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

Concept:

The characteristic impedance \(Z_0\) of a lossless transmission line is given by:

\(Z_0 = \sqrt{\frac{L}{C}}\)

Where,

• \(L\) = Inductance per unit length (in H/m)
• \(C\) = Capacitance per unit length (in F/m)

Given:

\(L = 100~\text{nH/m} = 100 \times 10^{-9}~\text{H/m}\)

\(C = 40~\text{pF/m} = 40 \times 10^{-12}~\text{F/m}\)

Calculation:

\(Z_0 = \sqrt{\frac{100 \times 10^{-9}}{40 \times 10^{-12}}} = \sqrt{\frac{100}{40} \times 10^3} = \sqrt{2.5 \times 10^3} = \sqrt{2500} = 50~\Omega\)

Hence, the correct answer is 3

The VSWR is mathematically related to the reflection coefficient (Γ), which quantifies the fraction of incident power reflected back due to impedance mismatch between the transmission line and the load. The reflection coefficient is given by:

Γ = (VSWR - 1) / (VSWR + 1)

Given in the problem:

• VSWR = 1.5
• Incident power = 50 W

Step 1: Calculate the Reflection Coefficient (Γ):

Substitute the given VSWR value into the formula:

Γ = (1.5 - 1) / (1.5 + 1)

Γ = 0.5 / 2.5

Γ = 0.2

Step 2: Calculate the Reflected Power:

The reflected power (P_reflected) can be calculated using the relation:

P_reflected = Γ² × P_incident

Substitute the values:

P_reflected = (0.2)² × 50

P_reflected = 0.04 × 50

P_reflected = 2 W

Correct Answer:

The reflected power is 2 W, which corresponds to Option 2.

Explanation:

Magic Tee

Definition: A Magic Tee, also known as a hybrid tee, is a specialized microwave junction used in waveguide technology. It combines the functionalities of two types of tees: the E-plane tee and the H-plane tee. The Magic Tee is widely used in microwave systems for splitting, combining, and analyzing signals. It is so named because of its unique ability to produce the sum and difference of two input signals simultaneously, which is critical in applications such as radar systems, mixers, and power dividers.

Working Principle: A Magic Tee has four ports:

• Port 1: The H-plane port, also known as the "sum port."
• Port 2: The E-plane port, also called the "difference port."
• Port 3 and Port 4: The collinear arms, where the input signals are usually applied.

The operation of a Magic Tee is based on the principle of constructive and destructive interference of electromagnetic waves. When signals are fed into the collinear arms (Port 3 and Port 4):

• The sum of the two signals is produced at the H-plane port (Port 1).
• The difference of the two signals is produced at the E-plane port (Port 2).

This unique property makes the Magic Tee an essential component in microwave circuits for signal processing tasks.

The correct answer is: 1) TEM

Explanation:

• Stripline is a type of transmission line used in high-frequency circuits, typically embedded between two ground planes with a dielectric in between.

• It supports pure TEM (Transverse Electromagnetic) mode, meaning:

  • Both electric and magnetic fields are entirely transverse to the direction of wave propagation.

  • There are no longitudinal field components (no Ez or Hz).

This is possible because the geometry of a stripline is completely symmetric and enclosed, allowing true TEM propagation — unlike microstrip, which supports quasi-TEM due to its asymmetric structure.

Explanation:

Matched Junction S-parameters

Definition: In RF and microwave engineering, S-parameters (scattering parameters) are used to characterize how RF signals behave in a network. For a matched junction, the term refers to a system where the impedance of the ports is perfectly matched, minimizing reflections.

• S₁₁ = 0: This indicates no reflection occurs at Port 1 due to perfect impedance matching.
• S₂₁ = 1: This indicates complete transmission of the signal from Port 1 to Port 2 without any loss.

In the case of a matched junction, the input signal is fully transmitted to the output port without any reflection or loss. This ideal condition is represented by S₂₁ = 1, meaning the transmission coefficient is unity (100% transmission). Matched junctions are often used in high-frequency systems to ensure maximum signal transfer and minimal signal degradation.

Derivation of S₂₁:

For a matched junction, the reflection coefficient S₁₁ is zero, which implies no signal is reflected back to the source. Consequently, all the power is transmitted to the output port, leading to the transmission coefficient S₂₁ being equal to 1.

The S-parameters are governed by the energy conservation principle:

• |S₁₁|² + |S₂₁|² = 1

For a matched junction:

• S₁₁ = 0
• |S₁₁|² = 0
• Thus, |S₂₁|² = 1
• S₂₁ = 1

This confirms that the correct option is S₂₁ = 1.

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°

Concept:

The characteristic impedance of a transmission line is defined as:

\({Z_0} = \sqrt {\frac{{R' + j\omega L'}}{{G' + j\omega C'}}} \)    ---(1)

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

\(\gamma = \alpha + j\beta = \sqrt {\left( {R' + j\omega C'} \right)\left( {G' + j\omega C'} \right)} \)    ---(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:

\(\frac{{R'}}{{L'}} = \frac{{G'}}{{C'}}\)

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

\({Z_0} = \sqrt {\frac{{L'}}{{C'}}} = \sqrt {\frac{{R'}}{{G'}}} \)

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

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

\(\gamma = \alpha + j\beta = \sqrt {R'G'} + j\omega \sqrt {L'C'} \)

\(\alpha = \sqrt {R'G'} \ne 0\)

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:

\({Z_0} = \sqrt {\frac{{L'}}{{C'}}} \)

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

\(\gamma = \alpha + j\beta = j\omega \sqrt {L'C'} \)

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).

