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Physics LibreTexts

3.8: Decibel Scale for Power Ratio

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In many disciplines within electrical engineering, it is common to evaluate the ratios of powers and power densities that differ by many orders of magnitude. These ratios could be expressed in scientific notation, but it is more common to use the logarithmic decibel (dB) scale in such applications.

In the conventional (linear) scale, the ratio of power P1 to power P0 is simply

G=P1P0   (linear units)

Here, "G'' might be interpreted as "power gain.'' Note that G<1 if P1<p0 and G>1 if P1>P0. In the decibel scale, the ratio of power P1 to power P0 is

G10log10P1P0   (dB)

where "dB'' denotes a unitless quantity which is expressed in the decibel scale. Note that G<0 dB (i.e., is "negative in dB'') if P1<P0>0 and G>0 dB if P1>P0.

The power gain P1/P0 in dB is given by Equation ???.

Alternatively, one might choose to interpret a power ratio as a loss L with L1/G in linear units, which is L=G when expressed in dB. Most often, but not always, engineers interpret a power ratio as "gain'' if the output power is expected to be greater than input power (e.g., as expected for an amplifier) and as "loss'' if output power is expected to be less than input power (e.g., as expected for a lossy transmission line).

Power loss L is the reciprocal of power gain G. Therefore, L=G when these quantities are expressed in dB.

Example 3.8.1: Power loss from a long cable

A 2 W signal is injected into a long cable. The power arriving at the other end of the cable is 10 μW. What is the power loss in dB?

Solution

In linear units:

G=10μW2 W=5×106   (linear units)

In dB:

\boldsymbol{G &= 10\log_{10}\left(5 \times 10^{-6}\right) \cong -53.0~\mbox{dB} \nonumber \\ L &= -G \cong \underline{+53.0~\mbox{dB}} \nonumber}

The decibel scale is used in precisely the same way to relate ratios of spatial power densities for waves. For example, the loss incurred when the spatial power density is reduced from S0 (SI base units of W/m2) to S1 is

L=10log10S0S1   (dB)

This works because the common units of m2 in the numerator and denominator cancel, leaving a power ratio.

A common point of confusion is the proper use of the decibel scale to represent voltage or current ratios. To avoid confusion, simply refer to the definition expressed in Equation ???. For example, let's say P1=V21/R1 where V1 is potential and R1 is the impedance across which V1 is defined. Similarly, let us define P0=V20/R0 where V0 is potential and R0 is the impedance across which V0 is defined. Applying Equation ???:

G10log10P1P0   (dB)=10log10V21/R1V20/R0   (dB)

Now, if R1=R0, then

G=10log10V21V20   (dB)=10log10(V1V0)2   (dB)=20log10V1V0   (dB)

However, note that this is not true if R1R0.

A power ratio in dB is equal to 20log10 of the voltage ratio only if the associated impedances are equal.

Adding to the potential for confusion on this point is the concept of voltage gain Gv:

Gv20log10V1V0   (dB)

which applies regardless of the associated impedances. Note that Gv=G only if the associated impedances are equal, and that these ratios are different otherwise. Be careful!

The decibel scale simplifies common calculations. Here's an example. Let's say a signal having power P0 is injected into a transmission line having loss L. Then the output power P1=P0/L in linear units. However, in dB, we find:

10log10P1=10log10P0L=10log10P010log10L

Division has been transformed into subtraction; i.e.,

P1=P0L   (dB)

This form facilitates easier calculation and visualization, and so is typically preferred.

Finally, note that the units of P1 and P0 in Equation ??? are not dB per se, but rather dB with respect to the original power units. For example, if P1 is in mW, then taking 10log10 of this quantity results in a quantity having units of dB relative to 1 mW. A power expressed in dB relative to 1 mW is said to have units of "dBm.'' For example, "0 dBm'' means 0 dB relative to 1 mW, which is simply 1 mW. Similarly +10 dBm is 10 mW, 10 dBm is 0.1 mW, and so on.


This page titled 3.8: Decibel Scale for Power Ratio is shared under a CC BY-SA license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .

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