researchpaper_Mode-Patterns-of-Parallel-plates-&Rectangular-wave-guides

International Journal of Scientific & Engineering Research Volume 3, Issue 8, August -2012 1 ISSN 2229 -5518 IJSER © 2012 http://www.ijser.org Mode Pattern s of Parallel plates &Rectangular wave guides Mr.K.Chandrashekhar, Dr.Girish V Attimarad Abstract -Parallel plate and rectangular waveguide both support Transverse Magnetic (TM) and Transverse Electric (TE) wave propagation and also it is well known that TM and TE modes have characteristic cutoff frequencies. TE/TM Waves of frequencies below the cutoff frequency of a particular mode cannot propagate through the waveguide. Computer simulations were performed to study this phenomenon, for various modes. Simulation study results show that only particular mode at a particular frequency will result in propagation. This study can be considered as a workbe nch for further designing of waveguides. Index Terms - Maxwell’s equations, Mode patterns, parallel plate waveguides, rectangular waveguide TE mode, TM Mode, wave propagation. 1 INTRODUCTION The C haracteristic s of the waves propagating along uniform guiding structures , Wave guiding structure may consists of two co -axial conductors or two parallel pl ates or it may be single hollow conductor called waveguide. Th ree types of transmission mode s are T rans verse Electric and Magnetic (TEM ) waves, Transverse Electric (TE) wave and Transverse Magnetic (TE) wave. A rectangular, Circular ,elliptical and hollow , Metallic waveguides supports only TE and TM modes but TEM waves cannot exist in a single - hollow conductor wave guide shape because TEM are characterized by E z=0, H z=0 and f c=0. [1] To support all TEM modes for a TEM wave two separate conductor struct ure is required , such as Co -axial cable, Parallel plate waveguide, Strip line and micro strip lines . The two conductor lines can be analyzed in terms of voltage, current and impedance by the distributed circuit theory [4]. The rest of the paper is organize d into some sections. Section II describes TM and TE wave equations in Parall el plate waveguide , Section 3 describes TM and TE wave equations in Rectangu lar plate waveguide , Section 4 discusses simu lation results finally section 5 Concludes. 2 PARALLEL PLATE WAVEGUIDE Parallel plate wave guide m ainly consist of two perfectly conducting plates separated by a dielectric medium with constituent parameters ε and µ. The plates are assumed to be of infinite length in x-direction as shown in figure .1. Let us supp ose that TM waves (Hz=0) propagate in the +z -direction. 2.1 TM waves Transverse magnetic (TM) wave do not have component of the magnetic field in the direction of propagation hence Hz=0. The behavior of the TM wave can be analyzed using the equation 1. Fig 1: parallel plate waveguide [3] [2] (1) The above equation satisfy the boundary condition We conclude that must be the following form (2) Where the amplitude An de pends on the strength of the particular TM wave . (3) (4) Propagation constant is given by Cut off frequency can be obtained by substituting fc= The instantaneous field expressions for TM 1 mode are obtained by multiplying equations (2) , (3) and (4) (5) (6) (7) International Journal of Scientific & Engineering Research Volume 3, Issue 8, August -2012 2 ISSN 2229 -5518 IJSER © 2012 http://www.ijser.org In the yz - plane E has both a y- and a z- component and the relation of the electric field lines at a given time can be obtain from the following equation . (8) Put t=0 in the above equation (8) and which can be rewritten as (9) Which gives the slope of the el ectric field lines and integrating above equation we get Several such electric field l ines are shown in the figure (3) Similarly (10) Since H has only an x -component the magnetic field lines are everywhere perpendicular to the yz -plane as shown in Fig 3 . 2.2 TE waves [3] Transverse electric waves, we solve the following equation for which is simplified (11) Consider the boundary conditions to be satisfied by are obtained (12) From equation (11) we obtain (13) Where Bn is the amplitude depends on the strength of exec ution of the particular TE wave,and we can also obtain other non zero field components (14) (15) The cut -off frequency for the TEn modes in a parallel plate wave guide is exactly the same as for the TM m mode. For n=0 both Hy and E x vanish; hence th e TE 0 mode doesn’t exist in a parallel plate waveguides. 3 RECTANGULAR WAVEGUIDE : Rectangular wavegu ides are the one of the type of transmission lines. They are used in many a pplications such as isolators, detectors, attenuators, couplers and slot ted lines and these waveguide components are available for various standard waveguide bands between 1 GHz to above 220 GHz. A rectangular waveguide supports TM and TE modes but not TEM waves. Shape of rectangular waveguide is as shown in Fig 2 . A mater ial permittivity e and permeability m fills the inside of the conductor. In a rectangular waveguide wave propagate s below some frequency called as cut-off frequency. Here, we will discuss TM wave propagation and TE wave propagation in rectangular waveguide s separately. Let’s start with the TM mode. Fig 2: Rectangular waveguide 3.1 TM waves [3] Consider the shape of the rectangular waveguide as shown above with dimensions a ,b (assume a>b) and the parameters e and m. For TM waves Hz =0 and E z can be solved from the equation given below 2 xy Ez0+h2Ez0=0 (16) Since Ez(x, y, z) =E z0 (x, y) e -gz , we get the following equation , ( + Ez0 (x,y) =0 (16) Using the method of separation of variables, that is Ez0 (x,y)=X(x),Y(y) we get, (18) Since the right side contains x terms only and the left side contains y terms only hence both are equal to a con stant. Calling that constant as k x2, we get (19) (20) Where =h2- Solve for X and Y from the previous equations. Also applying the following boundary conditions, International Journal of Scientific & Engineering Research Volume 3, Issue 8, August -2012 3 ISSN 2229 -5518 IJSER © 2012 http://www.ijser.org From all these, we conclude that X(x) is in the form of sin kx x, where kx=mp/a, m=1, 2, 3… Y(y) is in the form of sin ky y, where ky=np/a, n=1,2, 3… (21) So the solution for is From =h2- , we have (22) (22) For TM waves, we have (23) (24) (25) From these above equations, we get (26) (27) (28) (29) Where (30) Where m and n represent possible modes and it is designated as the TM mn .m denotes the number of half cycle variations of the fields in the x -direction and n denotes the number o f half cycle variations of the fields in the y -direction. Observing the above equations it can be concluded that TM modes in rectangular waveguides will not exist for m or n zero. This is because of the fact that the expressions are identically zero if ei ther m or n is zero. Therefore, the lowest possible values of m and n in a rectangular waveguide are 1 and 1; i.e. TM mode is TM 11. Cut-off wave number and propagation constant is given by (31) The cut -off frequency is at the point (32) Since l=u/f, we have the cut -off wavelength, (33) At a given operating frequency f, only those frequencies, which satisfy the condition fc <f will propagate. The mode with the lowest cut -off frequency is called the dominant mode . Similarly modes for rectangular waveguides start from TM 11 mode, the dominant frequency (34) The wave impedance is defined as the ratio of the transverse elec tric and magnetic fields. Therefore, we get from the expressions for E x and H y (see the equations above) ZTM (35) The guide wavelength is defined as the distance between two equal planes in the waveguide and it’s equal to > = (36) This is thus greater than I, the wavelength of a plane wave in the filling medium. International Journal of Scientific & Engineering Research Volume 3, Issue 8, August -2012 4 ISSN 2229 -5518 IJSER © 2012 http://www.ijser.org The phase velocity is given by (37) This is thus greater than the speed of light (plane wave) in the filling material 3.2 TE waves For TE waves Ez =0 and Hz can be solved from the equation given below Ñ2 xy Hz + h2 Hz=0 Since Hz (x, y,z) = Hz 0(x,y)e -gz, the following equation, (38) Use the method of separation of variables, tha t is we get, (39) Since the right side contains x terms only and the left side contains y terms only; hence they are both equal to constant. Calling that constant as kx2, we get; (40) (41) Where =h2- Solve for x and y from the preceding equations. Also we have following boundary conditions: at x=0 (42) at x=a (43) at y=0 (44) at y=b (45) From all these, we get (46) From =h2- , we have; (47) For TE waves, we have (48) (49) (50) (51) From the above equations, we can obtain the following equations . (52) (53) (54) (55) Where (56) Where m and n represent possible modes and it is shown as the TE mn mode. m denotes the number of half cycle variations of the fields in the x -direction and n the number of half cycle variations of the fields in the y -direction. Here, the cut -off wave nu mber and propagation is given by (57) International Journal of Scientific & Engineering Research Volume 3, Issue 8, August -2012 5 ISSN 2229 -5518 IJSER © 2012 http://www.ijser.org (58) The cut -off frequency is given by (59) Since l=u/f, we have the cut -off wavelength, (60) At a giv en operating frequen cy f, which have f> fc will propagate. The modes with f<fc will not propagate. The mode with the lowest cut -off frequency is called the dominant mode. Since mode is the minimum possible mode that gives nonzero field expressions for rect angular waveguides, it is the dominant mode of a rectangular waveguide with a>b and dominant frequency is (61) The wave impedance is defined as the ratio of the transverse electric and magnetic fields. Therefore, we get from the expre ssions for Ex and Hy ( refer above equations) ZTE (62) The guide wavelength is defined as the distance between two equal phase planes the waveguide and it’s equal to > = (63) The phase velocity is (64) This is thus greater than the speed of plane wave in the filling material . 4 RESULTS Parallel plate waveguides TM Modes Fig 3: Field lines for TM 1 Mode in parallel plate waveguides Parallel plate waveguides TE Modes Fig 4: Field lines for TE 1 Mode in parallel plate waveguides Mr.K.chandrashekhar is pursuing his doctoral degree fromVTU,Belgaum,India.e -mail:[email protected] Dr.Girish V Attimarad is a professor & Head DSCET,Bangalore,India.e -mail:[email protected] Rectangular waveguide TM modes Fig 5: Field lines for TM 11 Mode in rectangular waveguide Rectangular waveguide TE 10 Mode International Journal of Scientific & Engineering Research Volume 3, Issue 8, August -2012 6 ISSN 2229 -5518 IJSER © 2012 http://www.ijser.org Fig 6: Field lines for TE 10 Modes in rectangular waveguide 5 CONCLUSION A rectangular, Circular, elliptical a nd hollow, Metallic waveguides supports only TE and TM modes but TEM waves don’t exist in a single -conductor hollow wave guide because TEM waves are characterized b y Ez=0, Hz=0 and fc=0. To support all TEM modes for a TEM wave two separate conductor structure is required, such as Co -axial cable, Parallel plate waveguide, Strip line and micro strip lines. The two conductor lines can be analyzed in terms of voltage, current and impedance by the distributed circuit theory. The methodology applied to derive field equations can be extended to other waveguides also. Obtained equations and Computer Simulation results can be compared to understand the field lines and the Pr opagation characteristics. REFERENCES [1] N.Marcuvitz “ Waveguide Handbook” The institution of Electrical Engineers. Dec-1985 [2] David.K.Cheng “Field and Wave Electromagnetics” 2nd Ed, 2006 Tsinghua University Press pp-520-560 [3] Annapurna Das,Ssir K Das “Microwave Engineering “ Mc-Grawhill HE, Sept 2006. [4] Constantine A Balanis “Advanced Engineering Electromagnetics” John Wiley and Sons 1989