# Is the antenna gain time dependent?

Premise

The antenna gain of a transmitting antenna relates its radiation intensity U(θ,ϕ) to its input power as shown in the following formula:

$$G(θ,ϕ)=\frac{U(θ,ϕ)}{P_{IN}/4\pi}$$

Where the radiation intensity in far field zone is related to the frequency domain poynting vector $$\vec{S}(r,θ,ϕ) = \frac{1}{2}\cdot \vec{E} \times \vec{H}^*$$ by the equation:

$$U(θ,ϕ)=|\vec{S}(r,θ,ϕ)|\cdot r^2$$

The radiation intensity does not depend on the distance because of the inverse square behaviour of the Poynting vector, so it is only a representation of how the antenna distributes the power in the different directions (θ,ϕ).

Observation

The Poynting vector is not only defined in the frequency domain, but also in the time domain. Its definition is this:

$$\vec{S(t)}(r,θ,ϕ) = \frac{1}{2}\cdot \vec{E(t)} \times \vec{H(t)}^*$$

It represents the instantaneous power flowing out from the antenna.

Instead, the frequency domain Poynting vector is a vector whose real part is the average power transmitted by the antenna. In the case of harmonic (sine wave) fields, there is the factor 1/2 that comes from the averaging operator. This explains the reason why the frequency domain Poynting vector is defined with such a factor.

Question

The antenna gain is defined from radiation intensity which is defined from the frequency domain Poynting vector. So, it seems to me that the antenna gain means how an antenna "amplifies" (or "attenuates", depending on if it's + dBi or -dB) the average input power along a direction (θ,ϕ).

But physically (not from the definition convention), is this concept of antenna gain a function of time?

The time dependence of the transmitting power (due to the time dependence of the time domain Poynting vector) represents the electromagnetic power flowing out from the antenna, like shown in the following picture (see here to watch the animation): So, is physically the antenna gain animated? Is it a time varying quantity?

The antenna gain is by definition time - independent (since it involves a frequency domain quantity,) but is the antenna ratio between instant transmitted power and instantaneous input power time dependent?

• No , I think not but the E/H (t) ratio ought to be constant as a function of Zo. Mar 27, 2021 at 9:51

If you had an amplifier with a gain of 5 and fed to its input a sinewave of 1 volt RMS, you would expect to see on the output a 5 volts RMS sinewave. The fact that a 1 volt RMS sinewave has instantaneous values ranging from -1.4142 volts to +1.4142 does not alter the fact that the amplifier gain remains constant and produces an output sinewave that ranges between -7.071 volts and +7.071 volts.

Is it a time varying quantity?

No.

The gain of an antenna is a ratio between the rms values of two sine waves. The instantaneous ratio between two sine waves which are out of phase is not meaningful.

The phase difference between what is applied to the feedpoint of an antenna (whether you are measuring voltage, current or power) and what is measured some distance away (whether it is E, M, or Poynting) will depend upon the distance. If that distance is not an exact multiple of the wavelength, the phase difference will be non-zero. Therefore, the "instantaneous" will be a non-meaningful value.

Choosing a measurement point where the phases are the same as for the feedpoint is essentially the same as taking the ratio between rms values.

Maxwell's equations, in electrical engineering, were solved under the following conditions:

1. One has to know the analytical expression of current sources as a function of x,y,z,t (*)

2. Current sources are not influenced by the E and H fields that them selves generate (**)

3. Current sources are sinusoidal in time and the frequency is a fixed parameter f.

Condition 3 allows us to Fourier transform the equations and solve them in the phasors domain.

In this domain, the term that contains time is:

exp(-j2pift)

That term, for valid mathematical reasons that I can't write here, is thought as a fixed parameter in the equations.

That allows us to solve only the spatial x,y,z part of the equations.

All the antenna radiation patterns that you see in books are only the spatial part. Often you see only E field patterns because H is proportional to E.

To have a complete picture of propagation you have to imagine that those patterns vary in amplitude at a frequency f.

At a fixed point, for example, the direction of E is fixed while the amplitude and the versus vary in time at frequency f.

What is an antenna gain?

Antennas are passive devices therefore they can't introduce signal or energy gain.

Picture this ideal experiment.

Think about all of the possible antennas you may design. They are infinite.

Connect a voltage or current source to the first antenna.

The source works at a fixed frequency f and inject into the antenna a certain power P.

Have the system start radiating E and H.

Take a picture at instant t1 of the radiation pattern at a distance R. It's a 2D picture folded though onto a sphere. It's a spherical picture.

The picture changes in time but at (t1 + 1/f) you will see the same picture again.

To be continued later...

The antenna gain is defined from radiation intensity which is defined from frequency domain Poynting vector. So, it seems to me that antenna gain means how an antenna "amplifies" (or "attenuates", depending on if it's + dBi or -dB) the average input power along a direction (θ,ϕ).

But physically (not from the definition convention), is this concept of antenna gain function of time?

In a way, as "time" is the frequency and the antenna's wave-guide is the load at that frequency, and at resonant frequency of the antenna, the lowest impedance of the antenna is observed and its highest current transfer, and thus the greatest electromagnetic wave is propagated. In other words, the losses in media change of an antenna is the inverse square product of the power against the antenna's physical xc and xl at the antenna's frequency resonance.

In multiple element antenna design, mutual inductance comes into play to logarithmic-ally add electromagnetic strength.