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: