appendix-decibel_notation.md

(decibel)=

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. {eq}eq:PA. The unit dB (decibel) is used.

$$ {\rm Power ,, Gain ,, (dB)} =10 {}^{10}\log \frac{P_{\rm out}}{P_{\rm in}} $$ (eq:C1)

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. {eq}eq:C1 can be written as:

$$ {\rm Power ,, Gain,, (dB)} = 10 {}^{10} \log \dfrac{ \left( V^{2}{\rm out} / R{\rm L} \right)}{ \left( V^{2}{\rm in} / R{\rm i} \right)}, $$ (eq:C2)

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. {eq}eq:C2 can be simplified to:

$$ \begin{split} {\rm Power ,, Gain ,, (dB)} & = 20 {}^{10}\log \frac{V_{\rm out}}{V_{\rm in}} \ & = 20 {}^{10} \log ({\rm Voltage ,, Amplification}). \end{split} $$ (eq:C3)

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:

$$ {\rm Voltage ,, Gain ,, (dB)} = 20 {}^{10} \log \frac{V_{\rm out}}{V_{\rm in}}. $$ (eq:C4)

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