Electromagnetic Waves -EC501- Module3( MAKAUT-Syllabus)

Today Our communication elements: -Transmission Lines, Equations of Voltage and Current on Transmission Lines, Propagation Constant and Characteristic Impedance, Reflection Coefficient and VSWR, Tap-free and lower-loss transmission lines, Power Transfer on transmission lines, Smith Chart, Admittance Smith Chart, Applications of Transmission Lines, Impedance matching, Using transmission line sections, such as circuit elements.

Transmission Lines: -

Transmission lines are special cables or structures designed to carry electrical signals or power from one point to another with minimal damage. They are secured in power systems, telecommunications, high existing signal transfers, and efficient and reliable energy or data transfer.

In electrical engineering, a transmission line refers to any conductor that has alternating current (AC) at frequencies where the length of the line is equal to the wavelength of the signal. At low frequencies, simple wires are sufficient, but at high frequencies the physical properties of the line – such as induction, capacitance, and resistance – play an important role in performance.

Types of transfer lines:

• Overhead lines – used in power distribution, these are suspended conductors supported by towers or bars.
• Underground cable – buried under the ground for safety and aesthetics, common in urban areas.
• Waves – hollow metal structures that conduct electromagnetic waves, used in microwaves and satellite communication.
• Coaxial cable is widely used for radio, TV, and internet signals.

Main parameter: 

• Special impedance (Z₀): It determines how the line matches the equipment connected to prevent reflections.
• Dissemination constants: Defines how to remove characters and change steps from the distance.
• Road factor: The ratio of signal speed in a line to light speed in a vacuum.

Application: 

Transfer lines are important:

• Power grid (transport of power from generational facilities to consumers)
• Communication system (voice, video, and data transferred)
• RF and microwave system (antenna connection, radar)

Equations of Voltage and Current on Transmission Lines: -

In the transmission line theory, the distance and over time vary with the voltage and the streamline parameters along the line. At high frequencies or long distances, it is necessary to describe these variations mathematically to understand the indication spread.

Modelling is done using distributed elements to a transmission line: per length of unit, per unit length, per unit length, flow (G) per unit length, and per unit of unit (C) per unit length. By implementing Kirchhoff's laws in an infinite part of the line, we get the equations of the original telegraph:

Here:

• $V(z,t)$ = voltage at position z and time t

• $I(z,t)$ = current at position z and time t

• $\omega$ = angular frequency ($2\pi f$)

Combining these equations leads to wave equations for voltage and current:

Where:  γ=α+jβ

is the propagation constant, with:

• $\alpha$ = attenuation constant (loss per unit length)
• $\beta$ = phase constant (phase change per unit length)

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The general solutions are:

Here $V^+$ and $V^-$ represent forward and backward travelling waves, and $Z_0$ is the characteristic impedance.

Propagation Constant and Characteristic Impedance: -

In the transmission line theory, two basic parameters define how the signals travel and interact with the line: scattering figures and characteristic impedance.

Propagation constant

The spreading constant describes how the voltage and power waves spread and change the stage as they move along the transmission line. This is a complex volume:  γ=α+jβ

Where:

α = decaying constant (per length per unit length), resistance, and indicates loss due to damage.

β = phase constant (per unit length per radian per unit), representing distance phase displacement per unit.

The usual formula for γ is: 

Here, r = per unit length per resistance, L = per unit length per unit induction, g = per unit length, c = copitans per unit length, and 

ω = 2πf is the angular frequency.

Characteristic impedance (z₀): 

The characteristic represents the voltage ratio of the wave running with the line without impedance reflections. It is given by:

For a defective line (R = 0, G = 0) simplifies the formulas:

Importance

• The spread continuously determines the meaning of the signal and the phase formation.
• The characteristic impedance is important to match the line with the lines to prevent reflections.

Reflection Coefficient and VSWR: -

In the transmission line theory, the signal reflection occurs when the characteristic of the load impedance does not match the impedance. There are two main parameters, reflection coefficients and voltage wave conditions (VSWR), used to determine these reflections.

Reflection coefficient

The reflection coefficient measures the fraction of the wave reflected back to the source due to a discrepancy in impedance. It is defined as this:

Where:

• Zl = load impedance
• Z0 properties of transmission line imports

0 to 1 area magazines:

• Γ = 0 → perfect match (no reflection)
• Γ = 1 → total reflection (complete discrepancy)

The reflected tension is proportional to the wave, and its step depends on the signal and the type of discrepancy

Voltage Standing Wave Ratio (VSWR)

VSWR represents the ratio of the minimum tension to the maximum in permanent wave patterns formed due to reflections. It is given by:

VSWR = 1 → Perfect matching; Higher VSWR → Greater mismatch and more power reflected.

