BLOG 21 - FUNDAMENTAL CIRCUIT

RMS VALUE or Effective

WHAT IS RMS VALUE?

When dealing with Alternating Voltages (or currents) we are faced with the problem of how we represent the signal magnitude. One easy way is to use the peak values for the waveform. Another common method is to use the effective value which is also known by its more common expression of Root Mean Square or simply the RMS value.

Note that the RMS value is not the same as the average of all the instantaneous values. The ratio of the RMS value of voltage to the maximum value of voltage is the same as the ratio of the RMS value of current to the maximum value of current. Most multi-meters, either voltmeters or ammeters, measure RMS value assuming a pure sinusoidal waveform. For finding the RMS value of non-sinusoidal waveform a “True RMS Multimeter” is required.

Having now determined the RMS value of an alternating voltage (or current) waveform, in the next tutorial we will look at calculating the “Average” value VAV of an alternating voltage and finally compare the two.

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• THERE TWO TYPE BASIC METHOD IN RMS VALUE

RMS Voltage Graphical Method

Whilst the method of calculation is the same for both halves of an AC waveform, for this example we will consider only the positive half cycle. The effective or RMS value of a waveform can be found with a reasonable amount of accuracy by taking equally spaced instantaneous values along the waveform.

The positive half of the waveform is divided up into any number of “n” equal portions or mid-ordinates and the more mid-ordinates that are drawn along the waveform, the more accurate will be the final result. The width of each mid-ordinate will therefore be no degrees and the height of each mid-ordinate will be equal to the instantaneous value of the waveform at that time along the x-axis of the waveform.

Graphical Method

 

Each mid-ordinate value of a waveform (the voltage waveform in this case) is multiplied by itself (squared) and added to the next. This method gives us the “square” or Squared part of the RMS voltage expression. Next this squared value is divided by the number of mid-ordinates used to give us the Mean part of the RMS voltage expression, and in our simple example above the number of mid-ordinates used was twelve (12). Finally, the square root of the previous result is found to give us the Root part of the RMS voltage.

Then we can define the term used to describe an RMS voltage (VRMS) as being “the square root of themean of the square of the mid-ordinates of the voltage waveform” and this is given as:

 

and for our simple example above, the RMS voltage will be calculated as:

 

So lets assume that an alternating voltage has a peak voltage (Vpk) of 20 volts and by taking 10 mid-ordinate values is found to vary over one half cycle as follows:

 

Voltage	6.2V	11.8V	16.2V	19.0V	20.0V	19.0V	16.2V	11.8V	6.2V	0V
Angle	18o	36o	54o	72o	90o	108o	126o	144o	162o	180o

The RMS voltage is therefore calculated as:

 

Then the RMS Voltage value using the graphical method is given as: 14.14 Volts.

RMS Voltage Analytical Method

The graphical method above is a very good way of finding the effective or RMS voltage, (or current) of an alternating waveform that is not symmetrical or sinusoidal in nature. In other words the waveform shape resembles that of a complex waveform. However, when dealing with pure sinusoidal waveforms we can make life a little bit easier for ourselves by using an analytical or mathematical way of finding the RMS value.

A periodic sinusoidal voltage is constant and can be defined as V(t) = Vm.cos(ωt) with a period of T. Then we can calculate the root-mean-square (rms) value of a sinusoidal voltage (V(t)) as:

 

Integrating through with limits taken from 0 to 360o or “T”, the period gives:

 

Dividing through further as ω = 2π/T, the complex equation above eventually reduces down too:

RMS Voltage Equation

Then the RMS voltage (VRMS) of a sinusoidal waveform is determined by multiplying the peak voltage value by 0.7071, which is the same as one divided by the square root of two ( 1/√2 ). The RMS voltage, which can also be referred to as the effective value, depends on the magnitude of the waveform and is not a function of either the waveforms frequency nor its phase angle.

From the graphical example above, the peak voltage (Vpk) of the waveform was given as 20 Volts. By using the analytical method just defined we can calculate the RMS voltage as being:

VRMS = Vpk x 0.7071 = 20 x 0.7071 = 14.14V

 

Note that this value of 14.14 volts is the same value as for the previous graphical method. Then we can use either the graphical method of mid-ordinates, or the analytical method of calculation to find the RMS voltage or current values of a sinusoidal waveform. Note that multiplying the peak or maximum value by the constant 0.7071, ONLY applies to sinusoidal waveforms. For non-sinusoidal waveforms the graphical method must be used.

LEARNING:

The r.m.s. value of an a.c. supply is the steady d.c. which would convert electrical energy to thermal energy at the same rate in a given resistance. In RMS  VALUE we need to use the formula of Irms = I/square root of 2 and Vrms = V/square root of 2. The RMS value have two type of basic method the graphical and analytical method.