Path Loss Calculator

2025

This calculator allows the path loss between two antennas defined in terms of either area or gain to be calculated. Other calculators online are based on gain which is not convenient if you are working with a parabolic or phased array antenna with a cerain effective area.


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Transmitting Antenna



Receiving Antenna




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Background

The IEEE definition of free space loss is "The loss between two isotropic radiators in free space, expressed as a power ratio." However, the loss through free space does not change over frequency. The effective area of the antennas changes over frequency when you assume that they are isotropic (or any other constant gain). Path Loss is the correct term to use when talking about the loss between two specific antennas because it doesn't have an assumption of isotropic antennas baked into the definition.

\[FSPL = \left(\frac{4 \pi d}{\lambda}\right)^2= \left(\frac{4 \pi d f}{c}\right)^2\quad (1)\]

Where:

I dislike how 'free space loss' is presented because it leads people to believe that loss through space itself has a frequency dependence (note that the commenters on that Reddit post are misusing the term FSPL by including gain above isotropic). I much prefer 'geometric spreading loss' (or, interchangeably, 'spreading loss') which exclusively captures the loss from geometric spreading. Surprisingly to me, while I see this term used in the communication satellite industry, it doesn't seem to be formally defined by the ITU or IEEE.

\[SL = \frac{1}{4\pi d^2}\quad (2)\]

Where:

Expressed in dB:

\[SL_{dB} = -10\log_{10}\left(\frac{1}{4\pi d^2}\right)=10\log_{10}(4\pi d^2) \quad (3)\]

Its sloppy, but generally accepted in practice, for loss terms expressed in dB to be negative because no one ever talks about negative loss when they actually mean positive gain, so if you see negative loss terms they can be assumed to just be the loss. Here, I try to be consistent and display loss in positive dB terms which is where the first negative sign in the equation comes from.

The product of EIRP (Effective Isotropic Radiated Power) and geometric spreading loss directly gives you the power flux density at the receiver in terms of Watts per square meter:

\[P_{flux} = \frac{EIRP}{4\pi d^2} \quad (4)\]

Where:

In terms of dBW, the equation becomes:

\[P_{flux} = EIRP - 10\log_{10}\left(4\pi d^2\right) \quad (5)\]

Where:

Note that, critically, this equation is independent of frequency. It is only dependent on the distance between the antennas. If your transmitter is isotropic at all frequencies and your receiver has a constant effective area regardless of frequency, then the path loss will be constant over frequency.

If instead the receiver is also isotropic, then its effective area will decrease proportional to the square of frequency. If both the receiver and transmitter maintain constant area vs. frequency, though, the loss will decrease with the square of frequency.

Does path loss change with frequency? It is entirely dependent on what you assume about the antennas. With constant gain antennas (which implies that the effective area varies over frequency), the loss will increase with frequency. If you instead assume constant effective area, path loss decreases with frequency. If one antenna is constant gain and one antenna is constant area, then loss is constant over frequency.

Path Loss vs. Frequency with Different Antenna Assumptions

All of the above happens because antenna gain is a function of effective area and frequency:

\[A_{eff} = G \frac{\lambda^2}{4\pi}=G \frac{c^2}{4\pi f^2} \quad (6)\]

Where:

At \(f=\frac{c}{\sqrt{4 \pi}}=84.57 MHz\), the effective area of an isotropic antenna is 1 m^2 which is why the path loss lines for the three assumed cases intersect at that frequency.

With this information, the antenna characteristics can be added to equation 1 to get the path loss:

\[P_r=P_{flux}\cdot A_r = \frac{EIRP}{4\pi d^2} \cdot A_r = \frac{P_t\cdot G_t}{4\pi d^2} \cdot A_r \quad (7)\] \[L=\frac{P_r}{P_t}=\frac{G_t}{4\pi d^2} \cdot A_r \quad (8)\]

Where:

This is an ugly combination of gain and area, but I think it is a useful intermediate step for intuitively understanding the physical origin of path loss since it still has no explicit frequency term. Geometric spreading loss is always there independent of frequency. The gain of the transmitting antenna determines how much power is concentrated in the direction of the receiving antenna, and the area of the receiving antenna determines how much of that power can be captured.

It is less intuitive, but since passive antennas are reciprocal you can also swap the two and assume you are transmitting from a constant area antenna into a constant gain antenna and get the same result. The trick is converting the constant area of the transmitter into gain and the constant gain of the receiver into area. This may seem obvious or pedantic, but I would say that the average new graduate of electrical engineering programs does not have a good intuition of why this is the case which is why I am going through this derivation exhaustively.

If you convert the area of the receiver into gain, set both gains to 1 (isotropic), and convert to dB, you get the conventional formulation of free space path loss:

\[A_r = G_r \frac{\lambda^2}{4\pi}=G_r \frac{c^2}{4\pi f^2} \quad (9)\] \[L=\frac{P_r}{P_t}=\frac{G_t}{4\pi d^2} \cdot A_r=\frac{G_t}{4\pi d^2} \cdot G_r \frac{c^2}{4\pi f^2} = \left(\frac{c}{4\pi df}\right)^2 \quad (10)\]

This equation is inverted compared to how it is often presented because here it is technically gain and not loss. linear loss of 2 is the same as a gain of 0.5 or a loss of 3dB.

Inverted Free Space Path Loss Equation on Wikipedia

Friis Transmission Equation

The Friis transmission equation calculates path loss for two antennas of known area or gain. Equation 8 is fundamentally a form of the Friis Transmission Equation. Friis first published this equation in terms of antenna area, and you can arrive at this equation from equation 8 by converting the transmitter gain to area:

\[G_t=\frac{4 \pi A_t}{\lambda^2}\] \[L=\frac{P_r}{P_t}=\frac{G_t}{4\pi d^2} \cdot A_r = \frac{A_r A_t}{d^2 \lambda^2} \quad (11)\]

The more common modern form of this equation uses gain instead of area:

\[L=\frac{P_r}{P_t}=G_r G_t\left(\frac{\lambda}{4\pi d}\right)^2 \quad (12)\]