Radio Broadcast (May 1928-Apr 1929)

No. 15 February, 1929 Radio Broadcast's HomeStudy Sheets The Transmission Unit WHEN comparing the voltage amplification or the power output of two or more amplifiers, it is convenient to use a unit of comparison that bears some relation to the sensitivity of the ear. For example, the difference in volume output between a full orchestra playing very loud and playing very softly is about one million times. And yet to the ear the difference is only about 60 times; that is, between these two extremes in level, there are about 60 steps which the ear can detect by which the volume may be increased. As another example, one amplifier may deliver 600 milliwatts of power into a loud speaker while another is capable of turning out one watt, or 1000 milliwatts. Off hand one would say that the second is a great deal better, but is it? Of two amplifiers having voltage gains of 50 and 60, the second is better, of course, but if it costs a great deal more, is it worth it? As a matter of fact the differences between these two amplifiers would be scarcely audible to the average ear. A convenient unit of comparison has been known as the Transmission Unit of Loss or Gain, and is now called the Decibel, abbreviated to db. It has been called the nj, for want of a better name, up to the present time. The db is one tenth of the internationally used unit, the Bel, named in honor of Dr. Alexander Graham Bell, the inventor of the telephone. The transmission unit of loss or gain was originated in the telephone industry which deals almost exclusively with differences in volume in which the ear plays a part, and so such a unit, which had some connection with the manner in which the ear hears, was necessary. The db is defined as '"Ten times the common logarithm of the ratio between any two powers." Ndb 10 logm P1/P2 (1) in which N is the number of db by which the two powers Pi and P2 differ. The db is such a unit that the trained ear can just distinguish the differences between two powers which differ by one db, or one unit of loss or gain. The table below gives some easily remembered values of db and the corresponding power and voltage or current ratios. Ndb Apphox. power Approx. voltage or RATIO CURRENT RJ 3 2.0 1.4 4 2.5 1.58 6 4.0 2.00 7 5.0 2.24 9 8.0 2.8 10 10 3.16 20 100 10.0 23 200 14.0 30 1000 31.6 The second advantage in the use of such a unit, which is a logarithmic unit, will be apparent in glancing at the above table. Every time the power is doubled, we add 3 db, and every time the power is multiplied by 10, we add 10 db. Thus a ratio of 2 gives 3 db, a ratio of 4 gives 6 db, a ratio of 8 gives 9 db, etc. AH power ratios between 100 and 1000 are included between 20 and 30 db. DB are to be added when power ratios are multiplied, and subtracted when power ratios are divided. Thus, if one amplifier has a gain of 25 and is to be used after another similar amplifier, the total voltage gain is 252 or 625, which is awkward. But if the gain of each amplifier is 25 db, the total gain is 50 db. In other words the db is a compressed unit, and neglects differences of power the ear cannot detect. Thus, when an engineer SDeaks of the superior power output of his amplifier as compared with another, one must be careful to translate the power ratios into db before taking him too seriously. Example: Let us consider an amplifier that is capable of turning out 100 milliwatts of power. By how much must we increase its output before the ear can just tell the difference? Solution: A table of db, or a logarithm table, tells us that 1.0 db corresponds to a power ratio of 1.25. Thus the power output to which 100 milliwatts must be increased before the difference is audible to the ear is, db = 1.0 when P1/P2 = 1.25 or Pi/100 = 1.25 or Pi = 125 and so the power output must increased to 125 milliwatts before the ear can tell the difference. Strictly speaking, the unit of loss or gain deals with power ratios only, but with a little juggling of our mathematics we can use it to express ratios of current or voltage. It is only necessary to convert these voltages and resistances to powers, get the ratio and convert to db, or to use the following formula when voltage ratios are involved: Ndb= 20 log El/V^ . . E2/VR2 or (wheu current ratios are involved) Ndb onl d')2 Rl ,„, I1VR1 20 loS /i_v, 1, = 20 log I2VR2 .