Pulses in transmission lines

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1 Pulses in transmission lines Physics 401, Fall 013 Eugene V. Colla

2 Definition Distributed parameters networ Pulses in transmission line Wave equation and wave propagation eflections. esistive load Thévenin's theorem eflection. Non resistive load Appendix. Error propagation

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 taen into account. Courtesy Wiipedia 3

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

5 Specification: Impedance: 53 Ω Capacitance: 83 pf/m Conductor: Bare Copper Wire (1/1.0mm) 5

6 reflected r out i(x,t) V(t) V(x,t) Z x V 0 forward 6

7 Wavete 81 Tetronix 301B Sync output Triggering input Signal output G8U oad 7

8 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 8

9 i x = -C V t V x i dt t x i tx C V t V x i xt (1) () Combining (1) and () i x C i t V x V t C 9

10 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 i( x, t) C ooing for solution V ( x, t) V0 sin t i( x, t) i0 sin t v 1 C Speed of wave propagation Z - characteristic Impedance Equivalent to Ohm s law equation x v x v 10

11 C = capacitance per unit length Z = C Cross-section of the coaxial cable = inductance per unitlength πε0εr C= (F/m) D ln d d D e r dielectric permittivity m r -magnetic permeability 1 ε = m mm 0 r D ln d -1 (F/m) -7 0 =4 10 (H/m) (H/m) Finally for coaxial cable: Z 138 D log ( Ohms ) e d r 11

12 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 G-8/U, G58U: 1 (s/m) 9 For polyethylene e r ~.5(up to 1GHz) e ( s / m) e ( ns / m) Inner Insulation Materials: Polyethylene Nominal Impedance: 5 ohm Delay time ~5ns/m r r 1

13 x reflected esistive load Z = Z forward V i V x solution for the traveling in opposite direction V C t For reflected wave V r =-Z i r At any point of the transmission line: V V V V Z i r i r or V r V ( x, t) V0 sin t i( x, t) i0 sin t Z V V V Z r i i i i r i Z V i i V x v x v V Z r 13

14 V V V V Z i r esistive load Z = i r or V r Z Z V i Open line = V r = V i Incident pulse eflected pulse End of the line 14

15 Attenuation (db per 100 feet) Theory: = V r = V i MHz G-58U Experiment G 58U incident reflected V i V r V i ATTN ( db) 0log Vr Important parameter for cable is attenuation per length 15

16 In our case: 4.18 Attn(00 ft) 0 log 1.46dB 3.54 Where it is coming from? i i C i G i+1 16

17 G-58U 4.18 Attn(00 ft ) 0 log 1.46dB 3.54 G-8U > Core ø=0.81 mm Core ø=.17 mm Dielectric ø=.9 mm 9/16/ Attn(00 ft ) 0 log 0.335dB 3.78 Dielectric ø=7. mm Fall

18 eflected pulse does not follow the shape of the incident pulse G-58U Frequency dependence of the attenuation G-58U cable 18

19 FFT Spectrum correction Incident pulse spectrum reflected pulse spectrum IFFT 19

20 V V V V Z i r esistive load Z = i r or V r Z Z V i Shorted line =0 V r = - V i Incident pulse V(V) eflected pulse time (ns) 0

21 esistive load Z = Vi Vr V V Z i r or V r Z Z V i V (V) Matching the load impedance Z ; V r 0 Incident pulse V(V) Incident pulse time (ns) -0. V(V) time (ns) eflected pulse time (ns) 1 End of the line

single series resistor r Hermann udwig Ferdinand von

22 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 ( ) 3 éon Charles Thévenin ( ) r E 1 E e + -

23 V V V i i i i r i V i r i Z V Z r i V i Z Z Vi From this equivalent equation we can find the maximum possible power delivered to V i P i Z P=P max if =Z (no reflection) 3

24 G 8U Pulse at the end of the line Vi Z incident reflected This experiment better to perform on G 8U cable because of lower attenuation =, amplitude of the pulse at the end of line is expected to be Vi, where Vi is the amplitude of the incident pulse 4

25 i V 0 V i Z Z Z di V i iz ; dt t i i0 1exp ; Z V i V V r = V - V i time 5

26 i V i Z Z di V i iz ; dt t i i0 1exp ; V 0 Z Z 50ns, =Z~. 5mH 6

27 i V i Z Z Z V 0 C Z C C 3. nf Z 7

28 8

29 1. The reports should be uploaded to the proper folder and only to the proper folder For example folder C ab eport_1 should used by students from 1 section only I would recommend the file name style as: 1_lab_student1 ab section ab number Your name. Origin template for this wee ab: \\engr-file-03\phyinst\ap Courses\PHYCS401\Common\Origin templates\transmission line\time trace.otp 9

30 limit s a serious to C circuit resistor Fitting parameters: (1) limit, () ω 0 = 1 C, (3) Q = 1 C Initial values can be estimated from circuit parameters. H 1 j limit j Q 0 limit 1 j Q j C 1 1 j Q1 0 Q 0 limit Q For fitting results and actual fitting function equation go to: \\engr-file-03\phyinst\ap Courses\PHYCS401\Common\Simple Examples\ ab 3 Frequency Domain Analysis_example.opj 30

31 Now to simplify the equation we can introduce the reduced frequency γ = ω ω 0 and transfer function components can be presented as: Q j 1 Q 1 H( ) Q ; 0 Q 1 H E Q Q 0 Q 1 ; H IM Q j 1 Q 1 0 Q 1 31

32 Two other components the modulus and the phase shift: H Q Q 0 Q 1 Q 1 1 ; arctan H H IM E arctan 1 Q 1 Q 3

33 X Y Data courtesy Tsung-in Hsieh, Physics 401, Fall

34 y = f(x1, x... xn) n f i i i i1 xi f ( x, x ) x 1.15 f(x i ) f±fx 1.10 x i ± x i x i 34

35 Derive resonance frequency f from measured inductance ± and capacitance C± C f (, C) C 10 1mH, C 10 μf 1 1 f f f (, C,, C) C C f f C 1 C C ; esults: f( 1,C 1 )= Hz f= hz f( 1,C 1 )=503±56Hz 35

10 1mH, C 10 1μF Where these numbers are

36 10 1mH, C 10 1μF Where these numbers are coming from? Using commercial resistors, capacitors, inductances C=500pF±5% =35mH±10%. Measuring the parameters using standard equipment SENCOE Z meter model C53 Capacitance measuring accuracy ±5% Inductance measuring accuracy ±% Agilent E4980A Precision C Meter Basic accuracy ±0.05% 36

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