1. The document discusses transmission line theory and parameters. Key topics covered include: - Telegrapher's equation and circuit model for transmission lines - Wave propagation and characteristic impedance calculations - Reflection coefficient and standing wave ratio definitions - Comparisons of transmission line, circuit, and field theories 2. Specific transmission line types are analyzed, including planar lines, coaxial cables. Equations are given for calculating the capacitance, conductance, inductance, resistance, and characteristic impedance of these common line configurations. 3. Simulation and modeling techniques for transmission lines are briefly mentioned, such as the transmission line matrix method for modeling microstrip lines in antennas and circuits.

1. Transmission
Line
Microwave Engineering
CHO, Yong Heui

2. Microwave Engineering
1. Tx line theory
Telegrapher’s equation
  Communication service: telegraph
Telegraph
Ocean
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3. Microwave Engineering
1. Tx line theory
World wide transmission lines
EM Wave Lab

4. Microwave Engineering
1. Tx line theory
Introduction
  The differential equations which the voltage or
current must satisfy on a uniform transmission
line.
 Circuit model: tie field theory and circuit theory
together.
 Our circuit model contain the inductance L,
capacitance C, shunt conductance G, and series
resistance R associated with an incremental
length of line.
 Coaxial transmission line containing a dielectric.
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5. Microwave Engineering
1. Tx line theory
Wave propagation
 V ( z , t ) = Vo cos(ωt − βz + φ )
= Re{Vo e j (ωt − βz +φ ) }
jφ − jβz j ωt
= Re{Vo e e e }
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6. Microwave Engineering
1. Tx line theory
Distributed element
  Lumped element: R, L, C
  Distributed element: tx line
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7. Microwave Engineering
1. Tx line theory
Circuit model
ˆ
 Propagation in the a z direction
Length ∆z : R∆z[ Ω], L∆z[ H ], G∆z[1 / Ω], C∆z[ F ]
lim: infinitesimal approach
∆z →0
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8. Microwave Engineering
1. Tx line theory
Transmission line equation
 Phasor : Vs ( z ) = Vo e jφ e − jβz
1 1 
Vs ( z ) =  R∆z + jω L∆z  I s ( z )
2 2 
1 1 
+  R∆z + jω L∆z ( I s ( z ) + ∆I s ( z )) + Vs ( z ) + ∆Vs ( z )
2 2 
∆Vs ( z ) 1 1 
= −( R + jωL) I s ( z ) −  R + j ωL ∆I s (z) : As ∆z → 0, ∆I s → 0
 ∆z 2 2 
∆I s ( z ) ∆Vs ( z )
− = (Vs ( z ) + )(G + jωC ) : As ∆z → 0, ∆Vs → 0
∆z 2
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9. Microwave Engineering
1. Tx line theory
Transmission line equation
dVs ( z )
 = − ( R + jω L ) I s ( z )
dz
dI s ( z )
 = −(G + jωC )Vs ( z )
dz Tx line
Tx line modeling
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10. Microwave Engineering
1. Tx line theory
Wave solution
  Traveling wave solution
+ −γz
- Voltage: Vs ( z ) = V0 e + V0− eγz
- Current: I s ( z ) = I 0+ e −γz − I 0− eγz
d 2Vs ( z )
2
= ( R + jωL)(G + jωC )Vs ( z )
dz
γ = α + jβ = ( R + jωL)(G + jωC )
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11. Microwave Engineering
1. Tx line theory
Characteristic impedance
  Important parameter in tx line: Z0
R + j ωL
- Z0 =
G + j ωC
V0+ V0−
- Z0 = + = −
I0 I0
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12. Microwave Engineering
1. Tx line theory
Lossless line
 γ = jβ = jω LC
L
 Z0 =
C
+ − jβz − − jβz
 Vs ( z ) = V e
0 +V e0
+ −
V − jβz V jβz
I s ( z) = 0
e − e0
Z0 Z0
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13. Microwave Engineering
1. Tx line theory
Low-loss line
 γ = α + jβ = ( R + jωL)(G + jωC )
 j R G 
≈ jω LC 1 −  + 
 2  ωL ωC  
1 R 
 α =  + GZ 0 
2  Z0



L
 Z0 ≈
C
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14. Microwave Engineering
1. Tx line theory
Distortionless line
R
γ = α + jβ = + jω LC
 Z0
L
 Z0 =
C
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15. Microwave Engineering
1. Tx line theory
Reflection coefficient
  Voltage wave continuity conditions
 Current wave continuity conditions
Vo− Z L − Z 0
 Γ11 =| Γ | e jφ = + =
V0 Z L + Z0
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16. Microwave Engineering
1. Tx line theory
Wave power
+ 2

1
P = Re VI =
2
*
V
( )
2Z 0
0
1− Γ
2
( )
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17. Microwave Engineering
1. Tx line theory
SWR (Standing Wave Ratio)
  SWR: field theory
 VSWR (Voltage SWR): tx line theory
V
s= max
V min
1+ | Γ |
=
1− | Γ |
Experiment
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18. Microwave Engineering
1. Tx line theory
Comparison with circuit theory
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19. Microwave Engineering
1. Tx line theory
Comparison with circuit theory
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20. Microwave Engineering
1. Tx line theory
Comparison with circuit theory
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21. Microwave Engineering
1. Tx line theory
Comparison with field theory
For comparison, ∇ × E s = − jωµH s
Set E s = E xs a x and H s = H ys a y
where E xs and H ys are functions of z only.
dE xs dVs
 = − jωµH ys ⇔ = − ( R + j ωL ) I s
dz dz
∇ × H s = (σ + jωε )E s
dH ys dI s
 = −(σ + jωε ) E xs ⇔ = −(G + jωC )Vs
dz dz
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22. Microwave Engineering
1. Tx line theory
Comparison with field theory
 A direct analogy : I s and H ys , G and σ , C and ε , Vs and E xs
The boundary conditions on Vs and E xs
are the same to those for I s and H ys .
 E xs = E x 0 e − jkz , jk = jωµ (σ + jωε )
⇔ Vs = V0 e −γz , γ = α + jβ = ( R + jωL)(G + jωC )
 For a lossless line ( R = G = 0)
2π ω
1 β= vp =
γ = jβ = jω LC , v p = λ β
LC
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23. Microwave Engineering
1. Tx line theory
Reflection coefficient
E x 0 − jkz V0 − jβz
 H ys = e ⇔ Is = e
η Z0
( Z 0 : characteristic impedance)
jωµ R + jωL
η= ⇔ Z0 =
σ + jωε G + jωC
E x−0 η 2 − η1 Vo− Z 02 − Z 01
 Γ11 = + = ⇔ Γ11 =| Γ | e jφ = + =
E x 0 η 2 + η1 V0 Z 02 + Z 01
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24. Microwave Engineering
1. Tx line theory
Input impedance
 The ratio of Vs to I s at z = −l
when Z = Z 2 for z > 0 and Z = Z1 for z < 0 :
η 2 cos β1l + jη1 sin β1l
 ηin = η1 η cos β l + jη sin β l
1 1 2 1
Z 2 cos β1l + jZ1 sin β1l
⇔ Z in = Z1
Z1 cos β1l + jZ 2 sin β1l
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25. Microwave Engineering
1. Tx line theory
TLM (Transmission Line Matrix)
 Equivalence: field theory and tx line theory
 Simulation tool: Micro-stripes based on TLM method
Bluetooth antennas
Current density
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26. Microwave Engineering
1. Tx line theory
Antenna impedance
  Antenna impedance (not infinity) matching
 No reflection, power efficiency
Handy phone antenna
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27. Microwave Engineering
2. Tx line parameters
Planar line
 Assume b >> d
εb σ σb
C = ,G = C =
d ε d
µd 2
Lext = ,R =
b σ c bδ
Lext µ d Cross section
 Z0 = =
C ε b
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28. Microwave Engineering
2. Tx line parameters
Coaxial line
ρL
 ∫S D • dS = Q 2πρLD = ρ L L D = 2πρ a ρ
ρL
E= aρ
2πε ' ρ
a ρL ρ ρ b
Vo = − ∫
a
dρ = − L ln ρ b = L ln
b 2πε ' ρ 2πε ' 2πε ' a
Cross section
2πε
 C= [ F/m]
ln(b / a )
Coaxial line + connector
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29. Microwave Engineering
2. Tx line parameters
Coaxial line
ε C ε σ 2πσ
 RC = , = , G = C ⇒ G = [1 / Ωm]
σ G σ ε ln(b / a)
µId b µ b
 Φ= ln , Lext = ln [ H / m]
2π a 2π a
1 1 1 1 1
 Rinner = , Router = ⇒R=  + [ Ω / m]
2πaδσ c 2πbδσ c 2πδσ c  a b 
The characteristic impedance of a coax

Lext 1 µ b
Z0 = = ln
C 2π ε a
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