ECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 7

Transcription

1 ECE 634 Intermediate EM Waves Fall 16 Prof. David R. Jackson Dept. of ECE Notes 7 1

2 TEM Transmission Line conductors 4 parameters C capacitance/length [F/m] L inductance/length [H/m] R resistance/length [Ω/m] G conductance/length [ /m or S/m] Ω

3 TEM Transmission Line (cont.) i Physical Transmission Line B x x x v i(,t) R L i(+,t) + v(,t) - G C + v(+,t) - 3

4 TEM Transmission Line (cont.) i(,t) R L i(+,t) + v(,t) - G C + v(+,t) - it (,) vt (,) v ( + t,) + itr (,) + L t v ( + t,) it (,) i ( + t,) + v ( + t,) G + C t 4

5 TEM Transmission Line (cont.) Hence v ( + t,) vt (,) it (,) Ri(,) t L t i ( + t,) it (,) v ( + t,) Gv( +,) t C t Now let : v i i Ri L t v Gv C t Telegrapher s Equations 5

6 TEM Transmission Line (cont.) To combine these, take the derivative of the first one with respect to : v Ri i L t v i i R L t i i R L t v R Gv C t v v L G C t t Switch the order of the derivatives. i Gv C v t Substitute from the second Telegrapher s equation. 6

7 TEM Transmission Line (cont.) v v v v R Gv C L G C t t t Hence, collecting terms, we have: v v v ( RG) v ( RC LG) LC + t t The same equation also holds for i. 7

8 TEM Transmission Line (cont.) Time-Harmonic Waves v v v ( RG) v ( RC LG) LC + t t dv d + ( ) RG V ( RC LG) jωv LC( ω ) V 8

9 TEM Transmission Line (cont.) dv ( ) ( RG V + jω( RC + LG) V ω LC ) V d Note that RG + jω RC + LG ω LC R + jωl G + jωc ( ) ( )( ) Z R+ jωl series impedance/length Y G+ jωc parallel admittance/length Then we can write: dv ( ZY ) V d 9

10 Artificial Transmission Line (ATL) This is one that is made by artificially cascading periodic sections of elements, which are arranged differently from what an actual (TEM) transmission line would have. p λ p Z 1 1 jωc p e Y 1 1 jωl p e C e C e... L e L e... We still have: dv ( ZY ) V d Z R+ jωl Y G+ jωc 1

11 Propagation Constant Let γ ZY Then dv ( γ ) d V Solution: γ V ( ) Ae + Be + γ γ is called the "propagation constant." Convention: + wave is V ( ) V e γ + + To see what this implies, use [( R j L)( G j C) ] 1/ ( ZY ) 1/ γ + ω + ω 11

12 Propagation Constant (cont.) R + jωl G+ jωc There are two possible locations for γ: ±γ Write: γ α + jβ γ Propagation constant Attenuation constant Phase constant 1

13 Propagation Constant (cont.) Consider a wave going in the + direction: V ( ) V e V e e V e e e + + γ + α jβ + jφ α jβ We require α The propagation constant γ is always in the first quadrant. Hence: γ γ 13

14 Propagation Constant (cont.) The principal branch of the square root function: We have the property that Branch cut Re( ) j re θ y j / re θ π < θ π π < θ π This is the branch that MATLAB uses. 1 1/ 1 x 14

15 Propagation Constant (cont.) Hence: ( R j L)( G j C) γ + ω + ω The principal branch ensures that α To be more general, γ ZY (This also holds for an ATL.) 15

16 Characteristic Impedance Z I + () + V + () - A wave is traveling in the positive direction. Z V I + + ( ) ( ) V ( ) V e + + γ I ( ) I e + + γ so Z V I + + (Z is a number, not a function of.) 16

17 Characteristic Impedance Z (cont.) Use the first Telegrapher s Equation: v i Ri L t so dv d RI ZI jωli (This form also holds for an ATL.) Hence γv e ZI e Z + γ + γ V I Ze + γ + γ γ e Recall : V ( ) V e + + γ I ( ) I e + + γ 17

18 Characteristic Impedance Z (cont.) From this we have: Z Z γ (This also holds for an artificial TL.) We can also write Z Z Z Z ± γ ZY Y Which sign is correct? Positive power flow in the direction Re Z Hence Z Z Y (This also holds for an artificial TL.) 18

19 Characteristic Impedance Z (cont.) For a physical transmission line we have: Z R+ G+ jωl jωc 19

20 Backward-Traveling Wave I - () + V - () - A wave is traveling in the negative direction. V ( ) Z V ( ) so Z I ( ) I ( ) Most general case: A general superposition of forward and backward traveling waves: V( ) V e + V e + γ + γ 1 I( ) V e V e Z + γ + γ

21 Power Flow + V + () - I + () Z A wave is traveling in the positive direction * 1 ( + + * ) 1 + P V( I ) ( ) ZI ( I ) ( ) Z I( ) < P > Re P Hence < P > 1 Re ( ) + Z I 1

22 Power Flow (cont.) + V () - I () Z Allow for waves in both directions: 1 ( ) * ( ) P VI * * ( ZI ( ) ZI ( ) )( I ( ) + I ( ) ) * 1 + * Z I ( ) Z I ( ) + ZI ( I ) ( ) ZI ( I ) ( )

