Transmission Lines MCQ Quiz - Objective Question with Answer for Transmission Lines - Download Free PDF

The primary factor responsible for distortion in a transmitted signal is: 3) Variations in the propagation speed of different frequencies
Explanation:
Distortion occurs when different frequency components of a signal travel at different speeds, altering the signal's shape. Here’s why the other options are less accurate:

• 1) External electromagnetic interference → Causes noise (not distortion).
• 2) Signal attenuation → Reduces signal strength but doesn’t distort its shape.
• 4) Excessive bandwidth usage → May cause interference but isn’t a direct cause of distortion.

Key Point:
Frequency-dependent propagation speed (e.g., dispersion in optical fibers or group delay in RF systems) warps the signal waveform, leading to distortion.

Explanation:

Distortion-less Transmission Line

Definition: A distortion-less transmission line is one where the signal transmitted from one end of the line to the other experiences no distortion. This means that the shape of the signal remains unchanged during transmission. In practical terms, it implies that the transmission line has no frequency-dependent attenuation or phase distortion, ensuring the integrity of the signal over distance.

Condition for Distortion-less Transmission Line: For a transmission line to be distortion-less, it must satisfy a specific condition relating to its primary constants. These primary constants are Resistance (R), Inductance (L), Conductance (G), and Capacitance (C) per unit length of the transmission line. The condition is:

R/L = G/C

This condition ensures that the phase velocity of the signal remains constant across all frequencies, and there is no phase distortion or frequency-dependent attenuation.

Derivation and Explanation:

The primary constants of a transmission line are defined as follows:

• R: Resistance per unit length (ohms per unit length)
• L: Inductance per unit length (henries per unit length)
• G: Conductance per unit length (siemens per unit length)
• C: Capacitance per unit length (farads per unit length)

The propagation constant (γ) of a transmission line is given by:

γ = α + jβ

Where:

• α: Attenuation constant (nepers per unit length)
• β: Phase constant (radians per unit length)

The propagation constant can also be expressed in terms of the primary constants as:

γ = √((R + jωL)(G + jωC))

For a distortion-less transmission line, the attenuation constant (α) must be frequency-independent, and the phase constant (β) must be a linear function of frequency (ω). This implies that the real and imaginary parts of the propagation constant must satisfy specific conditions.

By expanding the expression for the propagation constant, we get:

γ = α + jβ = √((R + jωL)(G + jωC))

Separating the real and imaginary parts, we find that for the line to be distortion-less, the condition R/L = G/C must hold. This ensures that the phase velocity (v_p) is constant and given by:

v_p = 1/√(LC)

and the characteristic impedance (Z_0) is given by:

Z_0 = √(L/C)

Advantages of Distortion-less Transmission Lines:

• Ensures the integrity of transmitted signals, maintaining their original shape.
• Prevents signal degradation over long distances.
• Reduces the need for complex equalization and signal processing at the receiver end.

Applications: Distortion-less transmission lines are crucial in high-speed data communications, such as in telecommunications, networking, and broadcasting. They are also essential in precision measurement systems and other applications where signal integrity is paramount.

Correct Option Analysis:

The correct option is:

Option 1: R/L = G/C

This option correctly represents the condition for a distortion-less transmission line, ensuring no frequency-dependent attenuation or phase distortion

Explanation:

To find the magnitude of the reflection coefficient (Γ) at the load, we need to use the formula:

|Γ| = |(Z_L - Z₀) / (Z_L + Z₀)|

Where Z_L is the load impedance and Z₀ is the characteristic impedance of the lossless transmission line. Given:

• Z_L = 3 + j4Ω
• Z₀ = 1Ω

Step-by-Step Solution:

1. Calculate the numerator (Z_L - Z₀):

Z_L - Z₀ = (3 + j4) - 1 = 2 + j4

2. Calculate the denominator (Z_L + Z₀):

Z_L + Z₀ = (3 + j4) + 1 = 4 + j4

3. Find the magnitude of the numerator and the denominator:

|Z_L - Z₀| = |2 + j4| = √(2² + 4²) = √(4 + 16) = √20 = 2√5

|Z_L + Z₀| = |4 + j4| = √(4² + 4²) = √(16 + 16) = √32 = 4√2

4. Calculate the magnitude of the reflection coefficient:

|Γ| = |(Z_L - Z₀) / (Z_L + Z₀)| = |(2 + j4) / (4 + j4)| = (2√5) / (4√2) = (2√5) / (2√2 * 2) = √5 / (2√2) = (√5 / √2) / 2 = √(5/2) / 2

By simplifying √(5/2):

√(5/2) ≈ 1.58

Therefore:

|Γ| ≈ 1.58 / 2 ≈ 0.79

Hence, the magnitude of the reflection coefficient at the load is approximately 0.79. Therefore, the correct answer is Option 2.

