The inductor is a circuit element. It can store electrical energy. An inductor does this in the form of electromagnetic energy. Also, it releases its stored energy as when required.
A common form of inductor generally consists of a solenoid of conductors. So it behaves as an electromagnet when current flows through it. This is how an inductor stores electrical energy in the form of electromagnetic energy.
Again when a time-varying current flows through it, the inductor causes an induced emf across it. If the current is alternating with steady RMS value, the induced emf across the inductor will also be constant. That means the emf will also have constant RMS. We know this emf as self-induced emf.
Definition of Self Inductance
The property due to which the emf induced in an inductor is called self-inductance.
Let us calculate the self-inductance. For that, we consider that there is a current flowing through the inductor. Also, we consider that this current is changing its value with time.
The induced emf across an inductor is directly proportional to the rate of change of current with respect to time. Hence,
We can rewrite the above equation as
This L is the constant of proportionality for the above relationship. This constant is known as the inductance of the solenoid.
Expression of Self Inductance
Expression of Flux Density
Before deriving the expression of self-inductance, we need to recall the relationship between the flux density and other parameters of an inductor.
Suppose, B is the flux density inside a solenoid. It is needless to say that
- The value of B is proportional to the current in the solenoid.
- Then it is proportional to the closeness of the turns in the solenoid. If the position of turns is closed then the magnetic flux density will also be high and vice versa. In other words, B is proportional to the number of turns per unit length. Where N is the total number of turns and l is the length of the solenoid.
- Also, B has a relation with the permeability of the medium inside the solenoid. If the permeability is high, B is high and vice versa. For this reason, as soon as we insert an iron core in the solenoid, the flux density increases. Because the permeability of iron is much higher than air.
From these three relations, we can derive that,
Where μo is the permeability of air. Since we have considered our inductor as an air-cored solenoid.
Relation between Flux and Current
As per the laws of electromagnetic induction, the emf induced across a current carrying solenoid is
Again, the self-induced emf across the inductor is
So, we can write,
From here, we can write
Again, if the cross-sectional area of the solenoid is A, then we can write the expression of entire flux as,
Finally, we can write the expression of inductance as
Let us apply a sinusoidal voltage across an inductor. The general expression of the voltage is
The self-induced emf across the inductor will be the same and opposite. So, we can write
From the above expression, we can write
Now, integrating both sides with respect to t, we get
From the above expression of current, it is clear that the maximum value or amplitude of the current waveform is
The definition of reactance is the ratio of the voltage amplitude to the current amplitude. So, we can write from the above expression,
So, we have found that the inductive reactance is directly proportional to the frequency of the electrical signal.