\(VSWR = \frac{{{{\rm{V}}_{{\rm{max}}}}}}{{{{\rm{V}}_{{\rm{min}}}}}} = \frac{{{{\rm{I}}_{{\rm{max}}.}}}}{{{{\rm{I}}_{{\rm{min}}.}}}}\)

VSWR is also given by:

\(VSWR = \frac{{1 + {\rm{|ρ| }}}}{{1 - {\rm{|ρ| }}}}\)

Rearranging the above, we get:

​\(\left| \rho \right| = \frac{{\left( {VSWR - 1} \right)}}{{\left( {VSWR + 1} \right)}}\)​

ρ = Reflection coefficient, defined as:

\({\rm{Γ }} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)

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).

\(VSWR = \dfrac{{{{\rm{V}}_{{\rm{max}}}}}}{{{{\rm{V}}_{{\rm{min}}}}}} = \dfrac{{{{\rm{I}}_{{\rm{max}}.}}}}{{{{\rm{I}}_{{\rm{min}}.}}}}\)

VSWR is also given by:

\(VSWR = \dfrac{{1 + {\rm{Γ }}}}{{1 - {\rm{Γ }}}}\)

Γ = Reflection coefficient, defined as:

\({\rm{Γ }} = \dfrac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)

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:

\({\rm{Γ }} = \dfrac{{{Z_0} - {Z_0}}}{{{Z_0} + {Z_0}}}=0\)

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:

\({Z_{in}} = {Z_o}\frac{{\left( {{Z_L} + j{Z_0}tanβ l} \right)}}{{\left( {{Z_0} + j{Z_L}tanβ l} \right)}}\)  ---(1)

Z0 = Characteristic impedance

ZL = Load impedance

Application:

For l = λ/8

\(β l=\frac{2\pi}{\lambda}\times \frac{\lambda}{8}\)

\(\beta l=\frac{\pi}{4}\)

Putting this Equation (1), we get:

\({Z_{in}} = {Z_o}\frac{{\left( {{Z_L} + j{Z_0}} \right)}}{{\left( {{Z_0} + j{Z_L}} \right)}}\)

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

\({Z_{in}} = {Z_o}\frac{{\left( {{0} + j{Z_0}} \right)}}{{\left( {{Z_0} + j{0}} \right)}}\)

\({Z_{in}} = j{Z_o}\)

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:

\({\rm{Γ }} = \frac{{{{\rm{Z}}_{\rm{L}}} - {Z_0}}}{{{Z_L} + {Z_0}}}\)

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

\({\rm{Γ }} = \frac{{{{\rm{Z}}_{\rm{0}}} - {Z_0}}}{{{Z_0} + {Z_0}}}=0\)

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

\(VSWR = \frac{{1 + {\rm{Γ }}}}{{1 - {\rm{Γ }}}}\)

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

\(VSWR = \frac{{1 + {\rm{0 }}}}{{1 - {\rm{0}}}}=1\)

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

\({\tau _r} = \frac{{{Z_C} - {Z_L}}}{{{Z_C} + {Z_L}}}\)

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

\({\tau _r} = \frac{{300 - 300}}{{300 + 300}} = 0\)

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,

\(V_A=\int^\infty_r{\frac{Q}{2\pi x\epsilon_0}}+\int^\infty_D{\frac{-Q}{2\pi x\epsilon_0}}\)

or, \(V_A=\frac{Q}{2\pi \epsilon_0}[{log_e\frac{\infty}{r}}-log_e\frac{\infty}{D}]\)

or, \(V_A=\frac{Q}{2\pi \epsilon_0}[{log_e\frac{D}{r}}]\)

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

\(V_B=\frac{-Q}{2\pi \epsilon_0}[{log_e\frac{D}{r}}]\)

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 = \(\frac{2Q}{2\pi \epsilon_0}[{log_e\frac{D}{r}}]\)

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

Hence,

C_AB = \(\frac{Q}{V_{AB}}=\frac{Q}{​​\frac{2Q}{2\pi \epsilon_0}[{log_e\frac{D}{r}}]}\)

or, C_AB = \(\frac{\pi\epsilon_0}{[{log_e\frac{D}{r}}]}\)

Hence, When two conductors between each of radius r are at a distance D, the capacitance between the two is Proportional to \(\dfrac{1}{\log_e\left(\dfrac{D}{r}\right)}\)

Concept:

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

\(VSWR = \frac{{\left( {1 + \left|\Gamma \right|} \right)}}{{\left( {1 - \left| \Gamma \right|} \right)}}\)   ---(1)

Γ = Reflection coefficient given by:

\(\Gamma_L= \frac{{{Z_L} - {Z_o}}}{{{Z_L} + {Z_o}}}\)

Z_L = Load Impedance

Z₀ = Characteristic impedance

Application:

Given: Z_o = 50 Ω and Z_L = 40 + j30

Reflection co-efficient will be:

\(\Gamma_L= \frac{{{Z_L} - {Z_o}}}{{{Z_L} + {Z_o}}}\)

\(\Gamma_L= \frac{{40 + j30 - 50}}{{40 + j30 + 50}}\)

\(\Gamma_L = \frac{{ - 10 + j30}}{{90 + j30}}\)

\(\left| \Gamma \right| = \sqrt {\frac{{{{\left( {10} \right)}^2} + {{\left( {30} \right)}^2}}}{{{{\left( {90} \right)}^2} + {{\left( {30} \right)}^2}}}} \)

\(\Gamma= \sqrt {\frac{{10}}{{90}}} \)

Now, the voltage standing wave ratio will be:

\(VSWR = \frac{{1 +\Gamma}}{{1 - \Gamma}}\)

\(VSWR= \frac{{1 + \frac{1}{3}}}{{1 - \frac{1}{3}}}\)