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Explanation:

Lossless Transmission Line and VSWR:

Definition: A lossless transmission line is a theoretical transmission line in which there is no dissipation of electrical energy as heat. This means that the conductors and the dielectric material between them are perfect, resulting in no energy loss as the signal propagates along the line.

VSWR (Voltage Standing Wave Ratio) is a measure of how efficiently RF power is transmitted from the power source, through a transmission line, into the load. It is defined as the ratio of the maximum voltage to the minimum voltage in a standing wave pattern along the transmission line.

The VSWR is given by:

VSWR = (1 + |Γ|) / (1 - |Γ|)

Where |Γ| is the magnitude of the reflection coefficient.

The reflection coefficient (Γ) represents the fraction of the incident power that is reflected back from the load. It is a measure of the impedance mismatch between the transmission line and the load.

Calculation of Reflection Coefficient:

Given that the VSWR is 2, we can use the VSWR formula to find the reflection coefficient.

VSWR = (1 + |Γ|) / (1 - |Γ|)

Substitute VSWR = 2 into the formula:

2 = (1 + |Γ|) / (1 - |Γ|)

To find |Γ|, we solve the equation step-by-step:

Step 1: Cross-multiply to eliminate the fraction:

2 * (1 - |Γ|) = 1 + |Γ|

Step 2: Distribute the 2 on the left-hand side:

2 - 2|Γ| = 1 + |Γ|

Step 3: Combine like terms to isolate |Γ|:

2 - 1 = 2|Γ| + |Γ|

1 = 3|Γ|

Step 4: Solve for |Γ|:

|Γ| = 1 / 3

Therefore, the reflection coefficient (|Γ|) is 1/3.

Correct Option Analysis:

The correct option is:

Option 1: 1/3

This option correctly represents the calculated value of the reflection coefficient when the VSWR is 2.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: 3/4

This option is incorrect. If we use 3/4 in the VSWR formula, we would get:

VSWR = (1 + 3/4) / (1 - 3/4) = 7 / 1 = 7

This does not match the given VSWR of 2.

Option 3: Thu Jan 02 2025 00:00:00 GMT+0530 (India Standard Time)

This option is clearly irrelevant to the calculation of the reflection coefficient.

Option 4: Mon Feb 03 2025 00:00:00 GMT+0530 (India Standard Time)

Similar to option 3, this option is irrelevant to the calculation of the reflection coefficient.

Conclusion:

Understanding the relationship between VSWR and the reflection coefficient is crucial for analyzing transmission line performance. The reflection coefficient (|Γ|) provides valuable insight into the impedance matching of the transmission line and the load. In this case, with a VSWR of 2, the reflection coefficient is accurately calculated to be 1/3, confirming that option 1 is correct.

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Concept:

The voltage standing wave ratio is defined as the ratio of the maximum voltage to the minimum voltage.

The general expression of the voltage across a transmission line is given by:

l = distance from the load

β = Phase constant

ΓL = Reflection coefficient at the load calculated as:

The maximum and minimum voltage is given by:

  ---(1)

Calculation:

Given,

Using Equation (1), we get:

Given: Impedance = j0.8 Ω, Current = 500 A, Voltage = 300 V

We know that

V_S = V_R + IZ_S

Here V_S = Sending voltage, V_R = Receiving end voltage, Z_S = Line impedance

V_S = V_R + IZ_S = 300 + 500 × 0.8j = 300 + 400j = 500∠53.13°

Power factor = cos 53.13° = 0.6 lagging

Concept:

The characteristic impedance of a transmission line is defined as:

    ---(1)

And the propagation constant of the transmission line is defined as:

    ---(2)

Where,

α is attenuation constant

β is phase constant

R’ = Resistance per unit length of the line

G’ is conductance per unit length of the line

L’ is the inductance per unit length of the line

C’ is the capacitance per unit length of the line

Analysis:

A distortion less line satisfies the following condition:

So, the characteristic impedance of the distortionless line will be:

∴ The characteristic impedance of both lossless and a distortionless line is real.

