Psychologists say that our sense of hearing is roughly logarithmic. In other words, they think that you have to increase the sound intensity by the same factor to have the same increase in loudness. The decibel (dB) is a logarithmic unit used to describe a ratio. The ratio may be power, or voltage or intensity or several other things. The varied sensitivity of the human ear lead to the development of the decibel scale.

The decibel scale is a little odd because the human ear is incredibly sensitive. Your ears can hear everything from your fingertip brushing lightly over your skin to a loud jet engine. In terms of power, the sound of the jet engine is about 1,000,000,000,000 times more powerful than the smallest audible sound. That's a big difference!

Though the decibel scale originated to be used to express ratios, you will often see a decibel representation of a single sound level--the table below is an example of this. When you see this representation, the decibel values are normally considered to be compared to the threshold of hearing (the minimum sound that can be heard). On the decibel scale, the smallest audible sound (near total silence) is 0 dB. A sound 10 times more powerful is 10 dB. A sound 100 times more powerful than near total silence is 20 dB. A sound 1,000 times more powerful than near total silence is 30 dB.

So, from this table we see that a whisper is considered to be 100 times louder than 'near total silence' (10²), and a normal conversation is a million times (10⁶) louder than 'near total silence', or 10,000 (10⁴)times louder than a whisper.

Let us look at a mathematical example. Suppose we have two loudspeakers, the first playing a sound with power P1, and another playing a louder version of the same sound with power P2, but everything else (such as distance from the speaker and frequency) is kept the same.

The difference in decibels between the two is given by

(Notice that we have multiplied by a factor of 10. Actually, there is a unit called the 'bel'. Multiplying by 10 results in whole numbers, which are easier to talk about. Since the prefix 'deci' means 10, decibel means 10 bels.)

If P2 produces twice as much power as P1, the difference in dB is

If the second had 10 times the power of the first, the difference in dB would be:

If the second had a million times the power of the first, the difference in dB would be

A negative dB rating refers to a loss.

Decibels are used extensively in electronics for specifying amplifier and transmission system characteristics. We typically use them to express how much the voltage, current, or power has been increased by an amplifier, or how much one of these characteristics has been attenuated (reduced) by a transmission system or some other load.

Power is a product of voltage and current.

If either voltage or current is increased, the power will increase. In some applications it is necessary to increase the voltage, and the amount of the current may not be important. For this, we might use a special amplifier that amplifies voltage only (maybe even at the cost of reducing current). Likewise, we may have a need for a current amplifier. Other times, we are only concerned with an increase in power; in which case, the amplifier can increase voltage, current, or a combination of the two.

Power can also be expressed in terms of the voltage or current that is delivered to a load.

where R represents the constant resistance of the load.

The point here is that if the voltage delivered is doubled, then the power is quadrupled. For this reason, when using the Decibel notation for comparing voltages (or currents), we normally use a multiplication factor of 2. Remember, multiplying a log by 2 is the same as squaring its argument:

Therefore, when we are comparing two voltages (Vout versus Vin), and using the Decibel system, we would calculate the Decibel value:

The same is true for Decibels and current comparisons:

It is essential to realize that decibels are a measure of the ratio between two quantities. Decibels are a measure of how one quantity compares to some reference quantity. Decibels are not an absolute measure. A statement that a voltmeter reads 10 dB is meaningless without knowing the reference voltage, for example.

Some reference quantities are quite prevalent, and decibel measures relative to these quantities have been given unique symbols. For powers, both 1 watt and 1 milliwatt are often encountered as reference powers. A decibel measure of a power relative to 1watt is abbreviated dBW. A decibel measure relative to I milliwatt is abbreviated dBm. A decibel measure of a voltage or EMF relative to 1 volt as the reference is abbreviated dBV.

