# Measuring Power

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## Measuring the Power of Electronic Signals

In electronics, power is given by the product of current and voltage:

$P=IV \,\!$

Coarsely, you can interpret this equation as the number of electrons per second times the energy released per electron, giving you energy per second, or power.

If Ohm’s Law applied (V = IR), power may alternately be expressed by:

$P=\frac{V^2}{R} \,\!$

### Units

For current (I) measured in amperes and voltage (V) measured in volts, the most natural unit for power is watts:

${\rm watt} = {\rm ampere}\cdot {\rm volt} \,\!$

However, the gain of amplifiers and attenuators are often given in decibels. Although it takes a while to get your head around them, decibels are useful. Decibels are logrithmic (10log10(P1 / P2)). As such, when chaining amplifiers and attenuators, you can simply add the decibels of gain in these components to find the total gain in decibels. A handy rule of thumb is that 3dB is very close to a factor of 2.

To trace the signal level through gain stages, it is useful to have an absolute (as opposed to relative) representation for power. For this, we have dBm, which is just decibels relative to 1 mW.

${\rm dBm} = 10 \log _10\left(\frac{P}{1~ {\rm mW}}\right) \,\!$

### Power of a DC (constant) signal

Measuring the power of a DC signal is pretty straightforward. You directly measure the voltage (using a voltimeter in a high-impedance setting so that it doesn’t load the circuit you are trying to measure, thereby changing the measured voltage), and then either measure the current or impedance of the circuit drawing the current. Using the above equations, you can derive P.

### Power of a sine wave signal

To measure the power of a sine wave, we need to deal with the fact that the measured voltage changes with time. For example, if you connect your voltimeter across the circuit while it is set in DC mode, you’ll just measure the center point of the sine wave, which is usually 0 V. If you flip your voltimeter to AC mode, or plot the signal with an oscilloscope, you can measure the amplitude of the sine wave, rather than the bias voltage.

If you are using an oscilloscope, you can read off the amplitude of the sine wave directly. We’ll call this Vamp, so that our signal f(t) is given by:

$f(t)=V_{amp}\sin (2\pi \nu t) \,\!$

Other voltages that are commonly given by multimeters or power meters are Vrms and Vpeaktopeak. Obviously Vpeaktopeak = 2Vamp. Slightly less obvious is that $V_{rms}=\sqrt {2}V_{amp}$. This is because sine and cosine carry equal power (one is just the time-delayed version of the other), and

$\sin ^2\theta +\cos ^2\theta =1 \,\!$

Hence, when averaging over a period or more

$V_{rms}^2=\langle V_{amp}^2\sin ^2(2\pi \nu t)\rangle = V_{amp}^2\frac12 \,\!$

where $\langle \dots \rangle$ just means “average".

### Power of white noise

Noise is called “white” if it has a flat frequency power spectrum. To determine the total power coming from noise, you need to add up the power from all of the frequency components of the noise. If your noise were perfectly white (i.e. it had the same power at all frequencies), you’d end up with infinite power. Obviously, this doesn’t happen in real life, so instead, we need to talk about noise that is flat over some range in frequency (i.e. bandwidth) and is zero elsewhere.

If the amplitude of the frequency power spectrum of noise were, for example, 1 μW/Hz, and our noise had a bandwidth of 1 MHz, we would integrate the frequency-domain amplitude (1 μW/Hz) over 1 MHz to get 1 W. (Note that while power grows linearly with bandwith, B, voltage grows as $\sqrt {B}$, because $P\propto V^2$).

If power is given in less convenient units, like -30 dBm/Hz, your best bet is probably to convert to mW. Otherwise, if you want to get fancy, you’ll note that 1 MHz divided by 1 Hz is 106 = 60 dB. Hence, we’ll end up with − 30 + 60 = 30 dBm. Which, of course, is 1 W.