Coefficient of Coupling MCQ Quiz - Objective Question with Answer for Coefficient of Coupling - Download Free PDF

Coefficient of coupling

The coupling coefficient can be defined as a magnetic flux produced between two different coils while managing the flux successfully.

\(k={M\over \sqrt{L_1L_2}}\)

where, k = Coefficient of coupling

M = Mutual inductance

L1 and L2 = Self-inductance of two coils

Calculation

Given, L1  5H &  L2 = 20 H

M = 8 H

\(k={8\over \sqrt{5\times 20}}\)

k = 0.8

Explanation:

To determine the mutual inductance (M) between two coils, we can use the formula:

Formula: \( M = k \sqrt{L_1 \times L_2} \)

Where:

• M is the mutual inductance.
• k is the coefficient of coupling.
• L₁ is the inductance of the first coil.
• L₂ is the inductance of the second coil.

Given the data:

• Inductance of the first coil, L₁ = 16 H
• Inductance of the second coil, L₂ = 4 H
• Coefficient of coupling, k = 1

Let's calculate the mutual inductance using the given values:

Step-by-Step Solution:

1. First, we identify the given values in the problem:

• Inductance of the first coil, L₁ = 16 H
• Inductance of the second coil, L₂ = 4 H
• Coefficient of coupling, k = 1

2. Using the formula for mutual inductance, \( M = k \sqrt{L_1 \times L_2} \), we substitute the given values:

• \( M = 1 \times \sqrt{16 \times 4} \)

3. Calculate the product inside the square root:

• \( 16 \times 4 = 64 \)

4. Calculate the square root of the product:

• \( \sqrt{64} = 8 \)

5. Finally, multiply by the coefficient of coupling (which is 1 in this case):

• \( M = 1 \times 8 \)

So, the mutual inductance M is 8 H.

Conclusion:

The mutual inductance between the two coils is 8 H, which matches Option 2.

Correct Option Analysis:

The correct option is:

Option 2: 8 H

This option correctly represents the calculated mutual inductance based on the provided inductances and the coefficient of coupling.

Additional Information

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

Option 1: 10 H

This option is incorrect. By following the same calculation steps, it is clear that the mutual inductance should be calculated as \( M = 1 \times \sqrt{16 \times 4} = 8 H \). Therefore, the value 10 H is not supported by the given data and formula.

Option 3: 12 H

This option is also incorrect. The mutual inductance calculation yields 8 H, not 12 H. The value 12 H does not align with the correct calculation using \( M = k \sqrt{L_1 \times L_2} \) with the given parameters.

Option 4: 16 H

This option is incorrect. The mutual inductance is calculated to be 8 H. The value 16 H does not match the correct calculation. It appears to be a misunderstanding or misapplication of the given formula.

Conclusion:

Understanding the calculation of mutual inductance is essential for accurately determining the interaction between two coupled coils. The mutual inductance is influenced by the inductance of each coil and the coefficient of coupling. By correctly applying the formula \( M = k \sqrt{L_1 \times L_2} \), we can ensure accurate results and avoid common pitfalls in the calculation process.

Coefficient of coupling

The coefficient of coupling can be defined as a magnetic flux that is produced between two different coils while managing the flux successfully.

\(k={M\over \sqrt{L_1L_2}}\)

where, k = Coefficient of coupling

M = Mutual inductance

L1 and L2 = Self-inductance of two coils

Explanation

The value of k lies between 0 to 1. The maximum value of k = 1.

\(1={M\over \sqrt{L_1L_2}}\)

\(M=\sqrt{\mathrm{L}_1 \times \mathrm{L}_2}\)

Explanation:

Mutual Inductance and Self-Inductance Relationship

Definition: Mutual inductance (M) is a measure of the ability of one coil to induce an electromotive force (EMF) in another coil through the magnetic field created by the current flowing through the first coil. Self-inductance (L) is a measure of the ability of a coil to induce an EMF in itself due to the change in its own current.

Equation Analysis:

For two closely coupled coils with turns N₁ and N₂, the mutual inductance M between them is related to their self-inductances L₁ and L₂ and the number of turns as follows:

Option 1: M = L × √(N₁ × N₂)

This equation correctly represents the relationship between self-inductance (L), mutual inductance (M), and the number of turns (N₁ and N₂) for two closely coupled coils. The mutual inductance is proportional to the geometric mean of the number of turns in each coil, which is a reasonable approximation for closely coupled coils.

