1 / 16

EM Waveguiding

EM Waveguiding. Overview Waveguide may refer to any structure that conveys electromagnetic waves between its endpoints Most common meaning is a hollow metal pipe used to carry radio waves May be used to transport radiation of a single frequency

kieu
Download Presentation

EM Waveguiding

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. EM Waveguiding Overview • Waveguidemay refer to any structure that conveys electromagnetic waves between its endpoints • Most common meaning is a hollow metal pipe used to carry radio waves • May be used to transport radiation of a single frequency • Transverse Electric (TE) modes have E┴kg (propagation wavevector) • Transverse Magnetic (TM) modes have B┴kg • Transverse Electric-Magnetic modes (TEM) have E, B┴kg • A cutoff frequency exists, below which no radiation propagates

  2. EM Waveguiding Electromagnetic wave reflection by perfect conductor E┴can be finite just outside conducting surface E|| vanishes just outside and inside conducting surface EI EI┴ D┴1= D┴2 ER EI┴ ER┴ ER┴ EI|| ER|| qr qi D┴1 = eoE┴1 - - - - - - - y y D┴2 = eoeE ┴2 E||1 = E||2 z z EoI+ EoR = 0 EI|| EI|| EI|| ER|| ER|| ER|| y EoT= 0 z

  3. EM Waveguiding Electromagnetic wave propagation between conducting plates Boundary conditions B┴1= B┴2 E||1 = E||2 (1,2 inside, outside here) E|| must vanish just outside conducting surface since E = 0 inside E┴ may be finite just outside since induced surface charges allow E = 0 inside (TM modes only) B┴ = 0 at surface since B1 = 0 Two parallel plates, TE mode k2 E2 E1 k1 x z b y q b

  4. EM Waveguiding E = E1 + E2 = exEoeiwt (ei(-kysinq + kzcosq) - ei(kysinq + kzcosq)) = exEoeiwte-ikzcosq2i sin( kysinq) Boundary condition E||1 = E||2 = 0 means that E = E||vanishes at y = 0, y = b E||(y=0,b) if ky sinq= np n = 1, 2, 3, .. Fields invacuum E1= exEoei(wt - k1.r) k1= -eyk sinq + ez k cosq k1.r= -kysinq+ kzcosq E2= -exEoei(wt - k2.r) k2= +eyk sinq + ez k cosq k2.r= +ky sinq + kzcosq

  5. EM Waveguiding Allowed field between guides is E = exEoeiwte-ikzcosq2i sin( kysinq) = exEoeiwt e-ikzcosq2isin(npy/b) Since The wavenumber for the guided field is kg = k cosqn = 1, 2, 3, .. Profile of the first transverse electric mode (TE1) Fields E1= exEoei(wt - k1.r) k1= -eyk sinq + ez k cosq k1.r= -kysinq+ kzcosq E2= -exEoei(wt - k2.r) k2= +eyk sinq + ez k cosq k2.r= +ky sinq + kzcosq Ex sin(npy/b) y

  6. EM Waveguiding Magnetic component of the guided field from Faraday’s Law  x E = -∂B/∂t = -iwB for time-harmonic fields B= i x E /w = 2Eo / w(0, ikgsin(npy/b), √(- kg)cos(npy/b) )ei(wt - kgz) The BC B┴1 = B┴2 = 0 is satisfied since By = 0 on the conducting plates. The E and B components of the field are perpendicular since Bx = 0. The phase velocity for the guided wave is vp = w / kg = c k / kg kg = Hence vp= c The group velocity for the guided wave is vg = ∂w/ ∂kg= c ∂k/ ∂kg = c kg/k vpvg = c2

  7. EM Waveguiding Frequency Dispersion and Cutoff cutoff when → 1 w= ck = 2pn n = = ncutoff= kg== b b q q’ 2 modes 1 propagating mode kg vacuum propagation

  8. EM Waveguiding Summary of TEn modes E= 2 Eo(i sin(npy/b), 0 ,0) ei(wt - kgz)kg= B= 2Eo / w(0, ikg sin(npy/b), √(- kg)cos(npy/b) )ei(wt - kgz) Phase velocity vp = w / kg = c k / kg E B Group velocity vg= ∂w/ ∂kg= c kg/ k ncutoff,n = = x x y y viewed along kg

  9. EM Waveguiding Electric components of TEn guided fields viewed along x (plan view) n = 1 n = 2 n = 3 n = 4 Magnetic components of TEnguided fields viewed along x (plan view) z z y y

  10. EM Waveguiding Rectangular waveguides Boundary conditions B┴1 = B┴2 E||1 = E||2 E|| must vanish just outside conducting surface since E = 0 inside E┴ may be finite just outside since induced surface charges allow E = 0 inside B┴ = 0 at surface Infinite, rectangular conduit 0 a x z y b

  11. EM Waveguiding TEmn modes in rectangular waveguides TEn modes for two infinite plates are also solutions for the rectangular guide E field vanishes on xz plane plates as before, but not on the yz plane plates Charges are induced on the yz plates such that E = 0 inside the conductors Let Ex = C f(x)sin(npy/b) ei(wt- kgz) In free space .E= 0 and Ez= 0 for a TEmnmode and ∂Ez/∂z = 0 Hence ∂Ex/∂x = -∂Ey/∂y f(x) = -np/ b cos(mpx/a) satisfies this condition • By integration • Ex= -C np / b cos(mpx/a) sin(npy/b) ei(wt - kgz) • Ey= C mp / a sin(mpx/a) cos(npy/b) ei(wt - kgz) • Ez = 0

  12. EM Waveguiding Dispersion Relation Substitute into wave equation(2 - 1/c 2 ∂ 2/∂t2 )E= 0 2Ex,y =Ex,y ∂ 2/∂t2Ex,y = - w2Ex,y - w2/ c2 = 0 kg = Magnetic components of the guided field from Faraday’s Law Bx = -C mp / a / wsin(mpx/a) cos(npy/b) ei(wt - kgz) By= -C np/ b / wcos(mpx/a) sin(npy/b) ei(wt - kgz) Bz= iC√) / w cos(mpx/a) cos(npy/b) ei(wt - kgz)

  13. EM Waveguiding Cutoff Frequency kg = ncutoff=

  14. EM Waveguiding Electric components of TEmn guided fields viewed along kg m = 0 n = 1 m = 1 n = 1 m = 2 n = 2 m = 3 n = 1 Magnetic components of TEmnguided fields viewed along kg x x y y

  15. EM Waveguiding Comparison of fields in TE and TM modes www.opamp-electronics.com/tutorials/waveguides_2_14_08.htm

  16. EM Waveguiding The TE01 mode Most commonly used since a single frequency ncutoff,02 > n >ncutoff,01 can be selected so that only one mode propagates. Example 3 cm radar waves in a 1cm x 2 cm guide ncutoff,01= c = 7.5 x 109 Hz ncutoff,01= c = 7.50 x 109 Hz ncutoff,10= c = 1.50 x 1010 Hz ncutoff,11= c = 1.68 x 1010 Hz

More Related