Chapter 1 Introductory Linear Circuit Analysis ---From time-domain analysis to frequency-domain analysis C1-5: Analysis of Sinusoidal Steady-state Cireuit (Complex Solution to Linear and Time-invariant Circuits) (Principles of Circuit Analysis) COmplex method and phasor method, impedance and admittance ts laws and theorems Introductory Linear Circuit Analysis The relationship between transfer function 1-6: Filter Lecture 4 s The definition and classification of filters 2009.09.24 First-order filter(low-pass, high-pass Second-order filte Active filter (just know about it) Key points c4 Time-domain analysis of dynamic同 C1-5: Analysis of Sinusoidal Steady-state Circuit Complex Solution to Linear and Time-invariant Cireuits i(1)=l D Complex method and phasor method, impedance and admittance he complex forms of elements, laws and theorems dv(n) +v2()=R The relationship between transfer function C1-6: Filter Analysis 4: i(0)= a The definition and classification of filters O)+-RL)"4+R First-order filter(low-pass, high-pass) Steady-state response Second-order filter(band-pass, band-stop) Active filter(just know about it) General solution: is related to the parameter, it is determined by the the outer source C1-4: Time-domain analysis of dynamic circuits Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis uations for the eireuit which has N independent dynamic nalysis of sinusoidal Steady-state Cireuit Complex Solution to Linear and Time-invariant Circuits) +a1+a)y()=x() aComplex method and phasor method, I The complex forms of elements, laws and theorems When input isx(n)=Age/e, the zero-state response is -state(for self-study) ility of networks, transfer function. ((ay+a1(o)-++a4()+a2 1-6: Filt Fw=I/Hw) The definition and classification of filters Y(o) X(@)=X(O)H(jo) irst-order filter (le Fo) Active filter (just know about it)p Second-order filter(band-pass, band-sto steady-state cireuit
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 第？讲： 复习 北京大学 wwhu 北京大学 《Principles of Circuit Analysis》 Introductory Linear Circuit Analysis Lecture 4 2009.09.24 Interest Focus Persistence Originality 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 C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） Complex method and phasor method, impedance and admittance The complex forms of elements, laws and theorems The power of sinusoidal steady-state (for self-study) The stability of networks, transfer function, The relationship between transfer function C1-6: Filter The definition and classification of filters First-order filter (low-pass、high-pass) Second-order filter (band-pass、band-stop) Active filter（just know about it） Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis 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 Key points： C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） Complex method and phasor method, impedance and admittance The complex forms of elements, laws and theorems The power of sinusoidal steady-state (for self-study) The stability of networks, transfer function, The relationship between transfer function C1-6: Filter The definition and classification of filters First-order filter (low-pass、high-pass) Second-order filter (band-pass、band-stop) Active filter（just know about it） 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 C1-4: Time-domain analysis of dynamic circuits I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0 0 /R 0 [ ( )] ( ) [ ( )] ( ) c c dit RC i t I dt dv t RC v t RI dt + = + = I0 RI0 Analysis 4： 0 / 0 0 0 / 0 0 ( ) ( ) ( ) ( ) v t V RI e RI I e I R V i t t c t = − + = − + − − τ τ Transient response Steady-state response Special solution: is related to the stimulation, it is forced by the outer source. General solution: is related to the network’s structure and elements’ parameter, it is determined by the network’s nature characteristics. τ τ 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 *** C1-4: Time-domain analysis of dynamic circuits ( ... ) ( ) ( ) 1 1 0 ( 1) 1 ( ) a y t x t dt d a dt d a dt d a n n n n n n + + + + = − − − When input is , the zero-state response is j j t x t A e e ϕ0 ω 0 ( ) = j t j j t n j n n n a j a j a j a Y e e A e e ϕ y ω ϕ ω ω ω ω 0 0 0 0 1 1 1 1 ( ( ) + ( ) +...+ ( ) + ) = − − j t j y t Y e e ϕ y ω 0 ( ) = The differential equations for the circuit which has N independent dynamic elements: Let: F(jw)=1/H(jw) Y(jw) X(jw) So: ( ) ( ) ( )( ) ( ) X j Yj Xj Hj F j ω ω ω ω ω = = Analysis of sinusoidal steady-state circuit phasor method (complex method) 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 C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） Complex method and phasor method, impedance and admittance The complex forms of elements, laws and theorems The power of sinusoidal steady-state (for self-study) The stability of networks, transfer function, The relationship between transfer function C1-6: Filter The definition and classification of filters First-order filter (low-pass、high-pass) Second-order filter (band-pass、band-stop) Active filter（just know about it） Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis
