北京大学：《电路分析原理 Circuit Analysis》课程教学课件（英文版）第一章 线性电路分析基础 第四节 线性电路的时域分析

Content C1-4: Time-domain analysis of Linear and Time- invariant Networks ( Analysis of First Order Circuit) independent source Customarily: zero-input response Yzi-> stimulation by initial value >energy storage element whose initial value is not zero zero-state response Yzs-> stimulation by independent source Customarily: zero-input network(zi): independent source=0 overall response: Y(tFYzi(t)+Yzs(t) zero-state network(zs): initial value =0 plified expression Simplified expression output output (stimulation) (response) (response)

第 ？讲： 复习 《Principles of Circuit Analysis》 Introductory Linear Circuit Analysis Lecture 3 2009.09.22 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 Content C1-4: Time-domain analysis of Linear and Time￾invariant Networks (Analysis of First Order Circuit)  dynamic state and steady state typical source signals (stimulating signal) definition of initial state (initial value, initial conditions) Time-domain analysis of dynamic circuits 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 form of elements, law and theorem The power of sinusoidal steady-state (self-study) The stability of networks, 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 Terms and diagrams of linear systems: systems circuits networks Ns Ns NL NN NL Control Station Up￾Link Down￾Link Back bone Net￾work Optical mm-wave WDM Sources M U X D E M U X λu1 λu2 ...λuN BS1 : EDFA λu1 λu2 ...λuN User Terminal Data Down Data Up BS2 BSn Mm-wave Wireless Link Photo Detector : : : : : λd1 λd2 ...λdN ROF communication system 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 Terms and diagrams of linear systems: Ns Ns NL NN NL N x(t) y(t) simplification input （ stimulation ） output （response） A τ D ∫ ∑ 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 Terms of linear system N x(t) y(t) Simplified expression input (stimulation) output (response) ‰ There is two origin of input signals (stimulation): ¾independent source ¾energy storage element whose initial value is not zero Customarily: zero-input network (zi): independent source=0 zero-state network (zs): initial value =0 Ns Ns NL NN NL *** 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 ‰ output (response) the response of load circuit (v(t),i(t)) Customarily: zero-input response Yzi -> stimulation by initial value zero-state response Yzs -> stimulation by independent source overall response: Y(t)=Yzi(t)+Yzs(t) N x(t) y(t) Ns Ns NL NN NL Terms of linear system *** Simplified expression input (stimulation) output (response)

C1-4: Time-domain analysis of Linear and Time-invariant Networks Content C1-4: Time-domain analysis of Linear and Time- Example: invariant Networks ( Analysis of First Order Circuit) ndependent a dynamic state and steady state typical source signals(stimulating signal definition of initial state(initial value, initial conditions) Energy storage element whose initial value is not zero Time-domain analysis of dynamic circuits C1-5: Analysis of sinusoidal Steady-state Circuit V(0)=0,1 \0: zero-input network, zero-input response (Complex Solution to Linear and Time-invariant Circuits) Complex method and phasor method, impedance and admittance 10, V(0)+0: zero-state network, zero-state response The complex form of clements, law and theorem The power of sinusoidal steady-state (self-study) he stability of networks, transfer functi 1-4: Time-domain analysis of Linear and Time-invariant Networks" 1-4: Time-domain analysis of Linear and Time-invariant Networks I DC signal(omitted) f(t)=A 2 sinusoidal signal(omitted) f(t)=Acos(o t+p) 2u(t-1 3 unit step signal u(t)or U(t)or I(t u(0)=? 1t20+ (Singular signal) 2u(t-1)-2u(t-2) 2u(t-2 t Switch function o12 Wrong diagram? Right diagram!! 1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time-invariant Networks Unit step signal u(t) Unit step signal u(t 2u(t-1)-2u(t-2) 2u(t-1) >to your homework u(t)

