2TM waves in rectangular waveguides Finding E and H components in terms of z, WG geometry, and modes.FromExpanding for z-propagating field for the lossless WG getswhere
3Method of separation of variables (1) Assumewhere X = f(x) and Y = f(y).Substituting XY givesand we can show thatfor lossless WG.
4Method of separation of variables (2) Letandthen we can writeWe obtain two separate ordinary differential equations:
5General solutions Appropriate forms must be chosen to satisfy boundary conditions.
6Properties of wave in rectangular WGs (1) in the x-directionEt at the wall = 0 Ez(0,y) and Ez(a,y) = 0and X(x) must equal zero at x = 0, and x = a.Apply x = 0, we found that C1 = 0 and X(x) = c2sin(xx).Therefore, at x = a, c2sin(xa) = 0.
7Properties of wave in rectangular WGs (2) 2. in the y-directionEt at the wall = 0 Ez(x,0) and Ez(x,b) = 0and Y(y) must equal zero at y = 0, and y = b.Apply y = 0, we found that C3 = 0 and Y(y) = c4sin(yy).Therefore, at y = a, c4sin(yb) = 0.
8Properties of wave in rectangular WGs (3) andtherefore we can write
9TM mode of propagationEvery combination of integers m and n defines possiblemode for TMmn mode.m = number of half-cycle variations of the fields in the x-directionn = number of half-cycle variations of the fields in the y-For TM mode, neither m and n can be zero otherwise Ezand all other components will vanish therefore TM11 is thelowest cutoff mode.
12Ex2 A rectangular wg having the interior dimension a = 2 Ex2 A rectangular wg having the interior dimension a = 2.3cm and b = 1cm filled with a medium characterized by r = 2.25, r = 1Find h, fc, and c for TM11 modeIf the operating frequency is 15% higher than the cutoff frequency, find (Z)TM11, ()TM11, and (g)TM11. Assume the wg to be lossless for propagating modes.
13TE waves in rectangular waveguides (1) Ez = 0FromExpanding for z-propagating field for a lossless WG getswhere
14TE waves in rectangular waveguides (2) In the x-directionSince Ey = 0, then fromwe haveat x = 0 and x = a
15TE waves in rectangular waveguides (3) In the y-directionSince Ex = 0, then fromwe haveat y = 0 and y = b
16Method of separation of variables (1) Assumethen we have
17Properties of TE wave in x-direction of rectangular WGs (1) in the x-directionat x = 0,at x = a,
18Properties of TE wave in x-direction of rectangular WGs (2)
19Properties of TE wave in y-direction of rectangular WGs (1) 2. in the y -directionat y = 0,at y = b,
20Properties of TE wave in y-direction of rectangular WGs (2) For lossless TE rectangular waveguides,
23A dominant mode for TE waves For TE mode, either m or n can be zero, if a > b, is a smallest eigne value and fc is lowest when m = 1 and n = 0 (dominant mode for a > b)
24A dominant mode for TM waves For TM mode, neither m nor n can be zero, if a > b, fc is lowest when m = 1 and n = 1
25Ex1 a) What is the dominant mode of an axb rectangular WG if a < b and what is its cutoff frequency? b) What are the cutoff frequencies in a square WG (a = b) for TM11, TE20, and TE01 modes?
26Ex2 Which TM and TE modes can propagate in the polyethylene-filled rectangular WG (r = 2.25, r = 1) if the operating frequency is 19 GHz given a = 1.5 cm and b = 0.6 cm?
27Rectangular cavity resonators (1) At microwave frequencies, circuits with the dimensioncomparable to the operating wavelength become efficientradiatorsAn enclose cavity is preferred to confine EM field, providelarge areas for current flow.These enclosures are called ‘cavity resonators’.There are both TE and TM modesbut not unique.bda
28Rectangular cavity resonators (2) z-axis is chosen as the reference.“mnp” subscript is needed to designate a TM or TE standingwave pattern in a cavity resonator.
29Electric field representation in TMmnp modes (1) The presence of the reflection at z = d results in a standingwave with sinz or cozz terms.Consider transverse components Ey(x,y,z),from B.C. Ey = 0 at z = 0 and z = d1) its z dependence must be the sinz type2)similar to Ex(x,y,z).
30Electric field representation in TMmnp modes (2) FromHz vanishes for TM mode, therefore
31Electric field representation in TMmnp modes (3) If Ex and Ey depend on sinz then Ez must vary according to cosz, therefore
32Magnetic field representation in TEmnp modes (1) Apply similar approaches, namelytransverse components of E vanish at z = 0 and z = d- require a factor in Ex and Ey as well as Hz.factor indicates a negative partial derivative with z.- require a factor for Hx and Hyfmnp is similar to TMmnp.
33Dominant modeThe mode with a lowest resonant frequency is called ‘dominant mode’.Different modes having the same fmnp are called degenerate modes.
34Resonator excitation (1) For a particular mode, we need toplace an inner conductor of the coaxial cable where theelectric field is maximum.introduce a small loop at a location where the flux of thedesired mode linking the loop is maximum.source frequency = resonant frequency
35Resonator excitation (2) For example, TE101 mode, only 3 non-zero components areEy, Hx, and Hz.insert a probe in the center region of the top or bottomface where Ey is maximum or place a loop to coupleHx maximum inside a front or back face.Best location is affected by impedance matchingrequirements of the microwave circuit of which theresonator is a part.
36Coupling energy method place a hole or iris at the appropriate locationfield in the waveguide at the hole must have a componentthat is favorable in exciting the desired mode in theresonator.
37Ex3 Determine the dominant modes and their frequencies in an air-filled rectangular cavity resonator fora > b > da > d > ba = b = d