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# Class Note for ECE 3317 with Professor Jackson at UH

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This 38 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 20 views.

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Date Created: 02/06/15

ECE 3317 Prof David R Jackson Notes Lines Time omain Disclaimer Transmission lines is the subject of Chapter 6 in the book However the subject of wave propagation in the time domain is not treated very thoroughly there Therefore the material from Notes 6 is not taken from the book A transmission line is a twoconductor system that is used to transmit a signal from one point to another point Two common examples coaxial cable twin line A transmission line is normally used in the balanced mode meaning equal and opposite currents and charges on the two conductors Here s what they look like in reallife coaxial cable twin line Another common example for printed circuit boards microstrip line 52 LED EZZSSSI microstrip line USB Hub Ports Ne01973 Debug Port Some practical notes Coaxial cable is a perfectly shielded system no interference Twin line is not a shielded system more susceptible to noise and interference Twin line may be improved by using a form known as twisted pair 5 239 twin line At coax I symbol Z Note We use this schematic to represent a general transmission line no matter what the actual shape of the conductors These are per unit length parameters C capacitancelength Fm L inductancelength Hm R resistancelength Qm G conductancelength Sm capacitance between the two wires inductance clue to stored magnetic energy resistance due to the conductors conductance clue to the filling material between the wires Z 39 39 AZ I I Cir Quit Model Ta conductivity of dielectric Sm 0m conductivity of metal Sm c meg Fm G We Sm b b 1n 1n a a L 2 lm 3 Hm R 1 1 1 Qm 27 a am 27m 27 Overview of derivation conductance per unit length 590d RC Analogy C gt G C 2728 Relation Between L and C C 27 5 0 3 Fm L 2 lm a Hm b 27 1n a Hence LC 2 2 This is true for ALL transmission lines Apply KVL and KCL laws to a small slice of line 21 RAZ LAZ IzAz z 0 M o W 1 GAZ CA2 VzAzt i Z zAZ Z l KVL Vzt Vz Azt 21RAZ LAZ 31 6VZ AZ 1 KCL IztzAztVzAztGAzCAz at Hence VQA0 Wz0 RH0 LaHL0 AZ 6t IzAzt Izt GVZAZ tCaVZAZZ AZ at NOW letAZ 9 0 Telegrapher s Equations TE R1L 82 at rm472V 82 at Take the derivative of the first TE with respect to 2 Substitute in from the second TE 3 V R L2 j 822 82 82 8t R L j 82 8t 82 R GV Ca V L Ga V Ca V 8t 8r 8f 3 V R GV Ca V L Ga V Ca V 822 8t 8r 8f Hence we have 82V 822 RGV RCLGa V LC8 V 0 t if There is no exact solution to this differential equation except for the lossless case Losse35 case a ViRGV7RCLG8 ViLC 8 V 0 822 8t 8t2 8V 7 LC 8V o 822 8t2 Note The current satisfies the same differential equation Solution VZt fZ cdtgzcdt where f and gate arbitrary functions This is called the D Alembert solution to the Telegrapher39s equation the solution is in the form of traveling waves General solution Vzt fz cdt gZCdt 8 29 fquotZ Cdf gquotzcdl Z a Vagfquot ch fquotz cdtc gquotzcdl It is seen that the differential equation is satisfied by the general solution Example fZ I Z Z0 VZt fz cdt II II VZ7l IZO l2l1gt0 l2l2gtfl i I I I I I I I I I I I Loss causes an attenuation in the signal level and it also causes distortion the pulse changes shape and usually gets broader II Vzz 20 zzlgt0 ll2gtl1 i 1 ii gtZ I These effects can be studied numerically fi rst I IE Our goal is to how solve for the our reht Oh the Iihe Assume the followihg forms 1Zt uZ Cat I VZ I Cdt The derivatives are 3VZt f39Z cdt I g39Z I cdt Cd u39Z cdt I cd V39Z I cdt This becomes f39z cdt g39z cdt L cd u39z cdt cd v39z cdt Equating like terms we have f39z 661 L cd u39z 631 g39z cdt L Cd v39z 051 Hence we have Observation about term LCdZLJL8LZCjg Define characteristic impedance ZO L ZO The units of ZO are Ohms C uztZiofzt vztZigzt 0 Then General solution Vzt fz cdtgzcdt Izt 22fz Cdt gzcdt For a forward wave the current waveform is the same as the voltage but reduced in amplitude by a factor of Zo For a backward traveling wave there is a minus sign as well pictut quotFor d fQIWdId tl dvglirig Wave VZt fz icdz 1 1Zt Ofzicdt The minus sign arises from the reference direction for the current 770 0 intrinsic impedance of free space 8 0 80 i 88541878x103912 Fm yo47rx10397 Hmi exact 770 3767303 Q a radius of Wires These are the common values used for TV 75300 Q transformer 300 Q twin line 75 Q coax twin line coaxial cable parallelplate formulas w ngogr wgtgth h h Lzy0 wgtgth w I strip thickness More accurate CAD formulas Z Z 1207 hgt1 0 W w h1393066711w h1444 w gf gr18r1 1 gr 1X h whzl 2 2 112hw 46 wh w z wL llnD 7r 2 Note the effective relative permittivity accounts for the fact that some of the fields are outside of the substrate in the air region The effective width w accounts for the strip thickness Transmissionline theory is valid at any frequency and for any type of waveform assuming an ideal transmission line Transmissionline theory is perfectly consistent with Maxwell39s equations although we work with voltage and current rather than electric and magnetic fields Circuit theory does not view two wires as a quottransmission linequot it cannot predict effects such as single propagation distortion etc

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