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 Linear and Time-invariant Networks Independent source Energy storage element whose initial value is not zero Vc(0) =0 ，Is=0: zero-input network, zero-input response Is=0 ，Vc(0) =0: zero-state network, zero-state response Example: Is + R - t＝0 C Vc V0 (t) + - IR(t) 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 C1-4: Time-domain analysis of Linear and Time￾invariant Networks (Analysis of First Order Circuit)  dynamic state and steady state typical source signals (stimulating signal) definition of initial state (initial value, initial conditions) Time-domain analysis of dynamic circuits 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 form of elements, law and theorem The power of sinusoidal steady-state (self-study) The stability of networks, 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 1 DC signal (omitted) 2 sinusoidal signal (omitted) 3 unit step signal u(t) or U(t) or 1(t) 1 0 u(t)= t≤0- t≥0+ t 0 u(0)=？ Is R t＝0 IR(t) Switch function f(t) = A f(t) = Acos(ω t ) +ϕ (Singular signal) *** Wrong diagram? C1-4: Time-domain analysis of Linear and Time-invariant Networks Is R t＝0 IR(t) Switch function Right diagram!!! 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 0 t 1 u(t) 0 t 2 2u(t) 0 t 2 2u(t-1) 1 0 t 2 2u(t-1)-2u(t-2) 1 2 0 t -2u(t-2) 2 + C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 0 t 1 u(t) 0 t 2 2u(t-1) 1 0 t 2 2u(t-1)-2u(t-2) 1 2 Unit step signal u(t) 0 t f(t) 0 t f(t)u(t) causal signal *** Rectangular pulse C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 0 t 1 u(t) 0 t 2 1 ? Æto your homework 3 Step signal 4 4 C1-4: Time-domain analysis of Linear and Time-invariant Networks Unit step signal u(t)

C1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time-invariant Networks 4 unit pulse (1)={ Obviously to t Expressed by u(t): P()=[(1)-(t-△ C1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time-invariant Networks unit impulse signal 5 unit impulse signal 0t≠0 0t≠0 (1) t=0 (1)= haracteristieI: 8(n=limP(o=lim a(-(I-a)=r(o characteristic 3: Sampling integral definition f(no()dt=f(o) f(o(t-to )dt=f(o) 1-4: Time-domain analysis of Linear and Time-invariant Networks"' C1-4: Time-domain analysis of Linear and Time-invariant Networks unit impulse signal Question 60={0≠0 t(t)=? t=0 integral definition 广)d=00M=1 f(t)6(t-t0)=f(t0)6(t-t0 f(od(t-to )dt=f(to)

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 Linear and Time-invariant Networks 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 ⎪ ⎩ ⎪ ⎨ ⎧ > Δ Δ < < Δ < Δ = t t t 0 1/ 0 0 0 P ∫ ∞ −∞ PΔ (t)dt = 1 4 unit pulse signal Expressed by u(t): Obviously: [ ( ) ( )] 1 ( ) − − Δ Δ PΔ t = u t u t P (t) Δ C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 unit impulse signal characteristic 1： ⎩ ⎨ ⎧ ∞ = ≠ = 0 0 0 ( ) t t S t ∫ ∫ ∫ + − − ∞ −∞ = = = 0 0 ( ) ( ) ( ) 1 0 0 t dt t dt t dt t t δ δ δ '( ) ( ) ( ) ( ) lim ( ) lim 0 0 u t u t u t t P t = Δ − − Δ = = Δ→ Δ Δ→ δ δ (t) *** integral definition C1-4: Time-domain analysis of Linear and Time-invariant Networks characteristic 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 ⎩ ⎨ ⎧ ∞ = ≠ = 0 0 0 ( ) t t δS(tt) characteristic 3: Sampling ⎩ ⎨ ⎧ = ≠ = (0) ( ) 0 0 0 ( ) ( ) f t t t f t t δ δ ⎩ ⎨ ⎧ − = ≠ − = 0 0 0 0 0 ( ) ( ) 0 ( ) ( ) f t t t t t t t f t t t δ δ ∫− = ς ς f (t)δ (t)dt f (0) ∫ + − − = ς ς δ 0 0 ( ) ( ) ( ) 0 0 t t f t t t dt f t C1-4: Time-domain analysis of Linear and Time-invariant Networks 5 unit impulse signal 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 unit impulse signal ⎩ ⎨ ⎧ ∞ = ≠ = 0 0 0 ( ) t t S t ∫ ∫ ∫ + − − ∞ −∞ = = = 0 0 ( ) ( ) ( ) 1 0 0 t dt t dt t dt t t δ δ δ δ (t) ∫ + − − = − = − ς ς δ δ δ 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 t t f t t t dt f t f t t t f t t t Samplin g *** Question: tδ (t) = ? C1-4: Time-domain analysis of Linear and Time-invariant Networks integral definition 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 practical impulse signal: C1-4: Time-domain analysis of Linear and Time-invariant Networks