C1-5: Analysis of Sinus Cl-5: Analysis of Si (Complex Solution to Linear and Time-invariant Circuits 1. Complex representation of s 2.The advance (lead)and lag of phase: 6=1-e os(ar+ pl Her formula: ee=cose-jisine The phase-advance ofA, to 4, is a 42 vcos(art p/=Re/Vm elfa-e/Re/ e/=Re/v vde/ The phase-lag ofA, to A, is a Mas of the phlasorai-vmet=n4 amplitude灬,p:甲 It is ok to say that the phase-advance ofA, to A is 2*& but it (thyhasor: s Vep= VIp relationship: Vn"y It is customary that(in engineering field: I 1 <180o The general representati ) Ithe phase-advance ofA, to A, is 90o, then: l0a=p=Vn∠p The cos90°jin90oj( actor90°) 12=jl4/4 C1-5: Analysis of Sinusoidal Steady-state Cireuit C1-5: Analysis of Sinusoidal Steady-state Circuit 3. Complex representation of circuit elements cc(n) ()=C-1(o)=jjoCv(o) resistance:Qv()=Ri(n) vGo)-RIGjo) he current leads the voltage by 90" vj⑩) vGo) EA 0) aracteristic and 0) R The current and Inductance- voltage are in the )=1d t(o)=oL·(o) same phase. The voltage leads the current by 90. vgo) joL Ig haracteristic and dimension of resistance C1-5: Analysis of Sinusoidal Steady-state Cireuit C1-5: Analysis of Sinusoidal Steady-state Circuit (Complex Solution to Linear and Time-invariant Circuit (Complex Solution to Linear and Time-invariant Circuits) Ohm's law (VCR law) he impedance and admittance in the two-terminal passive network The time domain The complex representation Vo)=ZGo)Go) v(tR-l(t) vgoFzGolGo I(0)=Y(o)(o) (o) I(tG.v(t) IgjoYGo'vgjo) ZER+jX impedance) admittaneelUe ZU Admittance:Y=G+jB ntcre-ns, the time. include R, L and C, follo a law similar to ohn’s Inductance JeL Q2:G=R?X=/B?⑧
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 C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） 1. Complex representation of sinusoidal signals sinusoidal signals: v(t) =Vmcos(ωt+ ϕ) three fundamental elements Euler's formula: ejθ =cosθ+jsinθ ∴ Vmcos(ωt+ ϕ) =Re[Vm ej(ωt+ϕ) ]= Re[Vm ejωt ]= Re[√2V ejωt ] Max of the phasor: Vm= Vmejϕ = Vm∠ϕ amplitude: Vm ，phase: ϕ virtual value of the phasor: V= Vejϕ = V∠ϕ relationship： Vm = √2V · · ϕ ωt Vm The general representation of complex V(jω)= Vmejϕ = Vm∠ϕ - · · *** V(jω) ϕ The figure of phasor: Vm 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 2.The advance (lead) and lag of phase: The phase-advance of A1 to A2 is θ. The phase-lag of A2 to A1 is θ. It is ok to say that the phase-advance of A2 to A1 is 2π-θ, but it is not customary. It is customary that (in engineering field):︱θ︱≤180° If the phase-advance of A1 to A2 is 90°, then： cos90°+jsin90°=j （ factor 90°） A1/A2 =j|A1m/A2m | ϕ2 ϕ1 θ A1 A2 θ = ϕ1 - ϕ2 C1-5：Analysis of Sinusoidal Steady-state Circuit * （Complex Solution to Linear and Time-invariant Circuits） 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 resistance: Q v t Ri t () () = ＋ － R I(jω) V(jω) I(jω) V(jω) V(jω)=RI(jω) 3. Complex representation of circuit elements: *** The current and voltage are in the same phase. C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） 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 Capacitance: ( ) ( ) dv t it C dt = ＋ － 1 j Cω Inductance: ( ) ( ) di t vt L dt = Capacitive Impedance 1 X C ωC = With the characteristic and dimension of resistance. Inductance Impedance X L L = ω ＋ － j L ω I(jω) V(jω) I(jω) V(jω) I(jω) V(jω) I(jω) V(jω) I( jω) = jωC •V ( jω) V( jω) = jωL • I( jω) *** The current leads the voltage by 90° The voltage leads the current by 90° C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） With the characteristic and dimension of resistance. 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 The complex representation Ohm’s law（VCR law） V(t)=R·I(t) I(t)=G·V(t) V(jω)=Z(jω)·I(jω) I(jω)=Y(jω)·V(jω) The time domain *** impedanceZ(jω) admittanceY(jω) Resistance R Capacitance C Inductance L R G 1/ jωC jωC jωL 1/jωL Deduction: Using complex representations, the timeinvariant circuits which include R, L and C, follow a law similar to Ohm’s Law. C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） 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 The impedance and admittance in the two-terminal passive network N0 I( ) jω V j( ) ω ＋ － Q1: Y=1/Z？ Q2: G=1/R? X=1/B? Impedance: Z = + R jX Admittance: Y G jB = + resistance reactance conductance susceptance ( ) ( ) ( ) ( ) ( ) ( ) ω ω ω ω ω ω I j Y j V j V j Z j I j = = Z ( jω) R X Inductive capacitive Net resistance Net reactance *** ☺ / C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits）
