# Why is it not standard practice to integrate the output of an antenna?

Why does a loop antenna not measure the differential magnetic field?

Essentially I just found out today that we are actually measuring the derivative of the time varying electric field with an antenna, rather than its magnitude at a given point in time. So my questions is if that is the case, why do we not normally need to integrate the output (measured) of a receive antenna? I know the integral of a sin is a cos etc. but there will usually be an amplitude term as well which seems important?

To clarify the question as there seems to be confusion in the comments, I am referring to time varying fields not static ones.

• We observe the derivative, it contains all the information, the integral isn't any more useful. Commented May 15, 2023 at 19:40

Most communication is done using signals that are sufficiently bandlimited that we simply don't have any reason to care. The derivative of the signal is the same thing as the signal, except for a term that's proportional to the freqency (d/dt sin(ωt) = ω cos(ωt)). As long as the fractional bandwidth of the signal is reasonably small, this transform doesn't matter at all. As for the extra ω in the magnitude, we just say that the antenna has a response that depends on frequency — but all realistic antennas have a response that depends on frequency, so again, whatever.

• This is the answer I was looking for. Thank you. Commented May 15, 2023 at 20:23
• (As for the difference between sin and cos, that's the same thing that happens if you move the antenna slightly closer to or farther away from the transmitter so nobody cares) Commented May 16, 2023 at 16:32
• @user253751 Sure for a typical communications type signal (CW) it's as you say unimportant but for a transient signal I would imagine the phase information becomes quite important Commented May 17, 2023 at 8:59
• @Christian no, it really isn't. A change in phase would potentially matter, but a constant phase offset is undetectable. It's the same, as user253751 says, as moving the receiver a foot closer or further from the transmitter. For GPS, you would have to calibrate it out because measuring that phase distance is the purpose of the system. For almost everything else, you would never notice. Commented May 17, 2023 at 13:33
• @hobbs if I am measuring an impulse (lets say with the shape of sinc(t)) then the shape of the derivative of that waveform is very different to the shape of the sinc(t) itself. I don't understand how we can say that this is not important, if you are interested in the shape of a waveform? There are application for example, lets say you want to measure the electric field from a lightning strike Commented May 17, 2023 at 14:22

why do we not normally need to integrate the output of an antenna

We don't care about the integral.

This sounds a bit banal, but it's really the case.

A transmitter needs to change a field to cause a signal to radiate – so the whole phenomenon is always about fields' changes, never about static fields.

Luckily, for very long parts, electric and magnetic field in air and free space are linear – so that the changing fields actually carrying a signal can overlay linearly with the static magnetic field of earth and the static electric field that typically exists between ground (as in: dirt) and upper atmospheric layers. We don't care about these static fields.

A static field would "always have been there", and contains no information. No power can be extracted from it, either, until it changes (and no longer is static). (This is generally true; of course, if the amount of energy you extract is very small compared to the energy in the field, you won't change the "pseudo-static" field enough to really matter – turning the needle of a compass doesn't demagnetize the earth in any significant way.)

• Surely if you wanted to measure the amplitude of the electric field then the integral is important? Even the power? Commented May 15, 2023 at 19:35
• but you don't measure the static amplitude of any field with an antenna. Antennas typically don't even have paths for DC to flow. Finite-sized antennas are infinitely inefficient at receiving 0-frequency fields. Commented May 15, 2023 at 19:36
• I'm referring to the maximum amplitude ot the time varying signal Commented May 15, 2023 at 19:37
• Electronic equipment probably has plenty of other sources of static or low-frequency E and B fields, besides the Earth's magnetic poles and atmospheric charge. It's desirable NOT to have these included in the output of the radio receiver. It isn't that they don't contain information, it's that the information they contain is, from the perspective of the communication, interfering noise not signal Commented May 15, 2023 at 19:37
• @Christian you'll notice that the time derivative of the B-field happens to be proportional (through material constants and vacuum speed of light) to the amplitude of the curl of the E-field – and the derivative of the E-field (plus the current density) proportional to the B-field (again, material constants and speed of light). So you're right, the amplitude does change – but what we measure with the antenna is always the derivative of the "other" field. Commented May 15, 2023 at 20:03

I have come to believe that some of the responses to this question are misleading or perhaps just incorrect. I have discovered a paper which sheds some light on this question:

Chapter 4. Time domain characterisation of antennas by normalised impulse response

Essentially in the case of an antenna designed to measure the electric field component, we are not measuring the time differential of the electric field. However in the case of a pair of identical antennas, the transmit antenna will radiate the time differential of the input signal, so that what we measure on the receive antenna is the time differential of the original signal. It is not however measuring the time differential of the radiated waveform itself.

I found it helpful to think about the relative phase of current and electric field to understand this better. In the case of the (resonant) transmit antenna, the current drives a potential difference which is out of phase with the driving current, and this potential difference is directly proportional to the electric field transmitted. In the case of the receive antenna, the electric field drives a current directly, and it is this that we measure.