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Two coils having selfinductances of 10 mH and 40 mH are mutually coupled. What is the maximum possible mutual inductance?
Concept:
The mutual inductance is given by:
where, M = Mutual inductance
L = Selfinductance
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
L_{2} = 40 mH
M = 20 mH
The coefficient of coupling between two coils is 0.45. The first coil has an inductance of 75 mH and the second coil has an inductance of 105 mH. What is the mutual inductance between the coils?
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 selfinductance L_{1} and L_{2,} then mutual inductance M between them then Coefficient of Coupling (k) is given by
Where,
N_{1} and N_{2} is the number of turns in coil 1 and coil 2 respectively
A is the crosssection area
l is the length
Calculation:
Given,
L_{1} = 75 mH
L_{2} = 105 mH
k = 0.45
From the above concept,
M = 39.93 mH
The self inductance of two coils are 4mH and 9mH respectively. If the coefficient of coupling is 0.5, the mutual inductance between the coils is _____
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 selfinductance L_{1} and L_{2}, then mutual inductance M between them then Coefficient of Coupling (k) is given by
Where,
N_{1} and N_{2} is the number of turns in coil 1 and coil 2 respectively
A is the crosssection area
l is the length
Calculation:
Given,
L_{1} = 4mH
L_{2} = 9mH
k = 0.5 H
From the above concept,
M = 3 mH
The total inductance of two coupled coils in the ‘series aiding’ and ‘series opposing’ connections are 1.4 × 10^{3} Henry and 0.6 × 10^{3} Henry, respectively. The value of mutual inductance will be:
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} = L_{1} + L_{2} + 2M
For mutual opposing (when polarity dots are present at the opposite ends)
L_{eq} = L_{1} + L_{2} − 2M
Calculation:
For series aiding, the equivalent inductance is 1.4 mH, i.e.
1.4 mH = L_{1 }+ L_{2} + 2M (1)
For series opposing, the equivalent inductance is 0.6 mH, i.e.
0.6 mH = L_{1} + L_{2} − 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
An aircore radiofrequency transformer as shown has a primary winding and a secondary winding. The mutual inductance M between the windings of the transformer is ______ μH.
(Round off to 2 decimal places.)
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 selfinductance L_{1} and L_{2}, magnetically coupled through coupling coefficient k.
Input and output voltage expressions are given as
V_{1} = jωL_{1}I_{1} + jωMI_{2} ...(1)
V_{2} = jωL_{2}I_{2} + jωMI_{1} ...(2)
Where,
ω = 2πf
M = Mutual inductance
L_{1} = Inductance of coil one
L_{2} = Inductance of coil two
Calculation:
In the given circuit secondary is opencircuited, so I_{2} = 0 A
Given secondary voltage V_{2} = 7.3 Vpp
The output voltage expression from equation(2) is given as
V_{2} = jωMI_{1} ....(3)
The given voltage across the 22 Ω resistor is 5 V_{pp}
So primary current is calculated as I_{1} = 5 / 22 A_{pp}
From equation(3)
V_{2} = 2π f M I_{1}
7.3 = 2 × π × 100 × 10^{3} × M × (5/22)
M = 51.12 μH
Determine the mutual inductance between two coils when a current changing at 20 A/s in one coil induces an EMF of 50 mV in the other.
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
Two inductive coils which are close to each other have a mutual inductance of 0.3 H. Current through one coil is increased from 1 A to 4 A in 0.03 s. The voltage induced in the other coil is:
Concept:
Let current through the two coils are I_{1} & I_{2} and the voltages are V_{1} & V_{2} 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
dI_{1} = 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
Two coupled coils have self inductances L_{1} = 10 mH and L_{2} = 20 mH.
The coefficient of coupling (K) being 0.75 in the air. Voltage in the second coil when the current in circuit is given by I = 2 sin (314t) A is _______
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 (dI_{1} / 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:
L_{1} = 10 mH
L_{2} = 20 mH
K = 0.75
Voltage in the second coil:
= 6.66 cos (314 t) volt
Two coils having selfinductance of L_{1} and L_{2}, respectively, are magnetically coupled. The maximum possible value of mutual inductance between the coils is
Concept:
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, L_{1}, L_{2} = Selfinductance of coil 1 and coil 2 respectively
The maximum possible value of mutual inductance is at K = 1
Mutual inductance between two closely coupled coils is 2 H. Now, if the number of turns in one coil is reduced by 50 percent and those of the other coil is doubled then, new value of mutual inductance is:
Concept:
The inductance of a coil is given by,
μ_{0} is magnetic permeability of free space
μ_{r} is relative permeability
N is number if turns
A is crosssectional 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.
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