VSWR is often used in RF and microwave techniques to assess the performance of the transmission line.

Importance

• A low reflection coefficient and low VSWR indicate effective power transmission.
• High values cause possible damage to components such as signal loss, heating, and amplifiers.

Tap-free and lower-loss transmission lines: -

The impedance change is a basic concept in transmission line theory so that engineers can match the impedance loaded from the source for maximum power transmission. This is especially important in RF systems, antennas, and communication networks.

1. Tapless transmission lines:

A disadvantage – a free transmission line contains zero resistance (R = 0) and zero driving (G = 0). The parameters are only induction (L) and capacitance (C) per unit length. Pedestrians continuously simplify:

The input impedance at a distance $l$ from the load is given by:

Key observations:

• Impedance repeats every half-wavelength ($\lambda/2$).

• A quarter-wavelength line ($\lambda/4$) transforms impedances according to:  

  
  This property is used in quarter-wave transformers for impedance matching.

2. Low-loss Transmission Lines:

There is a small but non-zero R and G in a low-cap line, where: 

R≪ωLandG≪ωC

The propagation constant is approximately:  γ≈α+jβ

and the characteristic impedance:

The impedance changes are similar to the lines of the equation, but it contains a small due factor due to α. Consequently:

The change is slightly less fast than the lines.

Long lines reduce the voltage and current size due to the long lines.

Application:

• RF and microwave circuit – for matching the antenna from the transmitters
• Cable TV – to maintain the strength of the signal at a distance
• Power system – to reduce reflection and ensure stable operations

Power Transfer on Transmission Lines: -

In the transmission line system, an effective design is required to ensure that the maximum energy reaches the load with minimal loss. This concept is important in applications such as RF communication, power distribution, and antenna systems.

Maximum power transfer position:

For maximum power transmission, the load impedance (ZL) must match the impedance (Z0) characteristic of the transmission line. When:  ZL​=Z0

• The reflections are finished.
• All events are absorbed by the power load.

If 𝑍𝐿 of the Z0 does not match, the part of the wave is reflected, which ​reduces the pure power given.

Power Flow Equations:

There is an instant force at some point on the line:  p(z,t)=v(z,t)⋅i(z,t)

For steady-state conditions, the average power is given to the load:

Where:

• $V^+$ = forward voltage amplitude

• $\Gamma$ = reflection coefficient

The term $(1 - |\Gamma|^2)$ shows that power transfer efficiency decreases with higher reflections.

Discrepancy effect

Less discrepant (small ∣Γ∣): Most power when the load.

High error passing (large ∣Γ∣): A significant portion of power is reflected, possibly causing standing waves, overheating, or damage to components.

Practical idea

In the RF system, impedance matching networks (such as quarterwave transformers or stubs) are often used to maximise the power transmission. In power transmission, voltage and current levels are also adapted to reduce resistance injuries in conductors.

Smith Chart:-

The Smith Chart is a powerful graphic tool used in electrical engineering, especially in RF and microwave technology, to solve problems related to transmission lines and matching networks. Philip H. in the 1930s. Developed by Smith, it simplifies complex impedance and reflection coefficient calculation by representing them on a generalised polar plot.

Objectives and features:

Smith diagrams are mainly used:

• Plot and impedance, and analysis of entry.
• Determine the reflection coefficient and voltage wave ratio (VSWR).
• Impedance matching networks are designed for maximum power transmission.

This maps the complex reflection coefficient on a circle so that engineers can imagine how the impedance of the transmission line changes.

Basic Structure:

• Horizontal axis: Pure resistance represents generalised impedance (real axis).
• Circular arches: Continuous resistance and continuous response represent the curves.
• Circist: The reflection coefficients represent the size = 1 (full discrepancy).
• The focal point matches the generalised impedance to 1 + J0 (perfect match).

Smith using the Smith Chart:

• Normalise impedance:   Z=ZL/Z0
• Find the point on the chart that fits the general impedance.
• Go with the constant. A circle represents changes in the situation with the transmission line.
• Read the values of impedance, entry, and VSWR directly from the chart.

Advantage

• Long complex number ends the requirement for calculation.
• The impedance provides a visual understanding of change.
• Useful for designing a plug, matching section, and setting circuit.

