Electromagnetic Fields : Waveguides

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

∇²H_z = –ω²μεH_z

For TE wave (E_z = 0)

∇²E_z = –ω²μεE_z

For TM wave (H_z = 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 E_z and H_z.
• Also applying Maxwell equations we can find E_x, H_x, E_y, H_y.

  … (i)

  … (ii)

  … (iii)

  … (iv)

Note: For TEM wave E_z = 0 and H_z = 0, putting these values in equations (I to IV), all the field components along x and y directions, E_x, E_y, H_x, H_y 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 TE_mn and TM_mn.

m = Half wave variation along wider dimension a

n = Half wave variation along narrow dimension b

TE Mode in Rectangular Waveguides: TE_mn modes in rectangular cavity are characterized by E_z = 0 i.e., z component of magnetic field H_z must be existing in order to have energy transmission in guide. TE_mn field equations in rectangular waveguide as,

E_z = 0

Propagation Constant: The propagation of the wave in the guide is assumed in positive z-direction. Propagation constant γ_g 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 TE_mn mode

There are three cases for the propagation constant γ_g in waveguide.

Case 1

• If ω²με = h², then γ_g = 0, hence there will be no wave propagation (evanescence) in the guide.
• Thus at a given operating frequency f, only those mode having f > f_c will propagate, and modes with f < f_c 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 γ_g is a real quantity.

• For free space/ loss less dielectric (α = 0)

• The phase velocity in the positive z-direction for the TE_mn

is the phase velocity in vacuum.

i.e., v_p = v_g = c (velocity of light).

• The characteristic wave impedance of TE_mn mode in the guide

• Characteristic impedance of free space is 377 Ω.

• All wavelengths greater than λ_c are attenuated and those less than λ_c 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, λ_g = Guide wavelength

λ₀ = Free space wavelength

λ_c = Cut-off wavelength

when λ₀ << λ_c ⇒ λ_g = λ₀

when λ₀ = λ_c ⇒ λ_g is infinite

at λ₀ > λ_c, λ_g is imaginary i.e., no propagation in the waveguide.

Phase Velocity (u_p):

v_p = λ_g ∙ f but c = f ∙ λ₀

For propagation of signal in the guide, λ_g > λ_0, so v_p is greater than velocity of light but this is contradicting as no signal travel faster than speed of light. However, v_p represents the velocity with which wave changes its phase in terms of guide wavelength i.e., phase velocity.

Group Velocity (u_g): 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 v_p = v_g and v_p∙v_g = c²v_g = 

Note: Te₁₀, TE_01, TE₂₀ etc. modes can exist in rectangular waveguide but only TM₁₁, TM₁₂, TM₂₁ etc. can exist.

Power Transmission in Rectangular Waveguide

for TE_mn mode

for TM_mn 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 H_z = 0 i.e., the z component of electric field E must exist in order to have energy transmission in the guide.

The TM_mn mode field equations are

H_z = 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

1. Losses in the dielectric
2. 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, 

but 

So, 

So attenuation constant

  dB/length

So this is the attenuation at f < f_c but for f > f_c there is very low loss.

f_c = 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]

R_s = Surface resistance (Ω/m²)

but 

ρ = Resistivity

σ = Conductivity in S/m

δ = Skin depth (corresponds to skin losses)

For free space μ = μ₀μ_r

μ_r = 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 E_y = 0, H_x = 0 and E_xH_y exist.
• TE₁₀ mode: m = 1, n = 0 E_x = 0, H_y = 0, E_y and H_x exist.
• TE₁₁ mode: m = 1 and n = 1; 

For TE₁₀ mode, λ_c10 = 2a

TE₀₁ mode, λ_c01 = 2b

TE₁₁ mode, 

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,

∇²H_z = 0 and ∇²E_z = 0

TE Modes in Circular Waveguide: Helmholtz equation of H_z in circular guide is given as

∇²H_z = γ² ∙ H_z

TE_mn modes in circular waveguide

E_z = 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 E_φ 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 TM_nm modes in a circular guide are defined as H_z = 0. But E_z ≠ 0, in order to transmit energy in the guide.

Helmholtz equation in terms of E_z in circular guide is

∇²E_z = γ²E_z

The field equation for TM_nm modes are given as

H_z = 0

n = 0, 1, 2, 3 and m = 1, 2, 3, 4

Key Points

• For TE wave  and  for TM waves
• TE₁₁ is the dominant mode in circular waveguide for TE₁₁,  So λ_c for  also for TM wave 

Note: TEM mode cannot exist in circular waveguide.

Power Handling Capacity: For rectangular waveguide:(in watt)

where, E_d = Dielectric strength of material, f_c = Cut off frequency for TE₁₀ mode, f = Operating frequency, and f_max = 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 TE_np modes is given by

• For a TM_np modes

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