☆ Cl-5: Analysis of Si (Complex Solution to Linear and Time-inmvariant Circuits) olution he complex representations of Thevenin's theorem and Norton's theorem Kirchhoffs Voltage Law(KVL) ∑v(t)=0 ∑viw)=0 Kirchhoffs Current Law(KCl) N.Voc(w) ocw) ∑(0)=0∑xw)=0 N Norton's circuit C1-5: Analysis of Sinusoidal Steady-state Circuit ◆界 IGo) DY,(n & R Solving differential equatio -100 or phasor methog 10j 10=5-5,Z4=(-10jm10)-5j=5-10j 1=65-5)×-5 ).real cireuits symbolic circuits(using complex representation 10j+5 2) sloving algebraic equations, get the complex representations of results ) transform the complex solutions to time-d -Complex Solution to Linear and Time-invariant Circuits) C1-5: Analysis of Sinusoidal Steady-state Cireuit C1-5: Analysis of Sinusoidal Steady-state Circuit ( Complex Solution to Linear and Time-invariant Cire (1)=10cos1000+2cos20001 (1)=10cos1000n+2cos20001 rcuit is a good way to avoid solving fferential equations 亡 But it should be paid great attention to the response's amplitude and phas characteristic of different frequency stimulations while using the symbolic 1.24∠297° Jn2=0.37∠122 2=124∠297crus i()=1.24cos(1000+29.7)+0.37cos(2000n+12.2) ∴()=1.24co(10007+297)+0.37c0s(20001+122°)
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 Kirchhoff's Voltage Law (KVL) ∑ii(t) = 0 ∑Vi( ) t = 0 ∑Vi( ) jw = 0 Kirchhoff's Current Law (KCL) ∑Ii( ) jw = 0 C1-5：Analysis of Sinusoidal Steady-state Circuit *** （Complex Solution to Linear and Time-invariant Circuits） 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 Zeq(jw) VOC(jw) + - Thévenin's circuits Norton's circuits ISC Zeq(jw) (jw) Ns Ns equivalent Ns Ns V(jw)=0 ISC(jw) + - Ns Ns I(jw)=0 VOC(jw) + - The complex representations of Thévenin's theorem and Norton's theorem C1-5：Analysis of Sinusoidal Steady-state Circuit *** （Complex Solution to Linear and Time-invariant Circuits） 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 j Z ( ) j j j j j Voc 10 5 5 , eq 10 //10 5 5 10 10 10 10 × = − = − − = − − − = ( ) 2.5 5 10 5 5 5 5 = − + = − × j V j o Example: a + - 10 ~ b + - Voc 10 -10j -5j Zeq a b -10j 10 -5j a + - 10 ~ b + - Voc 10 -10j -5j a + - 10 ~ b + - Voc a + - 10 ~ b + - Voc + - 10 ~ b + - Voc 10 -10j -5j Zeq a b -10j 10 -5j Zeq a b -10j 10 -5j a + - 10 -10j 5 ~ b 10 -5j + - Vo=? Thévenin's theorem + - 5 ~ b Voc a Zeq 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 Symbolic circuits Real circuits Complex solution or phasor method Solving algebraic equations Complex solution to networks---symbolic circuits (circuits using phasor models) ＋ － ( ) s v t i t( ) C R L *** 1 jωC R j L ω ＋ － ( ) V j s ω I j ( ) ω ( ) i t( ) s v t & Real circuits Using time-domain method Solving differential equations V j s ( ) ω I j ( ) ω & Symbolic circuits transform inverse transform Steps for analysis of the response of networks using complex solution (phasor method): 1).real circuits → symbolic circuits (using complex representation) 2).sloving algebraic equations, get the complex representations of results 3).transform the complex solutions to time-domain solutions Steps for analysis of the response of networks using complex solution (phasor method): 1).real circuits → symbolic circuits (using complex representation) 2).sloving algebraic equations, get the complex representations of results 3).transform the complex solutions to time-domain solutions C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） 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 Example: ＋ － ( ) s v t i 500μF 2i 4mH 3Ω ＋ － ( ) 10cos1000 2cos 2000 s vt t t = + Q: i t( ) 3 ＋ － 10 − j2 j4 ＋ － 1 2 mI 1 1.24 29.7 mI = ∠ o 3 ＋ － 2 − j j8 ＋ － 2 2 mI 2 0.37 12.2 mI = ∠ o ω1 ω2 ∴it t t ( ) 1.24cos(1000 29.7 ) 0.37cos(2000 12.2 ) = ++ + o o ω1 ω2 i t( ) C1-5：Analysis of Sinusoidal Steady-state Circuit *** （Complex Solution to Linear and Time-invariant Circuits） 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 Example: 3 ＋ － 10 − j2 j4 ＋ － ＋ － ( ) s v t i 500μF 2i 4mH 3Ω ＋ － ( ) 10cos1000 2cos 2000 s vt t t = + 求: i t( ) 1 2 mI 1 1.24 29.7 mI = ∠ o 3 ＋ － 2 − j j8 ＋ － 2 2 mI 2 0.37 12.2 mI = ∠ o ω1 ω2 ∴it t t ( ) 1.24cos(1000 29.7 ) 0.37cos(2000 12.2 ) = ++ + o o ω1 ω2 i t( ) *** Correspond to time-domain circuits with a certain sinusoidal frequency, symbolic circuit is a good way to avoid solving differential equations. Correspond to time-domain circuits with a certain sinusoidal frequency, symbolic circuit is a good way to avoid solving differential equations. But it should be paid great attention to the response’s amplitude and phase characteristic of different frequency stimulations while using the symbolic circuits. But it should be paid great attention to the response’s amplitude and phase characteristic of different frequency stimulations while using the symbolic circuits. C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits）
Example for graphic method using phasor Tea break/ Q: V=? 1=? homework 1.47,1.49,1.51,1.55 JI Dont need too much time. Just know Object: Having further understandin of the phase relationship between the mple for graphic method Chapter 1 Introductory Linear Circuit Analysis l From time-domain analysis to frequency-o IIOA(measured). VI=100V(measured). +Q: V -?1-? (Complex Solution to Linear and Time-invariant Circuits) l=1 COmplex method and phasor method, i j5 ty of networks, transfer function. nship between transfer function C1-6: Filter j10 2 The definition and classification of filters First-order filter(low-pass, high-pass) Second-order filter(band-pass, band-stop Active filter (just know about it) V,VR, VL717 C1-5: Analysis of Sinusoidal Steady-state Cireuit C1-5: Analysis of Sinusoidal Steady-state Circuit Complex solution to Linear and Time-invariant Circuit t Sol variant circuits N-order differential equation set up by dynamic circuits N-order differential equation set up by dynamic circuits d"+a4-“+…+a+a)y()=x0) d+an+…+a1a+4)y(0)=x(0) ++a1S+ Secular equation: a S"+aS+.+a,S +ao=0 Eigenvalue: S= S S1 eneral solution: y()=Ke”+Ke4+…+Kne solution:y(t)=K, e+K e++K,e LIf Si is real. the solution is I. If there is one Re(si>o in the si i 2.If Si is imaginary, the solution is: K, cos(or)+K, sin(or) 2. If there is multiple imaginary roots: Ke '+K,te,Unstable 3. If Si is complex, the solution is: K,ecos(or)+K,e sin(or) If all the Re(si)o, 4. If Si has multiple roots, the solution is: Ke"+K,tet stable circuit+Sinusoidal signal Sinusoidal stimulation Steady-state Cire
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 Example for graphic method using phasor: + - + - ＝100V 5 j5 -j10 V (measured) 1 V0 . . V2 + - Xc I0 . Ic =10A（measured）Q :Vo=？ Io=？ Attention: It’s virtual value here! Don’t need too much time. Just know it. ☺It’s ok if you don’t understand it.☺ Object: Having further understanding of the phase relationship between the voltage and current. C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） 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 Tea break！ Tea break！ *homework: 1.47, 1.49, 1.51, 1.55 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 Step: V1,I1ÆVR ,VLÆ I3Æ I0Æ V2Æ V0 Example for graphic method using phasor: 5 j5 -j10 + V1 - V2 + - Xc I1 I1=10A(measured),V1=100V(measured), Q:Vo=?Io=？ I0 + - V0 I3 V1=100 I1=10 VR 50√2 VL 50√2 V2=10 0 I3 10√2 I0=10 V0 100√2 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 C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） Complex method and phasor method, impedance and admittance The complex forms of elements, laws and theorems The power of sinusoidal steady-state (for self-study) The stability of networks, transfer function, The relationship between transfer function C1-6: Filter The definition and classification of filters First-order filter (low-pass、high-pass) Second-order filter (band-pass、band-stop) Active filter（just know about it） Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis 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 N-order differential equation set up by dynamic circuits: ( ... ) ( ) ( ) 1 1 0 ( 1) 1 ( ) a y t x t dt d a dt d a dt d a n n n n n n + + + + = − − − ... 0 1 0 1 + 1 + + + = − − a S a S a S a n n n n s t n s t s t n y(t) = K e + K e +...+ K e 1 1 1 1 n s s ,s ,...,s = 1 2 1.If Si is real, the solution is: 2.If Si is imaginary, the solution is: 3.If Si is complex, the solution is: 4.If Si has multiple roots, the solution is : cos( ) sin( ) 1 2 K ωt + K ωt Secular equation: Eigenvalue： {General solution: General Solution s t s t K e K te 1 1 1 + 2 s t i i K e cos( ) sin( ) 1 2 K e t K e t at at ω + ω e-t/τ C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） 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 ( ... ) ( ) ( ) 1 1 0 ( 1) 1 ( ) a y t x t dt d a dt d a dt d a n n n n n n + + + + = − − − 1。If there is one Re(Si)>0 in the Si 2。If there is multiple imaginary roots: 3。If all the Re(Si)=0, 4。If all the Re(Si)<0, s t s t K e K te 1 1 1 + 2 Emanative Unstable Convergent, stable stable circuit + Sinusoidal Steady-state Circuit C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） N-order differential equation set up by dynamic circuits: ... 0 1 0 1 + 1 + + + = − − a S a S a S a n n n n s t n s t s t n y(t) = K e + K e +...+ K e 1 1 1 1 n s s ,s ,...,s = 1 2 Secular equation: Eigenvalue： {General solution: General Solution Sinusoidal signal stimulation Convergent, stable