Admittance Smith Chart: -

The input Smith diagram is a variant of traditional blacksmith diagrams used in RF and microwave techniques, to represent input (reciprocity of impedance) rather than impedance. Expressed as an entrance:

Where:

• $G$ = conductance (real part)

• $B$ = susceptance (imaginary part)

Objective:

While the Standard Smith diagram is designed for impedance (Z), many network design problems, especially with parallel components, are solved using a more natural approach. The entry provides a direct way to deal with such problems without changing the Smith diagram Z and Y.

structure:

• The entrance Smith diagram looks like the image version, but continuous operation and continuous-existing circles are marked for generalized entry (y = y/y0), where y = 1/z0 is the characteristic entry.
• The focal point matches Y = 1+J0 .
• The transfer clockwise to the Admittance chart corresponds to continuing along the transmission line against the load towards the direction of the image map.

Conversion Between Charts:

An impedance point on the Smith diagram can be converted to an entry by rotating about 180 ° around the center. This property allows engineers to use a single chart as both an impedance and an entrance map by just using this rotation.

Application:

• Stubmatching: Practical for especially parallel stubbonation.
• Antenna matching: Fast conversion between chain and parallel network ideas.
• Microwave Design: To analyze parallel resonance circuits.

Applications of Transmission Lines: -

Transfer lines are important components of electrical and communication systems, which enable the effective transmission of power or signals over distances. Their applications are scattered across different engineering fields:

1. Power transfer

High voltage lines and underground cables transmit electrical energy from stations to distribution networks and consumers, and ensure the minimum power loss.

2. Telecommunication

Coaxial cables, twisted pairs, and fiber-optic cables act as transmission lines to carry speech, videos, and data in telephone and internet systems.

3. RF and microwave system

The radar, which is used to add a transmitter, receiver, and antenna to satellite communication and wireless networks, ensures impedance matching for minimal reflections.

4. Antenna feeds

Feed lines such as coaxial cables or waves provide power from RF equipment to high-efficiency antennas.

5. Instrumentation and measurement

In the existing test layout, the transmission lines maintain signal integrity between the signal generators, oscilloscope, and network analyzers.

6. Electronics and PCB design

Controlled timber marks on printed circuit boards act as short transmission lines for high-speed digital signals.

Impedance matching: -

Impedance matching is the process of making the load impedance equal to the source or transmission line to ensure maximum power transmission and minimum signal reflection. It is an important concept in the power system, RF communication, and high-speed electronics.

Why impedan's matching case

When the characteristics of the Load Impedance (ZL) source or transmission line do not match the impedance (𝑍0), a part of the signal is reflected back to the source. This reflection reduces the power distribution efficiency, creates steep waves, and sensitive components can cause intervention or damage.

The maximum power transfer theorem says:  ZL​=Z0

where Z0​ is the complex conjugate of the source impedance for AC circuits.

Technology for impedance matching:

• Quarter-wave transformer transmission line with a specific length (λ/4) uses a transmission line and impedance to change the impedance to match 𝑍0.
• Stub employs openly owned or short-circulated sticks that are kept at a distance calculated with the matching line.
• L, T, and Pi Network - Use inductors and capacitors to match the impedance in the RF circuit.
• Automatic tuner - Adjust the impedance dynamics in modern communication systems.

Application:

• Antennas - Ensure maximum power transmission from the transmitter to the antennas.
• Sound system - matching speakers to amplifiers for optimal sound quality.
• Prevent current losses in microwave oven heights.
• Maintain signal integrity in data transfer in high-speed digital systems.

Using transmission line sections, such as circuit elements: -

Transfer lines are not only used to move characters or power to distance - they can also act as circuit elements in RF and microwave. By choosing specific lengths and characteristic impedance, engineers can design transmission line sections that act as inductors, capacitors, transformers, or resonators.

1. Short-Circuited and Open-Circuited Stubs:

A short length of the transmission line, finished in a short circuit or an open circuit, can behave as a reactive item:

• Short -circulated stub → behaves as an initiative or capacitor depending on length.
• Open circulated stub → also behaves reactively, but with opposite phase properties.

By adjusting the length, these stumps are used for impedance matching and filter design.

2. Quarter-wave transformer:

A transmission line section with a length of λ/4 can be changed according to a load impedance. Using transmission line sections as circuit elements:

It is used in the matching network for maximum power transmission.

3. Delay lines

By choosing a specific line length, a transmission line can introduce a controlled time delay for a signal, used in phase matrices and time circuits.

4. Resonators

Sections of transmission lines, such as λ/2 or λ/4 lines, can act as resonances in oscillators, filters, and amplifiers, defining specific operating frequencies.

Application:

• Stub tuner in a microwave circuit
• Filter for RF signal processing
• Antenna matching networks
• Charan -Changes in Communication Systems

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