C1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time-invariant Networks 7 unit ramp signal square wave (t)=t(1) f()=(1)-a(t-1)+l(-2)-l(t-3)+ t≥0 aw-tooth wave r(1)=l(1)+t6(1)=l(1) f(1)=r(1)-l(t-1)-(1-2)-(-3) triangular wave f(t)=r(t)-2r(t-1)+2r(t-2)-2r(t-3)+ 1-4: Time-domain analysis of Linear and Time-invariant Networks Terms of linear systems unit step response s(y) and unit impulse response h(o) Unit step response(s(t)is the response of a circuit whose input is a unit step signal, Unit impulse response(h(t))is the response 0)[(0) put is a unit impulse signal. s(t)and h(t)are both zero-state response u(t-T 8(t)=dlu(t)]/dt h(t)=d[s(t)]/dt system N Content C1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time- invariant Networks DynamIc (Analysis of First Order Circuit) a dynamic state and steady state lOV typical source signals(stimulating signal) 10v(R R definition of initial state(initial value, initial conditions) Time-domain analysis of dynamic circuits C1-5: Analysis of sinusoidal Steady-state Circuit Complex Solution to Linear and Time-invariant Circuits) X X The complex form of elements, law and theor The power of sinusoidal steady-state (self-study) The stability of networks, transfer function Static circuits (DC Static 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 7 unit ramp signal ⎩ ⎨ ⎧ ≥ < = = 0 0 0 ( ) ( ) t t t r t tu t r'(t) = u(t) + tδ (t) = u(t) tgα = 1 C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 square wave saw-tooth wave triangular wave f (t) = u(t) − u(t −1) + u(t − 2) − u(t − 3) + ..... f (t) = r(t) − u(t −1) − u(t − 2) − u(t − 3) − .... f (t) = r(t) − 2r(t −1) + 2r(t − 2) − 2r(t − 3) + ... C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 A τ D ∫ ∑ system C1-4: Time-domain analysis of Linear and Time-invariant Networks ∫ ∫ δ(t) u(t) r(t) r(t) u(t) u(t −τ ) δ (t −τ ) D τ D 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 ‰ unit step response s(t) and unit impulse response h(t) Ns Ns NL NN NL Terms of linear systems Definition: Unit step response (s(t)) is the response of a circuit whose input is a unit step signal. Unit impulse response (h(t)) is the response of a circuit whose input is a unit impulse signal. N u(t) S(t) s(t) and h(t) are both zero-state response. N δ(t) h(t) δ(t)=d[u(t)]/dt h(t)=d[s(t)]/dt prove it。。。 *** Simplified expression 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 C1-4: Time-domain analysis of Linear and Time￾invariant Networks (Analysis of First Order Circuit)  dynamic state and steady state typical source signals (stimulating signal) definition of initial state (initial value, initial conditions) Time-domain analysis of dynamic circuits 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 form of elements, law and theorem The power of sinusoidal steady-state (self-study) The stability of networks, 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 Dynamic circuits? + - 10V C R + - v(t) t=t0 + - - + 1Ω 2Ω 2Ω 1V 2Ω 1V + - 10V R R + - v(t) t=t0 X X X ☺ Static circuits (DC Static circuits ) C1-4: Time-domain analysis of Linear and Time-invariant Networks