23 Power Flow (cont.) + V () - I () Z * * P Z I ( ) Z I ( ) + ZI ( I ) ( ) ZI ( I ) ( ) 1 P P + P + Z I I I I + + * * ( ) ( ) ( ) + ( ) pure imaginary 3

24 Power Flow (cont.) + V () - I () Z 1 P P + P + Z I I I I + + * * ( ) ( ) ( ) + ( ) pure imaginary For a lossless line, Z is pure real: Z R+ jωl L G+ jωc C In this case, Re P Re P + Re P + so P P + P + (orthogonality) 4

25 Traveling Wave Let's look at the traveling wave in the time domain. { jωt} + + v( t, ) Re V ( e ) Re Re { + α jβ jωt V } e e e { + jφ α jβ jωt V } e e e e V e cos( ωt β+ φ) + α 5

26 Wavelength v( t, ) V e cos( ωt β+ φ) + + α t λ g V + e α 6

27 Wavelength (cont.) The wave repeats when: βλg π Hence: β π λ g 7

28 Phase Velocity Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest. v p (phase velocity) v( t, ) V e cos( ωt β+ φ) + + α 8

29 Phase Velocity (cont.) Set ωt β d ω β dt d ω dt β constant Hence vp ω β 9

30 Phase Velocity (cont.) In general, v p ω Im ( R+ jωl)( G+ jωc) vp f( ω)( ) function of frequency This is dispersion, resulting in waveform distortion 3

31 Distortion In general, waveform distortion is caused by either of two things: 1) Phase velocity v p is a function of frequency (dispersion) ) Attenuation α is a function of frequency In general, both effects arise when loss is present on a transmission line. 31

32 Lossless Case R, G γ α + jβ ( R+ jωl)( G+ jωc) jω LC so α β ω LC v p ω 1 ( ω) constant β LC no dispersion + no attenuation no distortion 3

33 Lossless Case (cont.) Z R G + + jωl jωc Z L C 33

34 Example Lossless coaxial cable ε r a b C L πε ε r b ln a µ b ln π a [ F/m] [ H/m] 34

35 Example (cont.) Using v p 1 L Z LC C We have v Z p 1 c µε ε η [ m/s] [ ] Ω π r r 1 b ln ε a 8 c [m/s] η µ ε / [ Ω] 35

36 Generaliation Generaliation to general lossless two-conductor transmission line with a homogeneous non-magnetic material filling: C L εεgf µ GF r Justification: GF geometrical factor Note: A proof of this independence is given in Notes 9. 1) We have ρ D nˆ and D ε E, with E independent of frequency and material. s ) The relation for L follows from the requirement that the phase velocity be equal to the speed of light in the filling material this is valid for any TEM mode in a lossless material, as discussed later in Notes 9. Hence: LC µε ε r 36

37 Generaliation (cont.) Consider next a lossy dielectric, but lossless conductors: C εε rc GF G ωεε rc GF µ L GF Justification of C and G formulas: Y G+ jωc jωε ( ) ( ) jωε jω ε jε ω ε + jε c c c c c ( G+ jωc) ω( ε + jε ) G ωε c C ε c c c (principle of effective permittivity) c Also, we have G ωc ε ε c c tanδ 37

38 Generaliation (cont.) Justification of L formula: LC µε c ε ε jε c c c This is proven in notes 9. 38

39 Determination of (L, G, C) Parameters Consider the general case of a lossy (dielectric loss only) transmission line: ε ε jε c c c We wish to calculate the parameters (G, L, C) in terms of the characteristic impedance of the lossless line and the complex permittivity of the filling material. Z lossless L C LC µε c G ωc ε ε c c tanδ For the calculation of the lossless Z, we set ε c ε c (ignore ε c ). From the first two equations we have (multiplying and dividing the two equations): µε L Z µε C Z lossless c c lossless 39

40 Determination of Parameters (cont.) Summary lossless Z L c C G ωc cz ε rc lossless ε ε rc rc ε tanδ rc These results tells us how to calculate the (L, G, C) line parameters from the characteristic impedance of the lossless line and the filling material. This information is what we would typically know about a line (e.g., from a vendor). 8 c [m/s] Note: Later we will see how to calculate R. (This involves the concept of the surface resistance of the conductors.) 4

41 Distortionless Case γ ( R+ jωl)( G+ jωc) R G L + jω C + jω L C Assume that the following condition holds: R L G C This is the "Heaviside condition," discovered by Oliver Heaviside. Then we have: R γ LC + L jω 41

42 Distortionless Case (cont.) v p R γ LC + jω L R α LC β ω LC L 1 ( ω) constant (no dispersion) LC There is then attenuation but no dispersion. Note: There will be some distortion in practice, since the Heaviside condition cannot be satisfied for all frequencies: R L G C G ω, R ω (assuming a fixed loss tangent) Also, there will be distortion since α is a function of frequency. 4

43 Distortionless Case (cont.) An example of loading a coaxial cable to achieve the Heaviside condition. For regular coax: R L > G C We can increase L to achieve the Heaviside condition. 43

44 Distortionless Case (cont.) Loading coils can also be placed periodically along the line. Loading coil 44

45 Distortionless Case (cont.) For an interesting history: Loading cables to improve performance was popular in the early 19 s, but declined after the 194s. The technology has been superseded by using digital repeaters on transmission lines. For long distances, transmission lines are usually replaced by fiber-optic cables (or wireless systems). 45