## Question File Number 14

### Decibels

• One cannot go very far in radio communications without encountering the term decibel.
• It is used to indicate the difference
• between two power levels, or
• between two signal levels.
• It can be used to indicate two things
• amplification (gain) or,
• loss (attenuation).
• It should be understood that this is a ratio based on the logarithmic scale.
• We are more used to thinking in linear units such as volts, ohms, amps, metres, litres.
• So, let's have a look at things logarithmic.

### Logarithmic changes

One of the fundamental laws of hearing and sight, and many other forms of physiological stimulus, is that the effect produced by an increase in the stimulus does not follow a simple arithmetic or linear scale. We have to keep doubling the stimulus to get an increase in unit steps. For example, if you stick two pins in your arm, you will find that it hurts twice as much as one pin. But it takes four pins to hurt three times as much and eight pins to hurt four times as much. (Don't bother to try this experiment!)

A system which measures unit steps in this way is the logarithmic scale.

### The Bel

We don't need to know much about logarithms but we need to be familiar with the following table:

 Unit Logarithm 1 0 2 0.3010 4 0.6021 8 0.9031

We don't even need to work to four decimal places. We can simplify the table to

 Unit Logarithm 1 0 2 0.3 4 0.6 8 0.9

Note that the first column doubles while the second column adds by + 0.3 each time.

We also need to know the following

 Unit Logarithm 1 0 10 1 100 2 1000 3

Observe the pattern shown by this table.

Note the number of zeros in the left-hand column and compare with the numbers in the right-hand column.

The bel was chosen to represent a power ratio of one logarithmic unit.

So a change of power level of 1 bel represents a power ratio of 10 times. (See the second line of the last table.)

Note too that a power gain of 100 times is 2 bel.

For many applications a bel is too large a unit and the decibel (dB) is used instead. So our two tables can be redrawn:

 Power ratio dB Power ratio dB 1 0 1 0 2 3 10 10 4 6 100 20 8 9 1000 30

A decibel (dB) is one-tenth of a bel.

### Power ratios

So, if we double the power, we can use the fact that the log of 2 is 0.3, and this would be a gain of 0.3 bel or 3 dB. Remember: Doubling the power is a 3 dB increase.

Similarly, if we halve the power, this would be 3 dB attenuation, which we can think of as -3 dB gain. A minus sign is used to indicate a decrease or negative gain.

### Some examples

Using the characteristics of logarithms, you will recognise that you can now multiply your gains by adding the logarithmic units. This is very convenient when you consider (say) a three-stage amplifier with 6 dB power gain in each stage.

The total power amplification will be (6 + 6 + 6) = 18 dB or 64 times.

(6 dB is a 4-times power increase - see tables above - and (4 x 4 x 4) is 64).

Another example: A power ratio of 20 times (i.e. 2 x 10) is the same as (3 dB + 10 dB) or 13 dB.

### Voltage ratios

In practice we often like to think in terms of voltage ratios rather than power ratios. We must remember that if we are measuring a signal voltage across an impedance, that if we double the voltage, the current also doubles, so the power has increased by FOUR times. We must remember too that the impedance where we are measuring must remain constant.

### Reference levels

Sometimes the dBW is used for the measurement and indication of power level. This means the measurement is compared to a reference level of 1 watt. The same rules apply:

1 watt represents 0 dBW

2 watt represents 3 dBW

4 watt represents 6 dBW

8 watt represents 9 dBW

16 watt represents 12 dBW

... and of course 10 watt represents 10 dBW.

### We can summarise with the following table

 Power ratio Voltage ratio dB 10,000:1 100:1 + 40 100:1 10:1 + 20 4:1 2:1 + 6 1:100 1:10 - 20

Note in the third line that doubling the voltage is equivalent to four times the power, this is a + 6 dB change.

### Suggestions

Learn the summary table given above and how it is derived.

Remember that RATIOS are involved - so look for the figures given in a question which have the same units (millivolts, watts, etc) from which you can work out a RATIO.

Be careful to use the correct column in the table for power and voltage ratios.

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Compiled Sun Nov 28 2010 at 8:41:44pm

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