EM Waveguiding Overview Waveguide may refer to any structure that conveys electromagnetic waves...
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Transcript of EM Waveguiding Overview Waveguide may refer to any structure that conveys electromagnetic waves...
EM WaveguidingOverview
• 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• 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
EM WaveguidingElectromagnetic wave reflection by perfect conductor
E┴ can be finite just outside
conducting surface
E|| vanishes just outside and
inside conducting surface
qi qr
EI
ER
z
y
EI||
EI┴
ER┴
ER||
z
y
EI┴ ER┴
- - - - - - -
D┴2 = eoeE ┴2
D┴1 = eo E ┴1
D┴1 = D┴2
z
y
EoI + EoR = 0
E||1 = E||2
EI|| ER||EI|| ER|| EI|| ER||
EoT = 0
EM WaveguidingElectromagnetic 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
b
E1E2k1
k2
y
xzb
q
EM WaveguidingE = E1 + E2
= ex Eo eiwt (ei(-ky sinq + kz cosq) - ei(ky sinq + kz cosq))
= ex Eo eiwt e-ikz cosq 2i sin( ky sinq )
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 in vacuum
E1 = ex Eo ei(wt - k1.r)
k1 = -ey k sinq + ez k cosq
k1.r = - ky sinq + kz cosq E2 = -ex Eo ei(wt - k2.r)
k2 = +ey k sinq + ez k cosq
k2.r = + ky sinq + kz cosq
EM WaveguidingAllowed field between guides is
E = ex Eo eiwt e-ikz cosq 2i sin( ky sinq )
= ex Eo eiwt e-ikz cosq 2i sin(npy/b)
Since
The wavenumber for the guided field is
kg = k cosq n = 1, 2, 3, ..
Profile of the first transverse electric mode (TE1)
Fields
E1 = ex Eo ei(wt - k1.r)
k1 = -ey k sinq + ez k cosq
k1.r = - ky sinq + kz cosq E2 = -ex Eo ei(wt - k2.r)
k2 = +ey k sinq + ez k cosq
k2.r = + ky sinq + kz cosq
Ex
y
sin(npy/b)
EM Waveguiding
Magnetic component of the guided field from Faraday’s Law
x E = -∂B/∂t = -iw B for time-harmonic fields
B = i x E / w = 2 Eo / w (0, ikg sin(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
vp vg = c2
EM Waveguiding
Frequency Dispersion and Cutoff
cutoff when → 1
w = ck = 2 pn n = =
ncutoff =
kg==
b b
q q’
0 1 2 3 4 5 6
1
2
3
4
5
6
kg
wc
n = 3
1 propagating mode
2 modes
n = 1
n = 2
vacuum propagation
EM Waveguiding
Summary of TEn modes
E = 2 Eo (i sin(npy/b), 0 ,0) ei(wt - kgz) kg =
B = 2 Eo / 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
y
x
y viewed along kg
EM WaveguidingElectric components of TEn guided fields viewed along x (plan view)
n = 1 n = 2 n = 3 n = 4
Magnetic components of TEn guided fields viewed along x (plan view)
z
y
z
y
EM WaveguidingRectangular 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
0b
y
xz
a
EM WaveguidingTEmn 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 TEmn mode 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
EM WaveguidingDispersion Relation
Substitute into wave equation (2 - 1/c 2 ∂ 2/∂t2 )E = 0
2Ex,y = Ex,y
∂ 2/∂t2 Ex,y = - w2 Ex,y
- w2 / c 2 = 0
kg =
Magnetic components of the guided field from Faraday’s Law
Bx = -C mp / a / w sin(mpx/a) cos(npy/b) ei(wt - kgz)
By = -C np / b / w cos(mpx/a) sin(npy/b) ei(wt - kgz)
Bz = i C√) / w cos(mpx/a) cos(npy/b) ei(wt - kgz)
EM WaveguidingElectric 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 TEmn guided fields viewed along kg
x
y
x
y
EM Waveguiding
Comparison of fields in TE and TM modes
www.opamp-electronics.com/tutorials/waveguides_2_14_08.htm
EM WaveguidingThe 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