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