# Henry Ott Consultants

Electromagnetic Compatibility Consulting and Training

# The Decibel

The decibel (dB) is a very commonly used and often misunderstood unit of measurement.  The dB is a logarithmic unit expressing the RATIO of two powers.  It is defined as:

Number of dB = 10 log (P2/P1).

The dB is not an absolute quantity, it is always a RATIO of two quantities.  The unit can be used to express power gain (P2>P1), or power loss (P2<P1) -- in the latter case the result will be a negative number.

The decibel  actually comes from a logarithmic unit of measurement called a "Bel", named after Alexander Graham Bell.  One Bel is defined as a power ratio of ten (or ten times the power).  It was originally used to measure acoustic power (sound)  ratios in telephony.  The Bel is a rather large unit, so the decibel (dB), which is 1/10 of a bel, is more commonly used.

Although the dB is defined with respect to power, it has become common practice to also use it to express voltage or current ratios, in which case it is defined as:

dB Voltage = 20 log (V1/V2),  or  dB Current = 20 log (I1/I2).
This is only correct, however, when V1 & V2 (I1 & I2) are measured across the same value of impedance.  See "Derivation of dB as a Voltage Ratio" below

Being a logarithmic unit it compresses the result, and allows us to easily measure or plot signals that cover a large dynamic range.  This can be a useful feature in the case of EMC measurements.  For example a voltage ratio of 1,000,000:1 could be expressed as 120 dB.

Since the definition of the decibel involves logarithms, it is appropriate to review some of the properties of logarithms. The common logarithm (log) of a number is the power to which 10 must be raised to equal that number.   Therefore,

if Y = log X,   then X = 10 raised to the power Y.

Some useful facts about logarithms are:

the log 1 = 0,
the log of numbers >1 are positive,
the log of numbers <1 are negative,
the log (AxB) = log A + log B,
the log (A/B) = log A - log B, and
the log of A to the N th. power = N log A.

As can be seen from the above, multiplication of numbers becomes addition of their logs.  This can be very useful for EMC measurements.  For example, let's assume that we want to know the voltage that would be measured by a meter connected by a long cable to an antenna exposed to an electromagnetic field.  We would have to multiply the field strength by  the antenna factor, then multiply it by the loss  of the cable connecting the antenna to the measuring instrument, and also multiply it by the amplifier or attenuator gain/loss, etc.  If all these numbers are expressed in dB, however, all we have to do is add them  together -- a much simpler task.

In EMC emission measurements we usually talk of dB microvolts (or dB microvolts/meter), which means the reference for the voltage ratio is a microvolt (or a microvolt/meter in the case of an electric field strength). Remember dB's are always ratios of numbers.

If  I measured a signal that was 40 dB microvolts, it would represent a 100 uV signal. An 80 dB microvolt signal would be a 10,000 uV signal.

Remember the dB is always a ratio not an absolute quantity. So to say that the voltage gain of an amplifier is 22 dB makes sense. It is 20 times the log of the ratio of the output voltage to the input voltage.

On the other hand to say that the signal level out of the amplifier is 40 dB makes no sense, since dB's are not an absolute measurement. We could, however, say that the signal level out of an amplifier is 40 dB millivolts.  The signal then is 40 dB greater than a millivolt, or 100 mV.

The dB can therefore be properly used in only two situations.  First, when it is obvious that we are talking about the ratio of two numbers,  such as "the gain of the amplifier" or "the attenuation of the filter."  Second, if we specify the reference to which a single number (or measurement) is being compared.  In the second case we usually add a subscript or additional letters to the term dB to specify the reference power, voltage, or current.  For example, dBm refers to a reference power of one milliwatt.  The following table lists some commonly used dB units and their reference levels and abbreviations.

Unit                 Reference

dBw                    1 watt
dBm                    1 milliwatt
dBu                     0.775 volts*
dBV                    1 volt
dBmV                 1 millivolt
dBuV                  1 microvolt
dBuV/m              1 microvolt/meter
dBSPL                   2x10^-6 Newtons/meter squared**

* Equal to a dBm if the impedance is 600 ohms. In other words, 0.775 volts across 600 ohms equals 1 milliwtt.  Typically used in audio work.

** SPL is an abbreviation for Sound Pressure Level.  Zero dBSPL is considered to be the threshold of hearing. .  Typically used in acoustics.

Derivation of dB as a Voltage Ratio

dB = 10 log (P2/P1)

Let P2 = (V2)^2/R2 and P1 = (V1)^2/R1, then

dB = 10 log [(V2/V1)^2] [R1/R2]  = 10 log [(V2/V1)^2]  + 10 log [R1/R2]

dB = 20 log (V2/V1)  + 10 log (R1/R2)

For R2 = R1, 10 log (R1/R2) = 0 and

dB = 20 log (V2/V1)

A similar derivation can be done for the case of dB as a current ratio by letting P1 = [(I1)^2] R1 and P2 = [(I2)^2] R2.

© 2002 Henry W. Ott                                                Henry Ott Consultants,  48 Baker Road  Livingston,  NJ  07039  (973) 992-1793

Henry Ott Consultants
48 Baker Road Livingston, NJ 07039
Phone: 973-992-1793,   FAX: 973-533-1442

December 7, 2003