5. Second-order filter (including two dynamic elements) 5. Second-order filter(including two dynamic elements) About Q factor(series resonant cireuit) About Q factor(Parallel Resonant Circuit) ORC vu∞)d VGo) “节+ -=-+ G.+G. Conclusion of the resonance frequency:w. without changing the structure of the resonant cireuit. the load or th Q factor: Q introduction of load or the factor:Q="G-L lower the Q factor of resonant with internal loss will lower the Q Content Laplace Transform -- Basic property 32-3 Solving the linear circuits using Laplace Transform Unique F(s)+f(t) A one-to-one relat L. Transform of basic laws(operational form) 2. Transform of branches(Vs, Is, R, L, C) Linearity a,f,(t)+a2f2(t)=a,F(s)+a,2F2(s) 3. Transform of passive single-port network eg-41 Apace t ransom or oam s linearity --the general operational form of ohm's Law (V(Sz(s)I(s)) v(n)=(s)(n)=I(s)R(n):R(s) 4. Transform of active single-port network v()=Ri(D)=(s)=RI() 2.3: Laplace Transform of KCL, KVL 5. Solution: transform and inverse transform Ifi1(t)=I1(s)L2(t)=12(s)… -4 S-domain description of the network transfer function L∑(t)∑工(s) L Definition(H(S)Y(S)F(S))2. Characteristic ∑1(t)=0[e∑()=0 Laplace Transform- Basic property evi Laplace Transform-- Basic property Differentiation f(t)=sF(s)-f(o) f(t)=sF(s)-f(0.) I(s)=Csv(s)-Cv(o. vit=L di(t transform v(s)=LsI(s)-LI(0.) v(o) V(s) =0+aao(它20)北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 5. Second-order filter (including two dynamic elements) VR(jω) + - Ii(jω) C R L V(jω) + - resonance frequency： Q factor： resonance frequency： Q factor： LC 1 ω0 = ω RC 1 R ω L Q 0 0 = = L 0 L L R ω L R → Q = ω R C 1 R Q 0 C C → C = About Q factor (series resonant circuit) : Conclusion: Without changing the structure of the resonant circuit, the introduction of load or the source with internal loss will lower the Q factor of resonant circuit. C L L C Q 1 Q 1 Q 1 R = R + R → = + If the loss of resonant circuit is only caused by the capacitors and inductors, then: 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 5. Second-order filter (including two dynamic elements) About Q factor (Parallel Resonant Circuit) : Conclusion: Without changing the structure of the resonant circuit, the introduction of load or the source with internal loss will lower the Q factor of resonant circuit. ω G L 1 G Q 0 L L → L = C 0 C C G ω C G → Q = resonance frequency： Q factor: resonance frequency： Q factor: LC 1 ω0 = ω GL 1 G ω C Q 0 0 = = C L L C Q 1 Q 1 Q 1 G = G + G → = + If the loss of resonant circuit is only caused by the capacitors and inductors, then: ( ) ( ) ( ) ( ) jωC Z jω R jωL V Z jω I jω H jω 1 1 = + + = = V(jω) I(jω) C G L + - 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Content §2－3 Solving the linear circuits using Laplace Transform 1. Transform of basic laws (operational form) 2. Transform of branches (Vs,Is,R,L,C) 3. Transform of passive single-port network --the general operational form of Ohm’s Law (V(S)=Z(S)I(S)) 4. Transform of active single-port network -- operational form of Thevenin's theorem and Norton’s theorem /Voc(S),Isc(S),Zeq(S) 5. Solution: transform and inverse transform §2－4 S-domain description of the network transfer function 1. Definition (H(S)=Y(S)/F(S)) 2. Characteristic 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 e.g.2：Laplace Transform of Ohm’s Law v t Ri t () () = V s RI s () () = vt V s () ( ) = it I s () ( ) = Ri t RI s () () = e.g.3：Laplace Transform of KCL、KVL If …… it Is 1 1 ( ) = ( ) it Is 2 2 ( ) = ( ) Linearity Linearity ∑it Is i i ( ) =∑ ( ) ∑it 0 i ( ) = Unique Unique ∑Ii(s) = 0 Linearity Linearity Unique Unique *** Unique F(s)↔ f(t) A one-to-one relationship Linearity α f (t ) α f (t ) () α F s α F (s) 1 1 + 2 2 = 1 1 + 2 2 Laplace Transform -- Basic property Review 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 ＋ CS － I( ) s V( ) s ( ) s V 0- ＋ － () () ( ) I s = CsV s − CV 0- ＋ － I( ) s V( ) s CS ( ) CV 0- () ( ) ( ) 0- f' t = sF s − f transform transform Equivalent Equivalent ( ) ( ) dt dV t I t = C ＋ － V( ) t I( ) t ( ) V 0- C ＋ － I( ) t ( ) V 0- ＋ － V( ) t C Equivalent Equivalent Laplace Transform -- Basic property *** Differentiation transform transform Review ∫ = + t i t d t C v t v 0 ( ) ( ) 1 ( ) (0) （t≥0） 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 ( ) ( ) dt dI t V t = L ＋ － V(t) I(t) ( ) I 0- L ＋ V(t) － I(t) ( ) I 0- L Equivalent Equivalent ＋ － ( ) s I 0- I( ) s LS V( ) s () () ( ) V s = LsI s −LI 0- ＋ LS － I( ) s ( ) LI 0- V( ) s － ＋ transform transform ( ) ( ) ( ) 0- f' t = sF s − f transform transform Equivalent Equivalent Laplace Transform -- Basic property *** Differentiation Review ∫ = + t v t d t L i t I 0 ( ) ( ) 1 ( ) (0) （t≥0）