**Transmission Lines and Waveguides**

**Image Credit: Matthew T Rader, https://matthewtrader.com, Transmission towers and lines at sunset in East Texas, CC BY-SA 4.0**

**Points of Discussion : Transmission lines and Waveguides**

*Introduction to Transmission lines and Waveguides**Types of Waveguides**Types of transmission lines**Parallel Plate Waveguide**Rectangular Waveguide**Circular Waveguide*

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**Introductions to Transmission Lines(TL) & Waveguide(WG)**

The invention and development of transmission lines and other waveguides for the low-loss transmission of power at high frequency are among the earliest milestones in the history of microwave engineering. Previously Radio Frequency and related studies were revolved around the different types of transmission medium. It has advantages for controlling high power. But on the other hand, it is inefficient in controlling at lower values of frequencies.

Two wires lines cost less, but they have no shielding. There are coaxial cables that are shielded, but it is difficult to fabricate the complicated microwave components. Advantage of Planar line is that it has various versions. Slot lines, co planar lines, micro-strip lines are some of its forms. These types of transmission lines are compact, economical, and easily integrable with active circuit devices.

Parameters like constant of propagation, characteristic impedance, attenuation constants consider how a transmission line will behave. In this article, we will learn about the various types of them. Almost all transmission lines (who have multiple conductors) are capable of supporting the transverse electromagnetic waves. The longitudinal field components are unavailable for them. This particular property characterizes the TEM lines and wave-guides. They have a unique voltage, current, and characteristic impedance value. Waveguides, having a single conductor, may support TE (transverse electric) or TM (transverse magnetic), or both. Unlike Now, Transverse Electric and Transverse Magnetic modes have their respective longitudinal field components. They are represented by that property.

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**Types of waveguides**

Though there are several types of waveguides, some of the most popular are listed below.

**Types of Transmission Lines**

Some of the types of transmission lines are listed below.

- Stripline
- Microstrip line
- Coaxial line

**Parallel Plate Waveguide**

Parallel plate waveguide is one of the popular types of waveguide, which are capable of controlling both Transverse Electric and Transverse magnetic modes. One of the reason behind the popularity of parallel plate waveguide is that they have applications in model making for the greater-order modes in lines.

The above image (Transmission lines and waveguides) shows the geometry of the parallel plate waveguide. Here, the strip width is W and considered more significant than the separation of d. That is how fringing field and any x variables can be cancelled. The gap between two plates is filled up by a material of permittivity ε and permeability of μ.

**TEM Modes**

**TEM Modes**

The solution of the TEM Modes is calculated with the help of solution of the Laplace’s equation. The equation is calculated considering the factor for the electrostatic voltage which lies in between the conductor plates.

Solving, the equation, the transverse electric field comes as:

**e ^{–} (x,y) = ∇_{t} ϕ (x,y) = – y^{^} V_{o} / d.**

Then, the total electric field is: **E ^{–} (x, y, z) = h^{– }(x, y) e^{– jkz} = y^{^} (V_{o} / d) * e^{-jkz}**

k represents the propagation constant. It is given as: **k = w √ (μ * ε)**

The magnetic fields’ equation comes as:

Here, η refers to the intrinsic impedance of the medium which lies in between the conductor plates of parallel plate waveguides. It is given as: η = √ (μ / ε)

**TM Modes**

**TM Modes**

Transverse magnetic or TM modes can be characterized by H_{z} = 0 and a finite electric field value.

**(∂ ^{2} / ∂y^{2} + k^{2}_{c}) e_{z }(x, y) = 0**

Here kc is the cut-off wavenumber and given by **k _{c} = √ (k^{2} − β^{2})**

After the solution of the equation, the Electric filed E_{X} comes as:

**E _{z} (x, y, z) = A_{n} sin (n * π * y / d) * e^{– jβz}**

The transverse field components can be written as:

**H _{x }= (jw ε / k_{c}) A_{n} cos (nπy / d) e^{– jβz}**

**E _{y} = (-jB/ k_{c}) A_{n} cos (nπy / d) e^{– jβz}**

**E _{x} = H_{y} = 0.**

The cut off frequency of TM mode can be written as:

*f*_{c}= k_{c} / (2π * √ (με)) = n / (2d * √(με))

The wave impedance comes as **Z _{TM} = β / ωε**

The phase velocity: **v _{p} = ω / β**

The guide wavelength: **λ _{g} = 2π / β**

*TE Modes*

*TE Modes*

**H _{z} (x,y) = B_{n} cos (nπy / d) e^{– jβz}**

Equations of the transverse fields are listed below.

The propagation constant **β = √ (k2 – (nπ/d ) ^{2})**

The cutoff frequency: *f*_{c} = n / (2d √ (με))

The impedance of the TM mode: **Z _{TE} = E_{x} / H_{y} = k_{n}/ β = ωμ/ β**

**Rectangular waveguide**

The rectangular waveguide is one of the primary types of waveguide used to transmit microwave signals, and still, they have been used.

