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# Waveguides Study Notes for Electronics and Communication Engineering

By BYJU'S Exam Prep

Updated on: September 25th, 2023

**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

**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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