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SUBJECT : ELEMENTS OF ELECTRICAL ENGINEERING

Topic :r.m.s. value , average value , form factor , peak factor

Created by :- akshat raval 151080106022 civil 2nd sem.

contents R.M.S VALUE :- DETERMINATION OF

R.M.S. VALUE1. Mid-Ordinate Method2.Analytical Method3.R.M.S. Value of A Half Wave Rectified A.C.

AVERAGE VALUE OF AN ALTERNATING QUANTITY :-

1.Mid-ordinate method2.Analytical Method3.Half wave rectified A.C.

Form Factor Peak Factor

Root mean square value The R.M.S. value of the a.c. current is also called the effective value. The R.M.S. value of an a.c. is the equivalent steady current which

when allowed to flow through a given period of time prouducess the same amount of heat as produced by the alternating current. When it flows through the same circuit for the same period of time.

We consider a simple example as in the figure which illustrates the significance of the R.M.S. value of an alternating current.

L is a metal filament lamp. S is a switch. Initially the switch S is closed on contact X, so that the lamp is connected to the a.c. supply. The brightness of the filament lamp is noted.

Next, the switch S is closed on contact Y and the resistor value is varied so that the lamp gives the same brightness as with a.c. When the lamp gives the same brightness, the reading on the moving coil ammeter A is noted. This reading gives nothing but the direct current that produces the same heating effect as produced by the alternating current.

If the ammeter reads 0.5 A when the equality of brightness is attained, the R.M.S. value of the alternating current 0.5 A.

Determination of r.m.s. value Mid-Ordinate Method:-

the wave is divide into m equal intervals. The instantaneous values of the current during these intervals are i1,i2,i3,…..im.

Thus, the heat produces in ‘t’ seconds on the application of the alternating current to the resistor R.

Now let I be the value of the direct current which when flowing through the same circuit of resistance R ohms for the same time ‘t’ seconds, produces the same amount of heat

Heat produced by d.c. =

But = Mean value of

Similarly R.M.S. value of an a.c. voltage, E= Mean value of

Mean value of

Analytical Method:-

The instantaneous value of the current i=Im sinθ Thus the mean value of (i)² over one complete cycle will be

Thus the R.M.S. value of the A.C. sinusoidal current. I=0.707*Im (max. value of the current)

R.M.S Value of a Half Wave Rectified A.C.:- A half wave rectified alternating current is one in which the current flows only during one half of a cycle and remains suppressed during the other half of that cycle. Figure shows a half wave rectified waveform.

the dotted path shows the suppressed half cycle So here the summation of the instantaneous values of the current is carried out only for the period for which the current flows i.e. from 0 to π but it would be averaged over the entire cycle.

Average Value Of An Alternating Quantity Average value of an alternating quantity is given by that steady

current which when flowing through a given circuit for a given time, transfers across that circuit the same amount of charge as is transferred by the alternating current when it flows through the same circuit for the same time.

The average value of an alternating voltage or current is given by the arithmetic mean of the ordinates of that quantity at equal intervals over a half cycle.

This is because the arithmetic mean found over one complete cycle of a symmetrical sinusoidal or non-sinusoidal waveform is zero. So, the average value of a symmetrical alternating quantity is found only over a half cycle.

Method to find average value

From the figure (a) & (b) we get,

Iav = i1 + i2 +……+in n

Analytical Method:-

Equation of the alternating current I = Im sinθ.

Thus the average value of a symmetrical a.c. is 0.637 Im.

Average Value of a Half Wave Rectified A.C.:-

The half wave rectified a.c. shown in the figure flows only over a half cycle i.e. its average value over a complete cycle will not be zero.

The summation of the currents is done only over the period for which the current flows. But it is averaged over the entire cycle.

Form factor Form factor is defined as the ratio of the R.M.S. value to the

average value of an alternating quantity. It is denoted by K f. Kf = R.M.S. value average value Thus for a symmetrical sinusoidal wave, Kf = .707 Im = 1.11 .637 Im

Having knowledge of the form factor, the R.M.S. value can be found from the average value and vice-versa.

Peak factor (Ka) Peak factor is defined as the ratio of the peak value to the R.M.S. value

of an alternating quantity. Thus for a symmetrical sinusoidal wave, Ka = Im = Im = 1.414 Irms .707 Im Peak factor is also called the crest factor or the amplitude factor. Normally the rms value of an alternating quantity is specified. If the

peak factor is known, the maximum value can be found. The knowledge of the peak factor is of importance :1. For measuring the iron losses as these losses depend upon the peak

value of the flux.2. In dielectric insulation testing, because the dielectric stresses during

insulation testing are proportional to the peak value of the applied voltage.

THANK YOU

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