Friday, September 16, 2016

Alternate Current through Resistor and Inductor and capacitor

Alternate current is not a flat current and it can be expressed as a function of the time. Over the average time period, the average value of the current becomes zero. At every half cycle, it reverses its direction.

The current in the circuit varies with time and it varies similar to SIN function of the trigonometry. Thus the alternate current can be expressed as a function of the time. The current is the product of maximum possible current and sin function of time. The maximum current is also known as peak value of the current. We can also express the corresponding EMF as a function of time and SIN function.

It is very clear that they vary with time. For each instant of time we can measure the value of instantiates current and voltage and we can also draw a graph s shown in the diagram below.

As the average value of the current and voltage values becomes zero over the entire cycle, to measure the average value of them is studied only for the half of the cycle. The average value is defined as the product of peak value with two divided with the value of pi value. We can also define one more parameter called RMS value as the peak value and square root of the two.

Alternate current through the resistor

Let us assume a resistor connected to an AC source. With the time, the voltage developed across it changes and we can express it with the help of the SIN function. The EMF can be written as the product of the current and resistance. Thus we can transform the equation in terms of current as shown in the diagram below. Thus we can understand that the voltage and current has no phase difference are they are in same phase.

When the alternate current passes through a pure resistor, there is no phase difference between the voltage and current are in the same phase.

Alternate current through a inductor

When a alternate current is passed through a inductor, an induced EMF is developed in the coil. This induced EMF opposes the current passing through the inductor.

For the current to pass in the circuit of the inductor, the applied voltage shall be at least equal to the induced EMF developed across the inductor.

Thus by equating both of them that is the developed EMF and induced EMF, we can get the equation of the current. We can express the induced EMF in terms of self inductance and rate o the change of the current. Thus we can integrate the equation and able to write the equation for the instantaneous current.

Basing on that it can be easily understood that the there is a phase difference between voltage and current passing through a inductor. We have proved that current is lagging when compared with the voltage by ninety degree. The other way the EMF is leading the current by 90 degree when alternate current is passing through a inductor.

Alternate current through capacitor

Let us assume that the alternate current is passing through a capacitor. We can express the capacity as the ratio of charge and voltage across it.

We can write the voltage of the alternate current and so that we can simplify as shown in the diagram below. We can simplify the equation and get the value of the charge. By integrate the charge equation with time, we do get the rate of change of charge and that is called the current. Thus we have prove that the current is leading the EMF by 90 degree. In the other way, the voltage across the capacitor is lagging by 90 degree when compared with the current.

The maximum current can be expressed as the ratio of maximum EMF and the corresponding opposition to the current. This opposition is called capacitive reactance.

Similarly we can also find the inductive reactance as the opposition offered by the inductor to the alternate current. Both of these opposition depend on the angular velocity of the alternate current passing through them.

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