Radio Broadcast (May 1928-Apr 1929)

OCTOBER, 1928 RADIO BROADCAST 357 O Radio Broadcast's Home Study Sheets Alternating Current Part I o o A N ALTERNATING current is one in which the magnitude and direction of flow of the current are continually changing. A direct current flows steadily in a given direction and at a more or less constant magnitude. The laws governing direct current phenomena and apparatus and the associated circuits are fairly simple; Ohm's Law will enable the experimenter to solve nearly all d.c. problems he runs into. The laws of a.c. circuits, on the other hand, are more complex — but for this very reason provide more enjoyment for the experimenter and those who like to solve problems. Home Study Sheet No. 3 shows how Ohm's Law is to be applied to some radio problems; this Sheet gives the fundamental facts about alternating currents. matically, using the data in Table 1? e/E, we shall have a ratio which is defined mathematically as the SINE of the angle, usually written sin <p. This is the factor which connects the length cf the arm and the veitical component. Thus sin <j> = e/E or e = E sin <*> The numerical values of the sines of a number of angles are given in Table 1, and with their use we have a means of calculating the instantaneous values of a voltage provided we know the maximum value and the phase angle in degrees. At 90 degrees the vertical component is equal to the arm E and so the instantaneous value of the voltage at this phase is the maximum value. Can you prove this mathe DEFINITIONS At regular intervals the direction of flow of an alternating current reverses, and therefore its variations in magnitude are as follows: the voltage starts at zero, rises to a maximum in one direction, decreases to zero, changes its direction, increases to a new maximum and then falls to zero, after which the CYCLE js repeated. Figure 1 is a representation of a single cycle of a.c. voltage. Such a picture is called a SINE WAVE. The number of times a second this cycle is repeated is called the FREQUENCY; the time required for one cycle is the PERIOD. House lighting currents are usually of 60 cycles although in some localities 25-cycle and 133-cycle circuits exist. So slowly do the alternations take place on a 25-cycle circuit that lights burning from them seem to flicker, although people who have never seen lights operated from circuits of higher frequency seem not to notice the unsteadiness of their own illumination. Audio-frequency currents have frequencies ranging from as low as the ear can hear, about 32 cycles per second, to as high as we can hear, about 15,000 cycles per second. Radio circuits have frequencies ranging from about 10,000 cycles to as high as 30,000,000 cycles. Longwave transoceanic communication takes place on the lower radio frequencies, broadcast transmissions on frequencies between 550,000 and 1,500,000 cycles, short-wave communication from 1,500,000 to 30,000,000 cycles. A kilocycle is one thousand cycles. PLOTTING AN A. C. CURRENT To show graphically what happens when an alternating current flows, let us look at Fig. 2 which consists of a circle in which is a rotating arm attached to the center and touching the circumference — a rotating radius. Suppose the circle moves to the right — carrying with it the rotating arm — at a constant speed such that it moves the distance of its diameter in the time it takes the rotating arm to make one complete rotation in a counterclockwise direction. Suppose a piece of chalk is attached to the end of the arm touching the circle. What sort of figure would it trace out as the two motions referred to take place? It would be a wavy form exactly like the alternating current curve in Fig. 1. The arm represents (mechanically) the rotating armature of an a.c. generator; the movement of the circle to the right represents the passage of time. The curve is a graphic representation of the changing values of an alternating current. PHASE Since a complete circle has 360 degrees, we may speak of the position of the arm in terms of the number of degrees it has rotated within the circle. When it is perpendicular to its starting position it has traversed one quarter of 360 degrees or 90 degrees; when it is parallel but pointing in the opposite direction, it has gone through 180 degrees, or one ALTERNATION, and so on. These various positions of the rotating arm are called its PHASES. Thus we speak of the 90-degree phase, and so on. Since the magnitude of the voltage in an a.c. circuit is continually changing, it becomes expedient to have a means of knowing what the voltage is at any particular instant. At 0 degrees it is zero, at 90 degrees it is maximum, at 180 degrees it is zero again, at 270 degrees it is maximum, but in the opposite direction, and at 360 degrees the cycle is completed, and the voltage is again zero. A.C— INSTANTANEOUS VALUE The INSTANTANEOUS value of an a.c. voltage or current is always referred to with regard to the maximum value. That is, if we multiply the maximum value by some factor which connects it and the phase, we shall have the instantaneous value. A measure of the instantaneous value is the vertical height of the end of the rotating arm above the horizontal axis. The vertical height is measured by the length of the line dropped perpendicularly from the end of the arm to the horizontal axis; it is known as the vertical component. Now let. us remove the vertical arm and its accessory lines from its circle and make what is known as a vector diagram at the 45-degree phase. In Fig. 3 let us label the arm, E (maximum voltage), the vertical component e (instantaneous voltage), and the angle which represents the phase, $>. Now if we divide the vertical component by the length of the arm, that is, EFFECTIVE OR R.M.S. VALUE Since an alternating current is reversing at a rapid rate, the needle and mechanism of an ordinary d.c. meter would indicate only an average value which would be zero. Some other means must therefore be provided for comparing an a.c. current with a d.c. current. We say, therefore, that an a.c. current is equal to a given d.c. current when they produce the same heating effect, and this value of the a.c. current is called its EFFECTIVE value. It is equal to the maximum value divided by the square root of 2, or _ I max. I eff. = 7=r = I x .707 and E eff, = l/2 E x .707 Since the heating effect of a current is proportional to the square of the current, we may obtain the effective or heating value over a complete cycle of alternating current by taking the average of the squares of several instantaneous values of _ current and extracting the square root. This value of current is then the square root of the average or mean squares of a number of values of current. This is abbreviated to "root mean square" or r.m.s., which is another term for effective value. In this expression "mean" and "average" have the same meaning. The maximum or "peak" value of an a.c. voltage is used in determining the C bias necessary for an amplifier; the r.m.s. value is used in all power problems. It may be obtained by dividing the maximum value by 1.4 or by multiplying the maximum value byO. 707. Meters for use on a.c. circuits indicate the effective or r.m.s. values. The form of the wave in well regulated a.c. power circuits is nearly a true sine wave, that is, one in which the relation between the length of the rotating arm (the maximum value E or I) and the vertical component (the instantaneous value e or i) is the sine of the angle between the arm and the horizontal axis. If the a.c. current is not a true sine wave these relations do not hold. PROBLEMS 1. Express in kilocycles the values of frequency given in paragraph four of this Study Sheet. 2. Assume that the maximum value of an alternating current is 10 amperes. On cross section paper plot its instantaneous values through one complete cycle by using the data in Table 1. 3. The effective value of an a.c. voltage is 110 volts. What is the maximum value? 4. What is the effective value of current in a circuit in which the maximum value of current is 10 amperes? 5. The maximum value of a certain current is 10 amperes. What is the phase when the instantaneous value is 5 amperes? 6. In a certain circuit the effective value of the voltage is 15 volts. What is the instantaneous value of the voltage at the 45 degree phase? 7. Check the relation between maximum and r.m.s. values by getting the square root of the average squares of several currents as plotted in Problem 2. 8. It power in watts is equal to (Ir.m.s.) 2 x R, what is the power used up in heating a resistance of 10 ohms when the peak voltage is 10? 9. Express by means of a vector diagram and in a formula the voltage in a circuit at phase 45 when the maximum value is 20. 10. Tell all you can about what the equation, e = 10 sin 30°, means. TABLE 1 Angle in Degrees FIG. 2 0 30 45 90 120 180 270 360 Sine 0.0 0.5 0.7 1.0 0.87 0.0 -1.0 0.0