Standing Wave Ratio MCQ Quiz - Objective Question with Answer for Standing Wave Ratio - Download Free PDF

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.

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:

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

Calculation:

VSWR and reflection coefficient are related by:

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

Γ = 0.8

\(= \frac{{1 + 0.8}}{{1 - 0.8}} = \frac{1.8}{0.2} \)

= 9

Proof:

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

\(VSWR=\frac{V_{max}}{V_{min}}=\frac{I_{max}}{I_{min}}\)

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

\(V\left( l \right) = {V^ + }{e^{j\beta l}}\left( {1 + \left| {{{\rm{Γ }}_{\rm{L}}}} \right|{e^{ - j2\beta z}}} \right)\)

l = distance from the load

β = Phase constant

ΓL = Reflection coefficient at the load calculated as:

\({{\rm{Γ }}_L} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)     -  -  - (1)

The maximum and minimum voltage is given by:

\({V_{max}} = \left| {{V^ + }} \right|\;\left( {1 + \left| {{{\rm{Γ }}_{\rm{L}}}} \right|} \right)\)

\({V_{min}} = \left| {{V^ + }} \right|\;\left( {1 - \left| {{{\rm{Γ }}_{\rm{L}}}} \right|} \right)\)

\(\therefore VSWR = \frac{{{V_{max}}}}{{{V_{min}}}} = \frac{{1 + {{\rm{Γ }}_L}}}{{1 + {{\rm{Γ }}_L}}}\)Vmax=|V+|(1+|ΓL|)" id="MathJax-Element-123-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">Vmax=|V+|(1+|ΓL|)Vmin=|V+|(1−|ΓL|)" id="MathJax-Element-124-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">Vmin=∣∣V+∣∣(1−|ΓL|)Vmin=|V+|(1−|ΓL|)VSWR=VSWR=VmaxVmin=1+ΓL1+ΓL" id="MathJax-Element-125-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">VmaxVmin=1+ΓL1+Γ

Concept:

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:

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

ZL = Load impedance

Z0 = Characteristic Impedance

Calculation:

\(\Gamma = \frac{1-5}{1+5} = \frac{-4}{6}\)

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

\(= \frac{75 - jX_c - 50}{75 - jX_c + 50}\)

\(= \frac{25 - jX_c}{125 - jX_c}\)

\(|\Gamma| = \frac{\sqrt{(25)^2 + X_c^2}}{\sqrt{(125)^2 + X_c^2}}= \frac{4}{6}\)

\((625 + X_c^2) \times 36 = 16 (15625 + X_c^2)\)

\(22500 + 36X_c^2 = 250000 + 16 X_c^2\)

\(20 X_c^2 = 227500\)

\(X_c^2 = 11375\)

X_c = 106.65 Ω

\(X_c = \dfrac{1}{\omega C} = 106.65\)

\(\omega C = \dfrac{1}{106.65}\)

\(C = \dfrac{10^{-6}}{2 \pi \times 250 \times 106.65}= 5.97 \ \rm{pF}\)

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 ∞)

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.

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

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:

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

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

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 

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

\(VSWR = \frac{{{V_{max}}}}{{{V_{min}}}} = \frac{{{I_{max}}}}{{{I_{min}}}}\)

Important Derivation:

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

\(V\left( l \right) = {V^ + }{e^{j\beta l}}\left( {1 + \left| {{{\rm{\Gamma }}_{\rm{L}}}} \right|{e^{ - j2\beta z}}} \right)\)

l = distance from the load

β = Phase constant

Γ_L = Reflection coefficient at the load calculated as:

\({{\rm{\Gamma }}_L} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)     -  -  - (1)

The maximum and minimum voltage is given by:

 \({V_{max}} = \left| {{V^ + }} \right|\;\left( {1 + \left| {{{\rm{\Gamma }}_{\rm{L}}}} \right|} \right)\)

\({V_{min}} = \left| {{V^ + }} \right|\;\left( {1 - \left| {{{\rm{\Gamma }}_{\rm{L}}}} \right|} \right)\)

Standing wave ratio (SWR) is a measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide.

Impedance mismatches result in standing waves along the transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the line.

Practical applications of SWR are following:

• SWR  have impact on the performance of microwave-based medical applications. In microwave electrosurgery an antenna that is placed directly into tissue may not always have an optimal match with the feedline resulting in an SWR. The presence of SWR can affect monitoring components used to measure power levels impacting the reliability of such measurements.
• SWR is  the ratio between transmitted and reflected waves. A high SWR indicates poor transmission-line efficiency and reflected energy, which can damage the transmitter and decrease transmitter efficiency.
• SWR is used as a measure of impedance matching of a load to the characteristic impedance of a transmission line carrying radio frequency (RF) signals.
• When a transmitter is connected to an antenna by a feed line, the driving point impedance of the antenna must match the characteristic impedance of the feed line in order for the transmitter to see the impedance it was designed for.

Concept:

• Standing wave Ratio (SWR) defines mismatch on the line.
• It is also the measure of the deviation of impedances or current or voltages from their central values.
•  \(VSWR = \frac{{{V_{\max }}}}{{{V_{\min }}}}\)
• The value of VSWR varies from 1 to ∞
•  (1 ≤ VSWR < ∞)

ISWR meter can measure current SWR and VSWR can measure voltage SWR.

In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space.

Important Points

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

\(VSWR=\frac{V_{max}}{V_{min}}\)

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

\(V\left( l \right) = {V^ + }{e^{j\beta l}}\left( {1 + \left| {{{\rm{\Gamma }}_{\rm{L}}}} \right|{e^{ - j2\beta z}}} \right)\)

l = distance from the load

β = Phase constant

ΓL = Reflection coefficient at the load calculated as:

\({{\rm{\Gamma }}_L} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)        ---(1)

The maximum and minimum voltage is given by:

\({V_{max}} = \left| {{V^ + }} \right|\;\left( {1 + \left| {{{\rm{\Gamma }}_{\rm{L}}}} \right|} \right)\)

\({V_{min}} = \left| {{V^ + }} \right|\;\left( {1 - \left| {{{\rm{\Gamma }}_{\rm{L}}}} \right|} \right)\)

\(\therefore VSWR = \frac{{{V_{max}}}}{{{V_{min}}}} = \frac{{1 + {{\rm{\Gamma }}_L}}}{{1 - {{\rm{\Gamma }}_L}}}\)

Concept:

VSWR (Voltage standing wave Ratio) is defined as \( = \frac{{\left( {1 + \left|\Gamma \right|} \right)}}{{\left( {1 - \left| \Gamma \right|} \right)}}\) where Γ = Reflection coefficient.

Case 1:

For Γ = 1; VSWR = ∞

 All the power is reflected back to the transmission line.

Case 2:

For Γ = 0; VSWR = 1

Transmission line is perfectly matched to the Antenna and no power is reflected back to the transmission line

So, 1 ≤ VSWR < ∞ (Range)

Calculation:

Given, VSWR = 5.8

\( \Rightarrow 5.8 = \frac{{\left( {1 + \left| \Gamma \right|} \right)}}{{\left( {1 - \left| \Gamma \right|} \right)}}\)

Solving, we get

|Γ| = 0.7

\(\Gamma = \frac{{{V_ - }}}{{{V_ + }}}\)

(Where V_- = Reflected voltage Amplitude at load)

(Where V₊ = Forward path voltage Amplitude)

% Power reflected \(= \frac{{V_ - ^2}}{{V_ + ^2}} = \Gamma ^2 = {\left( {0.7} \right)^2} = 0.49 \times 100\) 

Calculation:

VSWR and reflection coefficient are related by:

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

Γ = 0.8

\(= \frac{{1 + 0.8}}{{1 - 0.8}} = \frac{1.8}{0.2} \)

= 9

Proof:

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

\(VSWR=\frac{V_{max}}{V_{min}}=\frac{I_{max}}{I_{min}}\)

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

\(V\left( l \right) = {V^ + }{e^{j\beta l}}\left( {1 + \left| {{{\rm{Γ }}_{\rm{L}}}} \right|{e^{ - j2\beta z}}} \right)\)

l = distance from the load

β = Phase constant

ΓL = Reflection coefficient at the load calculated as:

\({{\rm{Γ }}_L} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)     -  -  - (1)

The maximum and minimum voltage is given by:

\({V_{max}} = \left| {{V^ + }} \right|\;\left( {1 + \left| {{{\rm{Γ }}_{\rm{L}}}} \right|} \right)\)

\({V_{min}} = \left| {{V^ + }} \right|\;\left( {1 - \left| {{{\rm{Γ }}_{\rm{L}}}} \right|} \right)\)

\(\therefore VSWR = \frac{{{V_{max}}}}{{{V_{min}}}} = \frac{{1 + {{\rm{Γ }}_L}}}{{1 + {{\rm{Γ }}_L}}}\)Vmax=|V+|(1+|ΓL|)" id="MathJax-Element-123-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">Vmax=|V+|(1+|ΓL|)Vmin=|V+|(1−|ΓL|)" id="MathJax-Element-124-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">Vmin=∣∣V+∣∣(1−|ΓL|)Vmin=|V+|(1−|ΓL|)VSWR=VSWR=VmaxVmin=1+ΓL1+ΓL" id="MathJax-Element-125-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0">VmaxVmin=1+ΓL1+Γ

Concept:

\({\rm{\Gamma }}\;\left( {Reflection\;coefficient} \right) = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}}\)

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

And power delivered at load = \(I_{rms}^2 \times {Z_L}\)

Calculation:

Given

P = 10k = \(I_{rms}^2\;{Z_L}\;and\;{Z_0} = 50\)

\({Z_L} = \frac{{10k}}{{I_{rms}^2}} = \frac{{10k}}{{20 \times 20}} = 25\;{\rm{\Omega }}\)

\({\rm{\Gamma }} = \frac{{{Z_L} - {Z_0}}}{{{Z_L} + {Z_0}}} = \frac{{25 - 50}}{{25 + 50}} = \frac{{ - 25}}{{75}} = \frac{{ - 1}}{3}\;\)