Chapter 1 Introductory Linear Circuit Analysis -invariant Circuits) ---From time-domain analysis to frequency-domain analysis Circuit: C1-5: Analysis of Sinusoidal Steady-state Cireuit (Complex Solution to Linear and Time-invariant Circuits) Imaginary axis omplex method and phasor method, impedance and admittance The relationship between transfer function C1-6: Filter The definition and classification of filters In linear time-invariant circuits, if all the natural frequencies a First-order filter(low-pass, high-pass) located in the left half-plane of the S-plane, the network will be Active filter (just know about it)p Second-order filter(band-pass, ba stable. The networks stable response to a sinusoidal signal stimulation is called sinusoidal steady-state response, the stable circuits is called Sinusoidal Steady-state Circuit C1-4: Time-domain analysis of dynamic cire 3. The amplitude and phase characteristic of the transfer function*k Simply: he differential equations for the eireuit which has N independent dynamic a1+a0)()=x(n) HGo When input is x(t)=Abe e, the stable response is 需需端 XGo)-/HGo)eJo( yn)=Yoe'e (a(o+a(+4(l)+4- Amplitude 为自变量 HGo): characteristic I/HGw) y(w) x(w) of network characteristic o 为自变量 Y(0)=X(0)H() () Phase characteristic characteristic of network function H(o)x(o) response curve Example for transfer function Example for transfer function Z,=R+joL+ (a) Go)=Is(je) I+ joC,Z, 。(jo jORC v, o) oRC+I H(o)-5()R Is Go) I+joC,Z2 oLC+ joRC+(1+C/C)
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 In linear time-invariant circuits, if all the natural frequencies are located in the left half-plane of the S-plane, the network will be stable. The network’s stable response to a sinusoidal signal stimulation is called sinusoidal steady-state response; the stable circuits is called Sinusoidal Steady-state Circuit. The definition of Sinusoidal Steady-state Circuit: S-plane： 0 imaginary axis real axis C1-5：Analysis of Sinusoidal Steady-state Circuit * （Complex Solution to Linear and Time-invariant Circuits） 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 C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） Complex method and phasor method, impedance and admittance The complex forms of elements, laws and theorems The power of sinusoidal steady-state (for self-study) The stability of networks, transfer function, The relationship between transfer function C1-6: Filter The definition and classification of filters First-order filter (low-pass、high-pass) Second-order filter (band-pass、band-stop) Active filter（just know about it） Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis 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 ( ... ) ( ) ( ) 1 1 0 ( 1) 1 ( ) a y t x t dt d a dt d a dt d a n n n n n n + + + + = − − − 0 ( ) x j j t x t Ae e ϕ ω = 1 1 1 1 00 0 ( ( ) ( ) ... ( ) ) y x j n n jt jt j n n a j a j a j a Ye e Ae e ϕ ω ϕ ω ωω ω − + ++ + = − j t j y t Y e e ϕ y ω 0 ( ) = Let: 1/H(jw) Y(jw) X(jw) So: Yj Xj Hj ( ) ( )( ) ω = ω ω *** C1-4: Time-domain analysis of dynamic circuits Review Introduction of transfer function： Y(jω) X(jω) H(jω) = The differential equations for the circuit which has N independent dynamic elements: When input is , the stable response is 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 H(jω) =Comp. rep. of res. Comp. rep. of sti. X(jω) Y(jω) = jΦ(ω) = |H(jω)|· e Simply: ∠Φ Simply: ∠Φ (ω) |H(jω)|： Φ(ω) : ω为自变量 ω为自变量 + = Frequency response curve Phase characteristic curve Amplitude characteristic curve ３.The amplitude and phase characteristic of the transfer function*** Amplitude characteristic of network Phase characteristic of network 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 C1 C2 L R + - Vo(jω) Is (jω) IR(jω) 2 2 1 j C Z R j L ω = + ω + () ()1 1 2 1 j C Z I j I j R S ω ω ω + = ( ) ( ) ( ) ( ) 1 1 1 2 2 1 2 1 1 LC j RC C C R j C Z R I j V j H j S o − + + + = + = = ω ω ω ω ω ω Example for transfer function: 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 H(jω) Vi (jω) Vo(jω) + + - - R 1 jωC ( ) ( ) ω ω ω ω V j j RC j RC V j o i + 1 = *** ωc 1 2 1 |H(jω)| 0 ω 0 ωc ω Φ(ω) 4 π 2 π Example for transfer function:
2. The relationship between H(o)and the network +k* 2. The relationship between Hg o)and the network Differential ans"+ans"+…+a 0 +a-c+-+ajv)-x() 1s1+…+a。=0 Transfer H(o) n-1 Characteristie s=s1, $2/Sn Nat freq of network O Compare the secular equation and the transfer function Re s,,.s,)<0 Located at the left of s-plane The polar point of HG@ ) is the zero point of the secular Comp.rep,of dif:em)=Guyei- equation, which is the natural frequency of network. sinsahleres:{aor+a(n-+-+a}Yo All the Relpolar point of Hgo))<0 is the necessary m= x(io 可 Transfer and sufficient condition of a stable network H fa)Yo Xa)a(ar+an1(a+…+a b Hg o)can fully describe the frequency characteristic f the netw (have nothing to do with input/output) C1-5: Analysis of Sinusoidal Steady-state Cireuit Chapter 1 Introductory Linear Circuit Analysis C1-5: Analysis of Sinusoidal Steady-state Circuit (Complex Solution to Linear and Time-invariant Circuits) Comp. rep of response Transfer function Hgo) COmplex method and phasor method, i Comp. rep of stimulation stability of networks transfer function. Time-domain Frequency-domain relationship between transfer function C1-6: Filter 2 The definition and classification of filters ) First-order filter(low-pass, high-pass) Second-order filter(band-pass, band-stop) y(t)=Fx(t), h(t)) YGo)=HGo). XGo) Active filter (just know about it) k 1. The definition of "Filter 2. Types of filter O Filter is a two-port network which processes the input signals. It transforms the input signal to output O According to selective characteris signal in frequency domain according to a certain b Low pass b Band pas b Band stop 垢吵) Yo x o HUGo ACcording to active or not b Passive filter(only R.L. Y(c)=H〔0)X() b Active filter (ineluding controlled souree) ACcording to realization method The transfer function can describe the frequency Analog filter Digital filter selective characteristic of a filter
