Presentation on theme: "Microwave Engineering"— Presentation transcript:
1Microwave Engineering ECEMicrowave EngineeringFall 2011Prof. David R. JacksonDept. of ECENotes 6Waveguides Part 3:Attenuation
2Attenuation 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 assumePhase constant for lossless wave guideattenuation constantAttenuation constant due to dielectric lossAttenuation constant due to conductor loss
3Attenuation due to Dielectric Loss: d Lossy dielectric complex permittivity complex wavenumber kNote:
4Attenuation due to Dielectric Loss (cont.) ThusThis 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:
5Approximate Dielectric Attenuation Small dielectric loss in medium:Use
6Approximate Dielectric Attenuation (cont.) Small dielectric loss:Use
7Attenuation due to Dielectric Loss (cont.) We assume here that we are above cutoff.
8Attenuation due to Dielectric Loss (cont.) For TEM modeWe can simply put kc = 0 in the previous formulas. Or, we can start with the following:
9Attenuation 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 .Note: 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.
10Surface ResistanceThis is a very important concept for calculating loss at a metal surface.CSPlane wave in a good conductorNote: 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.
11Surface Resistance (cont.) AssumeThe we haveNote thatHence
12Surface Resistance (cont.) Denote“skin depth” = “depth of penetration”Then we haveAt 3 GHz, the skin depth for copper is about 1.2 microns.
14Surface Resistance (cont.) zxSfields evaluated on this plane= time-average power dissipated / m2 on S
15Surface Resistance (cont.) Note: To be more general:Inside conductor:where“Surface resistance ()”“Surface impedance ()”
16Surface Resistance (cont.) Summary for a Good Conductor
17Surface Resistance (cont.) conductor“Effective surface current”Hence we haveFor 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.
19Surface Resistance (cont.) We then haveIn 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.HencePEC limit:
20Perturbation Method for c Power flow along the guide:z = 0 is calculated from the lossless case.Power loss (dissipated) per unit length:
21Perturbation Method: Waveguide Mode There is a single conducting boundary.CSFor these calculations, we neglect dielectric loss.Note :On PEC conductorSurface resistance of metal conductors:
22Perturbation Method: TEM Mode There are two conducting boundaries.For these calculations, we remove dielectric loss.Note :On PEC conductorSurface resistance of metal conductors:
23Wheeler 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 Z0 is already known.The formula is applied for each conductor and the conductor attenuation from each of the two conductors is then added.In this formula, dl (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.
24Example: TEM Mode Parallel-Plate Waveguide Previously, we showedOn the top plate:On the bottom plate:
25Example: TEM Mode PPW (cont.) (equal contributions from both plates)The final result is then
26Example: TEM Mode PPW (cont.) Let’s try the same calculation using the Wheeler incremental inductance rule.From previous calculations:dw
27Example: TMz/TEz Modes of PPW Results for TM/TE Modes (above cutoff): (derivation omitted)TMn modes of parallel-plateTEn modes of parallel-plate