And the propagation constant of the distortion less line will be:

Therefore, the attenuation constant of distortion less line is not zero but it is real.

As the line is distortionless, if a series of pulses are transmitted they arrive undistorted.

Important Points

For a lossless line:

R’ = G’ = 0

So, the characteristic impedance of a lossless transmission line using Equation (1) will be:

And the propagation constant of a lossless transmission line using Equation (2) will be:

α = 0

Therefore, the attenuation constant of the lossless line is always zero (real).

The voltage standing wave ratio is defined as the ratio of the maximum voltage (or current) to the minimum voltage (or current).

VSWR is also given by:

Rearranging the above, we get:

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ρ = Reflection coefficient, defined as:

ZL = Load impedance

Z0 = Characteristic Impedance

For ΓL varying from 0 to 1, VSWR varies from 1 to ∞.

Concept:

The voltage standing wave ratio is defined as the ratio of the maximum voltage (or current) to the minimum voltage (or current).

VSWR is also given by:

Γ = Reflection coefficient, defined as:

Z_L = Load impedance

Z₀ = Characteristic Impedance

For Γ_L varying from 0 to 1, VSWR varies from 1 to ∞.

Application:

Given Z_L = Z₀

The reflection coefficient is calculated to be:

VSWR for Γ = 0 equals 1, which is the minimum value (because it varies from 1 to ∞)

Concept:

The input impedance of a transmission line is given by:

  ---(1)

Z0 = Characteristic impedance

ZL = Load impedance

Application:

For l = λ/8

Putting this Equation (1), we get:

For short circuit load (ZL= 0), the input impedance becomes:

Since Impedance has a positive imaginary part, the transmission line behaves as an inductive Transmission Line.

The reflection coefficient is used to define the reflected wave with respect to the incident wave.

The reflection coefficient of the transmission line at the load is given by:

Z_L = Load impedance

Z₀ = Characteristic Impedance

It is always desirable to have a perfectly matched load for maximum power transfer, i.e. Z_L = Z₀

For Z_L = Z₀

The Voltage Standing Wave Ratio (VSWR) is defined as:

The voltage standing wave ratio for a perfectly matched load (which is always desirable) is obtained by putting Γ = 0 in the above equation.

Capacitance in Transmission line:

• Capacitance in a transmission line results due to the potential difference between the conductors.
• The conductors of the transmission line act as a parallel plate of the capacitor and the air is just like a dielectric medium between them.  
• The conductors get charged in the same way as the parallel plates of a capacitor.
• The capacitance between two parallel conductors depends on the size and the spacing between the conductors.
• The capacitance of a line gives rise to the leading current between the conductors.
• It depends on the length of the conductor. The capacitance of the line is proportional to the length of the transmission line.
• The capacitance effect is negligible on the performance of short (having a length less than 80 km) and low voltage transmission line.
• In the case of high voltage and long lines, it is considered as one of the most important parameters.
• Therefore a long transmission line has a considerable shunt capacitance

Concept:

Formula for reflection coefficient of transmission line at load end is

Where,

Z_C is surge impedance of transmission line

Z_L is load impedance at the end transmission line

Calculation:

Given that,

Surge impedance of transmission line Z_C = 300 Ω

Load impedance at the end transmission line Z_L = 300 Ω

Therefore, reflection coefficient of transmission line at load end is

Capacitance of a Single-Phase Two-wire Line:

Consider a single-phase overhead transmission line consisting of two parallel conductors A and B spaced D meters apart in the air.

Suppose that radius of each conductor is r meters.

Let their respective charge be + Q and − Q coulombs per metre length.

The total potential difference. between conductor A and neutral “infinite” plane is,

or, 

or, 

Similarly, total potential difference. between conductor B and neutral “infinite” plane is,

Both these potentials are with respect to the same neutral plane.

Since the unlike charges attract each other, the potential difference between the conductors is,

V_AB = 2V_A = 

We know that the capacitance is the ratio of charge to the voltage (C =Q/V),

Hence,

C_AB = 

or, C_AB = 

Hence, When two conductors between each of radius r are at a distance D, the capacitance between the two is Proportional to 

Concept:

Voltage standing wave ratio (VSWR) is mathematically defined as:

   ---(1)

Γ = Reflection coefficient given by:

Z_L = Load Impedance

Z₀ = Characteristic impedance

Application:

Given: Z_o = 50 Ω and Z_L = 40 + j30

Reflection co-efficient will be:

Now, the voltage standing wave ratio will be:

VSWR = 2