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 Complex representation Time-domain analysis 2. The relationship between H(jω) and the network Comp. rep. of diff.: ( ) ( ) j ω t n j ω t n n e j ω e dt d = ⋅ Sin. Stable res.: {a () () jω a jω a } Y( ) jω X(jω) 0 n 1 n 1 n n + + + ⋅ = − − L Transfer function: ( ) ( ) ( ) () () 0 n 1 n 1 n an jω a jω a 1 X jω Y jω H jω + + + = = − − L a s a s a 0 0 n 1 n 1 n n + + + = − − L Secular equation: a y() () t x t dt d a dt d a n 1 0 n 1 n n 1 n n + + + = − − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ L Differential equation: Stable conditions: Re { } s 1 ,K s n < 0 Located at the left of s-plane Characteristic root: 1 2 s n s = s , s ,K Nat. freq. of network *** 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 Polar point : which makes denominator zero. wwhu 北京大学 Zero point : which makes numerator zero. Compare the secular equation and the transfer function: Secular equation: ( ) () () 0 n 1 n 1 n a n jω a jω a 1 H jω + + + = − − L Transfer Function: The polar point of H(jω) is the zero point of the secular equation, which is the natural frequency of network. All the Re{polar point of H(jω)}<0 is the necessary and sufficient condition of a stable network H(jω) can fully describe the frequency characteristic of the network. (have nothing to do with input/output) a s a s a0 0 n 1 n 1 n n + + + = − − L *** 2. The relationship between H(jω) and the network 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 Transfer function H(jω) = Comp. rep. of response Comp. rep. of stimulation H(j H(jωω)) X(jω) Y(jω) Frequency-domain h(t) h(t) x(t) y(t) input (stimulation) output (response) Time-domain y(t) = F{ x(t), h(t) } Y(jω) = H(jω) · X(jω) C1-5：Analysis of Sinusoidal Steady-state Circuit *** （Complex Solution to Linear and Time-invariant Circuits） input (stimulation) output (response) 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 C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） Complex method and phasor method, impedance and admittance The complex forms of elements, laws and theorems The power of sinusoidal steady-state (for self-study) The stability of networks, transfer function, The relationship between transfer function C1-6: Filter The definition and classification of filters First-order filter (low-pass、high-pass) Second-order filter (band-pass、band-stop) Active filter（just know about it） Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis 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 H(j H(jωω)) X(jω) Y(jω) 1. The definition of “Filter” Filter is a two-port network which processes the input signals. It transforms the input signal to output signal in frequency domain according to a certain requirement. Y(jω) = H(jω) · X(jω) NN X(jω) Y(jω) *** The transfer function can describe the frequency selective characteristic of a filter. input (stimulation) output (response) input (stimulation) output (response) 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 ２. Types of filter According to selective characteristic Low pass High pass Band pass Band stop All pass According to active or not Passive filter (only R, L, C) Active filter (including controlled source) pass pass pass pass pass stop stop stop stop stop According to realization method pass Analog filter Digital filter ***
2. Types of filter 3. The terms of filter H川 Low-pass Bandwidth(3d B, 20dB) Transition filter st+cost d half power point P Stop (3B point) band band +cost 0.1cos5t+cos10t B. +cost +cos10t =10lg1-3[dB 4. First-order filter(including one dynamic element) 4. First-order filter(including one dynamic element) O The impedance and frequency response of capacitance DExample(First-order high-pass RC filter) (high-pass elemen →团→ w+ H(u)=R+1/joC-1+joRC 1 low freqopen high freq.short V,(o) .o)HGoX joc 1-0 d The impedance and frequency response of inductance H(川 6→∞|(→1aa)→0 moa-0|2-0a∞2→∞ 五 z=joL low freq short high freq.open HHGe)=0 do)= 5. Second-order filter(including two dynamic elements* 5. Second-order filter(including two dynamic elements) O Example: series resonant ciret About quality factor(Q-factor): v The physical defn En Q=2 M(jo) Some experien When the resonance frequeney or the structure of (frequeney response the resonance cireuit is changed, the quality factor HOW will change because of the change of the svstem's energy, period and dissipation. Usually, the changed 日orQ="=-1 Q-factor will be smaller than Land Q c w.RC 甲{u)=- arctan
