北京大学：《电路分析原理 Circuit Analysis》课程教学课件（英文版）第七章链式网络与传输线第三节均匀无耗传输线

团购合买资源类别：文库，文档格式：PDF，文档页数：8，文件大小：945.67KB

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第？讲：复习北京大学北京大学《Principles of Circuit Analysis》 Chapter 7: Uniform Lossless Transmission Line (Chain Network and Transmission Line) Lecture 3 2009.12.08 Interest Focus Persistence Originality 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 §7-1 Transmission Line: Some history §7-2 Chain Network and Transmission Line (conceptions) §7-3 Uniform Lossless Transmission Line (key points) (key points) natural impedance, transmission constant incident wave, reflected wave, reflectance terminal: open, short, matching §7-4 step response of transmission line Chapter 7: Transmission Line Properties of wave along the line Time-domain 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 Z Assuming R,G,L,C are the parameters of unit length *** -kZ kZ -kZ kZ e Z V e Z V I(z) V(z) V e V e + + − + − + = − = + C C (R j L)(G j C) k α jβ = + ω + ω = + attenuation constant Phase constant 2 Definition of wavelength : π λ β = Review 1 v ω β = = LC The velocity of wave in transmission line: Chain circuit → Transmission Line →Uniform transmission line Wavelength: The unit length of wave which is transmits on the transmission line. Definition 1: transfer constant G j C R j L - I V I V ZC ω ω + + = = = − − + + Definition 2: natural impedance 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 k = jβ 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 − − = + − = + + Review 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: reflection on the line *** ZL x x k ZC k ZC x k ZC k ZC x V(t)+ - incident wave reflected wave incident wave reflected wave absorption Transmissive wave incident wave reflected wave Transmissive wave incident wave reflected wave Voltage/current reflection coefficient: reflected wave incident wave ρV I (x) = =−ρ (x) Review ρV(x) ≤1 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 *** x - + V1 - + V2 I1 I2 k, Zc ZL Zin x 0 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 − − = + − = + + φ β β − = = + − = = + L C j V V L C L C -j2x -j2 x V V L C Z Z ρ (0) ρ (0) e Z Z Z Z ρ (X) e ρ (0)e Z Z voltage reflection coefficient: Uniform lossless transmission line: reflectance = reflection/incidence incident wave reflected wave Review

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 *** x - + V1 - + V2 I1 I2 k, Zc ZL Zin x 0 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 − − = + − = + + β β β β ββ + + + + LC LC j x -j2 x j x 1 V2 V 2 L L LC LC j x -j2 x j x 1 V2 V 2 c c ZZ ZZ V = [1+ρ (x)]V e = [1+ρ (0)e ]V e 2Z 2Z ZZ ZZ I = [1-ρ (x)]I e = [1-ρ (0)e ]I e 2Z 2Z Uniform lossless transmission line: reflectance = reflection/incidenceReview 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 − − = + − = + + 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 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 − − = + − = + + kx - ZC + V1 - + V2 I1 I2 ZC Zi =Zc *** Zc k ZC Zi x Zi Zi =ZC Zc k ZC x incident wave incident wave No reflected wave No reflected condition: Zi =ZC=ZL Uniform lossless transmission line: terminal open, short, matched 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 x 稳态：电压和电流 0 V(x) I(x) kx - ZC + V1 - + V2 I1 I2 ZC ZL = ZC k ZC x *** x β I(x), V(x) β = = j x 1 2 j x 1 2 V eV I eI t I(t), V(t) Signals at any point along the line Æ Signals at any point along the line at any timeÆ Uniform lossless transmission line: terminal open, short, matched Voltage and current 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 Zc k ZC Zi x Zi Zi =ZC Zc k ZC Zi x Zi Zi =ZC β β = = j x 1 2 j x 1 2 V eV I eI e.g.: =ZC2 =ZC2 ZC1 ZC2 ZC2 Zi 3λ+ λ/2 =? l L C L C i C jZ tg x Z Z jZ tg x Z Z + + = β β Uniform lossless transmission line: terminal open, short, matched 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 ZC1 ZL≠ ZC1 L 2 C2 C2 L L C2 i C2 Z Z 4 λ λ 2 Z jZ tg 4 λ λ 2 Z jZ tg Z Z = + + = π π Introducing a λ/4 transmission line between the two ZC1 ZC2 ZL Zi For impedance matched, we need: ZC2 = ZC1ZL Application example of uniform lossless transmission line: λ/4 impedance converter L C L C i C jZ tg x Z Z jZ tg x Z Z + + = 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 − − = + − = + + 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 k ZC x 1 ρ (0) Z Z Z Z V V ρ (0) I L C L C V = = − + − = = + − I I I 0 V V V 2V = + = = + = + − + − + V(0): max I(0): min t t V(t) I(t) *** incident wave Æ reflected wave Æ synthesis wave Æ Uniform lossless transmission line: terminal open, short, matched 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 Input impedance x 0 open short open short open short open capacitance inductance capacitance inductance capacitance inductance capacitance λ_ 4 λ_ 2 Z jZ ctg x Z i C L = − β → = ∞ x Standing-wave phenomenon of wave phenomenon of voltage and current 0 V(x) I(x) λ_ 4 λ_ 2 Uniform lossless transmission line: terminal open, short, matched 1 2 c 1 2 I (x) jsin xV Z V (x) cos xV 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 slotted line probe Using the standingwave phenomenon to measure wavelength and ZL slotted line Using the standingwave phenomenon to measure wavelength and ZL Uniform lossless transmission line: terminal open, short, matched 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 − − = + − = + + 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 ZL =0 k ZC x *** V(0): min I(0): max 1 ρ (0) Z Z Z Z V V ρ (0) I L C L C V = − = − + − = = + − + − + + − = + = = + = I I I 2I V V V 0 x x t t I(t) V(t) Standing-wave phenomenon |I(x)| |V(x)| Uniform lossless transmission line: terminal open, short, matched

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 x 0 V(x) I(x) λ/2 λ/4 Steady state: input impedance x 0 open short open short open short inductance capacitance inductance capacitance inductance capacitance short inductance Z jZ tg x Z i C L = β → = 0 Steady state: Steady state: Standing-wave phenomenon of wave phenomenon of voltage and current Uniform lossless transmission line: terminal open, short, matched 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 e.g.: ZL k ZC x λ/4 λ/4 λ/4 ZZin in =?=? =ZL =ZL Uniform lossless transmission line: terminal open, short, matched ZC=75 ohm x f=600MHz ZZin in ÆÆ0.7589uH 0.7589uH Shortest length?=? Shortest length?=? =0.123m =0.123m e.g.: 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 Tea break！ Tea break！ Homework: 7-7,10,12,14 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 Vi (t) ZC τ + - ρ (0) -1 ρ (0) 1 V V = Terminal open: = Terminal shorted: ZL=∞ e.g.: τ V+ V- 2τ 3τ V+ V- V0(t) …… Application example of uniform lossless transmission line: time delay, impulse response Analysis: V0(t)=? t t t Isn’t it interesting? Vi (t) A short pulse t 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 Uniform lossless transmission line: R=G=0, so: 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 ⋅ = Complex method Laplace Transform: j Laplace Transform: jωωÆÆSS jωτ ⇒ sτ Known time delay Known length Definition 1: transfer constant Definition 2: natural impedance 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 step response of transmission lines *** Complex method → solving the sinusoidal steady-state response on the transmission line (send the sine wave signal): Laplace transform → solving the any response on the transmission line (send any signal): 2 j x L c -1 2 L j x L c -1 V1 (2ZL) (Z Z )e V (2Z ) (Z -Z )e V β − β = + + 2 x L c -1 2 C x L c -1 I1 (2ZC) (Z Z )e I - (2Z ) (Z -Z )e I jβ − jβ = + 2 j L c -1 2 L j L c -1 V1 (2Z L ) (Z Z )e V (2Z ) (Z - Z )e V ωτ − ωτ = + + L c 2 -1 L c 2 C -1 I1 (2Z C ) (Z Z )e I - (2Z ) (Z - Z )e I jωτ − jωτ = + L c 2 -1 L c 2 L -1 V1 (2Z L ) (Z Z )e V (2Z ) (Z - Z )e V sτ − sτ = + + L c 2 -1 L c 2 C -1 I1 (2Z C ) (Z Z )e I - (2Z ) (Z - Z )e I sτ − sτ = +

