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1 Lectue 2 Date: Definition of Some TL Paametes Examples of Tansmission Lines

2 Tansmission Lines (contd.) Fo a lossless tansmission line the second ode diffeential equation fo phasos ae: LC 2 d I 2 d V(z) 2 V( z) 2 dz (z) 2 I( z) 2 dz V + and V ae complex constants ( ) V z V e V e j z jz Similaly the cuent phaso fo a lossless line can be descibed: 1 dv ( z) 1 d I( z) V e V e jl dz jl dz I( z) V e V e L jz jz jz jz Gives the Definition of Chaacteistic Impedance

3 Tansmission Lines (contd.) Z L L L LC C V V I( z) e e Z Z jz jz Completely Dependent on L and C Chaacteistic Impedance fo a Lossless Line is Real Opposite Signs in these Tems Gives a Clue about Cuent Flow in Two Diffeent Diections The time dependent fom of the voltage and cuent along the tansmission line can be deived fom phasos as: jt j( zt ) j( zt) v( z, t) Re V ( z) e Re V e V e V V i( z, t) Re I( z) e Re e e Z Z jt j( zt ) j( zt)

4 Tansmission Lines (contd.) Fo the simple case of V + and V being eal, the voltage and cuent along the tansmission line can be expessed as: v( z, t) V cos( t z) V cos( t z) V V i z t t z t z Z Z (, ) cos( ) cos( ) V cos( t z) V cos( t z) Wave Functions Let us examine the wave chaacteistics of v ( z, t) V cos( t z) 1

5 Tansmission Lines (contd.) Fo fixed position z and vaiable t: V + v z t 1, Fo fixed time t and vaiable position z V + v z t 1, V + π ωt t=t =2π 2π ωt V + π 2π βz z=λ =2π βz We can deduce: t 2 tt We can deduce: z 2 z Time Peiod of Wave 2 1 T f 2 Wavelength

6 Tansmission Lines (contd.) What is the physical meaning of β φ Let us conside once again: V cos( t z) Appaently β epesents the elative phase of this wave function in space (ie, function of tansmission line position) In pinciple, the value of β must have units of (φ/z) Radians/mete Theefoe, if the values of β is small, we will need to move a significant distance z down the tansmission line in ode to obseve a change in the elative phase of the oscillation Convesely, if the value of β is lage, a significant change in elative phase can be obseved if taveling a shot distance z down the tansmission line

7 Tansmission Lines (contd.) Fo example, in ode to obseve a change in elative phase of 2π, the distance z is: 2 ( z z ) ( z) z z2 2 z 2 λ: Wave Length Can t we call it spatial countepat?

8 Tansmission Lines (contd.) v z t Fo vaiable position z and vaiable time t 1, t = t 1 t = t 2 z = z 1 z = z 2 v = velocity z [m] It is appaent that the phase of both these ae identical and hence: v ( z, t ) v ( z, t ) Speed of Popagation z t z t cos( z t ) cos( z t ) z t z t Phase Velocity (v p ) v p LC 1 LC

9 Tansmission Lines (contd.) Simplified Expession fo Wavelength: i.e, the wavelength is the distance taveled by the wave in a time inteval equal to one peiod Let us examine this expession: 2 1 z t vp vt p LC f z t t 2 > t 1 and ω β is a positive quantity this implies that z 2 z 1 must be positive o z 2 > z 1 It ensues that the point of constant phase moves towads ight (i.e, towad the load in the tansmission line) In othe wods, the wave function V + cos(ωt βz) epesents a taveling wave moving at a velocity v p towads the load This wave is called outgoing wave when seen fom the souce and incident wave when viewed fom the load

10 Tansmission Lines (contd.) Similaly, the analysis of V cos(ωt + βz) will show that this function epesents a taveling wave at a velocity v p to the left (i.e, towads the souce in a tansmission line) This wave is called incoming wave when seen fom the souce and eflected wave when viewed fom the load V + e jβz is called incident wave (phaso fom) and V e jβz is called eflected wave (phaso fom) In geneal, the voltage and cuent on a tansmission line is composed of incident and eflected wave The quantity βz is known as electical length of the line Theefoe: V ( z) V ( z) V ( z) V e V e j z jz V jz V jz V ( z) V ( z) I( z) e e Z Z Z

