Calculating Coefficient of Coupling for Coupled Coils

Understanding the relationship between self-inductance, mutual inductance, and the coefficient of coupling is fundamental in the study of coupled circuits. This problem requires us to calculate the coefficient of coupling (k) given the individual inductances of two coils and their mutual inductance.

Understanding Inductance and Coupling

• Self-Inductance (\(L\)): This property of a coil describes its ability to oppose changes in current flowing through it. It's measured in Henrys (H).
• Mutual Inductance (\(M\)): When two coils are placed close to each other, a change in current in one coil can induce an electromotive force (EMF) in the other coil. This phenomenon is quantified by mutual inductance, also measured in Henrys (H). It represents the magnetic coupling between the two coils.
• Coefficient of Coupling (\(k\)): This dimensionless parameter indicates how tightly two coils are magnetically coupled. Its value ranges from 0 to 1.
  • If \(k = 0\), there is no magnetic coupling between the coils.
  • If \(k = 1\), the coils are perfectly coupled, meaning all the magnetic flux produced by one coil links with the other.
  • Values between 0 and 1 indicate partial coupling.

Coefficient of Coupling Formula

The coefficient of coupling (\(k\)) is defined by the following formula:

\[k = \frac{M}{\sqrt{L_1 L_2}}\]

Where:

• \(M\) is the mutual inductance between the two coils.
• \(L_1\) is the self-inductance of the first coil.
• \(L_2\) is the self-inductance of the second coil.

Step-by-Step Calculation

Let's use the given values to calculate the coefficient of coupling.

• Self-inductance of the first coil, \(L_1 = 0.2 \, \text{H}\)
• Self-inductance of the second coil, \(L_2 = 2.45 \, \text{H}\)
• Mutual inductance, \(M = 0.14 \, \text{H}\)

Step 1: Calculate the product of the self-inductances (\(L_1 L_2\)).

\[L_1 L_2 = 0.2 \, \text{H} \times 2.45 \, \text{H} = 0.49 \, \text{H}^2\]

Step 2: Calculate the square root of the product of the self-inductances (\(\sqrt{L_1 L_2}\)).

\[\sqrt{L_1 L_2} = \sqrt{0.49} = 0.7 \, \text{H}\]

Step 3: Apply the coefficient of coupling formula.

\[k = \frac{M}{\sqrt{L_1 L_2}} = \frac{0.14 \, \text{H}}{0.7 \, \text{H}}\]

\[k = \frac{0.14}{0.7} = \frac{14}{70} = \frac{1}{5} = 0.2\]

Conclusion on Coupling Coefficient

The calculated coefficient of coupling \(k\) is 0.2. This value indicates that there is some magnetic coupling between the two coils, but it is not perfect coupling (which would be \(k=1\)). This type of calculation is crucial in designing and analyzing transformers, induction motors, and other magnetically coupled circuits.