Unit 5 Waveguides P a g e 1

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Unit 5 Waveguides P a g e Syllabus: Introduction, wave equation in Cartesian coordinates, Rectangular waveguide, TE, TM, TEM waves in rectangular guides, wave impedance, losses in wave guide,

1 Unit 5 Waveguides P a g e Syllabus: Introduction, wave equation in Cartesian coordinates, Rectangular waveguide, TE, TM, TEM waves in rectangular guides, wave impedance, losses in wave guide, introduction to circular waveguide. CONVENTION [NOT IMP] Te oter convention for incident, reflected and transmitted field components at boundary between mediums and is as follows. Te capital letter represents te field. For example, E represents electric field intensity and H represents magnetic field intensity. A vector field is represented by an arrow over te field, like and. Te first subscript represents component of field. For example, represents component of field in X direction. : Te subscript 's' represents pasor form of te field component. : Subscript or represent te medium in wic te field component is referred. : Superscripts ' ' or ' ' represent te wave propagation in positive or negative reference direction respectively. For example, : represents pasor component of electric field intensity in X direction in medium, propagating in positive (Y or Z) direction. : represents scalar component of magnetic field intensity in Y direction in medium, propagating in negative (X or Z) direction. RECTANGULAR WAVE GUIDE 7-8

2 Unit 5 Waveguides P a g e TRANSVERSE COMPONENT OF ELECTRIC & MAGNETIC FIELD (IN TERMS OF COMPONENT IN Z DIRECTION OF PROPAGATION) [NOT IMP] Maxwell s st and nd equation in pasor form for perfect dielectric medium ( ) is given by ---() ---() Expanding equation () / / / Equation field components on bot sides ---(3a) ---(3b) ---(3c) Expanding equation () / / / Equation field components on bot sides ---(4a) ---(4b) ---(4c) We know from te solution of wave equation Were, is te complex propagation constant in te direction of propagation, it is given as, it is measured in te unit of Differentiating above equation w.r.t z 7-8

3 Unit 5 Waveguides P a g e 3 ( ) ( ),,, Putting te values of above four equations in equation 3 and 4 ---(5a) ---(5b) ---(5c) ---(6a) ---(6b) ---(6c) Putting te value of from equation (5a) into (6b) to get, tus on solving, we get, Tis topic is incomplete TEM WAVE Wy TEM wave cannot be propagated in Rectangular waveguides? (S-4, S-5/4M) Wy TEM wave does not exist in rectangular waveguide? (W-5/3M)(W-6/4M) Te transverse components of Electric and Magnetic fields in a rectangular waveguide are given in terms of components in direction of propagation (i.e. & ) as ---() ---() ---(3) ---(4) For TEM wave in rectangular waveguide te Z components of Electric and Magnetic wave are zero... & Putting tese values in equations (), (), (3) & (4), we get ; ; ; Tus, all te transverse components of Electric and Magnetic fields become zero. Hence, TEM wave cannot be propagated in Rectangular waveguides (or Hence, TEM wave does not exist in rectangular waveguide). 7-8

4 Unit 5 Waveguides P a g e 4 CUTOFF FREQUENCY, CUTOFF WAVELENGTH, PHASE VELOCITY For Transverse Electric (TE) and Transverse Magnetic (TM) waves, we know, Also, we know, Also, we know, Tis equation defines about propagation constant of TE and TM waves in Rectangular waveguide. At ig frequency, te term in above equation becomes larger tan, tus negative sign will dominate under te root. Hence, te solution of root will give complete imaginary value. We also know tat, Tus and For tis frequency, te velocity of propagation is known as Pase Velocity. It is given as 7-8

5 Unit 5 Waveguides P a g e 5 / Wen frequency is reduced, a point is reaced wen, Tus, was previously zero. Tis frequency is called as cut-off frequency of te waveguide. It is denoted by frequency, te propagation of wave ceases.. At tis It is given as, Above equation sows cutoff frequency for mode. Te wavelengt corresponding to cutoff frequency is known as cutoff wavelengt It is given as / / Above equation sows cutoff wavelengt for mode. For frequencies less tan cutoff frequency, < Tus becomes completely real. 7-8

6 Unit 5 Waveguides P a g e 6 Define te following: (i) Pase Velocity, (ii) Group velocity, (iii) Cutoff frequency, (iv) Caracteristics impedance, (v) Cutoff wavelengt & (vi) Wave impedance. (S-6/6M) (S-4/6m) Te velocity of propagation of equipase surface of te wave along a conductor is called as Pase Velocity. Or in oter words, it is te rate at wic te pase of te wave propagates in space. Or it is te velocity at wic te pase of any one frequency component of te wave travels. Te velocity at wic electromagnetic energy propagates in te direction of propagation of wave is called as Group Velocity. Or in oter words, it is te velocity wit wic te overall sape of te wave's amplitudes propagates troug space. Te frequency of a wave at wic tere is neiter a wave propagation nor any attenuation of te field components is called as cut-off frequency of waveguide. Te wavelengt of a wave corresponding to cut-off frequency, at wic tere is neiter a wave propagation nor any attenuation of te field components is called as cut-off wavelengt of waveguide. Te ratio of mutually perpendicular electric field component to magnetic field component as seen in te direction of propagation is called as Wave Impedance Z. Define te following caracteristics associated wit te waveguide: (i) Cutoff frequency, (ii) Cutoff wavelengt, (iii) Pase velocity (6M/S-4) Define pase and group velocity. Derive an expression for group velocity or pase velocity in rectangular waveguide. (S-4, S-5/8M) Wat is pase velocity? Derive an expression for pase velocity of wave in rectangular waveguide. (W-6/4M) Sow tat/prove tat geometric mean of pase velocity and group velocity is equal to velocity of ligt. (W-4/8M, W-6/7M) Te pase wavelengt is given by te geometric mean of two numbers, is just te square root of teir product 7-8

7 Unit 5 Waveguides P a g e 7 Te group wavelengt is given by Multiplying and, we get, Tus, geometric mean of pase velocity and group velocity equals velocity of ligt. RECTANGULAR WAVEGUIDE Explain ow rectangular waveguide acts as a ig pass filter. (W-5/4M) Sow tat te rectangular waveguide act as ig pass filter. (S-5/5M) Also derive an expression for cutoff frequency (W-4/7M) Te propagation constant in a rectangular waveguide is given as For frequencies less tan cutoff frequency,.. <,.. < will be purely real,, Te imaginary becomes zero. Tis means pase sift does not occur wit respect to distance covered at lower frequencies ( < ), tus ceases propagation in rectangular waveguide. For frequencies greater tan cutoff frequency,.. > 7-8

8 Unit 5 Waveguides P a g e 8... > will be purely imaginary,, Te real term becomes zero. Tis means pase sift occurs wit respect to distance covered at iger frequencies ( > ), tus propagation in rectangular waveguide takes place. Hence, Rectangular waveguide acts as Hig pass filter. Derive te following expression for a rectangular waveguide Were Guide wavelengt; Cutoff wavelengt, wat is guide wavelengt? (S-6/7M) Te wavelengt is given by Te pase constant is given as terefore, / also / / / 7-8

9 Unit 5 Waveguides P a g e 9 WAVE IMPEDANCE Wave Impedance of TE Wave (most important topic) Wat is wave Impedance? Sow tat impedance for TE waves in rectangular waveguide is always greater tan free space impedance. (S-5/5M) wat is wave impedance? Derive te expression for wave impedance for TE wave in rectangular waveguide. (W-5/6M) (W-4/7M) (W-6/6M) Wave impedance is an impedance experienced by electromagnetic fields wile propagating in free space. It is defined as te ratio of te transverse electric and magnetic fields... Te electric and magnetic fields of TE wave in rectangular waveguide in pasor form are as below cos sin sin cos cos sin sin cos cos cos cos sin sin cos cos sin sin cos 7-8

10 Unit 5 Waveguides P a g e cos sin sin cos sin cos cos sin As Tis is te wave impedance of TE wave in rectangular waveguide. Were, is te intrinsic impedance for free space or free space impedance. As te value of is less tan unity, >. i.e. impedance for TE waves in rectangular waveguide is always greater tan free space impedance. Wave Impedance of TM Wave Wave impedance is an impedance experienced by electromagnetic fields wile propagating in free space. It is defined as te ratio of te transverse electric and magnetic fields... Te electric and magnetic fields of TM wave in rectangular waveguide in pasor form are as below cos sin sin cos cos sin sin cos sin sin 7-8

11 Unit 5 Waveguides P a g e cos sin sin cos sin cos cos sin cos sin sin cos sin cos cos sin As Tis is te wave impedance of TM wave in rectangular waveguide. Were, is te intrinsic impedance for free space or free space impedance. Find te wave impedance for dominant mode in air filled rectangular guide of dimension 7cm x 4cm operating at a frequency of 4 GHz. (S-4/4M) Given: and 4 4.4, 4 Te Wave impedance is given as For Dominant Mode,..,, Te frequency is given as Te Mode wit lowest cut-off frequency is called as Dominant Mode. For TE, te dominant mode is and for TM te dominant mode is 7-8

