## Tuesday, September 7, 2021

### THE BASIS OF ALTERNATING CURRENT

ALTERNATING CURRENT

In the previous discussion on electricity and magnetism, we saw that current is generated in a wire loop rotating in a magnetic field flowed first in one direction and then in the other. In order to prevent similar reversals in the current flowing in the external circuit, we made a connection with the loop through a two-section commutator.

If we had not been concerned about the nature of the current in the external circuit, we could have used a pair of insulated slip rings, such as those shown above, instead of the commutator, and arranged for one of the brushes to bear on each ring. The current in the external circuit would then have kept on reversing in the direction in the same way as that in the wire loop.

Current of this kind is termed alternating current or a.c. it is simpler to generate and transmit than direct current and for most purposes is just as useful. Since, however, it must necessarily be always dying away in one direction and then building up again in the other, the effects of inductance and capacitance described before are of much greater importance than in the case of direct current.

GENERATION OF ALTERNATING CURRENT

In order to fix our ideas, let us consider the rotating in a magnetic field of the single conductor. We can, if we like, imagine that this conductor forms one side of a wire loop; it will not affect the argument. The path in which the conductor moves is in the direction of the rotation (counter-clockwise). The direction of the field is supposed to be from left to right, so that (as we can verify by applying the right-hand rule) the direction of current is into the paper.

Since the generation of current depends upon the cutting of lines of force, the part of the circular movement in which we are most interested in the motion across the field, i.e., the “up-and-down” motion that would be seen by a spectator away on the right-hand side of the page. Let such a spectator plot a curve showing how the position of the conductor changes from time to time as it moves across the field.

The curve that he will obtain is shown on the right of the figure. Horizontal distances represent time, and vertical distances the position of the conductor on either side of the horizontal centerline from which it starts. Let us suppose that one complete revolution takes 4 seconds, so that the horizontal divisions marked 0, 1, 2, etc., represent seconds. In the diagram, the vertical “timeline” are drawn for each third of a second.

Starting with the conductor in the position shown, the movement is upward for the first a quarter of a revolution, the highest point is reached at the 1-second line. Intermediate points on the curve are found by drawing horizontal lines from the position of the conductor in the circle to the corresponding timeline.

For the second quarter of the revolution, the movement is downward, so that at the 2-second line the conductor is back again at the centerline. For the next quarter, the downward movement continues, until at the 3-second line the lowest position is reached. For the fourth quarter, the movement is upward, and at the 4-second line the conductor is once more on the centerline. In the figure the curve is continued for a further second, this part representing the beginning of the next revolution.

The e.m.f generated in the conductor is dependent upon the rate of cutting the lines of force, that is, upon the speed at which it moves across the field. The distance that it moves across the field is represented by vertical heights on the curve, and since represented by horizontal distances, the rate at which the lines are being cut at any moment is indicated by the slope of the curve

Thus, at the start, the conductor is moving straight across the field and the rate of cutting is a maximum.

The curve is therefore steep. As the conductor moves round the first quarter of the circle, the rate diminishes until the top is reached, when there is no cutting of the field, and for a moment the curve is horizontal. Then, during the next quarter of the circle, the rate of cutting again increases to a maximum, this time in the opposite direction, as indicated by the increasing downward steepness of the curve. The rate then diminishes, and the curve becomes less steep until it is horizontal for a moment as the conductor passes the bottom of the circle. Finally, the rate of cutting increases to a maximum again in the first direction as the conductor returns to its original position.

The curve a which we have been considering is known to mathematicians as a sine curve. One of the properties of a  sine curve is that its steepness from point to point is represented by the height of an exactly similar curve displaced horizontally by one-quarter of a revolution. The dotted line b is such a curve. It crosses the centerline when curve a is momentarily horizontal, i.e., when the rate of cutting lines of force is zero. It reaches its greatest distance from the centerline when curve a is sloping most steeply, i.e., when the rate of cutting lines of force is a maximum. Moreover, its position above or below the centerline is an indication of whether curve a is sloping upwards or downwards, that is, of the direction in which the conductor is moving across the field.

Since curve, b gives a complete picture of the rate and direction of motion of the conductor across the field, it also represents the magnitude from moment to moment of the induced e.m.f.

Commercial a.c generators, like commercial d.c generators, use many conductors instead of one. Nevertheless, the e.m.f which they produce is usually not very different from that represented by a sine curve, and as this curve lends itself to fairly simple mathematical treatment, it forms the basis of most calculations.

