example of uniform lossless transmission li time delay, impulse response time delay, impulse response v.Y(e"y2-e":°-ax。-y L. Evaluate the voltage reflection coefficient of the end terminals (2-2)P 0)and the alculate the time the transmission lin 1+∑(e-") t)=[(-t1)+l(t-(z+r2) T=T+T=1+ (t-(3-r)+l(-(3r+3)]+ Application example of uniform lossless transmission lin Uniform lossless transmission line time delay impulse response jB V2=ZI V, 八 Zeshi chex人2 z,chkx+zcshkx +jzctg Z, shox+ Scchk记tgx+zc [(ZL+Z)e (z1-2ev2J/ -z)er2]/2z。 Uniform lossless transmission line: terminal open, short, uniform los- transmission line Z V-z Z+jzct9 Zc I,jz,tgAx+Z R+1aLG+」a 1+只,(0)e2 k=jB=jo Z-L/C p(0)= V=[(z1+z)ev2+(z-z)e^M2/22 [(z1+z)er2-(z1-z)e2]/22 Terminal matched:Z=ZcP(O)=0ZiN=Zc Laplace Transform: jo >Smth m0可P0=12=如 Jt→Sr Terminal open: Z P,(O)=1ZN-jzcctgAx北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 Solution: 3) 1 2 4 1 4 3 τ = τ + τ =τ +τ (e e ) 2 V V 2 x 2 2 sτ − sτ = + (e e ) 2 V V (2Z ) (Z Z )e V (2Z ) (Z - Z )e V 2 L c 2 -1 L c 2 L -1 1 L τ τ τ τ s s s s − − = + = + + (e e ) (e e ) Vx V1 τ τ τ τ s s s s − − + + = 2 2 Application example of uniform lossless transmission line: time delay, impulse response The length of transmission line is l = 300m, the natural impedance is 300Ω, the wave velocity is 3×108 m/s。 1.Evaluate the voltage reflection coefficient of the end terminals (2-2‘) ρv(0) and the voltage reflection coefficient of input terminals (1-1‘) ρv1. Calculate the time delay τ when the signals passes by the transmission line. 2.Evaluate the step response of the end terminals. 3.Get the voltage waveform of the position within 0 ~ 5μs, which is 75 m away from the end terminals. x l O u(t) + - 1' 2' 1 2 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 τ 2τ 3τ 4τ 5τ 1 2 O t Vx 4 3τ 距离2-2'端口75米处的电压波形 τ 2τ 3τ 4τ 5τ 1 2 O t Vx 4 3τ 距离2-2'端口75米处的电压波形 22 22 2 2 2 2 1 ss ss s ss s s ss s k ττ ττ τ τ τ τ τ ττ τ − − − − − − − ∞ − = + + = = + + + = + ∑ 1 x k V (e e ) (e e )e 1 V (e e ) s (1 e ) (e e )e [1 (-e ) ] s 21 2 1 1 ( ) [ ( ) ( ( ))] [ ( (3 )) ( (3 ))] ... v t ut ut ut ut τ τ τ ττ ττ = − + −+ − −− +−+ + 2 2 Application example of uniform lossless transmission line: time delay, impulse response the voltage waveform of the position within 0 ~ 5μs, which is 75 m away from the end terminals (2-2’) 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 Solution: e.g.: As shown in Fig 6-1, an ideal voltage source is connected to an ideal operational amplifier with a pair of uniform lossless transmission line. The natural impedance is Zc=75Ω. The resistances in the circuits are R1=150Ω, R2=1.5kΩ. The transmission time of signals from 1-1’ to 2-2’ is 2us. Since the output of the ideal voltage source is a 1-V and 1-us rectangular-shaped pulse which started from t=0, as shown in Fig 6-2. The switch is closed at t=4us. (It is opened before.) Please draw out the waveform of the output signal Vo from 0 to 12us. 1 75 75 2 = ∞ + ∞ − = + − = L c L c v Z Z Z Z ρ Zc 1 1' 2 2' Vs (t) 图 6-1 R1 + - R2 Vo K + - Zc 1 1' 2 2' Vs (t) 图 6-1 R1 + - R2 Vo K + - Vs (t) O t 1 μs 1V 图 6-2 Vs (t) O t 1 μs 1V Vs (t) O t 1 μs 1V 图 6-2 t (μs) Vo (V) 12 67 10 11 12 -40/9 40/3 t (μs) Vo (V) 12 67 10 11 12 -40/9 40/3 1/ 3 150 75 150 75 2 = + − = + − = L c L c v K Z Z Z Z ρ 1 0 75 0 75 2 = − + − = + − = L c L c v Z Z Z Z ρ Application example of uniform lossless transmission line: time delay, impulse response 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 V2 = ZLI2 kx - ZC + V1 - + V2 I1 I2 ZL Zin ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − 2 2 1 C C 1 1 I V Z shkx chkx chkx Z shkx I V *** k = jβ Uniform lossless transmission line x - + V1 - + V2 I1 I2 k, Zc ZL Zin x 0 L C L C C L C L C C 1 1 in jZ tg x Z Z jZ tg x Z Z shkx Z chkx Z chkx Z shkx Z I V Z + + = + + = = β β Zin changes along the transmission line by half wavelength. V and I change along the transmission line by one wavelength Summary 2 c j x 2 L c j x 1 L c 2 L j x 2 L c j x 1 L c I [(Z Z )e I (Z -Z )e I ]/2Z V [(Z Z )e V (Z -Z )e V ]/2Z β β β β − − = + − = + + 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 kx - ZC + V1 - + V2 I1 I2 ZL ZiN *** -j2 x -j2 x C L C L C C 1 1 iN 1- (0)e 1 (0)e Z jZ tg x Z Z jZ tg x Z I V Z β β ρ ρ β β v + v = + + = = = ∞ = = L L L C Z Z 0 Z Z ρ (0) 1 ρ (0) -1 ρ (0) 0 V V V = = = + = + − = V V Z Z Z Z ρ (0) - L C L C V Z -jZ ctg x Z jZ tg x Z Z iN C iN C iN C β β = = = Uniform lossless transmission line: terminal open, short, matched Terminal matched: Terminal short: Terminal open: 2 c j x 2 L c j x 1 L c 2 L j x 2 L c j x 1 L c I [(Z Z )e I (Z -Z )e I ]/2Z V [(Z Z )e V (Z -Z )e V ]/2Z β β β β − − = + − = + + Summary 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 uniform lossless transmission line *** k = jβ = jω LC Zc = L/C (R j L)(G j C) k α jβ = + ω + ω = + G j C R j L - I V I V ZC ω ω + + = = = − − + + G j C R j L - I V I V ZC ω ω + + = = = − − + + x - + V1 - + V2 I1 I2 k, Zc kx - ZC + V1 - + V2 I1 I2 β ω ω jωτ v x j x = j LCx = j ⋅ = Laplace Transform: j Laplace Transform: jωωÆÆSS jωτ ⇒ sτ summary λ π β 2 = Definition 1: transfer constant Definition 2: natural impedance Uniform lossless transmission line: R=G=0, so: Complex method Known time delay Known length kx - ZC + V1 - + V2 I1 I2