Tuesday, March 18, 2008

Decibel

Decibels are measured in Logarithms which is the inverse to exponentials.  
They measure a RATIO - ValueA/Value(Reference) and they are then multiplied by LOG

Value Reference is the faintest sound a human can hear and is measure by the following:

1*10-12 W/m2    (which is defined as 0dB)

This is referred to the Threshold of Hearing.
If the sound level increases by 100 times, then we have a value of 20dB ( since 10 base to the power of 2)   -->
10 * (Log100)
Log of 100 = 2
10 * 2 = 20dB

Similary,an increase of sound of 1,000,000 from the threshold of hearing, gives us 60dB
10 * (Log1,000,000)
Log of 1,000,000 = 6
10 * 6 = 60dB


POWER = BASE10 (10(LOGX))  where logx is   Xa/Xb

Take for example, 10 exp3 (10 to the power of 3),  = 1,000

The inverse of this is the LOG - The Decibel uses base 10 logs (the factor of 10 puts the deci in the decibel)

  10Log1000
= 10*(Log1000)
= 30

E.g If speaker A produces 20 times more power than speaker B, we can measure this in decibels:

10*(Log20)
=10 * (1.3)
=13 Decibels (dB) difference between the two speakers.  

Another example
To calculate the ratio of 1 kW (one kilowatt, or 1000 watts) to 1 W in decibels

10 * (Log (1000/1))
10 * Log 1000
10 * 3
=30 dB

SOUND, AMPLITUDE, VOLTAGE AND CURRENT = BASE20 (20(LOGX))

Electric power through a resistor is defined as the square of the voltage (or current)
The square of x is written as x to the power of 2

Since the ratio between two levels are now squared, the formula becomes
20*(LogX)

Just say circuit A produces 20 Volts, and circuit B produce 10 volts
20*(Log (B/A))
20*(Log(10/20))
20*(Log(0.5))
= -6 dB
Circuit A produces 6dB less than circuit B

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