How To Calculate Flux Linkage: Detailed Insight And Facts

by

Flux linkage is an important concept in electromagnetism that helps us understand the relationship between magnetic fields and circuits. It quantifies the amount of magnetic flux that passes through a given coil or circuit. In this blog post, we will explore how to calculate flux linkage in detail, providing step-by-step instructions, formulas, and examples to ensure a thorough understanding.

The Mathematical Approach to Flux Linkage

Flux Linkage Formula

Before diving into the calculations, let’s understand the basic formula for flux linkage. Flux linkage ((\lambda)) is defined as the product of the number of turns in a coil ((N)) and the magnetic flux ((\Phi)) passing through it. Mathematically, it can be expressed as:

[ \lambda = N \cdot \Phi ]\tag{1}

where:
– (\lambda) represents the flux linkage,
– (N) is the number of turns in the coil, and
– (\Phi) denotes the magnetic flux passing through the coil.

Total Flux Linkage Formula

In some cases, a circuit may have multiple coils or loops. To calculate the total flux linkage in such a scenario, we need to sum up the individual flux linkages for each coil. The total flux linkage ((\Lambda)) is given by:

[ \Lambda = \sum \lambda_i ]\tag{2}

where:
– (\Lambda) represents the total flux linkage, and
– (\lambda_i) is the flux linkage for the (i)-th coil.

How to Calculate Flux Linkage

Now that we understand the formulas, let’s go through a step-by-step guide to calculate flux linkage.

Step-by-step Guide to Calculate Magnetic Field Flux Linkage

Step 1: Determine the number of turns ((N)) in the coil.
Step 2: Measure the magnetic flux ((\Phi)) passing through the coil.
Step 3: Use the formula (\lambda = N \cdot \Phi) to calculate the flux linkage.

Let’s work through an example to solidify our understanding.

Example 1:
Suppose we have a coil with 100 turns, and the magnetic flux passing through it is 0.05 Weber. To calculate the flux linkage, we can use the formula (\lambda = N \cdot \Phi):

[ \lambda = 100 \cdot 0.05 = 5 \text{ Weber-turns} ]

Therefore, the flux linkage for this coil is 5 Weber-turns.

How to Calculate Change in Magnetic Flux Linkage

In certain situations, the magnetic flux passing through a coil may change. To calculate the change in magnetic flux linkage, we can use the formula:

[ \Delta \lambda = N \cdot \Delta \Phi ]\tag{3}

where:
– (\Delta \lambda) represents the change in flux linkage,
– (N) is the number of turns in the coil, and
– (\Delta \Phi) denotes the change in magnetic flux passing through the coil.

Worked-out Examples on Flux Linkage Calculation

Let’s work through a couple of examples to further solidify our understanding.

Example 2:
Consider a coil with 50 turns. The magnetic flux passing through the coil changes from 0.02 Weber to 0.05 Weber. To calculate the change in flux linkage, we can use the formula (\Delta \lambda = N \cdot \Delta \Phi):

[ \Delta \lambda = 50 \cdot (0.05 - 0.02) = 1.5 \text{ Weber-turns} ]

Therefore, the change in flux linkage for this coil is 1.5 Weber-turns.

Example 3:
Let’s consider a scenario where a coil has 200 turns, and the magnetic flux passing through it remains constant at 0.1 Weber. In this case, since there is no change in magnetic flux, the change in flux linkage ((\Delta \lambda)) would be zero.

Advanced Concepts in Flux Linkage

How to Determine Maximum Flux Linkage

To determine the maximum flux linkage ((\lambda_{\text{max}})) in a coil, we need to consider the maximum value of magnetic flux ((\Phi_{\text{max}})) passing through the coil. We can use the formula (\lambda_{\text{max}} = N \cdot \Phi_{\text{max}}) to calculate it.

Practical Applications of Flux Linkage Calculations

Flux linkage calculations have various practical applications. They are extensively used in the design and analysis of electrical transformers, electric motors, and generators. Understanding flux linkage is crucial for optimizing the performance and efficiency of these devices.