The addition of two signals to obtain their total power in decibels is performed by first adding the powers and then calculating the decibels. For example, if P₁ were the power of the first signal and P₂ were the power of the second signal, the total power (assuming that the signals added in phase, or coherently, and did not cancel each other).would be P₁ + P₂. The decibels relative to some reference power Pr would then be calculated from 10 log₁₀(P₁ + P₂)/P_r

Some practical examples might be helpful in understanding decibels and their use.

An input voltage of 1 volt to an amplifier gives an output of 100 volts. The voltage gain of the amplifier in decibels is

20 log₁₀100/1 = 20 log₁₀10² = 20 × 2 = 40 dB.

If an input voltage of 100 volts to some circuit gave an output of 1 volt, the gain would be

20 log₁₀ 1/100 = 20 log₁₀10^-2 20 × (-2) = -40 dB.

Clearly, if the output were less than the input, a loss occurred as the minus sign in the preceding example indicates. So, a loss is a negative gain. If a gain of 40 dB were followed by a loss of 40 dB, the net effect would be an overall gain of 40 - 40 = 0 dB.

A doubling in voltage represents about a 6-dB change, and a doubling in power represents about a 3-dB change. This is because the logarithm of 2 is about 0.3. Remembering this can make decibels easier to calculate. Consider a voltage increase by a factor of 20. Twenty equals two times ten. A factor of 2 in voltage corresponds to 6 dB, and a factor of 10 corresponds to 20 dB. Simply adding the two numbers, remembering that the logarithm of a product is the sum of the individual logarithms, gives 26 dB as the decibel figure corresponding to a factor of 20.

In analyzing filters, the decibel (db) unit is often used to describe the amount of attenuation offered by the filter. In basic terms, the decibel is a logarithmic expression that compares two power levels. Expressed mathematically,

where

= gain or loss in decibels

= input power

= output power

If the ratio is greater than one, the value is positive, indicating an increase in power from input to output. If the ratio is less than one, the value is negative, indicating a loss or reduction in power from input to output. A reduction in power, corresponding to a negative value, is referred to as attenuation.

A certain amplifier has an input power of 1 W and an output power of 100 W. Calculate the db power gain of the amplifier.

The input power to a filter is 100 mW, while the output power is 5mW. Calculate the attenuation, in decibels, offered by the filter.

Let us calculate a practical example for a stereo audio system. The output from a moving-magnet phonograph cartridge is about 2 mV. This signal is the input to the preamplifier, which boosts the voltage to about 0.2 volts. This amplification of voltage corresponds to

The stereo loudspeaker has an impedance of 4W , which is mostly pure resistance at audio frequencies. This loudspeaker consumes about 1 watt in producing a comfortable sound level. The voltage at the loudspeaker's terminals corresponding to 1 watt is obtained from the equation for power (P = E²/R) and is 2 volts. The output voltage of 0.2 volts from the preamplifier then needs to be amplified further by a factor of 10, or 20 dB, for driving the loudspeaker.

The output from the preamplifier does not have much associated current, and so will not have enough current to generate the power needed by a loudspeaker. There is then a need for further amplification in both voltage and current to drive the low-impedance loudspeaker. A power amplifier accomplishes this additional amplification. The input impedance to the power amplifier is customarily about 50kW . The 0.2 volts at its input from the output of the preamplifier would then correspond to a power of (0.2V)²/50W = 0.8 x 10^-12 watts. A factor of nearly one million to one, or a power gain of 10 log₁₀ 10⁶ = 60 dB amplifies this extremely small amount of power to 1 watt, which is an increase in power.

The overall voltage gain and power gain from the output of the phonograph cartridge to the input of the loudspeaker can also be calculated. The phonograph cartridge output voltage of 2 mV is amplified to 2 volts, which is a gain by a factor of 1,000 or 60 dB. The power from the cartridge at the input to the preamplifier is (0.002V)2/50kW = 80 x 10^-12 watts or 80 picowatts. This extremely small power is amplified to 1 watt, which is a power gain by a factor of nearly 10¹⁰, or 100 dB.

1. 2W
2. 5W
3. 10W
4. 100W