Lecture Notes Transmission Lines Part 2 (TL Formulas)

1. Prof. David R. Jackson
Dept. of ECE
Notes 3
ECE 5317-6351
Microwave Engineering
Fall 2019
Transmission Lines
Part 2: TL Formulas
1
Adapted from notes by
Prof. Jeffery T. Williams

2.  In this set of notes we develop some general formulas
that hold for any transmission line.
 We first examine the coaxial cable as an example.
Overview
2

3. Coaxial Cable
Here we present a “case study” of one particular transmission line, the coaxial cable.
Find C, L, G, R
We will assume no variation in the z direction, and take a length of one meter in
the z direction in order to calculate the per-unit-length parameters.
3
For a TEMz mode, the shape of the fields is independent of frequency, and
hence we can perform the calculation of C and L using electrostatics and
magnetostatics.
,
r d
 
a
b

4. Coaxial Cable (cont.)
-l0
l0
a
b
r

0 0
0
ˆ ˆ
2 2 r
E
 
 
    
 
 
   
 
   
Find C (capacitance / length)
Coaxial cable
h = 1 [m]
r

From Gauss’s law:
0
0
ln
2
B
AB
A
b
r
a
V V E dr
b
E d
a



 
  
 
   
 


4
0 (C/m)
  line charge density on the inner conductor

5. Coaxial cable
 
0
0
0
1
ln
2 r
Q
C
V b
a


 
 
   
   
 
 
Hence
We then have:
0
F/m
2
[ ]
ln
r
C
b
a
 

 
 
 
Coaxial Cable (cont.)
5
r

 
1 m
h 
r

0
l

0
l


a
b

6. ˆ
2
I
H 
 
 
  
 
Find L (inductance / length)
From Ampere’s law:
Coaxial cable
0
ˆ
2
r
I
B   
 
 
  
 
(1)
b
a
B d

 
 
Magnetic flux:
Coaxial Cable (cont.)
6
Note:
We ignore “internal inductance” here, and only
look at the magnetic field between the two
conductors (accurate for high frequency.
r

I
 
1 m
h 
z
S
Center conductor
I
I
1 [ ]
h m


7.   0
0
0
1
2
ln
2
b
r
a
b
r
a
r
H d
I
d
I b
a

   
  

 



 
  
 


0
1
ln
2
r
b
L
I a

 

 
   
 
0
H/m
ln [ ]
2
r b
L
a
 

 
  
 
Hence
Coaxial Cable (cont.)
7
r

I
 
1 m
h 

8. 0
H/m
ln [ ]
2
r b
L
a
 

 
  
 
Observations:
0
F/m
2
[ ]
ln
r
C
b
a
 

 
 
 
 
0 0 r r
LC     
 
This result actually holds for any transmission line that is
homogenously filled* (proof omitted).
Coaxial Cable (cont.)
8
(independent of frequency)
(independent of frequency)
*This result assumes that the permittivity is real. To be more general, for a lossy
line, we replace the permittivity with the real part of the permittivity in this result.

9. 0
H/m
ln [ ]
2
r b
L
a
 

 
  
 
For a lossless (or low loss) cable:
0
F/m
2
[ ]
ln
r
C
b
a
 

 
 
 
0
L
Z
C

0 0
1
ln [ ]
2
r
r
b
Z
a


 
 
 
 
 
0
0
0
376.7303 [ ]



  
Coaxial Cable (cont.)
9

10. 0 0
0
ˆ ˆ
2 2 r
E
 
 
    
 
 
   
 
   
Find G (conductance / length)
Coaxial cable
From Gauss’s law:
0
0
ln
2
B
AB
A
b
r
a
V V E dr
b
E d
a



 
  
 
   
 


Coaxial Cable (cont.)
10
d

 
1 m
h 
r

0
l

0
l


a
b

11. d
J E


We then have leak
I
G
V

 
0
0
(1)2
2
2
2
leak a
d a
d
r
I J a
a E
a
a
 
 

 

 
 




 
  
 
0
0
0
0
2
2
ln
2
d
r
r
a
a
G
b
a

 
 

 
 
 
 

 
 
 
2
[S/m]
ln
d
G
b
a


 
 
 
or
Coaxial Cable (cont.)
11
d

0
l

0
l


a
b

12. Observation:
F/m
2
[ ]
ln
C
b
a


 
 
 
d
G C


 
  
 
2
[S/m]
ln
d
G
b
a


 
 
 
0 r
  

Coaxial Cable (cont.)
12
*This result assumes that the G term arises only from conductivity, and not polarization loss.
This result actually holds for any transmission line that is
homogenously filled* (proof omitted).

13. d
G C


 
  
 
Hence:
tan
d
d
G
C


 
 
 
 
 
tan d
G
C



Coaxial Cable (cont.)
As just derived,
13
This is the loss tangent that would
arise from conductivity.
*This result is very general, and allows the G term to come from either conductivity or polarization loss.
or
This result actually holds for any transmission line that is
homogenously filled* (proof omitted).

14. Complex Permittivity
Accounting for Dielectric Loss
14
The permittivity becomes complex when there is
polarization (molecular friction) loss.
 
j
  
 
 
Loss term due to polarization (molecular friction)
Example: Distilled water heats up in a microwave oven, even though there is
essentially no conductivity!

15. Effective Complex Permittivity
c j

 

 
   
 
Effective permittivity that accounts for conductivity
15
c
c
 

 

 

 
 
  
 
The effective permittivity accounts for conductive loss.
 
c
c c
j j
j

  

 
 
 
    
 
 
 
Hence We then have
Accounting for Dielectric Loss (cont.)

16. Most general expression for loss tangent:
c c c
j
  
 
 
tan c
c


 



 
 
  
 


Loss due to molecular friction Loss due to conductivity
16
c
c
 

 

 

 
 
   
 
Accounting for Dielectric Loss (cont.)
The loss tangent accounts for both molecular friction and conductivity.

17. For most practical insulators (e.g., Teflon), we have
tan c
c
 



 
 


17
0
 
Note: The loss tangent is usually (approximately) constant for
practical insulating materials, over a wide range of frequencies.
Typical microwave insulating material (e.g., Teflon): tan = 0.001.
Accounting for Dielectric Loss (cont.)

18. 18
Accounting for Dielectric Loss (cont.)
r r
 

In most books, is denoted as simply
Important point about notation:
 
1 tan
rc r j
  
 
In this case we then write:
 
r rc

 

Means real part of
(We will adopt this convention also.)
The effective complex relative permittivity

19. Find R (resistance / length)
Coaxial cable
Coaxial Cable (cont.)
a b
R R R
 
1
2
a sa
R R
a

 
  
 
1
2
b sb
R R
b

 
  
 
1
sa
a a
R
 

1
sb
b b
R
 

0
2
a
ra a

  

0
2
b
rb b

  

Rs = surface resistance of metal
19
 
1 m
h 
(This is discussed later.)
 = skin depth of metal
,
b rb
 
d

,
a ra
 
a
b
Inner conductor
Outer conductor

20. General Formulas for (L,G,C)
 tan d
G C
 

0 0 0
lossless
r r
L Z    

0 0 0
/ lossless
r r
C Z
   

The three per-unit-length parameters (L, G, C) can be found from
20
0 , , tan
lossless
r d
Z  
0
lossless L
Z
C

 characteristic impedance of line
lossless
These values are usually known from the manufacturer.
These formulas hold for
any homogeneously-
filled transmission line.

21. General Formulas for (L,G,C) (cont.)
The derivation of the previous results follows from:
21
0 0
r r
LC    

 
2
0
lossless
L
Z
C

Multiply and divide
.
r r
 
denotes
Here

22. General Formulas for (L,G,C) (cont.)
Example:
22
A transmission line has the following properties:
 
2.1
r
  Teflon
0 50
lossless
Z  
tan 0.001
 
 
 
 
7
11
3
2.4169 10 H/m
9.6677 10 F/m
6.0744 10 S/m
L
C
G



 
 
 
Results:
 
10 GHz
f  (frequency is only needed for G)
Note:
We cannot determine R without
knowing the type of transmission line
and the dimensions (and the
conductivity of the metal).

23. Wavenumber Formulas
23
General case (R,L,G,C):
( )( )
z
k j j R j L G j C
   
     
Lossless case (L,C):
0
z r r
k LC k k
     
    
Dielectric loss only (L,C,G):
0
z c r rc
k k k jk k
   
 
     (please see next slide)

24. Wavenumber (cont.)
24
Dielectric loss only (L,C,G):
( )( )
( )( tan )
( )( tan )
( )(tan )
(1 tan )
(1 tan )
(1 tan )
z
d
d
d
d
d
d
c
k j j L G j C
j j L C j C
j jL C jC
j LC j j
j LC j
LC j
j
k
 
   
 
 
 
 
  
 
 
  
  
  
   
 

 


Note:
The mode stays a perfect
TEMz mode if R = 0.
kz = k for any TEMz mode.
0
R 
tan
c
c
c
 



 




Notes:

25. Common Transmission Lines
0 0
1
ln [ ]
2
lossless r
r
b
Z
a


 
 
 
 
 
Coax
1 1
2 2
sa sb
R R R
a b
 
   
 
   
   
25
 
0 0 0
0 0 0
/
tan
lossless
r r
lossless
r r
d
L Z
C Z
G C
R R
   
   
 




1
sa
a a
R
 

1
sb
b b
R
 

0
2
a
ra a

  

0
2
b
rb b

  

,
r r
 
a
b
a
b




conductivity of inner conductor metal
conductivity of outer conductor metal
tan ( )
d
 dielectric
(skin depth of metal for inner or outer conductors)
(surface resistance of metal for inner or outer conductors)

26. Common Transmission Lines
Twin-lead
1
0
0 cosh [ ]
2
lossless r
r
h
Z
a
 
 
  
 
 
 
2
1 2
1
2
s
h
a
R R
a h
a

 
 
 
 
 
 
  
 
 

 
 
 
 
26
 
0 0 0
0 0 0
/
tan
lossless
r r
lossless
r r
d
L Z
C Z
G C
R R
   
   
 




1
s
m
R
 

0
2
rm m

  

Two identical conductors
m
  conductivity of metal
,
r r
 
a a
h
tan ( )
d
 dielectric
(skin depth of metal)
(surface resistance of metal)

27. Common Transmission Lines (cont.)
Microstrip ( / 1)
w h 
27
Approximate CAD formula
r
w t
h
0
60 8
ln
4
eff
r
h w
Z
w h

 
 
 
 
1 1 1
2 2
1 12
eff r r
r
h
w
 

 
 
 
   
 

 
 

28. Common Transmission Lines (cont.)
Microstrip
   
 
 
 
 
0 0
1 0
0
0 1
eff eff
r r
eff eff
r r
f
Z f Z
f
 
 
 

  
 

 
 
     
 
0
120
0
0 / 1.393 0.667ln / 1.444
eff
r
Z
w h w h



 
 
  
 
( / 1)
w h 
2
1 ln
t h
w w
t

 
 
     
 
 
 
28
Approximate CAD formula
r
w t
h
Note:
Usually w/h > 1 for a
50  line.

29. Common Transmission Lines (cont.)
Microstrip ( / 1)
w h 
 
2
1.5
(0)
(0)
1 4
eff
r r
eff eff
r r
f
F
 
  
 

 
 
 

 
 
 
1 1 1
1 /
0
2 2 4.6 /
1 12 /
eff r r r
r
t h
w h
h w
  

 
    
   
 
  
    
 
   
  
 
2
0
4 1 0.5 1 0.868ln 1
r
h w
F
h


 
   
 
    
 
   
 
 
 
 
   
29
Approximate CAD formula
r
w t
h