Peak and RMS Value of Alternating Current,Voltage

Alternating current is a type of current whose direction changes periodically through a load,  that is,  a complete cycle of an alternating current consists of a negative cycle and a positive cycle.  When graphically represented,  a positive cycle begins from the axis, reaches the maximum positive value, upwards, and comes back to zero (axis) again,  similarly, a negative cycle begins from the axis, reaches the maximum negative value downside, and comes back to zero (axis)  again.  Such graphical representation is known as a sinusoidal waveform.

Peak value and RMS value

A sinusoidal waveform consists of various values namely,  peak value,  average value,  RMS value.

Peak Value is defined as the maximum value that the alternating quantity (current or voltage) reaches in one cycle (either positive or negative).

RMS value (root mean square)  stands for the square root of means of squares of instantaneous values.  The RMS value can be determined by either graphical or analytical methods.

Equation

In the definition of alternating current,  the reverting of direction periodically and the change in the magnitude with respect to time can be represented in form of the equation,

I = Io sin(ωt)

I = Io cos(ωt)

Io = Im = peak value of an alternating current

Graph representing peak, RMS and average value

From the above equation, we can infer that the current can be changing at any instantaneous time,  so much so that in a circuit when the current  is passed it is assumed that it would have been constant only for a small time,  which can be represented as “dt” ,

The small charge flowing through the circuit during this small time dt,  is represented as “dq”. Hence,  the current can now be represented as 

I = Io sin(ωt)

dq = I dt

dq = Io sin(ωt) dt

We are aware that the alternating current has a positive cycle as well as a negative cycle,  each of the cycles is considered half, thus,  if T,  if the time period for a complete cycle then,  T/2,  becomes the time period for half of the cycle.  on integrating,  the equation above,  from 0 to T/2, we get a value of charge,

 q = Io T/

From this charge we can calculate the mean value of alternating current, to be,

 q = Iav . T/2

 hence, Iav = 2 Io /π = 0.636 Io

If we try to find out the value of current for the entire cycle having a time period of T,  then the average value will come out to be zero,  the negative cycle part and the positive cycle path cancel each other.

Similarly, we can find out the RMS,  root mean square value of the current,  which is greater than the average value,  similar to the average value of current. The RMS value of current is also calculated for half cycle,

 Irms = Iv = Io/ √2 = 0.707 Io

 Irms = Iv = rms value of the current

This current leads to the generation of heat,  the small amount of heat that is produced in a small amount of time dt,  is represented as,  as”dH”,

 I = Io sin(ωt)

dH = I2 R dt

dH = (Io)2 R (sin ωt)2 dt

In order to find the heat from the above small heat equation,  for half cycle off time period T/2,  we have to integrate the above equation of heat it under the limits 0 to T/2,  after doing this we get the value of heat,

 H = (Io)2 R . T/2

 H = (Irms)2 R . T/2

where H= heat produced

R= resistance posed by the circuit

RMS value

The RMS or effective value of the alternating current is a representation of the study current flowing through the resistance posed in the circuit,  for a given period of time.

Conclusion

Thus,  we can infer from the above piece of information that the maximum value attained by an alternative quantity,  voltage, or current,  during its half cycle is called its peak for maximum value,  this value is also known as amplitude or crest value.  The sinusoidal alternating quantity,  as shown above, attains its peak at 90°. From this peak value, we can find out the average of all the instantaneous values of the alternating quantity over one complete cycle, the Average value. The average value is calculated without consideration of the signs of the positive or negative cycle,  as for both the cycles,  the average value is equivalent,  and for a complete cycle the average value comes out to be zero. We also calculate The RMS or effective value of the alternating current,  which is a representation of the study current flowing through the resistance posed in the circuit,  for a given period of time. As a result of this current certain amount of heat is produced in the circuit.