In general, there is no direct relationship between gain and frequency bandwidth. However, there is a relationship between gain and frequency. You can find it here: https://www.lairdconnect.com/resources/white-papers/antenna-design-gain-and-range

It depends on the type of antenna. If you are looking at small antennas, in the region of half wavelength across or less, then often gain falls with bandwidth. This always happens for antennas much smaller than this. See books and papers on Small Antennas, and articles quoting Chu and Wheeler.

For larger antennas, such as dishes or phased arrays, there is no connection between gain and bandwidth. The gain is proportional to the aperture area measured in square wavelengths, and will probably change across the band, but does not depend on bandwidth.

Yes, from radiation pattern of a certain wave length (frequency relation) to the effective area to antenna gain to the response, a relation can be found from curve fitting.

If you are working on Small antennas, according to FANO's bandwidth limit: increasing the bandwidth of your antenna means that you are decreasing the quality factor of the antenna, and that will result in decreasing the antenna efficiency.

Sounik Kiran Sir thank you for your reply but don't any equation give a clear explanation to my question

Malcolm White Sir thank you for your replay , clear explanation

Eanass Al-Shabkhoon please Miss, how I can get on this carve?

Abdenasser Lamkaddem , Sir this pretty convincing answer

thank you a lot of

Antenna gain versus frequency should be taken from workspace to cftool in Matlab

Conservation of energy that requires that the integral of the gain over 4 pi sterradians must be unity:

Integral[Gain(theta,phi) * d2Omega)]/(4*pi) = 1

where d2Omega is sin(theta) dtheta dphi and theta and phi are the elevationa and azimuth angles, respectively. What conservation of energy requires is that the engery per unit time (or power) through a sphere travelling outward from a transmitter at the speed of light must be constant without losses in the propagation medium. This integral then follows [see, for example, Dana, R. A., "Electroncially Scanned Arrays (AESAs) and K-Space Gain Formulation," Springer Nature, 2019].

As a check on this expression, the integral of unity gain in all directions (actually physically impossible) is unity, as one would expect.

This expression is valid for any one frequency of the signal. But now consider a modulated signal with a non-zero bandwidth being transmitted. Conservation of energy still applies, so now the expression above must be integrated over frequency to capture all of the transmitted signal. As that energy is spread over a larger bandwidth, the gain at any one frequency f must be lower to keep the integral over solid angle and frequency equal to unity. That is, to conserve energy it must be true that

Inegral[Integral[Gain(f,theta,phi) * d2Omega)] df]/(4*pi) = 1

(By the way, a consequence of the first equation for a uniformly weighted aperture with 100% efficiency is that the maximum gain is Gmax = 4*pi*Area/Lambda^2 where Lambda is the wavelength of the signal.)

Dear Amjad Abdulsatar Al-Rahmah ,

welcome!

The gain of the antenna by definition is the directional gain multiplied by the electrical efficiency. The directivity gain is = 4pi/ radiation angle of the antenna.

The electrical efficeincy is = Po/ Pi where po is the radiated power amd Pi is the input power.

Assuming the radiation resistance of the antenna =Rr and its loss resistance Rloss

Then Po= Vi^2/ Rr

Pi= Vi^2/ Rr in parallel with the Rloss

So, as you you see as Rloss decreases the electrical efficiency decreases.

On the other side considering resonant antenna with quality factor Q, then

Rloss = f0/BW where fo is the resonance frequency of the antenna and BW its 3 dB bandwidth.

As the bandwidth increases the losses of the antenna increases reducing its gain.

Best wishes.

Horn antennas with and without lenses, and dish antennas, can have wide bandwidth with very high gain. A lensed horn or dish can have gain of 20 dB or 30 dB across 25% bandwidth or more.

The general rule is that wide bandwidth is harder to get than narrower bandwidth, high gain harder to get than low gain, high efficiency harder to get than low efficiency, low sidelobes harder to get than high sidelobes. The compromises and which is easier to do depends on the type of antenna, but for any antenna type increasing one of these may result in the others being harder to do.

For antennas that need to fit in a sphere less than 1/6 wavelength radius the integral under the efficiency curve plotted against wavelength cubed is proportional to the volume of the sphere that the antenna, and its image in its ground, will fit in (Wheeler), so efficiency goes down fast with bandwidth. They all have basically no directivity (similar to small dipole/monopole), so the max gain is nearly the same as the efficiency.

If there is no limitation on size, fundamentally one can achieve any combination of bandwidth and gain e.g. imagine extremely long leaky traveling wave antenna. Or simply consider parabolic dish / optical mirror telescope which is essentially also antenna and with extremely high gain can cover from UV, visible, infrared to microwave.