Introduction in logarithm and decibel

1. INTRODUCTION TO LOGARITHM AND
DECIBEL

2. THE DECIBEL
• The decibel, db is a logarithmic scale used for comparing two
physical quantities especially in electronics.
• It is the logarithmic measurement of the ratio of one power to
another or one voltage to another.
• Is a logarithmic expression used in filter applications as a
unit of gain or loss.
• It is a unit of logarithmic gain measurement and is
commonly used to express amplifier response.

3. LOGARITHM
• Logarithm, the exponent or power to which a base must be raised to yield a given number.
expressed mathematically,
x is the logarithm of n to the base b
bx = n EXPONENTIAL FORM
𝑙𝑜𝑔𝑏 n=x LOGARITHM FORM
• EXAMPLE: 53 = 125
𝑙𝑜𝑔5125 = 3
LOG TO THE BASE 5 OF 125 IS 3

4. PROPERTIES OF LOGARITHM

5. Decibel Applications
• the decibel, db is widely used in many applications. it is used
within a wide variety of measurements in the engineering and
scientific areas, particularly within electronics, acoustics and also
within control theory.
• typically the decibel, db is used for defining amplifier gains,
component losses (e.g. attenuators, feeders, mixers, etc), as well
as a host of other measurements such as noise figure, signal to
noise ratio, and many others.

7. DECIBEL FORMULA FOR POWER
COMPARISONS
• The most basic form for decibel calculations is a comparison of power levels. as might
be expected it is ten times the logarithm of the output divided by the input. the factor ten
is used because decibels rather than bels are used.

8. DECIBEL FORMULAS FOR VOLTAGE &
CURRENT
• although the decibel is used primarily as comparison of power levels, decibel current equations
or decibel voltage equations may also be used provided that the impedance levels are the same.
in this way the voltage or current ratio can be related to the power level ratio.
• when using voltage measurements, it is easy to make the transformation of the decibel formula
because power = voltage squared upon the resistance:

11. • if the voltage gain(𝐴𝑉) is greater than 1, the dB gain is positive. if it
is less than 1, the dB gain is negative which is usually called
attenuation.
• in this way a positive number of decibels implies a gain, and where
there is a negative sign, it implies a loss.
• if the value of p2 is greater than p1, then the result is given as a gain, and
expressed as a positive value, e.g. +10db. where there is a loss, the decibel
equation will return a negative value, e.g. -15db.

12. EXAMPLE
Given:
P2(output power)=100W
P1(input power)=100mW=100𝑥10−3 = 0.1𝑊
𝐴𝑝 𝑑𝐵 =?
Solution:
𝐴𝑃 𝑑𝐵 = 10𝑙𝑜𝑔10
𝑃2
𝑃1
𝐴𝑃 𝑑𝐵 = 10𝑙𝑜𝑔10
100
0.1
𝑨𝑷 𝒅𝑩 =30dB
An audio amplifier
delivers 100 watts into an
8-ohm speaker load when
fed by a 100mw input
signal. calculate the
power gain of the
amplifier in decibels.

13. EXAMPLE
A PASSIVE RESISTIVE
NETWORK IS USED TO
PROVIDE AN
ATTENUATION (LOSS)
OF 10DB, WITH AN
INPUT VOLTAGE IS
12V. WHAT WILL BE
THE NETWORKS
OUTPUT VOLTAGE
VALUE.
given:
𝐴𝑉 𝑑𝐵 = −10𝑑𝐵 𝑙𝑜𝑠𝑠
v1(input voltage)=12v
v2(output voltage)=?
solution:
𝐴𝑉 𝑑𝐵 = 20𝑙𝑜𝑔10
𝑉2
𝑉1
-10db= 20𝑙𝑜𝑔10
𝑉2
12
-0.5= 𝑙𝑜𝑔10
𝑉2
12
0.3162=
𝑉2
12
𝑽𝟐=3.79v

14. • A POWER AMPLIFIER HAS AN OUTPUT OF 10 W
AND A GAIN OF 36 DB. WHAT IS THE POWER INTO
THE AMPLIFIER, INPUT POWER?

15. NEPER
• THE NEPER IS A LOGARITHMIC SCALE BASED ON NATURAL LOGARITHMS TO BASE E.
• ALTHOUGH DECIBELS ARE THE MAIN UNIT FOR EXPRESSING THE RATIOS OF GAIN AND LOSS
LIKE ELECTRICAL POWER IN ELECTRONIC AND OTHER SYSTEMS, THE NEPER IS ALSO USED.
• THE NEPER IS NOT PARTICULARLY WIDELY USED BUT HAS ADVANTAGES IN SOME INSTANCES
MATHEMATICAL ADVANTAGES CAN BE USED.
• THE NEPER USES THE SYMBOL NP AND IS A LOGARITHMIC UNIT FOR RATIOS OF MEASUREMENTS
FIELD AND POWER QUANTITIES, SUCH AS GAIN AND LOSS OF ELECTRONIC SIGNALS.