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 channel channel cos5t cost+cos2t 0.1cos5t+cos10t BPF BPF HPF HPF LPF LPF cos10t cos5t cos2t f(t) cost + + + = -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 4 5 系列1 系列3 LPF -3 -2 -1 0 1 2 3 4 5 系列1 系列2 BPF -3 -2 -1 0 1 2 3 4 5 系列1 系列4 HPF ２. Types of filter 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 A0 2 A 0 |H(jω)| 0 ω B0 Pass band Stop band Transition band Low-pass filter Low-pass filter half power point (3dB point) Cut-off frequency bandwidth(3dB,20dB) transition band ( ) 3 [dB] 2 1 10lg A H jω 10lg ω ωc 2 0 = ≈ − = *** 3. The terms of filter ωc ωc ωl 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 4. First-order filter (including one dynamic element) The impedance and frequency response of capacitance (high-pass element) jω C 1 Z = ω→ω→0 |Z| 0 |Z|→∞ ω→∞ →∞ ω→∞ |Z| |Z|→→00 low freq. open high freq.short I→0 V→0 The impedance and frequency response of inductance (low-pass element) Z = jωL ω→ω→0 |Z| 0 |Z|→→00 ω→∞ ω→∞ |Z| |Z|→∞→∞ low freq. short high freq. open V→0 I→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 4. First-order filter (including one dynamic element) Example (First-order high-pass RC filter)： ωc 1 2 1 |H(jω)| 0 ω 0 ωc ω Φ(ω) 4 π 2 π ω→∞ ω=1/RC ω= 0 ω→∞ ω=1/RC ω= 0 H( jω) →1 ( ) 2 1 H jω = H( jω) = 0 Φ(ω)→0 ( ) 4 π Φ ω = ( ) 2 π Φ ω = Vi(jω) Vo(jω) + + - - ( ) 1 j ωRC j ω RC R 1 j ω C R H j ω + = + = ( ) ( ) ( ) ( ) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = − + = − arctan ωRC 2 Φ ω 1 ωRC 1 H j ω 2 π *** 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) + - ( ) ( ) V ( ) j ω V j ω H j ω R = Transfer function Resonance frequency： quality factor (Q-factor): Resonance frequency： quality factor (Q-factor): LC 1 ω0 = ω RC 1 R ω L Q 0 0 = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ω ω - ω ω 1 jQ 1 0 0 Amplitude and phase (frequency response) ( ) 2 0 0 2 ω ω- ω ω 1 Q 1 H j ω ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − ω ω- ω ω φ ω arctanQ 0 0 jωC 1 R jωL R + + = *** Example: series resonant circuit 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 quality factor (Q-factor): The physical definition of quality factor Ｑ＝２π Energy Stored Energy dissipated per cycle Some experience: When the resonance frequency or the structure of the resonance circuit is changed, the quality factor will change because of the change of the system’s energy, period and dissipation. Usually, the changed Q-factor will be smaller thanＱＬandＱＣ
5. Second-order filter(including two dynamic elements) 5. Second-order filter(including two dynamic elements) About quality factor(series resonant circuit R→Q1=9止 Hgo w From the half power point (+40-1) Vda band-pass filter With the structure of resonance cireuits nt Relative frequency: bandwidth Q Q-factor:QR=Rc owE uree with essential resistance will er the Q-fac 5. Second-order filter(including two dynamic elements) 5. Second-order filter(including two dynamic elements) About quality factor(shunt-resonant cireuit) eries resonant cireuiteduality relation shunt-resonant cireuit G1→Q: c resistance of capacitance an ductance. we has QQQ 20o)vu)4“)B(/(Uo)1 quality facto th essential joc re YUo)=G+ joC+- Q Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis The power of sinusoidal steady-state C1-5: Analysis of Sinusoidal Steady-state Cireuit (Complex Solution to Linear and Time-invariant Circuits 1. instantaneous power Reflecting the instantaneous and phasor method, impedance and admittance steady-state (for self-study) z roo v(0=m cos(ot+p)=N2V cos(ot +p) C1-6: Filter othing to do with a The definition and classification of filters First-order filter(low-pass, high-pass Second-order filter(band-pass, band-stop) P(t=v(t)i(t)=lv cos +lv cos(2ot +o) Active filter (just know about it) between the network an aused by and o
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) 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 quality factor (series resonant circuit): Conclusion: With the structure of resonance circuits unchanged, the introduction of load or source with essential resistance will lower the Q-factor. C L L C Q 1 Q 1 Q 1 R = R + R → = + When the dissipation is induced by the resistance of capacitance and inductance, we have: + - I(jω) C R L V(jω) 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) 0 ω0 ω φ(ω) 2 π ω0 1 2 1 |H(jω)| 0 ωl ωh ω From the half power point: ( ) 1 4Q 1 2Q ω ω 0 2 l = + − ( ) 1 4Q 1 2Q ω ω 0 2 h = + + The 3 dB bandwidth of the band-pass filter Q ω ω ω ω 0 Δ = h − l = Relative bandwidth Q 1 ω ω 0 = Δ *** 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) series resonant circuitÅduality relationÆshunt-resonant circuit + - I(jω) C R L V(jω) ( ) ( ) ( ) ( ) ( ) jωC Z jω R jωL V jω Z jω I jω H jω 1 1 = + + = = ( ) ( ) ( ) ( ) ( ) jωL Y jω G jωC I jω Y jω V jω H jω 1 1 = + + = = V(jω) ÅÆ I(jω) Z(jω) ÅÆ Y(jω) L ÅÆ C R ÅÆ G ( ) ( ) ( ) ( ) 