FREQUENCY

In the above for example, curve b, like curve a, goes through a complete cycle of changes during each revolution of the conductor. The number of cycles that occur in a second is known as the frequency of the current. One cycle per second is called one hert(abbreviated Hz).

Although, for the sake of simplicity, we assumed that one revolution took 4 seconds, and the alternating current which took such a long time to complete one cycle would be of little value, and in practice, the length of a cycle is only a small fraction of a second. The frequency of most alternating-current power supplies in this country is fifty cycles per second or 60Hz.

EFFECTIVE VALUE OF CURRENT

We cannot expect a current that is continually dying away and building up again to be as effective as a steady current equal to its maximum strength. Evidently, we need some sort of average value of the alternating current before we can find its direct-current equivalent.

There are different kinds of average. Consider the four squares below, and suppose that the lengths of the sides are 2,3,4, and 5 meters respectively.

The average length of a side is then 2+3+4+5/4 = 3.5meters

The areas of the squares are 2×2, 3×, 4×4, 5×5 square meters, 9, 16, and 25 square meters, respectively. The average area is, therefore, 4+9+16+25/4 square meters = 13.5 square meters.

A square of this area has a side which measured in meters equals the square root of 13.5, and this is not 3.5, but about 3.67. we have therefore found two kinds of average: one the simple average of the side squares, and the other the side of a square equal to the average area. Both are equally correct, and either might be of greater interest in any particular case were putting fences round square fields for an instant should be concerned with the first average, but if we were buying turf with which to cover them, we should be concerned with the second.

We have investigated this point because the effective value of a current is proportional to the square of its strength. To find the value of, say, one half-cycle of alternating current, we might draw a large number of vertical lines, add them together, and divide the result by the number of lines. If the rise and fall of the current following a sine curve, the result would be 0.637 .ly, however, this is not the sort of average we want. To find the effective value, we must square (multiply by itself) the length of each vertical line, divide the total by the number of lines to find the average square, and then find the square root of the result to obtain the effective current. For the sine curve, this sort of average is not 0.637 but 0.707 of the maximum value. It is called the root-mean-square (r.m.s) value, because it is found by taking the square root of the mean (average) of all the squares. Another name for it is virtual value.

The diagram above will make this important but rather a difficult point clearer. Curve c  is a sine curve representing two cycles of alternating current. Curve s is obtained by taking a number of points on curve c,  squaring their distances or below the centerline, and plotting the results to any convenient vertical height on curve c above the curve center line represents the current flowing while depths below it represent current flowing in the other, it is permissible to consider the former as positive values and the latter as negative.

Note that curve s  lies wholly above the centerline; this is because the square of both positive and negative quantities is positive.

Since heights on curve s represent the squares of the current values, the shaded area represents the sum of these squares. It is evidently equal to half the area under the line sm, which represents the square of a steady current cm equal to the maximum value of the current. We may say, therefore, that

Effective value of current2 = Maximum value of current2/2

To come back to curve c, we take the square root of these quantities and find that

Effective value of current = Maximum value of current/Square root of 2.

To find the effective value of an alternating current, we, therefore, divide the maximum value by the square root of 2, or 1.414. this is the same as multiplying it by 0.707, the figure was already given. The same rule applies in the case of an alternating e.m.f.

Example: what is the effective value of an alternating current which varies in accordance with a sine curve and has a maximum strength of fifteen amperes/

Solution

Effective value = 0.707 ×maximum value

= 0.707 ×15amps. = 10.6 amps.

WAVEFORM

Although it is convenient for purposes of calculation to assume that the rise and fall of an alternating current can be represented by a sine curve, this is not always the case in practice. The actual shape of the curve is known as the waveform. The diagram below shows two possible waveforms, one more flat-topped than a sine curve and one more peaked.

ELECTRICAL ANGLES

When, as in the simple case we have been considering, one revolution of the conductor represents one cycle of e.m.f, it is natural to divide the cycle into 360 degrees. In practical generators of alternating current one revolution usually represents more than one cycle, but the practice of dividing the cycle into 360 degrees is so convenient for mathematical purposes that it is still followed. As a reminder that the angles measured by these degrees do not necessarily correspond to the angles actually turned through by the rotating part of the generator, the former is sometimes spoken of as electrical angles and the degrees as electrical degrees.