Transmission Lines MCQ Quiz in मल्याळम - Objective Question with Answer for Transmission Lines - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

Velocity factor

• The velocity of light and all other electromagnetic waves depends on the medium through which they travel.
• It is very nearly 3 × 10⁸ m/s in a vacuum and slower in all other media.

The velocity of light in a medium is given by

\(v = \frac{{{v_c}}}{{\sqrt k }}\)

v: velocity of the wave in the medium

v_c: velocity of the wave in free space

k: dielectric constant

The velocity factor of a dielectric substance, and thus of a cable, is the velocity reduction ratio and is therefore given by

\(vf = \frac{1}{{\sqrt k }}\)

The dielectric constants of materials commonly used in transmission lines range from about 1.2 to 2.8, giving corresponding velocity factors from 0.9 to 0.6

NOTE:

since ν = fλ and f is constant, the wavelength λ is also reduced by a ratio equal to the velocity factor.

Example

If a section of 300-Ω twin lead has a velocity factor of 0.82, the speed of energy transferred is 18 per cent slower than in a vacuum.

Calculation:

Given dielectric constant is 2.

The velocity factor of the line is

\(vf = \frac{1}{{\sqrt 2 }}\)

vf = 0.707

vf = 70.7%

Concept:

Characteristic Impedance of a transmission line is defined as:

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

R = Resistance per unit length of the line

G = Conductance per unit length of the line

L = Inductance per unit length of the line

C = Capacitance per unit length of the line

and the propagation constant of the line is given by:

\(\gamma=\sqrt{(R+jω L)(G+jω C)}\)   ---(2)

This can be written as:

γ = α + jβ

α = attenuation constant

β = phase constant

For a lossless line,

R = G = 0

Using Equation (1), the characteristic impedance of the lossless transmission line will become:

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

Using Equation (2), the propagation constant of the lossless transmission line will become:

\(\gamma =j\omega \sqrt{LC}\)

\(\alpha=0,~\beta=\omega \sqrt{LC}\)

Observation:

• The characteristic impedance of a lossless line is real.
• The attenuation constant of a lossless transmission line is always zero.

The Characteristic impedance of the transmission line is given as:

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

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

Concept:

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

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

ZL = Load impedance

Z0 = Characteristic Impedance

For a perfectly matched load:

ZL = Z0

The reflection coefficient is:

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

This indicates that all the power is transferred to the load and there is no reflected power, i.e. all the energy sent will be absorbed by the load.

Concept:

Voltage standing wave ratio: It is the ratio of maximum voltage to minimum voltage.

If the voltage minima occur at load ZL < Z0 then

\(VSWR = \frac{{{Z_0}}}{{{Z_L}}}\)

If the voltage maxima occur at load ZL > Z0 then

\(VSWR = \frac{{{Z_L}}}{{{Z_0}}}\)

Analysis:

Z₀ = 75 Ω 

When the load is shorted the voltage minima occur at the load, and it does not change even when short is replaced by R_L: 

\(VSWR = \frac{Z_0}{R_L}\) and R_L < Z₀ 

VSWR = 3

\(\frac{75}{R_L} = 3\)

R_L = 25 Ω 

Concept:

If a transmission line will characteristic impedance Z₀ is terminated at a load impedance Z_L then the reflection coefficient at the load is given by:

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

Z_L = Load Impedance

Z₀ = Characteristic Impedance

Calculation:

Given:

Since a voltage signal of 100V with 1 GHz frequency is applied to the transmission line with Z₀ = 100 Ω, we get:

\(I = \frac{{100\;V}}{{100\;{\rm{\Omega }}}} = 1\;A\)

For the antenna, this current will be the incident current.

But due to the impedance mismatch between the transmission line and the antenna, part of the current will be reflected.

Reflection coefficient will be:

\({{\rm{\Gamma }}_L} = \frac{{{Z_A} - {Z_0}}}{{{Z_n} + {Z_0}}} = \frac{{75 - 100}}{{75 + 100}} = - \frac{1}{7}\)

The peak value of the reflected current will be:

\({I_{ref}} = \left| {{{\rm{\Gamma }}_L}} \right| \times {I_{inc}}\)

\( = \frac{1}{7} \times 1\;A \approx 0.142\;A\)

• The load impedance, ZL at the end of the transmission line must match to its characteristic impedance, Z0 Otherwise there will be reflections from the transmission line’s end.
• A quarter-wave transformer is a component that can be inserted between the transmission line and the load to match the load impedance ZL to the transmission line’s characteristic impedance Z0.
• The input impedance of a quarter-wave transformer is given as:

           \({{\rm{Z}}_{{\rm{in}}}} = \frac{{{\rm{Z}}_0^2}}{{{{\rm{Z}}_{\rm{L}}}}}\;\).

Observation:

Concept:

The maximum and minimum impedance across a transmission line is defined as:

Maximum impedance:

Zmax = Z0 × VSWR

Minimum impedance:

\({Z_{min}} = \frac{{{Z_0}}}{{VSWR}}\)

VSWR = Voltage standing wave ratio given by:

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

Γ = Reflection Coefficient

Calculation:

Given:

Z₀ = 50Ω 

Z_L = 100Ω 

\(\left| {\rm{\Gamma }} \right| = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)

\(\left| {\rm{\Gamma }} \right| = \frac{{100 - 50}}{{100 +50 }}\)

\(\left| {\rm{\Gamma }} \right| = \frac{1}{3}\)

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

\(= \frac{{1 + \frac{1}{3}}}{{1 - \frac{1}{3}}} = 2\)

Zmax = Z0 × VSWR

Zmax = 50 × 2 = 100Ω

\({Z_{min}} = \frac{{{Z_0}}}{VSWR} = \frac{{50}}{2}\)

Zmin = 25Ω 

∴ The minimum Impedance on line is 25Ω.

Concept:

The propagation constant of a transmission line is a complex quantity given by:

γ = α + jβ

α = Attenuation constant, related to the line parameters as:

\(\alpha = \sqrt {RC}\)

β = Phase constant, related to the line parameters as:

\(\beta = {\rm{ω }}\sqrt {{\rm{LC}}} \)

For a loss lossless line, there is no attenuation, i.e. α = 0. The propagation constant will become:

γ = jβ

|γ| = |β|

Attenuation constant:

• Attenuation constant gives attenuation of an electromagnetic wave propagating through a medium per unit distance from the source.
• It is the real part of the propagation constant and is measured in nepers per metre.

Phase Constant:

• The phase constant is the imaginary component of the propagation constant for a plane wave
• It represents the change in phase per unit length along the path travelled by the wave at any instant
• Phase constant is measured in radians per unit length

The velocity of propagation is given as:

\({{\rm{v}}_{\rm{p}}} = \frac{1}{{\sqrt { {{\rm{LC}}}} }}\)

R = Resistance of the line per unit length.

L = Inductance of the line per unit length.

C = Capacitance per unit length.

Note:

Propagation constant is given by:

\({\rm{\gamma }} = \sqrt {\left( {{\rm{R}} + {\rm{j\;\omega L}}} \right)\left( {{\rm{G}} + {\rm{j\;\omega }}C} \right)}\)

Phase shift constant is given by:

 \({\rm{\beta }} = {\rm{\omega }}\sqrt {{\rm{LC}}}\)

Characteristic impedance is given by: