In general, a waveguide consists of a hollow metallic tube of a rectangular or circular shape used to guide an electromagnetic waves by successive reflections from the inner walls of tube.
- In the waveguide, no Transverse Electromagnetic (TEM) wave/mode can exist, but Transverse Electric (TE) and Transverse Magnetic (TM) waves can exist.
- The dominant mode in a particular guide is the mode having the lowest cut-off frequency.
Types of Waveguide: The waveguides can be classified based on these shapes given below
Rectangular Waveguide: Rectangular waveguide is situated in the rectangular coordinate system with its breadth along x-direction, width along y-direction and z-indicates direction of propagation.
Vector Helmholtz equations
∇2Hz = –ω2μεHz
For TE wave (Ez = 0)
∇2Ez = –ω2μεEz
For TM wave (Hz = 0), γ = α + iβ
γ = Propagation constant
β = Phase constant
α = Attenuation constant
γ2 + ω2με = h2
(for TE wave)
(for TM wave)
- Solving above equations we find Ez and Hz.
- Also applying Maxwell equations we can find Ex, Hx, Ey, Hy.
Note: For TEM wave Ez = 0 and Hz = 0, putting these values in equations (I to IV), all the field components along x and y directions, Ex, Ey, Hx, Hy vanishes and hence TEM wave cannot exist inside a waveguide.
TE and TM Modes: The electromagnetic wave inside a waveguide has an infinite number of patterns, called as modes. Generally two types of mode (TE and TM) are present in the waveguide. These modes are denoted as TEmn and TMmn.
m = Half wave variation along wider dimension a
n = Half wave variation along narrow dimension b
TE Mode in Rectangular Waveguides: TEmn modes in rectangular cavity are characterized by Ez = 0 i.e., z component of magnetic field Hz must be existing in order to have energy transmission in guide. TEmn field equations in rectangular waveguide as,
Ez = 0
Propagation Constant: The propagation of the wave in the guide is assumed in positive z-direction. Propagation constant γg in waveguide differs from intrinsic propagation constant γ of dielectric.
is cut-off wave number
For lossless dielectric γ2 = –ω2με,
Cut-off Wave Number
The cut-off wave number h is defined by
for TEmn mode
There are three cases for the propagation constant γg in waveguide.
- If ω2με = h2, then γg = 0, hence there will be no wave propagation (evanescence) in the guide.
- Thus at a given operating frequency f, only those mode having f > fc will propagate, and modes with f < fc will lead to imaginary β (or real α).
- Such modes are called evanescent modes. The cut-off frequency is
Case 2 If ω2μ2ε > h2
Case 3 If ω2μ2ε < h2
Note: So wave cannot propagate through waveguide as γg is a real quantity.
- For free space/ loss less dielectric (α = 0)
- The phase velocity in the positive z-direction for the TEmn
is the phase velocity in vacuum.
i.e., vp = vg = c (velocity of light).
- The characteristic wave impedance of TEmn mode in the guide
- Characteristic impedance of free space is 377 Ω.
- All wavelengths greater than λc are attenuated and those less than λc are allowed to propagate through waveguide (acts as high pass filter).
Guide Wavelength: It is nothing but distance travelled by wave in order to undergo phase shift of 2π radian.
where, λg = Guide wavelength
λ0 = Free space wavelength
λc = Cut-off wavelength
when λ0 << λc ⇒ λg = λ0
when λ0 = λc ⇒ λg is infinite
at λ0 > λc, λg is imaginary i.e., no propagation in the waveguide.
Phase Velocity (up):
vp = λg ∙ f but c = f ∙ λ0
For propagation of signal in the guide, λg > λ0, so vp is greater than velocity of light but this is contradicting as no signal travel faster than speed of light. However, vp represents the velocity with which wave changes its phase in terms of guide wavelength i.e., phase velocity.
Group Velocity (ug): If any modulated signal is transmitted through guide, then modulation envelope travels at slower speed than carrier and of course slower than speed of light.