C1-4: Time-domain analysis of Linear and Time-invariant networks Cl-4: Time-domain analysis of Linear and Time-invariant Networks Dynamic circuits Sinusoidal steady-state circuit static response when sinusoidal Dynamic 10平v(+CR signals stimulate RIv( process state v(ti dynamic 1. Including dynamic elements St tatIc i State phasor method state 2. Switch the circuit Vt (complex method) Content Tea break/ C1-4: Time-domain analysis of Linear and Time invariant Networks (Analysis of First Order Circuit a dynamic state and steady state typical source signals(stimulating signal) definition of initial state(initial value, initial conditions) Time-domain analysis of dynamic circuits ( Complex Solution to Linear and Time-invariant Circuits) and admitance The complex form of clements, law and theorem The power of sinusoidal steady-state (self-study) The stability of transfer functon 1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time-invariant Networks initial state(initial value, initial conditions) the state of network at t=t+ The state of network at t". (or ac)(steady state): When the dvnamic circu to steady state. there is no exchange of electromagnetic energy open a the inductance means short circuit. That is: v,(tFi(4-0 Static Static The determination of state state initial state→ ules for switching: If the cireuit is switched at t-t and the vdt)and i(t) are continuous while switching. That is: vdt. C 1. According to rules for switching 2. According to KCL and KvL

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 Dynamic circuits t0 t v(t) Dynamic process Static state switch + - 10V C R + - v(t) t=t0 1. Including dynamic elements 2. Switch the circuit *** C1-4: Time-domain analysis of Linear and Time-invariant Networks Static state 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 t0 t v(t) + - C R L cos(ωt) + - Sinusoidal steady-state circuit -- static response when sinusoidal signals stimulate. *** i(t) v(t) phasor method (complex method) VR(jω) Ii(jω) V(jω) C1-4: Time-domain analysis of Linear and Time-invariant Networks Static state Dynamic process switch Static state 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！ 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 C1-4: Time-domain analysis of Linear and Time￾invariant Networks (Analysis of First Order Circuit)  dynamic state and steady state typical source signals (stimulating signal) definition of initial state (initial value, initial conditions) Time-domain analysis of dynamic circuits 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 form of elements, law and theorem The power of sinusoidal steady-state (self-study) The stability of networks, 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 Static state initial state (initial value, initial conditions) : the state of network at t=t0+ + - 10V C R + - v(t) t0 t=t0 t t=t0- t=t0+ v(t) The determination of initial stateÆ C1-4: Time-domain analysis of Linear and Time-invariant Networks switch Dynamic process Static state 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 + - 10V C R + - v(t) t=t0 The state of network at t=t0-(or ∝)(steady state)： When the dynamic circuit comes to steady state, there is no exchange of electromagnetic energy. The capacitance means open circuit. And the inductance means short circuit. That is: vL(t0-)= ic(t0-)=0 The determination of initial state: 1。According to Rules for switching 2。 According to KCL and KVL Rules for switching: If the circuit is switched at t=t0, and the current of capacitance/ the voltage of inductance is limited, the vC(t) and iL(t) are continuous while switching. That is: vc(t0- )=vc(t0+) ，i L(t0-)=iL(t0+) C1-4: Time-domain analysis of Linear and Time-invariant Networks

C1-4: Time-domain analysis of Linear and Time-invariant Networks Content C1-4: Time-domain analysis of Linear and Time- When te ( Analysis of First Order Circuit) lR(0-)=0 a dynamic state and steady state typical source signals(stimulating signal) When t=t,+ definition of initial state(initial value, initial conditions) Time-domain analysis of dynamic circuits V(0+=V0-)=V C1-5: Analysis of sinusoidal Steady-state Circuit (Complex Solution to Linear and Time-invariant Circuits lg(0+)=V(0+)R=V Complex method and phasor method, impedance and admittance The complex form of clements, law and theorem The power of sinusoidal steady-state (self-study) he stability of networks, transfer functi 1-4: Time-domain analysis of Linear and Time-invariant Networks 1-4: Time-domain analysis of Linear and Time-invariant Networks Time-domain analysis of first order dynamic circuits Time-domain analysis of first order dynamic circuits Establish equation. t20+ rt (1)=Ri() Determination of initial state hen r-ta (s d+i(0=l -order differential equation with constant The capacitance means open cireuit And the inductance means short circuit coefficients. = Independent dynamic elements 0-=Va,(0-=0 f(0+)=。/R When 9 According to Rules for switching rdt Fv(L),i(te)i(t, 3. Solution foundations of mathematics t 9 According to KCL, KVL, VCR Solution of first-order differential equations with constant v(0+)=V此0+=VR coefficients (general solution, specifie solution) 1-4: Time-domain analysis of Linear and Time-invariant Networks Cl-4: Time-domain analysis of Linear and Time-invariant Networks Time-domain analysis of first order dynamic circuits VaR 3. solution l0(1-e-") Characteristic equation )=e"+Rl-c") +f()=l Characteristic Value: S=-I/RCE-1/T i(0+)=V/R Definition: time constant T eRO D> equals to t≥0+ ematical expression i(0)= +l0( )=e"+l0(1-e (t20+)

