# Does array gain violate the laws of physics or not?

I am a bit disturbed lately since I don't know the answer this basic problem.

Say we have a standard isotropic antenna with some fixed parameters (load impedance, etc), and we feed this antenna with a sinusoidal current of the form: $$I(t) = Acos(2\pi f_{c}t)$$ where $f_{c}$ is the carrier frequency (typically in the GHz range). Assuming no circuit mismatches or losses, the power delivered to the antenna (and radiated) is $P \propto (A^2)/2$. Since the antenna is isotropic this gives a electric far field strength of $|E| \propto I$ in every angle at some distance $d$ from the antenna.

Now let's say that we have 2 similar isotropic antennas "really" close together (compared to the wavelength), but we feed each one with sinusoidal current half of the amplitude, i.e. $$I_d(t) = \frac{A}{2}cos(2\pi f_{c}t)$$ and the power delivered to each one antenna will be $P_d\propto (A^2)/8$. The electric fields will sum up constructively in all angles at distance $d$ from this 2 element antenna array the field strength is $|\frac{E}{2}+\frac{E}{2}|=|E|$, i.e. equal to the single antenna case. The problem is that the sum of the powers feed to the 2 antennas is $(A^2)/8 + (A^2)/8 = (A^2)/4$ which is smaller than in the first case.

Thus there might be something wrong here, because if one takes this approach one step forward, I would a field with infinite magnitude (hence infinite power) using an infinite amount of antennas for a given input power $P$. Where is the error in my approach?

I agree that in a practical system, there will be coupling between antennas thus their efficiencies will decrease, etc. But this cannot be the fundamental explanation for this because in standard textbook in antennas as the one of Balanis, the superposition principle is assumed and everything should be coherent from this point-of-view.

Joao

$$I(t) = \frac {A}{\sqrt {2}} cos(2\pi f_c t). \tag{i}$$
About the electric field beware, it is as you say, $\vec E = \vec E_1 + \vec E_2$, but if you follow my $\text {(i)}$, it becomes
$$\vec E = \frac {\vec E_1 + \vec E_2}{\sqrt {2}}. \tag{ii}$$