For free space vp = vg and vp∙vg = c2vg =
Note: Te10, TE01, TE20 etc. modes can exist in rectangular waveguide but only TM11, TM12, TM21 etc. can exist.
Power Transmission in Rectangular Waveguide
for TEmn mode
for TMmn mode
where a and b are the dimensions of waveguide and is intrinsic impedance of free space.
TM Waves/Modes in Rectangular Waveguide
For TM mode Hz = 0 i.e., the z component of electric field E must exist in order to have energy transmission in the guide.
The TMmn mode field equations are
Hz = 0
Some of the TM mode, characteristic equations are same as that of TE mode but some are different and they are given as
Power Loss in a Waveguide: There are two ways of power losses in a waveguide as given below
- Losses in the dielectric
- Losses in the guide walls
If the operational frequency is below the cut-off frequency, propagation constant y will have only the attenuation term u, i.e., β will be imaginary implying that no propagation but total wave attenuation.
So attenuation constant
So this is the attenuation at f < fc but for f > fc there is very low loss.
fc = cut off frequency
Also attenuation due to non-magnetic dielectric is given by,
- δ–loss tangent of the dielectric material is given as,
- The attenuation constant due to imperfect conducting walls in TE10 mode is given as
η0 = Intrinsic impedance for free space [η0 = 377Ω]
Rs = Surface resistance (Ω/m2)
ρ = Resistivity
σ = Conductivity in S/m
δ = Skin depth (corresponds to skin losses)
For free space μ = μ0μr
μr = 1 and μ0 = 4π × 10–7 H/m for free space.
TE Modes in Rectangular Waveguide
- TE00 mode : m = 0, n = 1 It cannot exist, as all the field components vanishes.
- TE01 mode: m = 0, n = 1 Ey = 0, Hx = 0 and ExHy exist.
- TE10 mode: m = 1, n = 0 Ex = 0, Hy = 0, Ey and Hx exist.
- TE11 mode: m = 1 and n = 1;
For TE10 mode, λc10 = 2a
TE01 mode, λc01 = 2b
Similarly for TM mode also, different modes represents different cut-off wavelength.
Circular Waveguide: A circular waveguide is a tabular circular conductor. Figure shows circular waveguide of radius a and length z, placed in cylindrical coordinate systems.
- A plane wave propagating through a circular waveguide results in TE and TM modes.
- The vector Helmholtz wave equation for a TE and TM wave travelling in a z-direction in a circular waveguide is given as,
∇2Hz = 0 and ∇2Ez = 0
TE Modes in Circular Waveguide: Helmholtz equation of Hz in circular guide is given as
∇2Hz = γ2 ∙ Hz
TEmn modes in circular waveguide
Ez = 0
= represent characteristic wave impedance in the guide,
when n = 0, 1, 2, 3 and m = 1, 2, 3, 4,…..
The first subscript n represents, number of full cycles of field variation in one revolution through 2π radian of φ, while second subscript m indicates the number of zeros of Eφ i.e., along the radius of a guide.
The phase velocity, group velocity and guide wavelength remains same as that of rectangular waveguide.
TM Modes in Circular Waveguide: The TMnm modes in a circular guide are defined as Hz = 0. But Ez ≠ 0, in order to transmit energy in the guide.
Helmholtz equation in terms of Ez in circular guide is
∇2Ez = γ2Ez
The field equation for TMnm modes are given as
Hz = 0
n = 0, 1, 2, 3 and m = 1, 2, 3, 4
- For TE wave and for TM waves
- TE11 is the dominant mode in circular waveguide for TE11, So λc for also for TM wave
Note: TEM mode cannot exist in circular waveguide.
Power Handling Capacity: For rectangular waveguide:(in watt)
where, Ed = Dielectric strength of material, fc = Cut off frequency for TE10 mode, f = Operating frequency, and fmax = Maximum frequency
- For circular waveguide:
Power Transmission in Circular Waveguide or Coaxial Lines
For a loss less dielectric:
where, = Wave impedance in guide, a Radius of the circular guide,
- The average power transmitted through a circular waveguide for TEnp modes is given by
- For a TMnp modes
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