.

von WALTER R. WESTPHAL

Although repeated many times in audio literature, the information about the "decibel" and its uses - and misuses - must be thoroughly understood by anyone who wishes to understand specifications.

The decibel has long been the subject of a hazy issue in electronics. It is used a dozen times in each issue of electronics and high-fidelity magazines and each time it seems to be used in a different way.

The reader imagines that a wealth of higher mathematics is needed to understand its use. And a lot of careless usage has made him wonder if anyone really understands the decibel either!

What does it mean to read that an amplifier has a frequency response of "5 cps to 160kc, within 1db" or that an oscilloscope response is "down only 3db at 200kc"? A microphone is said to have an "output of -60db." An amplifier has a "hum level -90db below 20 watts."

What do these statements mean? Are these different uses of the decibel or is there some common denominator present in all of these statements'? The decibel is an important concept in the analysis of electronic equipment and one which should not be left to the imagination.

In every case the decibel is a ratio of two electrical powers. This ratio, like any other ratio, is really a fraction having a numerator and a denominator. The two values of power then become the numerator and denominator of this fraction.

One of these values is known as the reference power and in any situation is either expressly stated or what is more common, is understood. In many cases the difficulty is in knowing what value of power is being used as the reference.

We now turn to a modern definition of the decibel and see how it is used in some examples.

Two power levels, P\ and P2, are said to differ by n decibels when the following equation holds:

n = 10 log P1/P2

(where n is the number of decibels)
P1 is the larger of the two powers and P2 is the smaller of the two powers.

Both P1 and P2 must be measured in the same units (watts, milliwatts, etc.). The ratio P1/P2 is always arranged so that its value is (allways must be) greater than 1.

To signify a power gain, we prefix a "+" sign and a power loss is expressed by prefixing a "-" sign. Contrary to some textbooks it is not necessary to remember two separate formulas and a little checking with numerical values will show this. It is only necessary to add the + or - sign depending upon whether a gain or loss is encountered and this can always be determined by inspection.

We can now put this equation to work by using it in a simple example.

Figure 1 illustrates an audio power amplifier whose power gain is to be measured. The output power is seen to be
P out = E hoch2/R = 8 hoch2/16 = 4 watts.

The input power can be found in a similar fashion once the input impedance (assumed to be a pure resistance) is known. The manufacturer usually supplies this data and in this case it is 1 megohm. Therefore, P in = E hich2/R = 2 hoch2/1,000,000 =
0.000004 watts.

The power gain is then
n = 10 log (4/0.000004)
n = 60db

A power gain is indicated here by output being greater than input. The fact that this amplifier has a gain of 60db is of limited value in comparing it with another amplifier or against certain arbitrary standards.

What is of greater importance is the relationship between the power output at some reference frequency (usually 1000 cps) and at many other frequencies above and below this. Let's illustrate this with another example using the same amplifier as shown in Fig. 1.

Suppose that the input signal remains constant at 2 volts but that the power output drops to only 1 watt at 50 cps.

Using the 4 watts (at 1000 cps) of our original example as the reference, n would indicate a power loss of approximately 6 db.

If we could calculate the power output at many different frequencies and plot these data on semi-logarithmic graph paper, our results would look like Fig. 2. Here zero db represents the power output at some (mostly one only) reference frequency (here 1.000 cps). Any output lower than this would represent a power loss (-db) and these values could be plotted versus frequency.

Figure 2 is known as a universal response curve and shows us something about the response of an amplifier without regard to the actual power output. This is convenient when comparing several amplifiers as to frequency response alone.

In fact, absolute values of power are of small importance in studying response curves.

The reader has probably noticed that in the above calculations power was found by using E hoch2/R. And when two values of power were being compared (to find their ratio), one value of E2/R was divided by another value of E2/R. With the two values of R being the same in both cases (16 ohms) we could have saved some effort by simply dividing E 1-2/E 2-2 to obtain the power ratio. This can be carried one step further because

10 log(E 1-2/E 2-2) = 20 log (E1/E2)

There are two very important points to notice in regard to this formula. First, it is simply a short-cut to finding power gain; the result is not "voltage gain." Secondly, the value of resistance across which the two voltages are measured must be equal. The neglect of these two restrictions has led to much of the confusion regarding decibels.

(Continued on page 77)

Fig. 1. The gain of an amplifier can be expressed in decibels if input and output power are known. The value of input resistance is usually supplied by the manufacturer.
Fig. 2. A universal response curve. See text tor details ot how the curve is determined.

