Lectre 7 Wavegides TC 41 Microwave Commnications RS 1
Review Impedance matching to minimie power relection rom load Lmped-element tners Single-stb tners Microstrip lines The most poplar transmission line Knowing the characteristic impedance and the relative dielectric constant o the material helps determine the strip line conigration and vice versa. Attenation condction loss dielectric loss radiation loss
A pair o condctors is sed to gide TM wave Microstrip Parallel plate Two-wire TL Coaxial cable 3
The se o wavegide Wavegide reers to the strctre that does not spport TM mode. They are nable to spport wave propagation below a certain reqency, termed the cto reqency. Figres (a) and (b) are metallic wavegide. The rectanglar one is or high-power microwave applications bt is limited in reqency range and tends to ser rom dispersion. The circlar wavegide has higher power handling capability than the rectanglar one. The dielectric wavegide has a smaller loss than metallic wavegide at high req. Optical iber has wider bandwidth and provides good signal isolation between adjacent ibers. 4
Rectanglar wavegide ndamentals - A typical cross section is shown. Propagation is in the + direction. The condcting walls are brass, copper or alminm. The inside is electroplated with silver or gold and smoothly polished to redce loss. - The interior dimensions are a x b, the longer side is a. - a determines the reqency range o the dominant, or lowest order mode. - Higher modes have higher attenation and be diiclt to extract rom the gide. - b aects attenation; smaller b has higher attenation. - I b is increased beyond a/, the next mode will be excited at a lower reqency, ths decreasing the sel reqency range. - In practice, b is chosen to be abot a/. 5
Rectanglar wavegide ndamentals - Wavegide can spport T and TM modes. - The order o the mode reers to the ield conigration in the gide and is given by m and n integer sbscripts, as Tmn and TMmn. - The m sbscript corresponds to the nmber o hal-wave variations o the ield in the x direction. - The n sbscript is the nmber o hal-wave variations in the y direction. - m and n determines the cto reqency or a particlar mode. c mn 1 m a n b The relative cto req or the irst 1 modes o wavegide with a = b are shown. 6
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Wave Propagation (a) A y-polaried TM plane wave propagating in the + direction. (b) Waveront view o the propagating wave Note: the waves propagate at a velocity U, where sbscript indicates media nbonded by gide wall. In air, U = c 9
Now a pair o identical TM waves, labeled as U+ and U- are sperimposed. The horiontal lines can be drawn on the sperimposed waves that correspond to ero total ield. Along these lines, the U+ wave is always 180 o ot o phase with the U- wave. Next, replace the horiontal lines with perect condcting walls. 10
- a is the wall separation and is determined by the angle and wavelength - For a given wave velocity U, the reqency is = U/ - I we ix the wall separation at a and change the reqency, we mst then also change the angle i we are to maintain a propagating wave. - The edge o a +o waveront (point A) will line p with edge o a o waveront (point B), and two ronts mst be / apart or the m = 1 mode. For any vale o m, we can write sin m a or a m sin 11
The wavegide can spport propagation as long as the wavelength is smaller than a critical vale that occrs at = 90 o, or c a sin 90 m where c is the cto reqency or the propagating mode. We can relate the angle to the operating reqency () and the cto reqency (c) by sin The time t AC it takes or the waveront to move rom A to C is l AD m cos c c t AD l AD p o t AC a m l AC m cos p c m Meanwhile, a constant phase point moves along the wall rom A to D. Calling the phase velocity, Up: so 1
Since the times t AD and t AC mst be eqal, we have p cos c 1 The phase velocity can be considerably aster than the velocity o the wave in nbonded media, tending toward ininity as approaches c. The phase constant associated with the phase velocity is 1 Where is the phase constant in nbonded media. The wavelength in the gide is related to this phase velocity by = c 1 c 13
The propagation velocity o the sperposed wave is given by the grop velocity U G. G cos 1 The grop velocity is slower than that o an ngided wave, which is to be expected since the gided wave propagates in a ig-ag path, boncing o the wavegide walls. c Wavegide Impedance The ratio o the transverse electric ield to the transverse magnetic ield or a propagating mode at a particlar reqency. Z T mn c 1 Z TM mn c 1 where is the intrinsic impedance o the propagating media. In air = o = 10 14
Wavegide impedance o the T11 and TM11 modes vs reqency or WR90 15
xample: determine the T mode impedance looking into a 0-cm long section o shorted WR90 wavegide operating at 10 GH. Soltion: at 10 GH, only the T10 mode is spported, so We can ind Zin rom Now is ond rom Z T 10 So, l in the Zin eqation is 1 10π Ω 6. 5GH 10GH 500 Ω Z IN jzo tan( l) or a shorted line. 1 c c c 9 ( 1010 H) 1 6. 56GH 10GH 158 8 310 m / s l 1 rad/m ( 158 rad/m)(0. m) 31.6 rad And Zin is then calclated as Z IN j( 500 Ω) tan(31.6) j100ω 16
General wave behaviors along niorm giding strctres (1) The wave characteristics are examined along straight giding strctres with a niorm cross section sch as rectanglar wavegides. We can write in the instantaneos orm as t t (, ) 0 cos( ) We begin with Helmholt s eqations: assme WG is illed in with a charge-ree lossless dielectric H H 0 0 17
General wave behaviors along niorm giding strctres () so that We can write and in the phasor orms as H ( x, y, ) ( a x a y a ) e ( xy ) ( xy ) xy ( ) 0 xy and H ( ) H 0. xy x y and H ( x, y, ) ( H a x H a y H a ) e. x y Note that y are nctions o x and y only. The - H x H x y H dependency comes rom the term e -. In other words, or example x y ( x, ( x, y, y, ) ) x y ( x, ( x, y) e y) e aˆ aˆ x y and so on. 18
Use Maxwell s eqations to show and H in terms o components (1) From we have jh and y y x jh y jh x y x jh x y H y H j H x H y H x j x y y H x j y x x H j 19
Use Maxwell s eqations to show and in terms o components () 0 We can express x, y, H x, and H y in terms o and H by sbstittion so we get or lossless media = j, and H x y x y H j j y x H j j x y H j j H y x H j j H y x
Propagating waves in a niorm wavegide Transverse lectromagnetic (TM) waves, no or H Transverse Magnetic (TM), non-ero bt H = 0 Transverse lectric (T), non-ero H bt = 0 1
Transverse lectromagnetic wave (TM) Since and H are 0, TM wave exists only when p, TM 0 Z TM 1 m / s x j H j y rad / m A single condctor cannot spport TM
Transverse Magnetic wave (TM) From xy ( ) 0 We can solve or and then solve or other components rom (1)-(4) by setting H = 0, then we have Z TM x y H H j Notice that or j or TM is not eqal to that or TM. y x 3
igen vales We deine h Soltions or several WG problems will exist only or real nmbers o h or eigen vales o the bondary vale problems, each eigen vale determines the characteristic o the particlar TM mode. 4
Cto reqency From h The cto reqency exists when = 0 or or We can write c h h H. h 1 ( ). c c 5
a) Propagating mode (1) 1 c or c h and is imaginary Then h j j j c 1 ( ) 1 ( ). This is a propagating mode with a phase constant : c 1 ( ) / rad m 6
a) Propagating mode () Wavelength in the gide, g 1 where is the wavelength o a plane wave with a reqency in an nbonded dielectric medim (, ) c m. 1 p m 7
a) Propagating mode (3) The phase velocity o the propagating wave in the gide is g p m / s c 1 The wave impedance is then Z TM c 1 8
b) vanescent mode 1 c or c h Then c 1 ( ) real Wave diminishes rapidly with distance. Z TM is imaginary, prely reactive so there is no power low 9
Transverse lectric wave (T) From xyh H ( ) 0 xpanding or -propagating ield gets where H x H y ( ) 0 H We can solve or H and then solve or other components rom (1)-(4) by setting = 0, then we have H H ( x, y) e. Notice that or j or T is not eqal to that or TM. Z T x y H H y x j 30
T characteristics Cto reqency c,, g, and p are similar to those in TM mode. Bt Propagating mode > c Z T 1 c vanescent mode < c 31
x Determine wave impedance and gide wavelength (in terms o their vales or the TM mode) at a reqency eqal to twice the cto reqency in a WG or TM and T modes. 3