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

∇^{2}H_{z} = –ω^{2}μεH_{z}

For TE wave (E_{z} = 0)

∇^{2}E_{z} = –ω^{2}μεE_{z}

For TM wave (H_{z} = 0), γ = α + iβ

γ = Propagation constant

β = Phase constant

α = Attenuation constant

γ^{2} + ω^{2}με = h^{2}

(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 γ^{2} = –ω^{2}με,

**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 ω
^{2}με = h^{2}, 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 ω^{2}μ^{2}ε > h^{2}

**Case 3 **If ω^{2}μ^{2}ε < h^{2}

**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

λ_{0} = Free space wavelength

λ_{c} = Cut-off wavelength

when λ_{0} << λ_{c} ⇒ λ_{g} = λ_{0}

when λ_{0} = λ_{c} ⇒ λ_{g} is infinite

at λ_{0} > 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 ∙ λ_{0}

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} = c^{2}v_{g} =

**Note:** Te_{10}, TE_{01,} TE_{20} etc. modes can exist in rectangular waveguide but only TM_{11}, TM_{12}, TM_{21} 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

- 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,

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 TE
_{10}mode is given as

η_{0} = Intrinsic impedance for free space [η_{0} = 377Ω]

R_{s} = Surface resistance (Ω/m^{2})

but

ρ = Resistivity

σ = Conductivity in S/m

δ = Skin depth (corresponds to skin losses)

For free space μ = μ_{0}μ_{r}

μ_{r} = 1 and μ_{0} = 4π × 10^{–7} H/m for free space.

**TE Modes in Rectangular Waveguide**

- TE
_{00}mode : m = 0, n = 1 It cannot exist, as all the field components vanishes. - TE
_{01}mode: m = 0, n = 1 E_{y}= 0, H_{x}= 0 and E_{x}H_{y}exist. - TE
_{10}mode: m = 1, n = 0 E_{x}= 0, H_{y}= 0, E_{y}and H_{x}exist. - TE
_{11}mode: m = 1 and n = 1;

For TE_{10} mode, λ_{c10} = 2a

TE_{01} mode, λ_{c01} = 2b

TE_{11} 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,

∇^{2}H_{z} = 0 and ∇^{2}E_{z} = 0

**TE Modes in Circular Waveguide: **Helmholtz equation of H_{z} in circular guide is given as

∇^{2}H_{z} = γ^{2} ∙ 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

∇^{2}E_{z} = γ^{2}E_{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
- TE
_{11}is the dominant mode in circular waveguide for TE_{11}, 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 TE_{10} 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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