Prof. David R. Jackson Dept. of ECE Notes 6 ECE Microwave Engineering Fall 2015 Waveguides Part 3: Attenuation 1

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 Attenuation constant Phase constant for lossless wave guide Attenuation constant due to conductor loss (ignore dielectric loss) Attenuation constant due to dielectric loss (ignore conductor loss) Attenuation on Waveguiding Structures 2

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

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

Small dielectric loss in medium: Approximate Dielectric Attenuation Use: 5

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

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

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

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. Attenuation due to Conductor Loss 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. 9

This is a very important concept for calculating loss at a metal surface. Surface Resistance 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. C S 10 Note: The tangential fields determine the power going into the metal.

Surface Resistance (cont.) Assume Note that The we have (for the metal): Hence 11 (for the metal)

Surface Resistance (cont.) Denote Then we have “skin depth” = “depth of penetration” At 3 GHz, the skin depth for copper is about 1.2 microns. 12

Surface Resistance (cont.) Example: copper (pure) 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] 13 Note: A value of 3.0  10 7 [S/m] is often assumed for “practical” copper.

Surface Resistance (cont.) = time-average power dissipated / m 2 on S z x S Fields evaluated on this plane 14

where Inside conductor: “Surface resistance (  )” “Surface impedance (  )” 15 Note: To be more general: Surface Resistance (cont.) Note:

Surface Resistance (cont.) Summary for a Good Conductor 16 (fields at the surface)

17 “Effective 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. Hence we have Surface Resistance (cont.) For the “effective” surface current density we imagine the actual volume current density to be collapsed into a planar surface current. Conductor

Surface Resistance (cont.) Example: copper (pure) Frequency RsRs 1 [Hz] 2.61  [  ] 10 [Hz] 8.25  [  ] 100 [Hz] 2.61  [  ] 1 [kHz] 8.25  [  ] 10 [kHz] 2.61  [  ] 100 [kHz] 8.25  [  ] 1 [MHz] 2.61  [  ] 10 [MHz] 8.25  [  ] 100 [MHz] [  ]6.6 1 [GHz] [  ] 10 [GHz] [  ] 100 [GHz] [  ] 18 Note: A value of 3.0  10 7 [S/m] is often assumed for “practical” copper.

Surface Resistance (cont.) In general, Returning to the power calculation, we have: For a good conductor, Hence This gives us the power dissipated per square meter of conductor surface, if we know the effective surface current density flowing on the surface. 19 PEC limit:

Perturbation Method for  c Power flow along the guide: Power P z = 0 is calculated from the lossless case. Power loss (dissipated) per unit length: 20

There is a single conducting boundary. Surface resistance of metal conductors: Note : On PEC conductor Perturbation Method: Waveguide Mode C S 21 For these calculations, we neglect loss when we determine the fields and currents.

Surface resistance of metal conductors: Perturbation Method: TEM Mode There are two conducting boundaries. 22 Note : On PEC conductor For these calculations, we neglect loss when we determine the fields and currents.

Wheeler Incremental Inductance Rule 23 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 Z 0. In this formula, (for a given conductor) is the distance by which the conducting boundary is receded away from the field region. The top plate of a PPW line is shown being receded. The formula is applied for each conductor and the conductor attenuation from each of the two conductors is then added. E E H. Wheeler, “Formulas for the skin-effect,” Proc. IRE, vol. 30, pp , 1942.

Example: TEM Mode Parallel-Plate Waveguide Previously, we showed (+z wave) On the top plate: On the bottom plate: 24 y z w d x  Hence, we have

Example: TEM Mode PPW (cont.) (equal contributions from both plates) The final result is then 25

Example: TEM Mode PPW (cont.) 26 Let’s try the same calculation using the Wheeler incremental inductance rule. From previous calculations: d w Hence, we have:

Results for TM/TE Modes (above cutoff): (derivation omitted) TM n modes of PPW: TE n modes of PPW: Example: TM z /TE z Modes of PPW 27 Note: Below cutoff, we usually do not worry about conductor loss. y z w d x 

FrequencyRG59 Coax WR975 WG WR159 WG WR90 WG WF42 WG WR19 WG WR10 WG 1 [MHz] 0.1NA 10 [MHz] 0.3NA 100 [MHz] 1.1NA 1 [GHz] 40.04NA 5 [GHz] 100.4NA 10 [GHz] 151.1NA 20 [GHz] 233.7NA 50 [GHz] OM*10NA 100 [GHz] OM*30 28 Comparison of Attenuation Approximate attenuation in dB for 10 meters *OM = overmoded Typical singe-mode fiber optic cable: 0.3 dB/km Typical multimode fiber optic cable: 3 dB/km