Pulses in transmission lines

Transcription

1 Pulses in transmission lines Physics 401, Spring 2017 Eugene V. Colla

2 Definition Distributed parameters network Pulses in transmission line Wave equation and wave propagation Reflections. Resistive load Thévenin's theorem Reflection. Non resistive load Appendix. Error propagation 2

3 Transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account. Courtesy Wikipedia 3

4 Simplified equalent circuit i-1 i i+1 Ideal case C i-1 C i C i+1 R i i Real situation C i G i+1 4

5 Coaxial cable Courtesy Analog Devices Twisted line Courtesy Wikipedia Twin lead Courtesy Wikipedia 5

6 Specification: Impedance: 53 Ω Capacitance: 83 pf/m Conductor: Bare Copper Wire (1/1.02mm) 6

7 reflected r out i(x,t) V(t) V(x,t) Z x V 0 forward 7

8 Wavetek 81 Tektronix 3012B Sync output Triggering input Signal output RG8U oad 8

9 V(t) r out i(x,t) V(x,t) x dx C = capacitance per unit length = inductance per unit length CdxV dq; V q C t t i V = -C x t i; dv V x di ( dx) ; dt i dt 9

10 i x = -C V t V x i dt t x 2 i tx C 2 V 2 t 2 V 2 x 2 i xt (1) (2) Combining (1) and (2) i x 2 2 C i t 2 2 V x V t 2 2 C

11 i x V x i C t i dt V x V C t Now substituting V(x,t) and i(x,t) in We can find V i 0 0 i x C V = -C t or V ( x, t) i( x, t) Z k i( x, t) C ooking for solution V ( x, t) V0 sin t i( x, t) i0 sin t v 1 C Speed of wave propagation Z k - characteristic Impedance Equivalent to Ohm s law equation x v x v 11

12 C = capacitance per unit length Z k = = inductance per unit length C ε = Cross-section of the coaxial cable 2πε0εr C= (F/m) D ln d d D e r dielectric permittivity m r -magnetic permeability 1 0 m mm 0 r D ln 2 d -12 (F/m) -7 0 =4 10 (H/m) (H/m) Finally for coaxial cable: Z k 138 D log ( Ohms ) e d r 12

13 1 = C Speed of wave propagation = 1 c c m m e e m e e 0 r 0 r r r r 1 Delay time RG-8/U, RG58U: 1 (s/m) 9 For polyethylene e r ~2.25(up to 1GHz) e ( s / m) e ( ns / m) Inner Insulation Materials: Polyethylene Nominal Impedance: 52 ohm Delay time ~5ns/m r r 13

14 reflected... Z V x solution for the traveling in opposite direction V C t x... forward V ( x, t) V0 sin t i( x, t) i0 sin t x v x v For reflected wave V r =-Z k i r 14

15 reflected... V r =-Z k i r At any point of the transmission line: x... Z forward V R i V V V r i i i i r i Zk i V V Z r k 1. Resistive load Z =R V V R V V Z i r i r k or V r R R Z Z k k V i 15

16 V V R V V Z i r Resistive load Z =R i r k or V r R R Z Z k k V i Open line R = V r = V i and V=V i + V r Incident pulse Reflected pulse End of the line 16

17 Attenuation (db per 100 feet) Theory: R = V r = V i MHz RG-58U Experiment RG 58U incident reflected V i V r V i ATTN ( db) 20log Vr Important parameter for cable is attenuation per length 17

18 This unit was named the bel, in honor of their founder and telecommunications pioneer Alexander Graham Bell The decibel (db) is one tenth of the bel (B): 1B = 10dB. Alexander Graham Bell ) ( db) 10log P 1 10 P2 ( db) 20log V 1 10 V2 power ratio voltage (current, field ) ratio In case of our transmission line: V i ATTN ( db) 20log Vr 18

19 In our case: 4.18 Attn(200 ft) 20 log 1.46dB 3.54 Where it is coming from? R i i C i G i+1 19

20 RG-58U 4.18 Attn(200 ft ) 20 log 1.46dB 3.54 RG-8U > Core ø=0.81 mm Core ø=2.17 mm Dielectric ø=2.9 mm 2/6/ Attn(200 ft ) 20 log 0.335dB 3.78 Dielectric ø=7.2 mm Spring

21 Reflected pulse does not follow the shape of the incident pulse RG-58U Frequency dependence of the attenuation RG-58U cable 21

22 FFT Spectrum correction Incident pulse spectrum reflected pulse spectrum IFFT 22

23 V V R V V Z i r Resistive load Z =R i r k or V r R R Z Z k k V i Shorted line R =0 V r = - V i Incident pulse V(V) Reflected pulse time (ns) 23

24 Vi Vr R Vi Vr Zk Resistive load Z=R R Z k V Vi r or R Z k Incident pulse time (ns) V(V) V (V) 1.0 V(V) Matching the load impedance R Zk; Vr 0 Incident pulse time (ns) Reflected pulse time (ns) 2/6/2017 Spring 2016 End of the line 24

25 Any combination of batteries and resistances with two terminals can be replaced by a single voltage source e and a single series resistor r Hermann udwig Ferdinand von Helmholtz ( ) R 2 R 3 éon Charles Thévenin ( ) r R 1 R 4 + E 1 E e

26 V V V i R i i i r i V i r i Zk V Z r k i 2V i R Z k Z k 2Vi R From this equivalent equation we can find the maximum possible power delivered to R 2 2V i R P i R R 2 Z P=P max if R =Z k (no reflection) 2 26

27 RG 8U Pulse at the end of the line 2Vi Z k R incident reflected This experiment better to perform on RG 8U cable because of lower attenuation R =, amplitude of the pulse at the end of line is expected to be 2V i, where V i is the amplitude of the incident pulse 27

28 i V 0 2V i Z Z Z k k C Z C k C 32. nf Z k t V 1 exp T 1 T tt 1 1 V 2Vi 1exp exp T 1 28

29 i 2V i Z Z k di 2V i iz k ; dt t i i0 1exp ; V 0 Z k Z k 50ns, =Zk~2. 5mH 29

30 i V 0 2V i Z Z k Z k di 2V i iz k ; dt t i i0 1exp ; Z k V i V V r = V - V i time 30

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32 1. The reports should be uploaded to the proper folder and only to the proper folder For example folder Frequency domain analys_1 should used by students from 1 section only I would recommend the file name style as: 1_lab3_student1 ab section ab number Your name You do not need to submit two copies in pdf and in MsWord formats 2. Origin template for this week ab: \\engr-file-03\phyinst\ap Courses\PHYCS401\Common\Origin templates\transmission line\time trace.otp 32

33 y = f(x1, x2... xn) 2 n f 2 i i i i1 xi f ( x, x ) x 1.15 f(x i ) f±fx 1.10 x i ± x i x i 33

34 Derive resonance frequency f from measured inductance ± and capacitance C± C f (, C) 2 C 10 1mH, C 10 2μF f f f (, C,, C) C C 2 2 f f C 1 C C ; Results: f( 1,C 1 )= Hz f= hz f( 1,C 1 )=503±56Hz 34

35 10 1mH, C 10 1μF Where these numbers are coming from? Using commercial resistors, capacitors, inductances C=500pF±5% =35mH±10% 2. Measuring the parameters using standard equipment SENCORE Z meter model C53 Capacitance measuring accuracy ±5% Inductance measuring accuracy ±2% Agilent E4980A Precision CR Meter Basic accuracy ±0.05% 35

36 Origin uses the evenberg Marquardt algorithm for nonlinear fitting From experiment you have the array (x i,y i ) of independent and dependent variables: x i (e.g. f- frequency) and y i (e.g. magnitude of the signal) and you have optimize the vector of fitting parameters b of your model function f(x,b) in order to minimize the sum of squares of deviations: m i1 b 2 S( b) y f ( x, ) i Important point is the choice of fitting parameters. In some cases the algorithm will work with b=(1,1 1), but in many situations the choice of more realistic parameters will lead to solution For details go to: K. evenberg. A Method for the Solution of Certain Non-inear Problems in east Squares.The Quarterly of Applied Mathematics, 2: (1944). 36 i

37 Transmission line. Unknown load simulation oad parameters Function generator parameters ine characteristic impedance X-axes scaling Expected load ocation: \\engr-file-03\phyinst\ap Courses\PHYCS401\ab Software And Manuals\abSoftware\Transmission lines 37

38 Transmission line. Unknown load simulation ocation: \\engr-file-03\phyinst\ap Courses\PHYCS401\ab Software And Manuals\abSoftware\Transmission lines 38