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 5. Second-order filter (including two dynamic elements) About quality factor (shunt-resonant circuit): Conclusion: With the structure of resonance circuits unchanged, the introduction of load or source with essential resistance will lower the Q-factor. ω G L 1 G Q 0 L L → L = C 0 C C G ω C G → Q = Resonance frequency： quality factor (Q-factor): Resonance frequency： quality factor (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 → = + ( ) ( ) ( ) ( ) jωC Z jω R jωL V Z jω I jω H jω 1 1 = + + = = V(jω) I(jω) C G L + - When the dissipation is induced by the resistance of capacitance and inductance, we have: 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 C1-5：Analysis of Sinusoidal Steady-state Circuit （Complex Solution to Linear and Time-invariant Circuits） Complex method and phasor method, impedance and admittance The complex forms of elements, laws and theorems The power of sinusoidal steady-state (for self-study) The stability of networks, transfer function, The relationship between transfer function C1-6: Filter The definition and classification of filters First-order filter (low-pass、high-pass) Second-order filter (band-pass、band-stop) Active filter（just know about it） Chapter 1 Introductory Linear Circuit Analysis ----From time-domain analysis to frequency-domain analysis 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 p(t) = v(t)i(t) = IV cosϕ + IV cos(2ωt +ϕ) The power of sinusoidal steady-state 1. instantaneous power Z v t( ) i t( ) + - ( ) cos 2 cos ( ) cos( ) 2 cos( ) m m it I t I t vt V t V t ω ω ω ϕ ωϕ = = = += + ϕz vi =ϕϕϕ − = Reflecting the instantaneous power of network Have nothing to do with t. This constant is caused by R. Reflecting the energy exchange between the network and source, caused by L and C
The power of sinusoidal steady-state The power of sinusoidal steady-state 1. Instantaneous power 2. Average power Unit: w(Walt) p(=v(t)i(t)=lv cos+l cos(2at+o) let(=0-Pure resistance network P= p(O)dr=IV cosp p(r)=/(1+cos 2at)20 Always dissipation Reflecting the dissipation of network: active power p=90->Pure reactance network P(o=-IV cos 2ot Can be positive Unit: w(Walt) coS p Power Factor, the angle of power factor So we can change P(o) egative, no dissipation. p(t)=/cos o(l+cos 2ot)-lV sin osin 2a n=cos o reflects the power efficiency which The angle of power factor is caused by l and C reactive power It doesn't cause the dissipation, but it changes the energy power required by the device The power of sinusoidal steady-state The power of sinusoidal steady-state 3. Reactive power Unit: VAR or var 4. Apparent power Unit: VA O=lV sin g power rating: the capacity of the device of the twngger Q-factor, the bigger stored energy in the dynamic elements S=I If comparing the Q-factor in a penod, the Q-factor in unit time reflects 5. Complex power regard to each dynamic element V- W, =-Ll P=P+jQ=S∠q 0=-2ow 0,=low expecting less exchange between the source and network: Q=0+ e method: parallel C with inductive loaded, parallel L with capacitive load
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 The power of sinusoidal steady-state 1. Instantaneous power 瞬时功率 letϕ = 0 →Pure resistance network p t IV t ( ) (1 cos 2 ) 0 =+ ≥ ω Always dissipation ϕ=90o →Pure reactance network p t IV t ( ) cos 2 = − ω Can be positive or negative, no dissipation. So we can change p t( ): p t IV t IV t ( ) cos (1 cos 2 ) sin sin 2 = +− ϕ ω ϕω Active power reactive power p(t) = v(t)i(t) = IV cosϕ + IV cos(2ωt +ϕ) 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 The power of sinusoidal steady-state 2. Average power 0 1 ( ) cos T p p t dt IV T = = ϕ ∫ Reflecting the dissipation of network: active power Unit: W（Walt）; cosϕ Power Factor, the angle of power factor ϕ reflects the power efficiency which source supplies to the load; The angle of power factor is caused by L and C. It doesn’t cause the dissipation, but it changes the energy power required by the device. η = cosϕ P Unit: W（Walt） *** 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 The power of sinusoidal steady-state 3. Reactive power Unit: VAR or var Q IV = sin ϕ 1、Reflecting the energy exchange between network and source; 2、The bigger Q-factor, the bigger stored energy in the dynamic elements of the two-terminal network; 3、If comparing the Q-factor in a period, the Q-factor in unit time reflects the energy exchange rate between source and network. Q 1 2 2 W CV c = 1 2 2 W LI L = ∴ 2 Q W C c = − ω 2 Q W L = ω L regard to each dynamic element: If expecting less exchange between the source and network: Q=0 Æ ϕ=0 Æ method: parallel C with inductive loaded, parallel L with capacitive load 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 The power of sinusoidal steady-state 4. Apparent power Unit: VA 2 2 S IV P Q == + power rating: the capacity of the device 5. Complex power Unit: none Pc = P +jQ = S∠ϕ