The effect of the wavelegth on the receiving power in Friis equation

Question

In the Friis transmission equation,

$$\frac{P_r}{P_t} = G_t G_r \left( \frac{\lambda}{4 \pi d} \right)^2,$$

it seems that if the frequency is halved while the other parameter values are fixed, the received power \${P_r}\$ will quadruple since the wavelength \$\lambda\$ is doubled. I mean, for example, if I work on a 2 GHz system and if I switch my signal to 1 GHz, is the power level of my received signal \${P_r}\$ going to become four times higher? Or do I miss something?

(I assumed the bandwidth of the antennas and other systems is suitable for both frequencies.)

Answer 1

That's right, but you will also have to scale your antennas by the same amount, because \$G_{t,r}\$ for any kind of aperture antenna is quadratic with the ratio of length of antenna to wavelength. Halving your \$f\$ doubled your \$\lambda\$ and hence divided your \$G_t\$ and \$G_r\$ by a factor of 4. So, to keep \$G_{t,r}\$ constant while doubling the wavelength, you need to double the sizes of your antennas as well. Often, that's not very attractive…

There's no free lunch! If you just reduce the frequency of your transmission without changing your antennas, you win exactly nothing (best case, really, usually, because antenna systems are typically optimized for one wavelength).