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: When t=t0- Vc(0-)= V0 IR(0-)= 0 When t=t0+ Is R Vc(0-) + - IR(t) Vc(0+)= Vc(0-)= V0 IR(0+)= Vc(0+)/R= V0/R Is + R - t＝0 C Vc V0 (t) + - IR(t) C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 C1-4: Time-domain analysis of Linear and Time￾invariant Networks (Analysis of First Order Circuit)  dynamic state and steady state typical source signals (stimulating signal) definition of initial state (initial value, initial conditions) Time-domain analysis of dynamic circuits 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 form of elements, law and theorem The power of sinusoidal steady-state (self-study) The stability of networks, 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 Time-domain analysis of first order dynamic circuits I0 R + Vc(t) - Vc(0)=V0 t≥0+ i(t) When t=t0-(steady state) Æ The capacitance means open circuit. And the inductance means short circuit. vc(0-)=V0 ，i(0-)=0 When t=t0+ Æ According to Rules for switching vc(t0+)=vc(t0-)，i L(t0+)=iL(t0-) Æ According to KCL,KVL,VCR 1.Determination of initial state: Vc(0+)=V0 i(0+)= V0/R Let Is =I0 Is + R - C Vc V0 (t) t=0 i(t) C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 R + Vc(t) - Vc(0)=V0 t≥0+ i(t) 2. Establish equation: dt dV t I i t C V t Ri t c c ( ) ( ) ( ) ( ) 0 = + = foundations of mathematics ： Solution of first-order differential equations with constant coefficients (general solution, specific solution) i V R i t I dt di t RC (0 ) / ( ) ( ) 0 0 + = + = 3.Solution: I0 ?-order differential equation with constant coefficients. = ? Independent dynamic elements C1-4: Time-domain analysis of Linear and Time-invariant Networks Time-domain analysis of first order dynamic 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 i V R i t I dt di t RC (0 ) / ( ) ( ) 0 0 + = + = general solution Characteristic equation: RCS+1=0 Characteristic Value: S=-1/RC=-1/τ Definition: time constant τ=RC st i(t) = Ke ( ) (1 ) / 0 0 t /τ t τ e I e R V i t − − = + − specific solution 0 i(t) = I 0 i ( t ) Ke I st solution = + I0 R + Vc(t) - Vc(0)=V0 t≥0+ i(t) (t≥0+) C1-4: Time-domain analysis of Linear and Time-invariant Networks Time-domain analysis of first order dynamic circuits 3.solution: 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − t>0 equals to t≥0+ ( ) (1 ) ( ) / 0 0 / e I e u t R V i t t t ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = + − − τ − τ t≥0- mathematical expression of solutionÆ Analysis 1：how to express the response at t=0 I0 RI0 *** C1-4: Time-domain analysis of Linear and Time-invariant Networks

