Appendix C: Decibel notation

Decibel notation

The electrical power gain of an amplifier can take on very high values. Values greater then \(10^{6}\) are very common. Therefore, the power gain is often shown in a logarithmic form and not in a form of eqn. (21). The unit dB (decibel) is used.

(169)\[ {\rm Power \,\, Gain \,\, (dB)} =10 {}^{10}\log \frac{P_{\rm out}}{P_{\rm in}} \]

An important consequence of the logarithmic notation is that, when a coupling is made with a number of systems after each other, the overall gain can be found as the sum of the individual gain of each systems expressed in decibels.

The power gain of + 3 dB and -3 dB correspond with doubling and halving of the power gain respectively. These two values will turn up in the course regularly.

Using Ohm’s law eqn. (169) can be written as:

(170)\[ {\rm Power \,\, Gain\,\, (dB)} = 10 {}^{10} \log \dfrac{ \left( V^{2}_{\rm out} / R_{\rm L} \right)}{ \left( V^{2}_{\rm in} / R_{\rm i} \right)}, \]

where \(R_{\rm L}\) is the load and \(R_{i}\) is the input resistance of the amplifier. For the special case that \(R_{\rm L}\) and \(R_{\rm i}\) are equal to each other eqn. (170) can be simplified to:

(171)\[\begin{split} \begin{split} {\rm Power \,\, Gain \,\, (dB)} & = 20 {}^{10}\log \frac{V_{\rm out}}{V_{\rm in}} \\ & = 20 {}^{10} \log ({\rm Voltage \,\, Amplification}). \end{split} \end{split}\]

Although the decibel notation is only applicable to power amplification. in literature, it is very common use it for voltage amplification as well, even when \(R_{\rm L}\) and \(R_{\rm i}\) are not equal to each other. The voltage gain in decibels is defined as:

(172)\[ {\rm Voltage \,\, Gain \,\, (dB)} = 20 {}^{10} \log \frac{V_{\rm out}}{V_{\rm in}}. \]

A voltage gain of + 3 dB or - 3 dB, means a voltage gain of \(\sqrt{2}\) and \(\dfrac{1}{2}\sqrt{2}\) respectively.