Presentation on theme: "Notes 7 ECE Microwave Engineering Waveguides Part 4:"— Presentation transcript:
1 Notes 7 ECE 5317-6351 Microwave Engineering Waveguides Part 4: Fall 2012Prof. David R. JacksonDept. of ECENotes 7Waveguides Part 4:Rectangular and Circular Waveguide
2 Rectangular Waveguide One of the earliest waveguides.Still common for high power and high microwave / millimeter-wave applications.It is essentially an electromagnetic pipewith a rectangular cross-section.For conveniencea b.the long dimension lies along x.Single conductor No TEM mode
6 TEz Modes (cont.) Therefore, From the previous field-representation equations, we can showNote:m = 0,1,2,…n = 0,1,2,…But m = n = 0is not allowed!(non-physical solution)
7 TEz Modes (cont.) Lowest cutoff frequency is for TE10 mode (a > b) Lossless Case TEmn mode is at cutoff whenLowest cutoff frequency is for TE10 mode (a > b)We will revisit this mode.Dominant TE mode (lowest fc)
8 TEz Modes (cont.)At the cutoff frequency of the TE10 mode (lossless waveguide):For a given operating wavelength (corresponding to f > fc) , the dimension a must be at least this big in order for the TE10 mode to propagate.Example: Air-filled waveguide, f = 10 GHz. We have that a > 3.0 cm/2 = 1.5 cm.
9 TMz Modes Subject to B.C.’s: Recall where (eigenvalue problem)Subject to B.C.’s:Thus, following same procedure as before, we have the following result:
11 TMz Modes (cont.) Therefore From the previous field-representation equations, we can showm=1,2,3,…n =1,2,3,…Note: If either m or n is zero, the field becomes a trivial one in the TMz case.
12 TMz Modes (cont.) (same as for TE modes) Lossless Case(same as for TE modes)Lowest cutoff frequency is for the TM11 modeDominant TM mode (lowest fc)
13 Mode Chart b < a/2 f b > a/2 f Two cases are considered: a > bSingle mode operationfThe maximum band for single mode operation is 2 fc10.b > a/2Single mode operationf
14 Dominant Mode: TE10 ModeFor this mode we haveHence we have
15 Dispersion Diagram for TE10 Mode Lossless Case(“Light line”)Phase velocity:Velocities are slopes on the dispersion plot.Group velocity:
16 Field Plots for TE10 ModeTop viewSide viewEnd view
17 Field Plots for TE10 Mode (cont.) Top viewSide viewEnd view
18 Power Flow for TE10 Mode Time-average power flow in the z direction: Note:In terms of amplitude of the field amplitude, we haveFor a given maximum electric field level (e.g., the breakdown field), the power is increased by increasing the cross-sectional area (ab).
19 Attenuation for TE10 Mode (calculated on previous slide)Recallon conductorSide walls
20 Attenuation for TE10 Mode (cont.) Top and bottom walls(since fields of this mode are independent of y)
21 Attenuation for TE10 Mode (cont.) Assume f > fc(The wavenumber is taken as that of a guide with perfect walls.)Simplify, usingFinal result:
22 Attenuation in dB/m Attenuation [dB/m] dB/m = 8.686 [np/m] Let z = distance down the guide in meters.Attenuation[dB/m]HencedB/m = [np/m]
23 Attenuation for TE10 Mode (cont.) Brass X-band air-filled waveguide(See the table on the next slide.)
24 Attenuation for TE10 Mode (cont.) Microwave Frequency BandsLetter DesignationFrequency rangeL band1 to 2 GHzS band2 to 4 GHzC band4 to 8 GHzX band8 to 12 GHzKu band12 to 18 GHzK band18 to 26.5 GHzKa band26.5 to 40 GHzQ band33 to 50 GHzU band40 to 60 GHzV band50 to 75 GHzE band60 to 90 GHzW band75 to 110 GHzF band90 to 140 GHzD band110 to 170 GHz(from Wikipedia)
25 Modes in an X-Band Waveguide Standard X-band waveguide (WR90)Modefc [GHz]50 mil (0.05”) thickness
26 Example: X-Band Waveguide Determine and g at 10 GHz and 6 GHz for the TE10 mode in an air-filled waveguide.
27 Example: X-Band Waveguide (cont.) Evanescent mode: = 0; g is not defined!
28 Circular Waveguide TMz mode: a z The solution in cylindrical coordinates is:Note: The value n must be an integer to have unique fields.
31 Circular Waveguide (cont.) Choose (somewhat arbitrarily)The field should be finite on the z axisis not allowed
32 Circular Waveguide (cont.) B.C.’s:HenceSketch for a typical value of n (n 0).xn1xn2xn3xJn(x)Note: Pozar uses the notation pmn.Note: The value xn0 = 0 is not included since this would yield a trivial solution:This is true unless n = 0, in which case we cannot have p = 0.
40 Cutoff Frequency: TEz x´np values TE11, TE21, TE01, TE31, …….. p \ n 1 123453.8321.8413.0544.2015.3175.4167.0165.3316.7068.0159.28210.52010.1738.5369.96911.34612.68213.98713.32411.70613.170TE11, TE21, TE01, TE31, ……..
41 TE11 Mode The dominant mode of circular waveguide is the TE11 mode. Electric fieldMagnetic field(From Wikipedia)TE10 mode ofrectangular waveguideTE11 mode ofcircular waveguideThe mode can be thought of as an evolution of the TE10 mode of rectangular waveguide as the boundary changes shape.
42 TE01 ModeThe TE01 mode has the unusual property that the conductor attenuation decreases with frequency. (With most waveguide modes, the conductor attenuation increases with frequency.)The TE01 mode was studied extensively as a candidate for long-range communications – but eventually fiber-optic cables became available with even lower loss. It is still useful for some high-power applications.
43 TE01 Mode (cont.) ac f fc, TE11 fc, TM01 fc, TE21 fc, TE01 TE01 TE21 P0 = 0 at cutoff
44 TE01 Mode (cont.) Practical Note: The TE01 mode has only an azimuthal ( - directed) surface current on the wall of the waveguide. Therefore, it can be supported by a set of conducting rings, while the lower modes (TE11 ,TM01, TE21, TM11) will not propagate on such a structure.(A helical spring will also work fine.)
45 TE01 Mode (cont.)From the beginning, the most obvious application of waveguides had been as a communications medium. It had been determined by both Schelkunoff and Mead, independently, in July 1933, that an axially symmetric electric wave (TE01) in circular waveguide would have an attenuation factor that decreased with increasing frequency . This unique characteristic was believed to offer a great potential for wide-band, multichannel systems, and for many years to come the development of such a system was a major focus of work within the waveguide group at BTL. It is important to note, however, that the use of waveguide as a long transmission line never did prove to be practical, and Southworth eventually began to realize that the role of waveguide would be somewhat different than originally expected. In a memorandum dated October 23, 1939, he concluded that microwave radio with highly directive antennas was to be preferred to long transmission lines. "Thus," he wrote, “we come to the conclusion that the hollow, cylindrical conductor is to be valued primarily as a new circuit element, but not yet as a new type of toll cable” . It was as a circuit element in military radar that waveguide technology was to find its first major application and to receive an enormous stimulus to both practical and theoretical advance.K. S. Packard, “The Origins of Waveguide: A Case of Multiple Rediscovery,” IEEE Trans. MTT, pp , Sept