C1-4: Time-domain analysis of Linear and Time-invariant Networks 1-4: Time-domain analysis of Linear and Time-invariant Networks Time-domain analysis of first order dynamic circuits it 0(1-e-) v(0+)=V。0+)=VR Analysis 2t about time )(c2 τ’ s dimension:se Determining the t discharging process of circuits p:(n)=le"+Rl1-e")(p>0) 4T,Ⅴ(=V/e=1.84%V ()=[ne"+l(l-e")jp(t) (20) 2()=loe+R(1-e-) general engineering purposes, if f4 TsT, discharging is over. C1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time-invariant Netw 00)=e"+la (1) e:+la(l-e v(=loe+Ro(l-e) v (n)=voe+Rlo(l-e) Analvsis 2t about time constant t Analysis 3t about the natural frequency of network: Ts dimension: second (s) 0 Determining the time of discharging process of circuits 8=-1/T, which has a dimension of frequency parameters, we call it the natural frequency of network. steady state steady state Switch C1-4: Time-domain analysis of Linear and Time-invariant Networks Cl-4: Time-domain analysis of Linear and Time-invariant Networks VaR v(=Ve/+Rl,(1-e-) ve()=loew/t Analysis 4: Analysis 5: Three-element method of first order circuits (TEM) ()=(6-R 0→(0+)=R,x)=c esponse 2()→FQ+)=la,F()=R,e the stimulation, it is forced by destination Time constant T TEM: when V.RIo no transient state?(attention: the network changed when y()=Dy(0+)-y(∞)"+y(∞) i the cireuit is switched)

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 Is + R - C + Vc(t) - V0 t=0 i(t) I0 R + Vc(t) - t≥0+ i(t) t≥0+ Vc(0+)=V0 i(0+)= V0/R Let Is =I0 ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − 0 t Vc(t) V0 0 t i(t) V0/R I0 RI0 （t≥0+） （t>0） 0 / / 0 / / 0 0 ( ) [ (1 )] ( ) ( ) (1 ) t t t t c V it e I e ut R v t V e RI e τ τ τ τ − − − − = +− = +− （t≥0） or or *** C1-4: Time-domain analysis of Linear and Time-invariant Networks Time-domain analysis of first order dynamic 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − I0 RI Analysis 2： about time constant τ 0 τ’s dimension: second (s) Determining the time of discharging process of circuits. When t=τ, Vc(t)=V0/e=36.8% V0 t=4τ, Vc(t)=V0 /e4=1.84% V0 t=5τ, Vc(t)=V0 /e5=0.68% V0 For general engineering purposes, if t=4τ～5τ, discharging is over. τ τ C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − I0 RI0 t0 t Dynamic state steady state steady state Switch t0+5τ τ τ *** C1-4: Time-domain analysis of Linear and Time-invariant Networks Analysis 2： about time constant τ τ’s dimension: second (s) Determining the time of discharging process of 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − I0 RI0 s=-1/τ, which has a dimension of frequency. Because s is determined by network’s structure and parameters, we call it the natural frequency of network. Analysis 3：about the natural frequency of network: s τ τ *** C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − I0 RI0 Analysis 4: steady-state or transient response 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 Q: when V0＝RI0, no transient state? (attention: the network changed when the circuit is switched) C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − I0 RI Analysis 5：Three-element method 0 of first order circuits (TEM) τ τ / 0 0 / 0 0 ( ) (0 ) , ( ) , ( ) (0 ) , ( ) , t c c c t V t V V V RI e i I e R V i t i − − → + = ∞ = → + = ∞ = TEM: Starting point destination destination Time constant τ e-t/τ *** ( ) [ (0 ) ( )] ( ) / = + − ∞ + ∞ − y t y y e y t τ C1-4: Time-domain analysis of Linear and Time-invariant Networks Starting point

Cl-4: Time-domain analysis of Linear and Time-invariant Network 1-4: Time-domain analysis of Linear and Time-invariant Networks diflerential circuit k ()=le“4+(1 Analysis 6t ch dischanging v1(t) v。(t) Overall response: Y(tFYzi(t)+ Yzs(t) C1-4: Time-domain analysis of Linear and Time-invariant Networks C1-4: Time-domain analysis of Linear and Time-invariant Netw Analysis 8: Q: the v(=loe+Ro(l-e) Analvsis 7: According to definition, let i(0=Ae/ unit-step response (n=u(n), then: Io=l,o=0 o 1+R·J Response ofrectangular signal oi let:s=lo cos(or)=Re(/) Ealer's formula: ee=cos e ising 口a()=Re(e-) C1-4: Time-domain analysis of Linear and Time-invariant Networks C1-s: Analysis of sinusoidal Steady-state Circuit (phasor method)o he differential equations for the circuit which has N independent dynamie (t) Analysis 8: Q: the steady- When input is x()=Aoee' i ene (e)"+an1 d'mi(n d"(oe) Fw)=I/HGw) Yw) XGw) o)"·le"=(a)yw(t) X(O)Hgo ()=L当O Y(jo)Fujo v(D)=joL·(1) v() steady-state circuit steady-state circuit m=(