Correct Option Analysis:

The correct option is:

Option 1: M = L × √(N₁ × N₂)

This option accurately describes the relationship between mutual inductance, self-inductance, and the number of turns in each coil. When two coils are closely coupled, the mutual inductance depends on the product of the turns of both coils and the square root of their self-inductances.

Additional Information

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

Option 2: L = M × N₁ × N₂

This option is incorrect because it suggests that the self-inductance is directly proportional to the mutual inductance and the product of the number of turns in both coils. This does not accurately represent the physical relationship between these quantities.

Option 3: L = M / (N₁ + N₂)

This option is incorrect as it implies an inverse relationship between self-inductance and the sum of the number of turns in each coil. This does not conform to the established principles of inductance in coupled coils.

Option 4: M = L / (N₁ × N₂)

This option is incorrect because it suggests that mutual inductance is inversely proportional to the product of the number of turns in each coil. This relationship does not hold true for closely coupled coils as it does not consider the magnetic coupling factor.

Conclusion:

Understanding the relationship between self-inductance, mutual inductance, and the number of turns in coupled coils is crucial for correctly identifying their operational characteristics. The correct relationship, as described in option 1, shows that mutual inductance is proportional to the geometric mean of the number of turns in each coil and their self-inductances. This understanding is essential for designing and analyzing coupled inductive components in various electrical and electronic applications.

Coefficient of coupling

The coefficient of coupling can be defined as a magnetic flux that is produced between two different coils while managing the flux successfully.

\(k={M\over \sqrt{L_1L_2}}\)

where, k = Coefficient of coupling

M = Mutual inductance

L1 and L2 = Self-inductance of two coils

The mutual inductance between the coils is maximum when k = 1

Calculation

Given, L1 = 20H and L2 = 320H

k = 1

\(k={M\over \sqrt{20\times 320}}\)

M = 80 H

The coefficient of coupling is a factor used when two coils are mutually coupled in a magnetic circuit.

The coupling coefficient K is a measure of the magnetic coupling between two coils.

The coupling coefficient is the amount of inductive coupling that exists between the two coils is expressed as a fractional number between 0 and 1.

The coupling factor between the coils or the coupling coefficient is given as

\(K = \frac{M}{{\sqrt {{L_1}{L_2}} }}\)

∴ \(K \propto \frac{1}{{\sqrt {{L_1}{L_2}} }}\)

Where M = Mutual inductance

L1 and L2 = Self-inductance

Important Points

0 ≤ K ≤ 1

K < 0.5 loosely coupled

K > 0.5 tightly coupled

K = 1 magnetically tightly coupled.

Concept:

Coefficient of Coupling (k):
The coefficient of coupling (k) between two coils is defined as the fraction of magnetic flux produced by the current in one coil that links the other.

Two coils have self-inductance L₁ and L₂, then mutual inductance M between them then Coefficient of Coupling (k) is given by 

\(k=\frac{M}{\sqrt {L_1L_2}}\)

Where,

\(M=\frac{N_1N_2\mu_o \mu_rA}{ l}\)

\(L_1=\frac{N_1^2\mu_o \mu_rA}{ l}\)

\(L_2=\frac{N_2^2\mu_o \mu_r A}{l}\)

N₁ and N₂ is the number of turns in coil 1 and coil 2 respectively
A is the cross-section area
l is the length

Calculation:

Given,

N₁ = N₂ = N

l₁ = l₂ = l

A₁ = X_a

A₂ = X_b

From the above concept,

\(M=\frac{N^2\mu_o \mu_rX_a}{ l}\)

\(L_1=\frac{N^2\mu_o \mu_r X_a}{l}\)

\(L_2=\frac{N^2\mu_o \mu_r X_b}{l}\)

We know that,

\(k=\frac{M}{\sqrt {L_1L_2}}=\frac{X_a}{\sqrt {X_aX_b}}\)

\(k=\frac{X_a}{\sqrt {X_aX_b}}\times\frac{\sqrt X_a}{\sqrt X_a}\)

\(k=\sqrt {\dfrac{X_a}{X_b}}\)

Concept:

Coefficient of Coupling (k):
The coefficient of coupling (k) between two coils is defined as the fraction of magnetic flux produced by the current in one coil that links the other.