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 τ ρ ρ s V s V -2 -2 i C 1 (0)e 1 (0)e Z Z − + = ωτ ωτ β β ρ ρ ρ ρ β β -2j -2j -j2 x C -j2 x C L C L C i C 1 (0)e 1 (0)e Z 1 (0)e 1 (0)e Z jZ tg x Z Z jZ tg x Z Z V V V V − + = − + = + + = step response of transmission lines Complex method → solving the sinusoidal steady-state response on the transmission line (send the sine wave signal): Laplace transform → solving the any response on the transmission line (send any signal): 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 *** 1.Laplace transform → get the operational form of any response on the transmission line Vi (t) ZC k τ或x + - ZL V0(t)=? Vi (s) ZC sτ + - ZL V0(s)=? V0(t)=? step response of transmission lines 2. Get the time-domain representation of any response on the transmission line using Inverse Laplace Transform 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 Vi (t) ZC τ It is a short pulse signal. + - ρ (0) -1 ρ (0) 1 V V = = ZL=∞ e.g.: τ V+ V- 2τ 3τ V+ V- Solution 1： V0(t)=? V0(t) …… τ Application example of uniform lossless transmission line: time delay, impulse response Terminal open: Terminal shorted: 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 ZC τ + - ZL=∞ V0(t)=? v t u t V s s v t u t u t i i i ( ) ( ) ( ) 1/ ( ) ( ) ( ) = → = = − − Δ L c 2 -1 L c 2 L -1 V1 (2Z L ) (Z Z )e V (2Z ) (Z - Z )e V sτ − sτ = + + V1 e V2 e V2 sτ − sτ = + 2 1 2 1 s(1 e ) 2e (e e ) 2V V 1 2 τ τ τ τ s s s s − 2 − − + = + = Using a expansion formula: ∑ ∞ = + = + k 1 x x k 1/(1 ae ) 1 (-ae ) Application example of uniform lossless transmission line: time delay, impulse response Vi (t) It is a short pulse signal. e.g.: Solution 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 ZC τ + - ZL=∞ e.g.: V0(t)=? v t u t V s s v t u t u t i i i ( ) ( ) ( ) 1/ ( ) ( ) ( ) = → = = − − Δ [1 (-e ) ] s 2e s(1 e ) 2e (e e ) 2V V 1 k 2 ∑ ∞ = − − − − − = + + = + = 1 2 2 k s s s s s s τ τ τ τ τ τ ... 3 5 7 9 s 2e s 2e s 2e s 2e s 2e V2 − sτ − sτ − sτ − sτ − sτ = − + − + (t) = 2u (t − τ ) − 2u (t − 3τ ) + 2u (t − 5τ ) − 2u (t − 7τ ) + 2u (t − 9τ )... V2 Application example of uniform lossless transmission line: time delay, impulse response Vi (t) It is a short pulse signal. Solution 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 ZC τ + - ZL=∞ V0(t)=? v t u t V s s v t u t u t i i i ( ) ( ) ( ) 1/ ( ) ( ) ( ) = → = = − − Δ 2 () 2 ( ) 2 ( 3 ) 2 ( 5 ) 2 ( 7 ) 2 ( 9 )... v t ut ut ut ut ut τ τ τ τ τ = −− − + −− −+ − τ V2(t) …… τ V2(t-∆) V0(t) …… τ Æ ( ) ( ) ( ) v 0 t = v 2 t − v 2 t − Δ Application example of uniform lossless transmission line: time delay, impulse response e.g.: Vi (t) It is a short pulse signal. Solution 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 u(t) ZC τ + - ZL=∞ τ V+ V- 2τ 3τ V+ V0(t)=? t t V0(t) V- V+ V- Solution 3: Get the unit step response using the terminal’s reflectance Application example of uniform lossless transmission line: time delay, impulse response e.g.: ρ (0) -1 ρ (0) 1 V V = Terminal open: = Terminal shorted: 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 Z C Z L =3Z C V (t) + - τ t V 0 (t) 10us 0 x=0 8 V x = l + - Solution 2: Determine the waveform of output signal Vo(t) (04G) ( ) 2 1 3 1 3 1 0 = + − = + − = L c L c v Z Z Z Z ρ 1 1 oc c v oc c ZZ Z ZZ Z ρ − − = = =− + τ = 20μs 2τ 4 -4 τ V+ V- 3τ V+ V- V0(t) …… τ 8 4 -4 -2 12 -6 3 t t 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解：1) 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 ( ) 1 300 300 0 = ∞ + ∞ − = + − = L c L c v Z Z Z Z ρ 1 0 300 1 0 300 o c v o c Z Z Z Z ρ − − = = =− + + ( )s v l τ 1 μ 3 10 300 8 = × = = 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 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. Solution: 2) x l O u(t) + - 1' 2' 1 2 ( ) ( ) ∑( ) +∞ = − − − − + − − = − + + = − + = 0 3 5 (2 1) 2 2 1 2 2 1 2 k s s s k s k s s e s e e e e s e s V s τ τ τ τ τ τ L v t u t V s s i i ( ) = ( ) → ( ) =1/ (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 − − = + = + + v t u t V s s i i ( ) = ( ) → ( ) =1/ (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 − − = + = + + 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 () ( ) ( ) ( ) [ ] ∑( )( ) +∞ = = × − − − = − − − + − + 0 2 2 1 2 2 3 5 k k u t k V t u t u t u t τ τ τ τ τ L v t u t V s s i i ( ) = ( ) → ( ) =1/ L c 2 -1 L c 2 L -1 V1 (2Z L ) (Z Z )e V (2Z ) (Z - Z )e V sτ − sτ = + + 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 Solution: 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: 3) x l O u(t) + - 1' 2' 1 2 τ τ τ 4 1 4 3 = + 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 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. the voltage waveform of the position within 0 ~ 5μs, which is 75 m away from the end terminals

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

点击进入文档下载页（PDF格式）

已到末页，全文结束

点击下载（PDF格式）

浏览记录