If we assume that sustituting (4) into (3), we have d H y A()e ;j (4) d +! ; Letting! ;, (5) ecomes d d + where the independent solutions are Hence, H

Transcription

1 W.C.Chew ECE 350 Lecture Notes. Innite Parallel Plate Waveguide. y σ σ 0 We have studied TEM (transverse electromagnetic) waves etween two pieces of parallel conductors in the transmission line theory. We shall study other kinds of waves etween two innite parallel plates, or planes. We have learnt earlier that for a plane wave incident on a plane interface, the wave can e categoried into TE (transverse electric) with electric eld polaried in the y-direction. Hence, etween a parallel plate waveguide, we shall look for solutions of TE type with E ^ye y, or TM (transverse magnetic) type with H ^yh y. We shall assume that the eld does not vary in the y-direction so We have shown earlier that if re 0, the equation for the E eld in a source region is If rh 0,the equation for the H eld I. TM Case, H ^yh y. (r +! )E 0: () (r +! )H 0: () in these two equations. In @ +! H y 0: (3)

2 If we assume that sustituting (4) into (3), we have d H y A()e ;j (4) d +! ; Letting! ;, (5) ecomes d d + where the independent solutions are Hence, H y is of the form where A() A() 0: (5) A() 0 (6) cos sin : (7) cos H y H 0 e ;j (8) sin +! (9) which are the dispersion relation for plane waves. We can also dene cos, sin so that (9) is automatically satised. To decide a viale solution from (8), we look at the oundary conditions for the E-eld at the metallic plates. From rh j!e, we H 0in the aove equation) or H y (0) E! H 0 cos H 0in the aove equation) or e ;j H H () E ; j! H 0 sin ; cos e ;j : (3)

3 The oundary conditions require that E ( 0)E ( ) 0. Only the rst solution gives E ( 0)0. Hence, we eliminate the second solution, or E ; j! H 0 sin( )e ;j : (4) In order for E ( ) 0,we require that or and consequently, sin 0 (5) m m 0 3 ::: (6) m m 0 3 ::: : (7) This is known as the guidance condition for the waveguide. Finally, we have m H y H 0 cos e ;j (8) E! H 0 cos E ; m j! H 0 sin m e ;j (9) m e ;j (0) where m! ; () which is the dispersion relation for the parallel plate waveguide. Equation (8) can e written as H y H 0 [ej + e ;j ]e ;j H 0 [ej;j + e ;j;j ]: () The rst term in the aove represents a plane wave propagating in the positive ^-direction and the negative ^-direction, while the second term corresponds to a wave propagating in the positive and directions. Hence, the eld in etween a parallel plate waveguide consists of a plane wave ouncing ack and forth etween the two plates, as shown. θ β ^ β + ^ β θ β ^ β + ^ β 3

4 Since we dene cos, sin, the wave propagates in a direction making an angle with the ^-direction. Since the guidance condition requires that m cos, the plane wave can e guided only for discrete values of. From (), we note that for dierent m's, will assume dierent values. When m 0,! p, E 0, and we have a TEM mode. When m > 0, we have a TM mode of order m we call it a TM m mode. Hence, there are innitely many solutions to Mawell's equations etween a parallel plate waveguide with the eld given y (8), (9), (0), and the dispersion relation given y () where m 0 3 :::. II. Cuto Frequency From (), for a given TM m mode, if! p < m, then is pure imaginary. In this case, the wave is purely decaying in the ^-direction, and it is evanescent and non-propagating. For a given TM m mode, we can always lower the frequency so that this occurs. When this happens, we say that the mode is cut o. The cuto frequency is the frequency for which a given TM m mode ecomes cuto when the frequency of the TM m mode is lower than this cuto frequency. Hence, When (m +)v! mc m p or f mc >f> mv > (m ; )v m p mv : (3) > (m ; )v >:::>0 (4) the TEM mode plus all the TM n modes, where 0 <n m are propagating or guided while the TM m+ and higher order modes are evanescent or cuto. For the parallel plate waveguide, there is one mode with ero cuto frequency and hence is guided for all frequencies. This is the TEM mode which is equivalent to the transmission line mode. The wavelength that corresponds to the cuto frequency is known as the cuto wavelength, i.e., mc v f mc m : (5) When < mc, the corresponding TM m mode will e guided. You can think of as some kind of the \sie" of the wave, and that only when the \sie" of the wave is less than mc canawave \enter" the waveguide. Notice that mc is proportional to the physical sie of the waveguide. IV. TE Case, E ^ye y. 4

5 The eld for the TE case can e derived similarly to the TM case. The electric eld is polaried in the ^y-direction, @ +! The elds can e shown in a similar fashion to e The oundary conditions are E y 0: (6) E y E 0 sin( )e ;j (7) H ;! E 0 sin( )e ;j (8) H ; j! E 0 cos( )e ;j : (9) E y ( 0)0 E y ( ) 0: (30) This gives m (3) as efore, where +!. Hence, the TE m modes have the same dispersion relation and cut-o frequency as the TM m mode. However, when m 0, 0,and (7){(9) imply that we have ero eld. Therefore, TE 0 mode does not eist. We say that TE m and TM m modes are degenerate when they have the same cuto frequencies. We can decompose (7) into plane waves, i.e., E y E 0 j [ej;j ; e ;j;j ] (3) and interpret the aove as ouncing waves. Compared to (), we see that the two ouncing waves in (3) are of the opposite signs whereas that in () are of the same sign. This is ecause the electric eld has to vanish on the plates while the magnetic eld need not. TM mode eld H-field 5 E-field

6 TE mode eld H-field E-field The sketch of the elds for TM and TE modes are as shown aove. For the TM mode, H 0, and E 6 0, while for the TE mode, E 0, and H 6 0. Tangential electric eld is ero on the plates while tangential magnetic eld is not ero on the plates. The aove is the instantaneous eld plots. E H is in the direction of propagation of the waves. III. Phase and Group Velocities. The phase velocity in the ^-direction of a waveina waveguide is dened to e v p!! h! ; ; i m (33) p f ; mc f which is always larger than the speed of light for f >f mc. The group velocity is v g d! d d d! ; h! ; ; m! which is always less than the speed of light. i ; f mc f p (34) ω TM and TE modes ω C β με ω c TM and TE modes TM TEM mode 0 ω c ω 0c v ω/β p v g dω dβ β 6

7 h Since! ; ; i! mc, a plot of! versus c! is as shown. When! 0, the group velocity ecomes ero while the phase velocity approaches innity. When!, or!!, the group and phase velocities oth approach the velocity of light in free-space which is the TEM wave velocity. 7