# 1946 Friis: A Note on a Simple Transmission Formula **Citation:** Friis, H.T. (1946). "A Note on a Simple Transmission Formula." *Proceedings of the I.R.E. and Waves and Electrons*, Vol. 34, No. 5, pp. 254-256. May 1946. DOI: 10.1109/JRPROC.1946.234568. **Author:** Harald T. Friis, Bell Telephone Laboratories, Holmdel, N.J. Fellow, I.R.E. **Manuscript received:** December 6, 1945. ## Overview Friis derives the standard radio transmission formula that relates received power to transmitted power in free space. The paper is 2.5 pages long. It defines the concept of "effective area" for an antenna, derives the transmission formula from first principles using an isotropic radiator, and states the formula's limitations. This equation is the foundation of every radio link budget calculation in telecommunications engineering and is the starting point for the flat Earth SNR calculation in the Knickebein analysis. --- ## The formula ![[Friis1946_p1.png]] Friis presents his formula as Eq. (1): $\frac{P_r}{P_t} = \frac{A_r A_t}{d^2 \lambda^2} \tag{1}$ > [!note] What this says in plain language > The ratio of received power to transmitted power equals the product of the two antenna areas, divided by the distance squared times the wavelength squared. Bigger antennas catch more signal. Greater distance means less signal. Shorter wavelength means more signal (tighter beam for a given aperture). ### Variable definitions | Variable | Friis's definition (his words) | Units | |---|---|---| | $P_t$ | "power fed into the transmitting antenna at its input terminals" | Watts | | $P_r$ | "power available at the output terminals of the receiving antenna" | Watts | | $A_r$ | "effective area of the receiving antenna" | m$^2$ | | $A_t$ | "effective area of the transmitting antenna" | m$^2$ | | $d$ | "distance between antennas" | m | | $\lambda$ | "wavelength" | m | > [!note] Friis wrote it in terms of effective areas, not gains > The modern "gain form" ($P_r/P_t = G_t G_r (\lambda/4\pi d)^2$) is a rearrangement using $G = 4\pi A/\lambda^2$. Friis himself preferred the area form because, as he writes at the end of the paper, "it has no numerical coefficients. It is so simple that it may be memorized easily." --- ## Effective areas ![[Friis1946_p2.png]] Friis defines the effective area of any antenna (Eq. 2 and 3): $A_{eff} = \frac{P_r}{P_0} \tag{2}$ $P_r = P_0 A_{eff} \tag{3}$ where $P_0$ is the power flow per unit area of the incident field at the antenna. > [!note] What "effective area" means > If a plane wave carrying $P_0$ watts per square metre hits an antenna, the antenna delivers $P_r = P_0 \times A_{eff}$ watts to the receiver. The effective area is how much of that wave the antenna "captures." A bigger effective area means a more sensitive antenna. Friis explicitly notes that "the definition does not impose the condition of no heat loss in the antenna." He then gives effective areas for specific antenna types: ### A. Small dipole (no heat loss) $A_{dip} = \frac{3\lambda^2}{8\pi} = 0.1193\lambda^2 \tag{4}$ ### B. Half wave dipole (no heat loss) $A_{0.5\lambda} = 0.1305\lambda^2 \tag{5}$ > [!note] The half wave dipole is only 9.4% (0.39 dB) better than the infinitesimal dipole > This is why the "small dipole approximation" works so well in practice. The difference between an ideal point source and a real half wave dipole is less than half a decibel. ### C. Isotropic antenna (no heat loss) $A_{isotr} = \frac{\lambda^2}{4\pi} \tag{6}$ > [!note] This is the key equation > The isotropic antenna (equal radiation in all directions) has effective area $\lambda^2/4\pi$. This is the reference point for all antenna gain measurements. When we say an antenna has "26 dBi gain," we mean its effective area is $10^{2.6} = 398$ times larger than $\lambda^2/4\pi$. ### D. Broadside arrays (pine tree antennas) $A_{pine-tree} \approx n \times 0.5\lambda \times 0.5\lambda \tag{7}$ where $n$ is the number of half wave dipoles in the front curtain. > [!note] For large arrays, the effective area equals the physical area > This is exactly the approximation we use for the Knickebein antenna. Friis found this using the method of Pistolkors (1929). For 6 rows of 17 dipoles, the calculated effective area was only 3% below this simple formula. He notes that "the heat loss in the connecting transmission lines will reduce the effective areas in actual antennas." ### E. Parabolic reflectors Effective area is approximately two thirds of the projected area of the reflector (experimentally determined). ### F. Electric horns For a very long horn: 81% of aperture area. For an optimum horn: approximately 50% of aperture area. --- ## Derivation Friis's derivation is elegant in its simplicity. Three steps: **Step 1.** Start with an isotropic transmitting antenna. The power flow per unit area at distance $d$ is: $P_0 = \frac{P_t}{4\pi d^2} \tag{8}$ > [!note] This is the inverse square law > The transmitter radiates $P_t$ watts equally in all directions. At distance $d$, that power is spread over a sphere of surface area $4\pi d^2$. The power per unit area is $P_t / 4\pi d^2$. Nothing else is happening here. No physics beyond geometry. **Step 2.** The receiving antenna captures power proportional to its effective area: $\frac{P_r}{P_t} = \frac{A_r}{4\pi d^2} \tag{9}$ **Step 3.