1) Why does power gain use a coefficient of 10 unlike the other two?

2) Why can't power gain be negative?

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# Amplifier Gain – How to Express Gain in Decibels

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1) Why does power gain use a coefficient of 10 unlike the other two?

2) Why can't power gain be negative?

## Best Answer

$$\begin{align} A_{v,dB} &= 10\cdot\log \frac{V_o^2}{V_i^2} \\ &= 10\cdot\log \left(\frac{V_o}{V_i}\right)^2 \\ &= 10\cdot 2 \cdot\log \frac{V_o}{V_i} \end{align} $$

EDIT: Power is proportional to voltage/current squared for all linear circuits. In AC circuit analysis, voltage, current, and power all become phasors. In nonlinear circuits, power may not be proportional to voltage/current squared, but the convention is still used for decibels. It even holds in an abstract field like signal processing, where the "power" of a signal is defined to be the average of the amplitude squared.

magnitudeof the gain. If you want to talk about power loss, you use negative decibels, which represents a fractional gain. For example:$$A_p = 0.01$$ $$A_{p,dB} = 10\cdot\log 0.01 = -20\:\mathrm{dB}$$

You can also have an actual negative gain, like what you get from an inverting amplifier. The negative there is described as a

phase shift. To fully describe an amplifier, you usually need both the magnitude and the phase of the gain. (This might be a bit advanced for you, but you could try looking up Bode Plots to see how this is used in real life.) Anyway, here's how to describe an inverting amplifier with a voltage gain of -2.4:$$A_v = -2.4$$ $$|A_v| = 20\cdot\log 2.4 \approx 7.6\:\mathrm{dB}$$ $$\angle A_v = 180^\circ$$