Unit:3 Alternating Quantities Notes (Second semester First parts) Diploma in  IT Engineering

Introduction to Alternating Quantity:  

Quantity which varies periodically with time is known as the alternating quantity. It can be voltage or current. Some waveform of alternating quantities is illustrated in Figure

Basic terms used in Alternating Quantity:

Here we take the instance of sinusoidal form of alternating current.

Fi

Cycle

It is the one complete set of positive & negative half of any alternating quantity.

Time Period

This is the time required in second to complete one cycle of any alternating quantity. Frequency

Number of cycles per second is known as the frequency of alternating quantity. This is the reciprocal of the time period and its unit is cycle per second or Hertz.

              f = 1/T (in Hertz)

Amplitude

This is the peak or maximum value of the alternating quantity. In Figure, I_m indicate the amplitude of the current wave.

Average and rms values of current and Voltage Average Value

* Definition: The average of all the instantaneous values of an alternating voltage and currents over one complete cycle is called Average Value.

If we consider symmetrical waves like sinusoidal current or voltage waveform, the positive half cycle will be exactly equal to the negative half cycle. Therefore, the average value over a complete cycle will be zero.

The work is done by both, positive and negative cycle and hence the average value is determined without considering the signs.

So, the only positive half cycle is considered to determine the average value of alternating quantities of sinusoidal waves. Let us take an example to understand it. 

Divide the positive half cycle into (n) number of equal parts as shown in the above figure

Let i₁, i₂, i₃…….. i_n be the mid ordinates

The Average value of current I_av = mean of the mid ordinates

 

* R.M.S Value

Definition: That steady current which, when flows through a resistor of known resistance for a given period of time than as a result the same quantity of heat is produced by the alternating current when flows through the same resistor for the same period of time is called R.M.S or effective value of the alternating current.

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In other words, the R.M.S value is defined as the square root of means of squares of instantaneous values.
Let I be the alternating current flowing through a resistor R for time t seconds, which produces the same amount of heat as produced by the direct current (I_eff). The base of one alteration is divided into n equal parts so that each interval is of t/n seconds as shown in the figure below. 

Let i₁, i₂, i₃,………..in be the mid ordinates

Then the heat produced in

Since Ieff is considered as the effective value of this current, then the total heat produced by this current will be

 

Now, equating equation (1) and (2) we will get

 

I_eff = square root of mean of squares of instantaneous values = R.M.S value

* Introduction to Alternating Quantities:

Assumed that alternating voltages and currents follow sine law and generators are designed to give emfs having sine waveform. The above said assumption makes the calculations simple. The method of representing alternating quantities by waveform or by the equations giving instantaneous values is quite cumbersome.

For solution of ac problems it is advantageous to represent a sinusoidal quantity (voltage or current) by a line of definite length rotating in counterclockwise direction with the same angular velocity as that of the sinusoidal quantity. Such a rotating line is called the phasor.

Consider a line OA (or phasor as it is called) representing to scale the maximum value of an alternating quantity, say emf i.e., OA = E_max and rotating in counter-clockwise direction at an angular velocity ω radians/second about the point O, as shown in Fig. 3.30. An arrow head is put at the outer end of the phasor, partly to indicate which end is assumed to move and partly to indicate the precise length of the phasor when two or more phasors happen to coincide.

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Figure 3.30 shows OA when it has rotated through an angle θ, being equal to ωt, from the position occupied when the emf was passing through its zero value. The projection of OA on Y-axis, OB = OA sin θ = E_max sin ωt = e, the value of the emf at that instant.

 

Thus the projection of OA on the vertical axis represents to scale the instantaneous value of emf.

* It will be seen that the phasor OA rotating in counterclockwise direction will represent a sinusoidal quantity (voltage or current) if:

(i)Its length is equal to the peak or maximum value of the sinusoidal voltage or current to a suitable scale.

(ii)It is in horizontal position at the instant the alternating quantity (voltage or current) is zero and increasing, and

(iii)Its angular velocity is such that it completes one revolution in the same time as taken by the alternating quantity (voltage or current) to complete one cycle.

* Phasor Diagram Using RMS Values:

Since there is definite relation between maximum value and rms value (E_max = n√2 E_rms), the length of phasor OA can be taken equal to rms value if desired. But it should be noted that in such cases, the projection of the rotating phasor on the vertical axis will not give the instantaneous value of that alternating quantity.

The phasor diagram drawn in rms values of the alternating quantities helps in understanding the behaviour of the ac machines under different loading conditions.