11 Chaacteistic Impedance (Z ) The chaacteistic impedance is defined as : Z = (incoming voltage wave) / (incoming cuent wave) = (outgoing voltage wave) / (outgoing cuent wave) Fo a geneic tansmission line: Z The incoming and outgoing voltage and cuent R jl waves ae position dependent the atio of G jc voltage and cuent waves ae independent of position actually is a constant an impotant chaacteistic of a tansmission line called as Chaacteistic Impedance Z is not an impedance in a conventional cicuit sense definition is based on the incident and eflected voltage and cuent waves this definition has nothing in common with the total voltage and cuent expessions used to define a conventional cicuit impedance Its impotance will be appaent duing the couse of this COURSE!!!

12 Example 1 A plane wave popagating in a lossless dielectic medium has an electic field given as E x = E cos(ωt βz) with a fequency of 5. GHz and a wavelength of 3. cm in the mateial. Detemine the popagation constant, the phase velocity, the elative pemittivity of the medium, and the intinsic impedance of the wave. The popagation constant: 2 The phase velocity: v p 2 f f m.3 vp m/ sec Lowe than the speed of light in fee medium

13 Example 1 (contd.) Relative pemittivity of the medium: v p c c v p Chaacteistic impedance of the wave: wave 377 wave

14 Line Impedance (Z) Hey, I know what this is! The atio of incoming voltage to incoming cuent wave. Right? NO!

15 Line Impedance (Z) contd. Actually, line impedance is the atio of total complex voltage (incoming + outgoing) wave to the total complex cuent voltage wave. Z( z) V( z) I( z) V ( z) V ( z) V ( z) V ( z) Z Z In most of the cases Howeve, the line and chaacteistic impedance can be equal if eithe the incoming o outgoing voltage wave equals ZERO! Say, if V z = then: Z( z) V ( z) V ( z) V ( z) V ( z) Z Z

16 Line Impedance (Z) contd. It appeas to me that Z is a tansmission line paamete, depending only on the tansmission line values R, L, C and G. Wheeas, Z(z) depends on the magnitude and the phase of the two popagating waves V + z and V z values that depend not only on the tansmission line, but also on the two things attached to eithe end of the tansmission line. Right? Exactly!!!

and opposite cuents (and chages) on the two conductos.

17 Example of Tansmission Lines Two common examples: coaxial cable z a b twin line A tansmission line is nomally used in the balanced mode, meaning equal and opposite cuents (and chages) on the two conductos. Hee s what they look like in eal-life: coaxial cable twin line coax to twin line matching section

18 Example of Tansmission Lines (contd.) C Twin Line a = adius of wies d d L 2a 1 d cosh 2a 1 cosh H/m Z cosh 2 d a F/m 1 2 x cosh ln 1 ln 2 x x x x Z 1 1 ln a d d a

19 Example of Tansmission Lines (contd.) Coaxial Cable d = conductivity of dielectic [S/m]. z a b m = conductivity of metal [S/m]. C L 2 b ln a b 2 a ln F/m H/m G R 2 d b ln a S/m /m 2 a 2 b m 2 m (skin depth of metal)

20 Example of Tansmission Lines (contd.) Anothe common example (fo pinted cicuit boads): w h micostip line Gound plane helps in peventing the field leakage and thus educes the adiation loss

21 Micostip Line (contd.) The seveity of field leakage also depends on the elative dielectic constants ε. Magnetic Field Lines Electic Field Lines It is appaent that the adiation loss could be minimized by using substates with high dielectic constants Altenative appoaches to educe adiation loss and intefeence ae shielded micostip line and multi-laye boads

22 Micostip Line (contd.) micostip line

23 Micostip Tansmission Lines Design h t w Simple paallel plate model can not accuately define this stuctue. Because, if the substate thickness inceases o the conducto width deceases then finging field become moe pominent (and theefoe need to be incopoated in the model). Case-I: thickness (t) of the line is negligible Fo naow micostips ( w h 1): Z f h w Z ln 8 2 w 4h Z / 377 wave impedance in fee space Whee, f eff eff 1 1 h w w h 1/2 2 Effective Dielectic Constant

24 Micostip Tansmission Lines Design (contd.) Fo wide micostips w h 1 : Z eff Z f w 2 w ln h 3 h Whee the effective dielectic constant is expessed as: eff 1 1 h w 1/2 The two distinct expessions give appoximate values of chaacteistic impedance and effective dielectic constant fo naow and wide stip micostip lines these can be used to plot Z and ε eff as a function of w h.

25 Micostip Tansmission Lines Design (contd.) Fo a desied chaacteistic impedance using known substate, the dimension w/h can be identified fom this cuve

26 Micostip Tansmission Lines Design (contd.) Once the line dimensions ae known, the effective dielectic constant can be identified

27 Micostip Tansmission Lines Design (contd.) The effective dielectic constant (ε eff ) is viewed as the dielectic constant of a homogenous mateial that fills the entie space aound the line. Theefoe: v p c f f Speed of Light eff Fee Space Wavelength eff The wavelength in the micostip line fo W h. 6 is: The wavelength in the micostip line fo W h. 6 is: ( 1) W / h ( 1) W / h 1/2 1/2

28 Micostip Tansmission Lines Design (contd.) In some specifications, wavelength is known. In that case following cuve can be used to identify the w/h atio. It is a good appoximation at lowe micowave fequencies. Howeve, at highe micowave fequencies this assumption is no moe valid.

29 Micostip Tansmission Lines Design (contd.) If Z and ε is specified o known, following expession can be used to detemine w/h: A w 8e Z Fo w/h 2: 2 A Whee: A 2.23 h e 2 Z 2 1 Fo w/h 2: f w B 1 ln(2b 1) ln( B 1).39 h 2 Whee: Z B 2Z Case-II: thickness (t) of the line is not negligible in this scenaio all the fomulas ae valid with the assumption that the effective width of the line inceases as: t 2x weff w 1 ln t f Whee x = h if w > h 2π o x = 2πw if h 2π > w > 2t

30 Example 2 A micostip mateial with ε = 1 and h = 1.16 mm is used to build a naow tansmission line. Detemine the width fo the micostip tansmission line to have a chaacteistic impedance of 5Ω. Also detemine the wavelength and the effective elative dielectic constant of the micostip line. Using the Fomulas: Let us conside the fist fomula: w h A 8e 2 A e 2 Z A Z f A Theefoe: w h e 8e (2.1515) Now: h = 1.16 mm =.116 cm =.116(1/2.54) mils = 4 mils w.9563* 4mils 38.2mils

31 Example 2 (contd.) ( 1)( w/ h) 1/ (1 1)(.9563) /2.387 v p c f f eff eff 2 eff 1 eff

32 Example 2 (contd.) Using the Design Cuves 1 Z 5 w h 1 h = 1.16 mm = 4 mils => w = 4 mils

33 Example 2 (contd.) Using the Design Cuves TEM eff w h 1

34 Example 3 a. Using the design cuves, calculate W, λ, and ε eff fo a chaacteistic impedance of 5Ω using RT/Duoid with ε = 2.23 and h =.7874 mm. b. Use design equations to show that fo RT/Duoid with ε = 2.23 and h =.7874 mm, a 5Ω-chaacteistic impedance is obtained with W h = Also show, ε eff = 1.91 and λ =.7236λ. W h 3.1 W=3.1 h = = 2.44mm

35 Example 3 (contd.) W Fo h 3.1 and ε = 2.23 λ λ TEM = 1.8 We know: λ = 1.8λ TEM λ TEM = λ ε λ =.723λ λ = λ Also: ε eff = 1.91 ε eff

36 Example 3 (contd.) Fo w/h 2: Theefoe: w B 1 ln(2b 1) ln( B 1).39 h 2 w B 1 ln(2b 1) ln( B 1).39 h Whee: Z B 2Z f Whee: Fo B W h. 6: w h ( 1) W / h (2.23 1) /2 1/2.724

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 8 Due on Oct. 19, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes.