Decibels

Voltage ratios, as used in the design of attenuators are often expressed in terms of decibels. The voltage ratio (K below) must be derived from the attenuation in decibels. Power ratios expressed as decibels are additive. For example, a 10 dB attenuator followed by a 6 dB attenuator provides 16dB of attenuation overall.

10 dB + 6 db = 16 dB

Changing sound levels are perceptible roughly proportional to the logarithm of the power ratio (P_I / P_O).

sound level = log₁₀(P_I / P_O)

A change of 1 dB in sound level is barely perceptible to a listener, while 2 db is readily perceptible. An attenuation of 3 dB corresponds to cutting power in half, while a gain of 3 db corresponds to a doubling of the power level. A gain of -3 dB is the same as an attenuation of +3 dB, corresponding to half the original power level.

dB = 10 log₁₀(P_I / P_O)

P_O = V _O I_O = V_O² / R = I_O² R

P_I = V _I I_I = V_I² / R = I_I² R

dB = 10 log₁₀(P_I / P_O) = 10 log₁₀(V_I² / V_O²) = 20 log₁₀(V_I/V_O)

dB = 10 log₁₀(P_I / P_O) = 10 log₁₀(I_I² / I_O²) = 20 log₁₀(I_I/I_O)

The two most often used forms of the decibel equation are:

dB = 10 log₁₀(P_I / P_O)      or         dB = 20 log₁₀(V_I / V_O)

We will use the latter form, since we need the voltage ratio. Once again, the voltage ratio form of equation is only applicable where the two corresponding resistors are equal. That is, the source and load resistance need to be equal.

Example: Power into an attenuator is 10 watts, the power out is 1 watt. Find the attenuation in dB.

dB = 10 log₁₀(P_I / P_O) = 10 log₁₀ (10 /1) = 10 log₁₀ (10) = 10 (1) = 10 dB

Example: Find the voltage attenuation ratio (K= (V_I / V_O)) for a 10 dB attenuator.

dB = 10= 20 log₁₀(V_I / V_O)

10/20 = log₁₀(V_I / V_O)

10^10/20 = 10^{log₁₀(V_I / V_O)}

3.16 = (V_I / V_O) = A_P(ratio)

Example: Power into an attenuator is 100 milliwatts, the power out is 1 milliwatt. Find the attenuation in dB.

dB = 10 log₁₀(P_I / P_O) = 10 log₁₀ (100 /1) = 10 log₁₀ (100) = 10 (2) = 20 dB

Example: Find the voltage attenuation ratio (K= (V_I / V_O)) for a 20 dB attenuator.

dB = 20= 20 log₁₀(V_I / V_O )

10^20/20 = 10 ^{log₁₀(V_I / V_O )}

10 = (V_I / V_O ) = K