Standing Wave Ratio MCQ Quiz in मल्याळम - Objective Question with Answer for Standing Wave Ratio - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

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

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

This can also be written as:

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

Γ = Reflection coefficient, defined as:

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

Z_L = Load impedance

Z₀ = Characteristic Impedance

Calculation:

Given:

Z₀ = 50 Ω

ZL = j50 Ω

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

\({\rm{Γ }} = \frac{{50j - 50}}{{50j + 50}} = \frac{j-1}{1+j}\)

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

Γ = j

Now:

\(VSWR = \frac{{1 + |j|}}{{1 - |j|}} = \infty\)

Concept:

The voltage at load is superposition of 2 waves

1) Forward travelling wave

2) Reflected wave

If V⁺ is the amplitude of forward travelling wave

Then voltage at load

V_L = V⁺ + V_R

V_L = V⁺ + Γ V⁺

V_L = V⁺ (1 + Γ)

Where V_R = Reflected wave voltage

Also:

V_max = |V⁺| (1 + (Γ(z)))

\(=\left| {{V}^{+}} \right|\left( 1+\frac{SWR-1}{SWR+1} \right)\)

\(=\left| {{V}^{+}} \right|\left( \frac{2~S\omega R}{S\omega R+1} \right)\)

Similarly,

\({{V}_{min}}=\left| {{V}^{+}} \right|\left( 1-\text{ }\!\!\Gamma\!\!\text{ }\left( z \right) \right)\)

\(=\left| {{V}^{+}} \right|\left( \frac{2}{SWR+1} \right)\)

Calculations:

Reflection coefficient (Γ_L)

\({{\text{ }\!\!\Gamma\!\!\text{ }}_{L}}=\frac{{{Z}_{L}}-{{Z}_{0}}}{{{Z}_{L}}+{{Z}_{0}}}\)

\(=\frac{50-100}{50+100}\)

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

\(=\frac{1}{3}{{e}^{j\pi }}\)

\(\left| {{\text{ }\!\!\Gamma\!\!\text{ }}_{L}} \right|=\frac{1}{3}~,~{{\theta }_{L}}=-\pi \)

\(SWR=\frac{1+\left| {{\text{ }\!\!\Gamma\!\!\text{ }}_{L}} \right|}{1-\left| {{\text{ }\!\!\Gamma\!\!\text{ }}_{L}} \right|}=\frac{3+1}{3-1}=2\)

To calculate minimum and maximum voltage first calculate forward voltage amplitude.

\({{V}_{L}}={{V}^{+}}\left( 1+{{\text{ }\!\!\Gamma\!\!\text{ }}_{L}} \right)\)

\({{V}^{+}}=\frac{{{V}_{L}}}{\left( 1+{{\text{ }\!\!\Gamma\!\!\text{ }}_{L}} \right)}=\frac{50}{\left( 1-\frac{1}{3} \right)}=\frac{50}{\frac{2}{3}}\)

= 75 V

\({{V}_{max}}=\left| {{V}^{+}} \right|\left( \frac{2~SWR}{SWR+1} \right)\)

\(=75\left( \frac{2\times 2}{2+1} \right)\)

Concept:

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

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

Calculation:

Given ZL= jZ0

\(So,\;{\rm{\Gamma }} = \frac{{j{Z_0} - {Z_0}}}{{j{Z_0} + {Z_0}}} = \frac{{\left( {j - 1} \right)\left( {j - 1} \right)}}{{\left( {j + 1} \right)\left( {j - 1} \right)}} = \frac{{ - 1 + 1 - 2j}}{{ - 2}} = j\)

Transmitted power = 75%

⇒ Reflected power = 25% = 1/4

Magnitude of reflection coefficient,

 \(\left| {\rm{\Gamma }} \right| = \sqrt {1/4} = 1/2\)

Also, \(\left| {\rm{\Gamma }} \right| = \frac{{VSWR - 1}}{{VSWR + 1}} = \frac{1}{2}\)

VSWR = 3

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.

Concept:

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:

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

Γ = 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 ∞.

Application:

Given ZL = 300 Ω

The reflection coefficient is calculated to be:

\({\rm{Γ }} = \frac{{{300} - {75}}}{{{300} + {75}}}=0.6\)

VSWR is now calculated as:

\(VSWR = \frac{{1 + {\rm{\Gamma }}}}{{1 - {\rm{\Gamma }}}} = \frac{{1 + 0.6}}{{1 - 0.6}} =4\)

Concept:

If the load is purely resistive for lossless line i.e. characteristic impedance must be purely resistive then;

I) VSWR \( = \frac{{{Z_L}}}{{{Z_O}}}\) when Z_L > Z_O

II) VSWR \( = \frac{{{Z_O}}}{{{Z_L}}}\) when Z_O > Z­_L

Normalized impedance ‘Z’ means \(\frac{Z}{{{Z_O}}}\) 

Calculation:

Given, normalized resistive load

i.e. \({\left( {{Z_L}} \right)_n} = \frac{{{Z_L}}}{{{Z_O}}} = n\)

but n < 1 ⇒ Z_O > Z_L

\(\therefore VSWR = \frac{{{Z_O}}}{{{Z_L}}}\)

\( = \frac{1}{n}\)

Note:

1) Value of VSWR lie from 1 < VSWR < ∞

2) For open circuit or for short circuit

3) Range of reflection coefficient (|Γ|) is;

0 < |Γ| < 1

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:

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

ZL = Load Impedance

Z0 = Characteristic impedance

Calculation:

Z_L = R + jX_L

= R + jωL

= 100 + j5 × 10⁸ × 0.6 × 10^-6

Z_L = 100 + j300 Ω

\({\rm{\Gamma }} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}} = \frac{{\left( {100 + j300} \right) - 120}}{{\left( {100 + j300} \right) + 120}}\)

= 0.62 + j0.52

= 0.808 ∠40°

\(VSWR = \frac{{1 + \left| {\rm{\Gamma }} \right|}}{{1 - \left| {\rm{\Gamma }} \right|}} = \frac{{1 + 0.808}}{{1 - 0.808}}\)

\(VSWR = 9.4\)

VSWR is given by:

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

VSWR = infinite means reflection coefficient |Γ| = 1

Γ = 1 or -1

Reflection coefficient is given by

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

For short circuit, Z_L = 0

Γ = -1 and S = ∞

For open circuit, Z_L = ∞

Γ = +1 and S = ∞

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.