Pulses in transmission lines

Pulses in transmission lines Physics 401, Fall 2018 Eugene V. Colla

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

1.Networks with distributed parameters 2.Propagation of pulses in transmission lines 3.Impedance matching 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 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 5

Coaxial cable Courtesy Analog Devices Twisted line Courtesy Wikipedia Twin lead Courtesy Wikipedia 6

Specification: Impedance: 53 Ω Capacitance: 83 pf/m Conductor: Bare Copper Wire (1/1.02mm) 7

reflected r out i(x,t) V(t) V(x,t) Z x V 0 forward 8

Wavetek 81 Tektronix 3012B Sync output Triggering input Signal output RG8U oad 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 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 2 2 11

i x V x 2 2 i C t 2 2 i dt V x 2 2 V C t 2 2 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 12

C = capacitance per unit length Z k = = inductance per unit length C ε =8.854 10 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 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) 3.33610 e ( s / m) 3.336 e ( ns / m) Inner Insulation Materials: Polyethylene Nominal Impedance: 52 ohm Delay time ~5ns/m r r 14

reflected... Z V x solution for the traveling in opposite direction V C t 2 2 2 2 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 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 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 = 2V i (on the load) Incident pulse Reflected pulse End of the line 17

Attenuation (db per 100 feet) Theory: R= Vr = Vi MHz 30 50 100 146 150 RG-58U 2.5 4.1 5.3 6.1 6.1 Experiment RG 58U Vi incident reflected Vr V ATTN (db ) 20 log i Vr Important parameter for cable is attenuation per length 9/17/2018 Spring 2016 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 1847 1922) ( 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 19

In our case: Attn(200 ft ) 20 log 4.18 1.46dB 3.54 Where it is coming from? Ri i Ci 9/17/2018 Spring 2016 Gi+1 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 9/17/2018 3.932 Attn(200 ft ) 20 log 0.335dB 3.78 Dielectric ø=7.2 mm Spring 2016 21

Reflected pulse does not follow the shape of the incident pulse RG-58U Frequency dependence of the attenuation RG-58U cable 9/17/2018 Spring 2016 22

FFT Spectrum correction Incident pulse spectrum reflected pulse spectrum IFFT 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 3.0 2.5 2.0 Incident pulse V(V) 1.5 1.0 0.5 0.0-0.5-1.0-1.5-2.0-2.5 Reflected pulse -200 0 200 400 600 800 time (ns) 24

Vi Vr R Vi Vr Zk Resistive load Z=R R Z k V Vi r or R Z k 1.5 0.5 24.8 179.9 100.2 149.6 50 0.8 0.6 0.4 Incident pulse 0.0-0.5-1.0-1.5 0 200 400 time (ns) 1.5 0.2 1.0 0.5 0.0 V(V) V (V) 1.0 V(V) Matching the load impedance R Zk; Vr 0 Incident pulse -0.2 0.0-0.5-1.0 0 100 200 300 400 500 600 700 800-1.5 time (ns) Reflected pulse 0 200 400 time (ns) 9/17/2018 Spring 2016 End of the line 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 (1821-1894) R 2 R 3 éon Charles Thévenin (1857 1926) r R 1 R 4 + E 1 E 2 - - + e + - 26

V V V i R i i i r i V i r i Zk V Z r k i R 2V i 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 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 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 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 30

i V 0 Z 2V i Z k Z k di 2V i iz k ; dt t i i0 1exp ; Z k 20 10 0 20 10 0-10 20 10 0 V i V V r = V - V i -10-10 0 10 20 30 40 50 60 time 31

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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 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 1 1.5 x i 34

Derive resonance frequency f from measured inductance ± and capacitance C± C f 0 1 1 (, C) 2 C 10 1mH, C 10 2μF 1 1 2 2 f f f (, C,, C) C C 2 2 f f C 1 C 4 1 3 2 2 1 C 4 1 3 2 2 ; Results: f( 1,C 1 )=503.29212104487Hz f=56.26977hz f( 1,C 1 )=503±56Hz 35

10 1mH, C 10 1μF Where these numbers are coming from? 1 1 1. 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% 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: http://en.wikipedia.org/wiki/evenberg%e2%80%93marquardt_algorithm K. evenberg. A Method for the Solution of Certain Non-inear Problems in east Squares.The Quarterly of Applied Mathematics, 2: 164-168 (1944). 37 i

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 38

Transmission line. Unknown load simulation ocation: \\engr-file-03\phyinst\ap Courses\PHYCS401\ab Software And Manuals\abSoftware\Transmission lines 39