1. Notes 5 ECE 5317-6351 Microwave Engineering
Fall 2015
Prof. David R. Jackson
Dept. of ECE
Notes 5
Waveguiding Structures Part 2:
Attenuation

2. Attenuation on Waveguiding Structures
For most practical waveguides and transmission lines the loss associated with dielectric loss and conductor loss is relatively small.
To account for these losses we will assume
Phase constant for lossless wave guide
Attenuation constant
Attenuation constant due to conductor loss
(ignore dielectric loss)
Attenuation constant due to dielectric loss
(ignore conductor loss)

3. Attenuation due to Dielectric Loss: d
Lossy dielectric  complex permittivity
 complex wavenumber k
Note:

4. Attenuation due to Dielectric Loss (cont.)
An exact general expression for dielectric attenuation is found from:
This is an exact formula for attenuation due to dielectric loss. It works for both waveguides and TEM transmission lines (kc = 0).
Remember: The value kc is always real, regardless of whether the waveguide filling material is lossy or not.
Note: The radical sign denotes the principal square root:

5. Approximate Dielectric Attenuation
First we approximate k:
Assume a small dielectric loss in medium:
Use:

6. Approximate Dielectric Attenuation (cont.)
For the wavenumber of the guided wave:
Assume a small dielectric loss:
Use:

7. Approximate Dielectric Attenuation (cont.)
We assume here that we are above cutoff.
(works for both WGs and TLs)

8. Approximate Dielectric Attenuation (cont.)
TEM mode
We can simply put kc = 0 in the previous formulas.
Or, we can start with the following:
(exact equations)
(approximate equation)

9. Summary of Dielectric Attenuation
TEM mode
Waveguide mode (TMz or TEz)

10. Attenuation due to Conductor Loss
Assuming a small amount of conductor loss:
We can assume fields of the lossy guide are approximately the same as those for lossless guide, except with a small amount of attenuation.
 We can use a “perturbation” method to determine c .
Notes:
Dielectric loss does not change the shape of the fields at all, since the boundary conditions remain the same (PEC).
Conductor loss does disturb the fields slightly.

11. The tangential fields determine the power going into the metal.
Surface Resistance
This is a very important concept for calculating loss at a metal surface.
Note:
The tangential fields determine the power going into the metal.
Plane wave in a good conductor
Note: In this figure, z is the direction normal to the metal surface, not the axis of the waveguide. Also, the electric field is assumed to be in the x direction for simplicity.

12. Surface Resistance (cont.)
Assume
(for the metal)
The we have (for the metal):
Note that
Hence

13. Surface Resistance (cont.)
Denote
“skin depth” = “depth of penetration”
Then we have
At 3 GHz, the skin depth for copper is about 1.2 microns.

14. Surface Resistance (cont.)
Frequency 
1 [Hz]
6.6 [cm]
10 [Hz]
2.1 [cm]
100 [Hz]
6.6 [mm]
1 [kHz]
2.1 [mm]
10 [kHz]
0.66 [mm]
100 [kHz]
21 [mm]
1 [MHz]
66 [m]
10 [MHz]
20.1 [m]
100 [MHz]
6.6 [m]
1 [GHz]
2.1 [m]
10 [GHz]
0.66 [m]
100 [GHz]
0.21 [m]
Example: copper (pure)
Note:
A value of 3.0107 [S/m] is often assumed for “practical” copper.

15. Surface Resistance (cont.)z x S
Fields evaluated on this plane
= time-average power dissipated / m2 on S

16. Surface Resistance (cont.)
Inside conductor:
where
“Surface resistance ()”
“Surface impedance ()”
Note: To be more general:
Note:

17. Surface Resistance (cont.)
Summary for a Good Conductor
(fields at the surface)

18. Surface Resistance (cont.)
Conductor
“Effective surface current”
Hence we have
For the “effective” surface current density we imagine the actual volume current density to be collapsed into a planar surface current.
The surface impedance gives us the ratio of the tangential electric field at the surface to the effective surface current flowing on the object.

19. Surface Resistance (cont.)
Frequency Rs
1 [Hz]
2.6110-7 []
10 [Hz]
8.2510-7 []
100 [Hz]
2.6110-6 []
1 [kHz]
8.2510-6 []
10 [kHz]
2.6110-5 []
100 [kHz]
8.2510-5 []
1 [MHz]
2.6110-4 []
10 [MHz]
8.2510-4 []
100 [MHz]
[]6.6
1 [GHz]
[]
10 [GHz]
[]
100 [GHz]
[]
Example: copper (pure)
Note:
A value of 3.0107 [S/m] is often assumed for “practical” copper.

20. Surface Resistance (cont.)
Returning to the power calculation, we have:
In general,
For a good conductor,
This gives us the power dissipated per square meter of conductor surface, if we know the effective surface current density flowing on the surface.
Hence
PEC limit:

21. Perturbation Method for c
Power flow along the guide:
Power z = 0 is calculated from the lossless case.
Power loss (dissipated) per unit length:
Note:
 = c for conductor loss

22. Perturbation Method: Waveguide Mode
There is a single conducting boundary.
For these calculations, we neglect loss when we determine the fields and currents.
On PEC conductor
Surface resistance of metal conductors:

23. Perturbation Method: TEM Mode
There are two conducting boundaries.
For these calculations, we neglect loss when we determine the fields and currents.
On PEC conductor
Surface resistance of metal conductors:

24. Wheeler Incremental Inductance Rule
The Wheeler incremental inductance rule gives an alternative method for calculating the conductor attenuation on a transmission line (TEM mode):
It is useful when your have a formula for Z0.
The formula is applied for each conductor and the conductor attenuation from each of the two conductors is then added.
In this formula, (for a given conductor) is the distance by which the conducting boundary is receded away from the field region. E
The top plate of a PPW line is shown being receded. E
H. Wheeler, “Formulas for the skin-effect,” Proc. IRE, vol. 30, pp , 1942.

25. Calculation of R for TEM Mode
From c we can calculate R (the resistance per unit length of the transmission line):
Assume conductor loss only.

26. Summary of Attenuation Formulas
Transmission Line (TEM Mode)
Method #1
Method #2
The two methods are related, since we have:

27. Summary of Attenuation Formulas (cont.)
Waveguide (TMz or TEz Mode)

28. Comparison of Attenuation
Approximate attenuation in dB per meter
Frequency
RG59 Coax
WR975 WG
WR159
WR90
WF42
WR19
WR10
1 [MHz]
0.01 NA
10 [MHz]
0.03
100 [MHz]
0.11
1 [GHz]
0.4
0.004
5 [GHz]
1.0
0.04
10 [GHz]
1.5
20 [GHz]
2.3
0.37
50 [GHz]
OM*
100 [GHz]
3.0
*OM = overmoded
Typical single-mode fiber optic cable: 0.3 dB/km
Typical multimode fiber optic cable: 3 dB/km
NA = below cutoff

29. Comparison of Waveguide with Wireless System (Two Antennas)
(attenuating wave)
Antenna:
(spreading spherical wave)
For small distances, the waveguide delivers more power (no spreading).
For large distances, the antenna (wireless) system will deliver more power.

30. Wireless System (Two Antennas)
Here we examine a wireless system in more detail.
Two antennas (transmit and receive):
Pt = power transmitted
Pr = power received
Matched receive antenna:
(from antenna theory)
Aer = effective area of receive antenna
Hence, we have:

31. Wireless System (Two Antennas) (cont.)
Friis transmission formula
Total dB of attenuation:
Hence, we have:

32. dB Attenuation: Comparison of Waveguiding system with Wireless System
Examples:
transmit
receive
D = diameter (choose 34 meters)

33. Comparison of Waveguiding System with Wireless System
dB Attenuation:
Comparison of Waveguiding System with Wireless System
1 GHz
RG59
Single Mode
Two Dipoles
34m Dish+Dipole
Distance
Coax
Fiber
Wireless
1 m
0.4
0.0003
28.2 NA
10 m 4
0.003
48.2
100 m 40
0.03
68.2
1 km
400
0.3
88.2
39.3
10 km
4000 3
108.2
59.3
100 km - 30
128.2
79.3
1000 km
300
148.2
99.3
10,000 km
3000
168.2
119.3
100,000 km
188.2
139.3
1,000,000 km
208.2
159.3
10,000,000 km
228.2
179.3
100,000,000 km
248.2
199.3