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 TEM R + - V1 V2 + - C V 0 t 10τ V 0 t 0.5τ Vi(t) Vo(t) + + - - t Vi (t) Vo(t) changing dischanging 1 5τ “differential circuit” C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − I0 RI Analysis 6： 0 Overall response: Y(t)=Yzi(t)+Yzs(t) Zero-input response Yzi(t) Zero-state response Yzs(t) C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 I0 R + Vc(t) - Vc(0)=V0 0 t Vc(t) V0 t≥0+ i(t) 0 t i(t) V0/R ( ) (1 ) ( ) (1 ) / 0 / 0 / 0 0 / τ τ τ τ t t c t t v t V e RI e e I e R V i t − − − − = + − = + − I0 RI Analysis 7： 0 unit-step response According to definition, let: ( ) (1 ) ( ) / i t e u t −t τ = − 0 1 2 t ( ) 2[(1 ) ( 1) (1 ) ( 2)] ( 1)/ ( 2)/ = − − − − − − − − − i t e u t e u t t τ t τ *** Response of rectangular signal or stair-step signal？ is (t) = u(t),then：I0 =1,V0 = 0 i (t) s solution 2 i (t) = 2[u(t −1) − u(t − 2)] s C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 Special solution at steady state j t i t Ae ω ( ) = j t j t j t RC j Ae Ae I e ω ω ω ω + = 0 • • R j C I A + • ω = 1 0 Is R + Vc(t) - Vc(0)=0 t≥0+ i(t) i t Is dt di t RC + ( ) = ( ) Euler's formula : ejθ =cosθ+jsinθ cos( ) Re( ) 0 0 j t Is I t I e ω let： = ω = ( ) Re( ) j t i t Ae ω = Algebra solution differential *** Analysis 8：Q: the steady￾state response of j t Is I e ω = 0 C1-4: Time-domain analysis of Linear and Time-invariant Networks 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 dv t i t C dt di t v t L ( ) ( ) ( ) ( ) = = ( ) ( ) ( ) ( ) i t j C v t v t j L i t = • = • ω ω Is R + Vc(t) - Vc(0)=0 t≥0+ i(t) i t Is dt di t RC + ( ) = ( ) ( ) ( ) ( ) 0 0 j I e j i t dt d I e dt di t j t j t = = ω • = ω • ω ω ( ) ( ) ( ) ( ) ( ) 0 0 ( ) ( ) j I e j i t dt d I e dt d i t n j t n n n j t n n = = ω • = ω • ω ω Analysis of sinusoidal steady-state circuit phasor method (complex method) C1-4: Time-domain analysis of Linear and Time-invariant Networks Analysis 8：Q: the steady￾state response of j t Is I e ω = 0 differential 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 + + + + = − − − 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: C1-5: Analysis of sinusoidal Steady-state Circuit (phasor method) 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)

C1-4: Time-domain analysis of Linear and Time- invariant Networks (Analysis of First Order Circuit) a dynamic state and steady state typical source signals(stimulating definition of initial state (initial value, initial Time-domain analysis of dynamic C1-5: Analysis of sinusoidal Steady-state Circuit (Complex Solution to Linear and Time-invarant CComplex method and phasor method, impedance and admi The complex form of elements, law and theorem The power of sinusoidal steady-state (self-study The stability of networks, 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 Coming next … C1-4: Time-domain analysis of Linear and Time￾invariant Networks (Analysis of First Order Circuit)  dynamic state and steady state typical source signals (stimulating signal) definition of initial state (initial value, initial conditions) Time-domain analysis of dynamic circuits 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 form of elements, law and theorem The power of sinusoidal steady-state (self-study) The stability of networks, transfer function