12 Unit 5 Waveguides P a g e Ω Find te wave impedance for te dominant mode in an air filled rectangular waveguide of dimension 8 cm x 5 cm operating at a frequency of 5 GHz. (S-5/4M) Given: and 5 5.5, 5 Te Wave impedance is given as For Dominant Mode,..,, Te frequency is given as Ω A parallel plate waveguide as a spacing a 4 cm and filled wit dielectric (error- ere it is ). Find intrinsic wave impedance for te TE and TM mode at a frequency of.5 GHz. (S-6/6M) Given: 4.4 ; 9;.5 For Mode of TE and TM,, Let (9) 5.58 Ω 7-8

13 Unit 5 Waveguides P a g e Wave impedance for : Ω Wave impedance for : Te dominant mode for TM wave is, ence, te mode will not propagate. Te wave impedance for tis mode may be calculated as below Ω A rectangular waveguide as cross section dimension a 7 cm, b 4 cm. Determine all te modes wic will propagate at a frequency of (i) 3 MHz, (ii) 5 MHz (W-5/6M) Given: 7.7 ; 4.4 ; To determine all modes, we must start from lowest to igest mode, i.e. from,,, and,, Part (i): Given: 3 3 For mode, m, n < As <, tis mode will propagate at 3 For mode, m, n > As >, tis mode will not propagate at 3 For mode, m, n 7-8

14 Unit 5 Waveguides P a g e > As >, tis mode will not propagate at 3 For mode, m, n > As >, tis mode will not propagate at 3 Part (ii): Given: 5 5 For mode, m, n < As <, tis mode will propagate at 5 For mode, m, n < As <, tis mode will propagate at 5 For mode, m 3, n > As >, tis mode will not propagate at 5 For mode, m, n < As <, tis mode will propagate at 5 For mode, m, n < As <, tis mode will propagate at 5 For mode, m, n 7-8

15 Unit 5 Waveguides P a g e > As >, tis mode will not propagate at 5 For mode, m, n > As >, tis mode will not propagate at 5 A rectangular waveguide of cross sectional dimensions of (.5x) cm is used to transmit a signal of 9 GHz. Determine te cutoff frequency for te dominant mode. Find te pase and group velocity of te dominant mode. (S-5/4M) Given:.5.5 ;. ; 9 ; Dominant mode for TE is and for TM is For mode, m, n ; For mode, m, n ; Pase Velocity is given as For dominant mode, 9 ( 9 ) ().5 () / Group Velocity is given as (3 )

16 Unit 5 Waveguides P a g e 6.5 / A rectangular waveguide of dimensions of (5 x ) cm is used to transmit a signal of 5GHz. Determine te cutoff frequency for TE mode. Also, find pase velocity and group velocity for TE mode. (W-6/5M) Given: 5.5 ;. ; 5 ; ; ; ;.5. 3 Pase Velocity is given as For dominant mode, 5 ( 5 ) ().5 () / Group Velocity is given as (3 ) / A rectangular waveguide as a dimension of (3X) cm is used to transmit a signal of 9 GHz. Determine cutoff wavelengt for dominant mode. Also, find (i) Guide wavelengt, (ii) Group Velocity (iii) Pase velocity (iv) Wave impedance (v) Pase constant. (W-6/7M) Given: 3.3 ;. ; 9 ; ; ; ;.3. 5 /

17 Unit 5 Waveguides P a g e 7 ( 5 ) / ().5 () / Group Velocity is given as (3 ) / Wave impedance for dominant mode : Ω Wave impedance for dominant mode : Ω Pase Constant / Wat will be te cutoff wavelengt for dominant mode in rectangular waveguide wose breadt is cm. For.5 GHz signal, calculate guide wavelengt, group and pase velocities and cutoff frequency. (W-4/6M) Given:. ;.5 ; 7-8

18 Unit 5 Waveguides P a g e 8 ; ; ;. /.5.. (.5 ) / (). () Pase velocity is given as.5 / Group Velocity is given as (3 ) / A ollow rectangular waveguide as inner dimension of 7 cms x 4 cms. Find cutoff frequencies in TE, TE, and TM modes. Wy is TE mode usually preferred? (S-4/6M) A ollow rectangular waveguide 3 x 4.5 cm internally and as a 9 GHz signal propagating in it. Calculate cutoff wavelengt, te guide wavelengt, pase velocity, group velocity and caracteristics impedance for te TE mode. Given: 4.5 ; 3 ; 9 ; For mode, i.e. ; (Since ) (a) Cut-off wavelengt: 7-8

19 Unit 5 Waveguides P a g e (b) Guide Wavelengt (Pase wavelengt): () () ( ) ( 9 ) 3 / (c) Pase Velocity: 3. / (d) Group Velocity: (3 ).78 / 3. (e) Caracteristics impedance: Ω 7-8

Photograph of the rectangular waveguide components

Waveguides Photograph of the rectangular waveguide components BACKGROUND A transmission line can be used to guide EM energy from one point (generator) to another (load). A transmission line can support