Peak and RMS Value of Alternating Current/Voltage

Introduction

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

Peak Value and RMS value

A sinusoidal waveform consists of various values such as peak value, average value and 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).

The average value is defined as the average of all the instantaneous values of an alternating quantity such as current or voltage over one complete cycle.

RMS value (root mean square) is the square root of means of squares of instantaneous values. The RMS value can be evaluated by either graphical or analytical method.

Equation

While defining alternating current, the periodical reverting of direction and the change in magnitude concerning time can be represented in the form of the equation,

I = Io sin (ωt)

I = Io cos (ωt)

Io = Im = peak value of an alternating current

From the above equation, we can infer that the current may change at any instantaneous time. Thus, when the current is passed in a circuit, it is assumed to have been constant only for a small time, 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 = I0 sin (ωt)

dq = I dt

dq = I0 sin(ωt) dt

 We are aware that the alternating current presents a positive and negative cycle. Thus, if T is the time period for a complete cycle, then T/2 becomes the time period for half of the cycle. After integration, in the above equation from 0 to T/2, we get a value of charge,

 q = I0 T /.

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

 q = Iav . T/2.

 Hence, Iav = 2 I0/π = 0.636 I0

The average value of current for the entire cycle with a time period of T will be zero, as the negative and positive cycle paths cancel each other.

Similarly, we can find out the current root mean square value (RMS), greater than the average value.The RMS value of current can also be calculated for half cycle,

 Irms = Iv = Io/ √2 = 0.707 Io

 Irms = Iv = rms value of the current

 This current leads to heat generation. A small amount of heat produced in a small amount of time dt, is represented as ‘dH’.

 I = Io sin(ωt)

dH = I2 R dt

dH = (I0)2 R (sin ωt)2 dt.

To find the heat from the above small heat equation, for half cycle off time period T/2, we need to integrate the equation under the limits 0 to T/2.

 H = (Io)2 R . T/2

 H = (Irms)2 R . T/2

where, H = heat produced

R = resistance posed by the circuit.

The RMS value of alternating current is similar to direct current (DC) value while flowing through a circuit. It releases the same amount of heat produced by the alternating current (AC) at a specific time. 

In simpler words, the effective value in an equivalent DC of current and voltage represents the amperes or volts of DC that a sinusoidal waveform is equal to and its ability to produce the same power.

 The RMS value can be determined by either graphical or analytical method. The graphical method is used to find the RMS value of any non-sinusoidal time-varying waveform by drawing various mid-ordinates of the waveform. In contrast, the latter is a mathematical procedure representing the effective or RMS value of any periodic voltage or current using calculus (integration and differentiation). 

In order to find the RMS, we will find the mean square value.

 Irms = √mean of squares of the instantaneous value

 Irms = √[(i1)2 + (i2)2 + (i3)2 +——-+ (in)2]

Conclusion

Thus, we can infer that the maximum value attained by an alternative quantity, voltage or current, is called its peak for maximum value during its half cycle. This value is also known as amplitude or crest value. The sinusoidal alternating quantity 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 considering the positive or negative cycle signs. The average value is equivalent in both cycles; the average value is zero for a complete cycle. We can calculate the RMS or effective value of the alternating current, which represents the study current flowing through the resistance posed in the circuit, for a given period. Thus, a certain amount of heat is produced in the circuit.