(2) in which the factor 20 appears because of the fact that the voltages in the above equation are squared. (When you square a number, you double its logarithm.) If the impedances into which two currents flow, or across which two voltages appear, are equal, the expression for db becomes, Ndb = 20 log or Ei2 20 log J-1 I2 (3) How to Use DB To convert power ratios to db look up the logarithm of the ratio and multiply it by ten. To convert current or voltage ratios to db, look up the logarithm and, if the impedances are equal, multiply this logarithm by 20. If the impedances are not equal, use formula (2). When looking up logarithms, remember that all numbers up to 10 have logs between 0 and 1, all numbers between 10 and 100 have logs between 1.0 and 2.0, all numbers between 100 and 1000 have logs between 2.0 and 3.0, etc. In other words the first figure of the log of all numbers between 100 and 1000 will be 2 and the next number tells us exactly where, between 100 and 1000, the number is. Thus, the power gain corresponding to 100 is 20 db and corresponding to 200 is 23 db — adding 3 db every time the power is doubled — and for 400 is 26 db. When converting db to power ratios, follow this example. What is the power ratio corresponding to 18 db? Dividing by 10 gives 1.8. The figure 1 tells us that the number lies somewhere between 10 and 100, and the figure 0.8, when looked up in a log table, is the log of 6.32. The ratio is, then, 63.2. If it were 28 db the figure 2 indicates that the number lies between 100 and 1000 and the "antilog" of 0.8 is 6.32 so the ratio is 632. Example: An amplifier has one volt applied to its input resistance of 10,000 ohms. Across its output resistance of 4000 ohms appears a potential of 40 volts. What is the power gain in db, the voltage gain in db, and the voltage gain expressed as a ratio? Would it be worth while to increase the amplification so that 50 volts appeared across the output? Solution: Power output watts Power ratio ■ Po = (E,) = _ 12 Ri ~ 10000 (Eo)= _ 402 Ro ~ 4000 10-1 watts 1600 4000 = .4 x 10" = 0.4. 4000 Power gain = 10 log 4000 = 36 db (because the log of 4 is 0.6 and because the first figure of the logs of all numbers between 1000 and 10,000 is 3 and the power gain ill db is 10 times the log of 4000) Voltage gain = 36 db = 20 log EoVbo Ei/i/RT Eo/i/Ro . 3 6 _ 1 It Voltage gain = 63 If Eo becomes 50, gain (between Eo = 50 and E0 = 40) = 20 log fg = 2.0 db. And so the difference between an output voltage of 50 and one of 40 would be hardly worth any trouble to get it. The solution to this example is characteristic of all such problems. The easiest way to learn to use the db chart in Fig. 1 is to look along the horizontal axis for the DB corresponding to a power ratio of 100 which is along the right vertical axis. This we know is 20 db. A power gain ratio of 20 corresponds to 13 db. A voltage gain ratio of 6 corresponds to 15.6 db. A power loss ratio of 0.2 corresponds to 7 db, a voltage loss ratio of 0.06 corresponds to 24.5 db, etc. Problems Problem 1. What in db corresponds to a voltage ratio of 100? Power ratio of 100? What voltage ratio corresponds to 100? What power ratio? Problem 2. A current of 0.006 amperes flows through a resistance of 1000 ohms. A switch reduces this current to 1.0 milliampere. How much is the current reduced in db? Problem 3. An amplifier has a normal output of 1 watt. A switch is provided so that its output can be reduced in 5 db steps. What is the output in watts when it is reduced by 5, 10, 20, and 25 db? Problem 4. A radio receiver has a voltage gain in its radio-frequency amplifier of 50 db. Express this in voltage ratio, and in power amplification provided that the same impedance closes the input and output of the amplifier. o 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 P I, vl il DB L0log,„ p"= i Olog.o I2VR2 /& tot Each Bon )B on 'actor he po he cu of K wer . rren adds cale £ scale nd 201 8 3 15 16 17 POWER SCALE 6 7 8 9 10 II 12 13 14 CURRENT SCALE 12 14 16 18 20 22 24 26 28 30 32 34 36 0 1 2 3 4 5 0 2 4 6 8 10 Fig. 1 — Chart of transmission units (DB) 19 20 38 40 • february, 1929 . . . Page 253 •