With miniaturization development, the waveguide has been replaced by planar transmission lines such as strip lines and microstrip lines. Applications which uses highly rated power, which uses millimeter wave technologies, some specific satellite technologies still use the waveguides.

As the rectangular waveguide has not more than two conductors, it is only capable of Transverse Magnetic and Transverse Electric Modes.

*TE Modes*

*TE Modes*

The solution for H_{z} comes as: **H _{z} (x, y, z) = A_{mn} cos (mπx/a) cos (nπy/b) e^{– jβz}**

Amn is a constant.

The field components of the TEmn modes are listed below:

The propagation constant is,

*TM Modes*

*TM Modes*

The solution for E_{z} comes as:** E _{z} (x, y, z) = B_{mn} sin (mπx/a) sin (nπy/b) e^{– jβz}**

Bmn is constant.

The field component of TM mode are calculated as below.

Propagation constant **:**

The wave impedance: **Z _{TM} = E_{x} / H_{y} = -E_{y} / H_{x} = bη * η / k**

**Circular Waveguide**

The circular waveguide is a muffled, round pipe structure. It supports both the TE and TM modes. The below image represents the geometrical description of a circular waveguide. It has an inner radius ‘a,’ and it is employed in cylindrical coordinates.

**E _{ρ} = (− j/ k^{2}_{c}) [ β ∂E_{z}/ ∂ρ + (ωµ/ρ) ∂ H_{z}/ ∂φ]**

**E _{ϕ} = (− j/ k^{2}_{c}) [ β ∂E_{z}/ ∂ρ – (ωµ/ρ) ∂ H_{z}/ ∂φ]**

**H _{ρ} = (j /k^{2}_{c}) [(ωe/ ρ) ∂E_{z} /∂φ − β ∂ H_{z}/ ∂ρ]**

**H _{ϕ} = (-j /k2_{c}) [(ωe/ ρ) ∂E_{z} /∂φ + β ∂ H_{z}/ ∂ρ]**

*TE Modes*

*TE Modes*

The wave equation is:

**∇ ^{2}H_{z} + k^{2}H_{z} = 0.**

**k: ω√µe**

The propagation constant: **B _{mn} = √ (k^{2} – k_{c}^{2})**

Cutoff frequency: **fc _{nm} = k_{c} / (2π * √ (με))**

The transverse field components are:

**E _{p} = (− jωµn /k^{2}_{c}ρ) * (A cos nφ − B sin nφ) J_{n} (k_{c}ρ) e^{− jβz}**

**H _{φ} = (− jβn/k^{2}_{c}ρ) (A cos nφ − B sin nφ) J_{n} (k_{c}ρ) e^{− jβz}**

The wave impedance is:

**Z _{TE} = E_{p} / H_{ϕ} = – E_{ϕ} / H_{p} = ηk / β**

*TM Modes*

*TM Modes*

To determine the necessary equations for the circular waveguide operating in Transverse magnetic modes, the wave equation is solved and the value of Ez is calculated. The equation is solved in cylindrical coordinates.

**[∂ ^{2} /∂ρ^{2} + (1/ρ) ∂/ ∂ρ + (1 /ρ^{2}) ∂^{2}/ ∂φ^{2} + k^{2}c] e_{z} = 0,**

TM_{nm} Mode’s Propagation Constant ->

**βnm = √ (k ^{2 }– kc^{2}) = √ (k^{2} − (p_{nm}/a)^{2})**

**Cutoff frequency: f _{cnm} = k_{c} / (2π√µε) = p_{nm} / (2πa √µε)**

The transverse fields are:

**E _{ρ} = (− jβ/ k_{c}) (A sin nφ + B cos nφ) J_{n}^{/ }(k_{c}ρ) e^{− jβz}**

**E _{φ} = (− jβn /k^{2}_{c}ρ) (A cos nφ − B sin nφ) J_{n} (k_{c}ρ) e^{− jβz}**

**H _{ρ} = (jωen /k^{2} _{c}ρ) (A cos nφ − B sin nφ) J_{n} (k_{c}ρ) e^{− jβz}**

**H _{φ} = (− jωe/ k_{c}) (A sin nφ + B cos nφ) Jn` (k_{c}ρ) e^{− jβz}**

**The wave impedance is Z _{TM} = E_{p} / H_{φ} = – E_{ϕ}/H_{p} = ηβ/k**

**Stripline**

One of the examples of planar type transmission line is Stripline. It is advantageous for incorporation inside microwave circuits. Stripline can be of two types – Asymmetric Stripline and Inhomogeneous stripline. As stripline has two conductors, thus it supports the TEM mode. The geometrical representation is depicted in the below figure.

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