Test: Mutual Inductance - 2

Concept:
The mutual inductance is given by:

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, L1 = 10 mH
L₂ = 40 mH

M = 20 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 L_2, then mutual inductance M between them then Coefficient of Coupling (k) is given by 

Where,

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 = 75 mH
L₂ = 105 mH
k = 0.45 
From the above concept,

M = 39.93 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 

Where,

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 = 4mH
L₂ = 9mH
k = 0.5 H
From the above concept,

M = 3 mH

Concept: 
The equivalent inductance of two inductors in a series connection can be calculated as
For mutual aiding (when polarity dots are present at the same ends)
L_eq = L₁ + L₂ + 2M
For mutual opposing (when polarity dots are present at the opposite ends)
L_eq = L₁ + L₂ − 2M

Calculation:
For series aiding, the equivalent inductance is 1.4 mH, i.e.
1.4 mH = L₁ + L₂ + 2M   ---(1)
For series opposing, the equivalent inductance is 0.6 mH, i.e.
0.6 mH = L₁ + L₂ − 2M   ---(2)
Subtracting the two equations, we get:
1.4 mH - 0.6 mH = 4 M
4 M = 0.8 mH
M = 0.2 mH = 0.2 × 10^-3 Henry

Concept:
Mutual Inductance:
When two coils are placed close to each other, a change in current in the first coil produces a change in magnetic flux, which cuts not only the coil itself but also the second coil as well. The change in the flux induces a voltage in the second coil, this voltage is called induced voltage and the two coils are said to have a mutual inductance.
Consider a pair of coupled inductors with self-inductance L₁ and L₂, magnetically coupled through coupling coefficient k.

Input and output voltage expressions are given as
V₁ = jωL₁I₁ + jωMI₂   ...(1)
V₂ = jωL₂I₂ + jωMI₁   ...(2)
Where,
ω = 2πf

M = Mutual inductance
L₁ = Inductance of coil one
L₂ = Inductance of coil two
Calculation:

In the given circuit secondary is open-circuited, so I₂ = 0 A
Given secondary voltage V₂ = 7.3 Vp-p
The output voltage expression from equation(2) is given as
V₂ = jωMI₁  ....(3)
The given voltage across the 22 Ω resistor is 5 V_p-p
So primary current is calculated as I₁ = 5 / 22 A_p-p
From equation(3)
|V₂| = 2π f M I₁
7.3 = 2 × π ×  100 × 10³ × M × (5/22)
M = 51.12 μH

Concept:
The coefficient of mutual induction of two coils is numerically equal to the magnetic flux linked with one coil when unit current flows through the neighboring coil.
If E is the induced emf in the coil at any instant of time, then from the laws of electromagnetic induction:

ϕ = M × I
M is the coefficient of mutual induction
I is the current.
Equation (1) becomes:

Calculation:
Given:
di/dt = 20
E = 50 mV
From equation (1) M can be calculated as:
50 × 10⁻³ = −M × 20
M = - 2.5 mH
As - sign indicates only induced EMF direction,
So M = 2.5 mH

Concept:
Let current through the two coils are I₁ & I₂ and the voltages are V₁ & V₂ respectively.
∴ The voltage induced in the 2^nd coil due to current in the first coil can be calculated as:

Where M = Mutual inductance between two inductive coils
dI₁ = change in current
dt = change in time
Calculation:
Mutual inductance (M) = 0.3
Change in current (dI1) = 4 – 1 = 3 A
Change in time (dt) = 0.03 sec

MUTUAL INDUCTION:
If two coils of wire are brought into close proximity with each other so that the magnetic field from one links with the other, a voltage will be generated in the second coil as a result.
This is called mutual induction: when voltage impressed upon one coil induces a voltage in another.
Mutual inductance formula-
V_m = M (dI₁ / dt)
where Vm = voltage induced in the secondary coil
I1 = is the current flowing in the primary foil
M = mutual inductance

where K = coupling coefficient
Given:
L₁ = 10 mH
L₂ = 20 mH
K = 0.75

Voltage in the second coil:

= 6.66 cos (314 t) volt

Concept:

• The inductor is an electrical component that is capable of storing electrical energy in the form of magnetic energy. 
•  The property of an electrical component that causes an emf to be generated by changing the current flow is known as inductance. Inductance is of two types
• Self-inductance: This is the phenomena in which change in electric current produce an electromotive force in the same circuit, and is given by

ϕ = L I 
Where ϕ  = Magnetic flux, L = Self inductance, I = Current
Mutual inductance: This is the phenomena in which change in flux linked with one circuit produce an emf in another coil and is given by
ϕ = MI
Where M = mutual inductance, ϕ  = magnetic flux, I = Current
The coupling coefficient is the ratio of mutual inductance to the maximum possible value of mutual inductance and is given by

Where M = Mutual inductance, L₁, L₂ = Self-inductance of coil 1 and coil 2 respectively
The maximum possible value of mutual inductance is at K = 1

Concept:
The inductance of a coil is given by,

μ₀ is magnetic permeability of free space
μ_r is relative permeability
N is number if turns
A is cross-sectional area
l is length
Calculation:

Given that M = 2 H
If the turns of one coil are decreased to half and the other is doubled.
Now the new value of the mutual inductance would be 2 H.