Two coils have self-inductance L1 and L2, then mutual inductance M between them then Coefficient of Coupling (k) is given by 

\(k=\frac{M}{\sqrt {L_1L_2}}\)

Where,

\(M=\frac{\mu_o \mu_rN_1N_2A}{ l}\)

\(L_1=\frac{\mu_o \mu_rN_1^2A}{ l}\)

\(L_2=\frac{\mu_o \mu_rN_2^2A}{l}\)

N1 and N2 is the number of turns in coil 1 and coil 2 respectively
A is the cross-section area
l is the length

Calculation:

Given,

L1 = 18 H

L2 = 2 H

M = 3 H

From the above concept,

\(k=\frac{M}{\sqrt {L_1L_2}}\)

\(k=\frac{4}{\sqrt {10\times10}}=\frac{4}{10}=0.4\)

The correct answer is option 1):(10 mH)

Concept:

The mutual inductance is given by: M = k\( \sqrt{L_1\times L_2} \)

where,

M = Mutual inductance

L = Self-inductance

k = Coefficient of coupling

The value of k lies between 0 and 1.

The maximum value of mutual inductance is possible for k = 1.

Calculation:

Given 

K =1

L₁ = 5 mH

L₂ = 20 mH

 M = k\( \sqrt{L_1\times L_2} \)

=  \(\sqrt{100} \)

= 10 mH

Concept:

Coefficient of Coupling (k): 
The coefficient of coupling (k) between two coils is defined as the fraction of magnetic flux produced by the current in one coil that links the other.

Two coils have self-inductance L1 and L2, then mutual inductance M between them then Coefficient of Coupling (k) is given by 

\(k=\frac{M}{\sqrt {L_1L_2}}\)

Where,

\(M=\frac{N_1N_2 \mu_o \mu_r A}{l}\)

\(L_1=\frac{\mu_o \mu_r N_1^2A}{ l}\)

\(L_2=\frac{\mu_o \mu_rN_2^2A}{ l}\)

N1 and N2 is the number of turns in coil 1 and coil 2 respectively
A is the cross-section area
l is the length

Calculation:

Given,

L1 = 2 H

L2 = 8 H

M = 4 H

From the above concept,

\(k=\frac{M}{\sqrt {L_1L_2}}\)

\(k=\frac{4}{\sqrt {2\times 8}}=1\)

• The term impedance matching is simply defined as the process of making one impedance look like another
• Impedance matching is a process in which the impedance of an electrical load is made equal to the source impedance to maximize the power transfer or minimize signal reflection from the load
• The Power amplifiers generally use transformer coupling because the transformer permits impedance matching
• The disadvantage of impedance matching is that it gives distorted output

The coefficient of coupling is a factor used when two coils are mutually coupled in a magnetic circuit.

The coupling coefficient K is a measure of the magnetic coupling between two coils.

The coupling coefficient is the amount of inductive coupling that exists between the two coils is expressed as a fractional number between 0 and 1. The coupling factor between the coils or the coupling coefficient is given as

\(K = \frac{M}{{\sqrt {{L_1}{L_2}} }}\)

0 ≤ K ≤ 1

K < 0.5 loosely coupled

K > 0.5 tightly coupled

K = 1 magnetically tightly coupled.

Coefficient of coupling

The coefficient of coupling can be defined as a magnetic flux that is produced between two different coils while managing the flux successfully.

\(k={M\over \sqrt{L_1L_2}}\)

where, k = Coefficient of coupling

M = Mutual inductance

L1 and L2 = Self-inductance of two coils

The mutual inductance between the coils is maximum when k = 1

Calculation

Given, L1 = 20H and L2 = 320H

k = 1

\(k={M\over \sqrt{20\times 320}}\)

M = 80 H

Coefficient of coupling

The coupling coefficient can be defined as a magnetic flux produced between two different coils while managing the flux successfully.

\(k={M\over \sqrt{L_1L_2}}\)

where, k = Coefficient of coupling

M = Mutual inductance

L1 and L2 = Self-inductance of two coils

Calculation

Given, L1  5H &  L2 = 20 H

M = 8 H

\(k={8\over \sqrt{5\times 20}}\)

k = 0.8

The coefficient of coupling is a factor used when two coils are mutually coupled in a magnetic circuit.

The coupling coefficient K is a measure of the magnetic coupling between two coils.

The coupling coefficient is the amount of inductive coupling that exists between the two coils is expressed as a fractional number between 0 and 1.

The coupling factor between the coils or the coupling coefficient is given as

\(K = \frac{M}{{\sqrt {{L_1}{L_2}} }}\)

∴ \(K \propto \frac{1}{{\sqrt {{L_1}{L_2}} }}\)

Where M = Mutual inductance

L1 and L2 = Self-inductance

Important Points

0 ≤ K ≤ 1

K < 0.5 loosely coupled

K > 0.5 tightly coupled

K = 1 magnetically tightly coupled.