Average Value of AC

What is an Alternating Current?

An alternating current is a type of electric current in which the direction and magnitude of the current change in time. The number of times this happens in 1 second is termed as the frequency of the alternating current. Alternating current is the mode in which current is transferred across large distances. This is because there is comparatively less loss in power in this mode of transportation compared to that of DC.

An alternating current can be of many types. It could be a sinusoidal, square wave, triangular wave etc. A sinusoidal voltage is usually represented by the equation,

V = V0 sin(ωt)

(Here V0 = amplitude of the voltage, ω = angular frequency and t = time )

Time average of a function:

The value of time average of function f(t) for time period of T seconds can be represented as

f(t)av = ( f(t1) + f(t2) + f(t3) + …… + f(tn))/(t1 + t2 + t3 + …… + tn )

Where t1 + t2 + t3 + …… + tn = T

This is the case where the function is discrete. But for the case of a continuous function, let’s say a sinusoidal function, we have to use integration across the time period to find the time average value of the function. We can mathematically represent it in the following way.

  f(t)av = 1T0Tf(t)dt

The Average value of Alternating Current:

The average value of alternating current (AC) can be obtained by finding the time average of current over a time period.

Let us now consider an alternating current with an amplitude I0 and angular frequency ω.

We can represent it as, I = I0sin(ωt) ( t = time )

We can see that the time period ‘T’ of the wave is 2π/ω.

Now, let’s take a small time element dt. We need to find out the charge transferred in the time period dt. So we use the equation

dQ = Idt         ( dQ = charge transferred, I = current )

Now, we need to find the total charge transferred, Q in one cycle of the alternating current.

So we will now integrate the current from t = 0 to t = T, where T is the total time period.

Therefore Q = 0TIdt

⇒ Q = t=0t=TI0sin (ωt)dt

At t = T, the value of ωt becomes 2π. So let’s replace ωt with a variable k. Therefore, the equation now becomes, 

⇒ Q =1ω 02πI0sin (k)dk

⇒ Q =1ω I002πsin (k)dk = 0

Therefore, we cannot integrate over the entire cycle to find the total charge transferred.

We can say this by looking at the graph, where the first half has the current to be positive and the second half has the current to be negative. So, their sum would become zero.

Since the total current transferred over the total time period is zero, we cannot find the average value of the alternating current. But we can use another method to calculate this value.

We know that the function of the alternating current is sinusoidal as well as symmetrical. So we can find the value of one half and multiply it but 2 to get the total charge transferred in one cycle. So, total charge transferred in a half cycle will be given by,

  Q’ =  t=0t=T/2I0sin(ωt)dt

⇒   Q’  =  1ω0πI0sin(m)dm (replacing ωt with m)

⇒   Q’  = (I0/ω) 0πsin(m)dm

⇒   Q’  = -(I0/ω) (cos(π) – cos(0))

⇒   Q’  = 2 x (I0/ω)

Now, we know that ω is given by ω = (2π)/T. Therefore, the equation now becomes,

  Q’  = 2 x (I0/(2π)/T)

  ⇒   Q’  = (I0T)/π

For both the halves, we will get the total charge transferred to be 

    Q = (2I0T)/π

Now, to find the time average value of the alternating current, we divide the total charge transferred by total time period. Therefore,

  = 2I0/π

Let us now consider a DC Id such that this current transfers the same amount of charge as the AC in the time interval T.

Therefore,   Id = Q/T

We can now equate both and Id to get,

Id = ((2I0T)/π)/T

⇒ Id = 2I0 / π

Now, the value of Id is called the average value of AC. We can represent it by the term Iav.

Therefore, Iav = 2I0 / π

Therefore, the average value of AC is given by multiplying the peak current value with 2/π.

Note: We can use this method to find the average value of AC for other types of functions as well. 

Conclusion: 

AC is a type of current where the value of voltage and current varies sinusoidally. In order to find the time average of a function, we integrate it over the time period and divide it by the time period. We cannot integrate over the total time period of a sinusoidal wave, instead, we integrate it over its half time period and multiply it by two. The average value of AC is given by the equation, Iav = 2I0 / π.