Study Notes: Waveguides | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

Waveguides (Single Lines)

• The term waveguide may refer to any linear structure that conveys electromagnetic waves between its end points. At frequencies more than 3 GHz losses in the transmission lines and cables become significant due to the losses that occur in the dielectric needed to support the conductor and within the conductor itself.
  
• 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:

1. 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

Rectangular coordinate system in Rectangular waveguide

• (for TE wave)
  (for TM wave)

• Solving above equations we find Ez and Hz. Also applying Maxwell equations we can find E_x, H_x, E_y, H_y.
  
  
  

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

Electric and magnetic fields in rectangular waveguide

TE₁₀ and TE₂₀ mode in rectangular waveguide

➢ 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 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 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 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, λ_g = Guide wavelength
  λ₀ = Free space wavelength
  λc = 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 (up)

• v_p = λ_g ∙ f but c = f ∙ λ₀
• For propagation of signal in the guide, λ_g > λ₀, 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 a 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.
  v_g = dω/dβ
• For free space v_p = v_g and v_p∙v_g = c²vg =
• Te₁₀, TE₀₁, 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:
(i) Losses in the dielectric
(ii) 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 R_s = ρ/δ = 1/σδ
  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.

2. 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
  Circular Waveguide

➢ 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 Eφ i.e., J'_n(P'_nmr/α 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
  H = Pnm/α, n = 0, 1, 2, 3 and m = 1, 2, 3, 4

➢ Key Points

• For TE wave P'nm/α and Pnm/α for TM waves
• 
• TE₁₁ is the dominant mode in circular waveguide for TE₁₁, P'nm = 1.841
  So λ_c for TE₁₁ = 2πα/1.841 also for TM wave λ_c = 2π/h = 2πα/P_nm

Note: TEM mode cannot exist in circular waveguide.

(a) Circular waveguide

TE₀₁ Mode in circular waveguide

(b) 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: