8.7.6.5.4.3.. Overdmped ytem 4 6 8 Time (ec) Ce : The ytem h two equl rel root t : j σ c t () t e ( t) : Chpter 5/8
EEE 87.5 Step epoe Criticlly dmped ytem Amplitude.5 Overdmped ytem Ce 3: < < 4 6 8 Time (ec) ± j or ± jd j j d θ σ j d Chpter 6/8
EEE 87 The lie betwee the origi d the pole i: d d ( ) The gle i coθ θ co ( ). c () t e t i d t t Expoetil term.8.6.4. -. -.4 Siuoidl term -.6 -.8-3 4 5 6 7 8 9 The evelope tht will be creted from the expoetil term i: Chpter 7/8
EEE 87.8.6.4. -. -.4 -.6 -.8-3 4 5 6 7 8 9 Ad their product:.8.6.4. -. -.4 -.6 -.8-3 4 5 6 7 8 9 Ad by ddig the cott fctor: Chpter 8/8
EEE 87.5.5 -.5-3 4 5 6 7 8 9 Chpter 9/8
EEE 87.5 Step epoe Uderdmped ytem Criticlly dmped ytem Amplitude.5 Overdmped ytem 4 6 8 Time (ec).8 Step epoe.6.4. Amplitude.8.6.4. 4 6 8 Time (ec) Chpter /8
EEE 87 Ce 4: j j θ σ j Note: The ytem i clled mrgilly tble becue the olutio do ot diverge to ifiity. Hece if the previou four ce re combied to oe grph:.8 Mrgilly tble Step epoe.6.4 Uderdmped. Amplitude.8.6 Overdmped.4 Criticlly dmped. 5 5 5 Time (ec) Ce 5: < Chpter /8
EEE 87 c () t e t i d t t 5-5 - 3 4 5 Sice < the gle θ defied i the -ple co ) h to be greter th 9 o : ( θ θ co ( ) j j d θ σ j d By combiig the previou -ple we hve: Chpter /8
EEE 87 j coθ ( ) (,) j ( ) ( ) co θ, j d θ σ > j d < j Chpter 3/8
EEE 87 j.5.4..8.6.4. 4 6 8.5 4 6 8 x 8 - - -3-4.5.5.75.8.6.8.6 8 6 4 8.4. 4 6 8.4. 4 6 8 4..4.6.8 6 4..4.6.8 σ Chpter 4/8
EEE 87 A geerl repoe i:.4..8.6.4. 3 4 5 6 Time tht the ytem eed to rech hlf of it fil vlue: ie time (%-9% or 5%-95% or %-%) t r π θ d Pe time: t p π d Mximum overhoot: Settlig time: Mp e 3 " π t 5 % d t % 4 Chpter 5/8
EEE 87.4 Step epoe..8 Amplitude.6.4. 3 4 5 6 Time (ec) Chpter 6/8
EEE 87 Extr pole d zero Geerl form of TF: m m m b b b C...... ) ( ) ( For tep iput: () ( ) r q j j j c b p C ( ) r q, i.e. combitio of firt d ecod order ytem. Exmple: ( )( ) e d f c b 3 > ( )( ) e d f c b 3 ( ) ( ) fe fd e f d c b 3 3 fe c fd e b f d The repoe of higher order ytem i the um of expoetil d dmped iuoidl curve. Aumig tht ll pole re t the left hd ide the the fil vlue of the put i ice ll expoetil term will coverge to. Chpter 7/8
EEE 87 Chpter 8/8 Let ume tht ome pole hve rel prt tht re fr wy from the imgiry xi> () t i t e t c d t > t e Overll performce i chrcteried by the iolted (fr wy from zero) pole tht re cloe to the imgiry xi. If we hve oly oe pole (or pir for complex root) tht i cloed to the rel xi the we y tht thi pole (or pir of pole) i (re) the DOMINANT pole() for the ytem. A imple rule i tht the domit pole mut be t let five to te time cloer to the imgiry xi th the other oe. () ( ) r q j j j c b p C () ( ) ( ) r t r t q j t p j t e c t e b e t c j i co The vlue of b (umertor coefficiet) determie the mplitude of the ocilltio of the ytem.