The points brought out in the preceding paragraph can best be illustrated by means of some examples.

Figure 3 shows a step-up transformer in which the primary voltage is 100 and the secondary Voltage is 500 volts. To use the above formula without regard to its true meaning and the restrictions imposed on it would lead to a "voltage gain" of almost 14db.

As there is no power gain, the use of the decibel is meaningless and shows what an indiscriminate use of a formula can lead to.

Many older textbooks still contain examples in which the voltage gain is calculated in this manner. Of course, this should not be confused with the voltage gain of an amplifier stage in which the output voltage is divided by the input voltage giving rise to a number representing the stage gain. This is perfectly legitimate; the trouble arises when the decibel is made to serve as a measure of stage gain.

Fig. 3. A transformer can step up a voltage but cannot produce a power gain. Hence, using n = 20 log E1/E2, to get the voltage gain in db is meaningless as the decibel is reserved as a measure

Figure 4 illustrates another example of the incorrect use of the formula. The input and output voltages are equal and improper use of n = 20 log E1/E2 would give zero gain as the result. But the input power can be seen to be 0.625 milliwatt and the output power is 6.25 watts, resulting in an actual gain of 40db.

The situations illustrated in Figs. 3 and 4 are typical of the results obtained when certain restrictions are disregarded. Decibels are always used to measure power gain; voltage gain in this sense is meaningless. The two expressions

n = 20 log E1/E2 and
n = 20 log I1/I2

are merely short-cuts to calculating-power gain and furthermore can only be used when the resistances across which the two voltages were measured or through which the two currents are flowing are equal.

This restriction was disregarded in a recent textbook problem on antenna gain. A folded dipole developed a signal of 150 microvolts while a rhombic antenna, receiving the same signal, developed a signal of 700 microvolts. Use of the formula in the preceding paragraph yields a gain of 13.4 db whereas the gain is actually only 10.4 db. The explanation lies in the fact that the characteristic impedance of the two antennas is not the same. A folded dipole has a characteristic impedance of 300 ohms while that of a rhombic is 600 ohms.
.

We can now turn with profit to some of the statements mentioned at the beginning of this article and interpret them in the light of what we have learned.

When amplifier specifications read: "5cps to 160kc, within 1 decibel", what is meant is that if the power output at any frequency in this range is compared to the power output at 1000 cps, they will not differ by more than 1 decibel. Or to say that the "response is down only 3db at 200 kc" means that the gain of the oscilloscope's amplifier at 200kc is only one-half that of its midfrequency range.

(A 3db loss means the same as half-power, and a 3db gain represents twice the reference power.) When the power output of a microphone is given, it always refers to a power of 1 milliwatt as the reference. For an output of -60db, the power output would be 10 hoch -9 watts.

Fig. 4. This stage has a gain of 40 db, but improper use of the same formula illustrated in Fig. 3 would indicate that the gain is 0 db. The text points out that the voltages must be measured across equal resistances for this formula to be valid.

When a manufacturer states that his amplifier has a hum level -90db below 20 watts, he means that when the output of the amplifier is 20 watts the power level of the hum signal is 0.02 microwatts. The hum present in the output of an amplifier is quite often measured in decibels and this might have been done in the circuit shown in Fig. 4 using the following procedure.

Suppose the signal voltage (at 1000 cps) across the plate-to-ground circuit measures 100 volts. And further that a harmonic wave analyzer set to 60cps measures 5 millivolts at full amplifier output. Then the hum is found to be down -80db.

Hence we see that the decibel is used as a means of comparing two values of power. It can be used to compare the input and output power of an amplifier, resulting in the db gain of the entire circuit.

Or, what is usually more important, it is used to compare the power output at some specified frequency with the power output at many other frequencies in its intended range.

Here the decibel is used to compare these values and the result, when plotted, becomes known as a universal response curve. Gain or loss refers to power levels above or below the reference power; input always remaining constant. The value of the decibel is in comparing power levels without the need to refer back to actual values of power.

When two voltages are measured across equal resistances they can be used to calculate the power gain. Much confusion has arisen in the past by calling the result "voltage gain." Actually, this is merely a short-cut to finding the power gain, measured in decibels.

The decibel is a useful tool in the analysis of electronic equipment. It enables comparisons to be made quite readily and "normalizes" the many variables present in a circuit under inspection. Its present use transcends the earlier concept of the logarithmic response of the human ear.

• Anmerkung : Der Artikel ist aus 1959 und manche Definition ist überholt !

.