What is the difference between magnetic flux and magnetic flux linkage?

The difference between magnetic flux and magnetic flux linkage lies in their definitions and applications. Magnetic flux refers to the total magnetic field passing through a surface, while magnetic flux linkage refers to the product of magnetic flux and the number of turns in a coil. To learn more about the distinction between these two concepts, you can visit the article on Difference between magnetic flux and magnetic flux linkage.

Numerical Problems on How to Calculate Flux Linkage

Problem 1:

A coil with 100 turns is wound around a magnetic core. The core has a magnetic field strength of 0.05 T and an area of 0.02 m². Calculate the flux linkage in the coil.

Solution:
To calculate the flux linkage, we can use the formula:

\text{Flux Linkage} = \text{Number of Turns} \times \text{Magnetic Flux}

The magnetic flux can be calculated using the formula:

\text{Magnetic Flux} = \text{Magnetic Field Strength} \times \text{Area}

Substituting the given values into the formulas, we get:

\text{Magnetic Flux} = 0.05 \, \text{T} \times 0.02 \, \text{m}^2 = 0.001 \, \text{Wb}

\text{Flux Linkage} = 100 \, \text{turns} \times 0.001 \, \text{Wb} = 0.1 \, \text{Wb}

Therefore, the flux linkage in the coil is 0.1 Wb.

Problem 2:

how to calculate flux linkage
Image by Nicholas D. Ward, Thomas S. Bianchi, Patricia M. Medeiros, Michael Seidel, Jeffrey E. Richey, Richard G. Keil and Henrique O. Sawakuchi – Wikimedia Commons, Licensed under CC BY-SA 4.0.

A solenoid has 500 turns and a magnetic field strength of 0.02 T. The length of the solenoid is 0.1 m. Calculate the flux linkage in the solenoid.

Solution:
To calculate the flux linkage, we can use the formula:

\text{Flux Linkage} = \text{Number of Turns} \times \text{Magnetic Flux}

The magnetic flux can be calculated using the formula:

\text{Magnetic Flux} = \text{Magnetic Field Strength} \times \text{Area} \times \text{Number of Turns}

The area of the solenoid can be calculated using the formula:

\text{Area} = \text{Length of Solenoid} \times \text{Turns per Unit Length}

Substituting the given values into the formulas, we get:

\text{Area} = 0.1 \, \text{m} \times \frac{500}{0.1 \, \text{m}} = 5 \, \text{m}^2

\text{Magnetic Flux} = 0.02 \, \text{T} \times 5 \, \text{m}^2 \times 500 = 10 \, \text{Wb}

\text{Flux Linkage} = 500 \, \text{turns} \times 10 \, \text{Wb} = 5000 \, \text{Wb}

Therefore, the flux linkage in the solenoid is 5000 Wb.

Problem 3:

A circular coil with a radius of 0.1 m is placed in a magnetic field with a magnetic flux density of 0.04 T. The coil has 200 turns. Calculate the flux linkage in the coil.

Solution:
To calculate the flux linkage, we can use the formula:

\text{Flux Linkage} = \text{Number of Turns} \times \text{Magnetic Flux}

The magnetic flux can be calculated using the formula:

\text{Magnetic Flux} = \text{Magnetic Flux Density} \times \text{Area}

The area of the circular coil can be calculated using the formula:

\text{Area} = \pi \times \text{Radius}^2

Substituting the given values into the formulas, we get:

\text{Area} = \pi \times (0.1 \, \text{m})^2 = 0.0314 \, \text{m}^2

\text{Magnetic Flux} = 0.04 \, \text{T} \times 0.0314 \, \text{m}^2 = 0.00126 \, \text{Wb}

\text{Flux Linkage} = 200 \, \text{turns} \times 0.00126 \, \text{Wb} = 0.252 \, \text{Wb}

Therefore, the flux linkage in the coil is 0.252 Wb.

Also Read: