Here we will learn more about the function of an electric circuit, where an AC voltage is applied to the inductor. In order to find out the equation, we will consider the circuit as shown in the figure below. According to the figure, we have an inductor and an AC voltage V, which is represented by the symbol ~. The voltage produces a potential difference across its terminals that varies using a sinusoidal equation. The difference that is, the AC voltage thus can be given as,
From the equation, we deduce that vm sis used to signify the amplitude for the oscillating potential to denotes the differences. The angular frequency is given by ω. The current can be calculated by using the Kirchhoff’s loop rule. the equation which forms is as under,
Using the above equation in the given circuit, we can write,
Here the second term is the self-induced Faraday emf of the inductor and the L is the term given to the self-inductance of the inductor. The above two equations, that is, the equation for the voltage across the inductor and that derived from the Kirchhoff’s law from the given circuit, give the following equation,
Here, from the above equation, we see that the equation for the change in current as a function of time is sinusoidal in nature and is in the same phase as the source voltage and its amplitude is given by Vm/L. We can calculate the current through the inductor by integrating the above quantity with respect to time as:
The above equation leads us to the value of current as given by,
The integration constant has the dimension of current and is time independent. Since the source has an emf which oscillates symmetrically about zero, the current it sustains also oscillates symmetrically about zero, so that no constant or time-independent component of the current exists. Therefore, the integration constant is zero.
Here, the amplitude of the current is given by
The quantity ωL can be said to be equivalent to the resistance of this device and is termed as the inductive resistance. We denote the inductive resistance of the device as XL.
Thus, we can say that the amplitude of current in this circuit is given as
In the given equations, the dimension of the inductive resistance is seen to be the same as that of the resistance and also, the SI unit of capacitance is given as ohm. The capacitive resistance restricts the passage of current in a purely inductive circuit the same way as resistance hinders the passage of current in a purely resistive circuit.
Here we say that the inductive resistance is directly proportional to the frequency and the inductance. We also see from the above equations that the current in an inductive circuit is π/2 behind the voltage across the capacitor.