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EEE445_Lectr06_lossy_rough_TEM

# EEE445_Lectr06_lossy_rough_TEM - 2.7 Lossy Transmission...

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2.7 Lossy Transmission Lines In practice, all transmission lines have some loss due to imperfect conductors and lossy dielectrics. Of course, we almost always want the loss to be as small as possible. In many situations, where small lengths of line are used, the assumption of lossless transmission lines does not degrade the usefulness of the solution. However, in some situations (e.g., transmission line resonator Q factor, long runs of line), we must precisely account for loss. At 60 GHz, coplanar lines have typical loss in 40 dB/m. EEE 591/445 Lecture 06 1

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Phase Velocity of Lossless and Lossy Line EEE 591/445 Lecture 06 2 The phase velocity is not constant versus  frequency (dispersion) – a pulse  propagating on the line experiences  distortion. 0 1 2 3 4 5 6 7 8 9 1 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 ϖ ( r a d i a n s p e r s e c o n d ) P h a s e V e l o c i t y ( m / s ) R = 5 , G = 0 , L = 1 , C = 1 0 1 2 3 4 5 6 7 8 9 1 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 ϖ ( r a d i a n s p e r s e c o n d ) P h a s e V e l o c i t y ( m / s ) R = G = 0 , L = 1 , C = 1 The phase velocity is constant versus  frequency – a pulse propagating on the line  experiences no distortion.
Low-Loss Line EEE 591/445 Lecture 06 3 0 0 0 1 2 R L G C R GZ Z LC R j L L Z G j C C ϖ ϖ α β ϖ ϖ ϖ << << + + = + (2.83) (2.85a) (2.85b) (2.86)

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Distortionless Line Line may have substantial loss, but the phase velocity is still constant because EEE 591/445 Lecture 06 4 0 real. R G L C R j L L Z G j C C j C R j LC L ϖ ϖ γ α β ϖ = + = = = + = + = + (2.87) (2.88)
Example 2.6 Attenuation of coaxial line. Solution . From (2.85), Using Example 2.1 EEE 591/445 Lecture 06 5 1 2 C C R G L L α + ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 0 2 2 2 2 2 0 0 2 2 2 0 1 ln / (H/m) 2 2 ' 1 2 ' (F/m) ln / ln / 1 1 1 1 (Ω/m) 2 2 '' 1 2 '' (S/m) ln / ln / b a b a s s b a L d d a b C d d b a b a R R R ad bd a b a b G d d b a b a π φ φ ρ π φ ρ π π φ φ π φ ρ μ μ ρ ρ φ ρ π π ε πε ρ ρ φ ρ φ φ π π ϖε πϖε ρ ρ φ ρ = = = = = = = = = = = = = + = + = = ∫ ∫ ∫ ∫ ∫ ∫

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we obtain EEE 591/445 Lecture 06 6 0 0 0 1 2 1 1 1 1 2 '' 2 2 ln / 1 1 1 '' 2 ln / where we used ln / ln / ln / 2 2 ' 2 ' 2 ln / s s c d C C R G L L R Z a b Z b a R b a a b a b L a b a b Z C b a α πϖε π ϖε η α α η μ μ π η πε π ε π + = + + = + + = + = = = =
Typical Transmission Line Circuit: Terminated lossy line EEE 591/445 Lecture 06 7 Z 0 , γ z=0 z Z L + - V L I L z=-l + - V in I in + - V g Z g Γ L Γ g

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Voltage and Current on the Line (Lossless Line)
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