** Replace the isotropic transmitter with a directional one. A directional transmitting antenna with effective area $A_t$ concentrates power by the ratio $A_t / A_{isotr}$ compared to isotropic: $\frac{P_r}{P_t} = \frac{A_r A_t}{4\pi d^2 A_{isotr}} \tag{10}$ Substituting $A_{isotr} = \lambda^2/4\pi$ from Eq. (6): $\frac{P_r}{P_t} = \frac{A_r A_t}{4\pi d^2 \times \lambda^2/4\pi} = \frac{A_r A_t}{d^2 \lambda^2}$ which is Eq. (1). > [!note] The modern "gain form" follows immediately > Since gain $G = 4\pi A / \lambda^2$, we can write $A = G\lambda^2/4\pi$. Substituting for both antennas: > $\frac{P_r}{P_t} = \frac{G_t \lambda^2}{4\pi} \cdot \frac{G_r \lambda^2}{4\pi} \cdot \frac{1}{d^2 \lambda^2} = G_t G_r \left(\frac{\lambda}{4\pi d}\right)^2$ > This is the form we use in the Knickebein link budget. The factor $(\lambda/4\pi d)^2$ is called the Free Space Path Loss (FSPL). In decibels: FSPL $= 20\log_{10}(4\pi d/\lambda)$. --- ## Limitations ![[Friis1946_p3.png]] Friis states three limitations explicitly: ### 1. Far field only > "In deriving (1), a plane wave front was assumed at the distance $d$. Formula (1), therefore, should not be used when $d$ is small." He cites W.D. Lewis's theoretical study: $d \geq \frac{2a^2}{\lambda} \tag{11}$ where $a$ is the largest linear dimension of either antenna. > [!note] For Knickebein: $d_{min} = 2 \times 99^2 / 9.517 = 2{,}059$ m $\approx 2$ km > The Knickebein antenna is 99 m wide, so the far field starts at about 2 km. Both paths (440 km and 694 km) are hundreds of times beyond this limit. The far field condition is trivially satisfied. ### 2. Free space only > "Formula (1) applies to free space only, a condition which designers of microwave circuits seek to approximate. Application of the formula to other conditions may require corrections for the effect of the 'ground,' and for absorption in the transmission medium, which are beyond the scope of this note." > [!critical] This is why we also compute the Weyl-van der Pol formula > Friis explicitly says his formula ignores the ground. On a flat conducting surface, the ground reflection adds +6 dB (Fock 1965, p. 201, Eq. 3.23). We use Friis (no ground) as the conservative baseline and note that including the ground makes the flat model stronger by 6 dB. ### 3. Seven years of validation > "Almost 7 years of intensive use has proved its utility in transmission calculations involving wavelengths up to several meters, and it may become useful also at longer wavelengths." > [!note] The paper was written in December 1945 > "Almost 7 years" means Bell Labs had been using this formula internally since approximately 1939. The formula predates its publication. It was validated operationally during the war years before Friis wrote it up. --- ## Friis's recommendations At the end of the paper, Friis makes two suggestions that became standard practice: 1. **Use power flow per unit area** ($P_t A_t / \lambda^2 d^2$) instead of field strength in volts per metre 2. **Characterize antennas by effective area** instead of power gain or radiation resistance > [!note] Both suggestions were adopted by the industry > Modern antenna specifications still use effective area (or equivalently, gain in dBi) as the primary figure of merit. The link budget framework Friis established here is used unchanged in 2026 by every telecommunications company, satellite operator, and radio engineer on Earth. --- ## Application to Battle of the Beams The Friis equation is the foundation of the flat Earth model in the Knickebein propagation analysis. The link budget chain: $P_{rx}(\text{dBW}) = P_{tx}(\text{dBW}) + G_{tx}(\text{dBi}) + G_{rx}(\text{dBi}) - \text{FSPL}(\text{dB})$ For Kleve to Spalding (440 km): | Step | Value | |---|---| | $P_{tx} = 10\log_{10}(3000)$ | 34.8 dBW | | $G_{tx} = 10\log_{10}(4\pi \times 2871 / 9.517^2)$ | 26.0 dBi | | $G_{rx}$ (small Yagi) | 3.0 dBi | | FSPL $= 20\log_{10}(4\pi \times 439541 / 9.517)$ | 115.3 dB | | **$P_{rx}$** | **-51.5 dBW** | | Noise floor (galactic + receiver, ITU-R P.372) | -137.2 dBW | | **SNR at beam peak** | **85.7 dB** | The flat model gives 85.7 dB of SNR at beam peak. Even after the -19 dB equisignal crossover correction (for a 500 yd corridor), the SNR is 66.7 dB. The signal is 372 million times more powerful than the noise at beam peak, and still nearly 5 million times above noise at the equisignal. Friis's own limitation ("applies to free space only") is satisfied on a flat surface with no curvature obstruction. The formula's assumption of unobstructed straight line propagation is exactly the condition that exists on a flat Earth. --- ## Cross references - [[1945_Fock_Diffraction_Radio_Waves_Earth]] -- Fock's sphere diffraction formula (Eq. 6.10, p. 209) and flat Earth formula (Eq. 3.23, p. 201) from the same book - [[Knickebein_Propagation_Null]] -- Full null hypothesis analysis using this equation - [[BotB-Diffraction-Analysis]] -- Technical analysis with all intermediate calculations - [[BotB-Wave-Propagation]] -- Comprehensive prezzie reference --- ## Citation Friis, H.T. (1946). "A Note on a Simple Transmission Formula." *Proceedings of the I.R.E. and Waves and Electrons*, Vol. 34, No. 5, pp. 254-256. May 1946. Bell Telephone Laboratories, Holmdel, N.J. DOI: 10.1109/JRPROC.1946.234568.