Radiometry and the Friis transmission equation Joseph A. Shaw Citation: Am. J. Phys. 81, 33 (2013); doi: 10.1119/1.4755780 View online: http://dx.doi.org/10.1119/1.4755780 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v81/i1 Published by the American Association of Physics Teachers Related Articles The reciprocal relation of mutual inductance in a coupled circuit system Am. J. Phys. 80, 840 (2012) Teaching solar cell I-V characteristics using SPICE Am. J. Phys. 79, 1232 (2011) A digital oscilloscope setup for the measurement of a transistor’s characteristic curves Am. J. Phys. 78, 1425 (2010) A low cost, modular, and physiologically inspired electronic neuron Am. J. Phys. 78, 1297 (2010) Spreadsheet lock-in amplifier Am. J. Phys. 78, 1227 (2010) Additional information on Am. J. Phys. Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html Downloaded 07 Jan 2013 to 153.90.120.11. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Radiometry and the Friis transmission equation Joseph A. Shaw Department of Electrical & Computer Engineering, Montana State University, Bozeman, Montana 59717 (Received 1 July 2011; accepted 13 September 2012) To more effectively tailor courses involving antennas, wireless communications, optics, and applied electromagnetics to a mixed audience of engineering and physics students, the Friis transmission equation—which quantifies the power received in a free-space communication link—is developed from principles of optical radiometry and scalar diffraction. This approach places more emphasis on the physics and conceptual understanding of the Friis equation than is provided by the traditional derivation based on antenna impedance. Specifically, it shows that the wavelength-squared dependence can be attributed to diffraction at the antenna aperture and illustrates the important difference between the throughput (product of area and solid angle) of a single antenna or telescope and the throughput of a transmitter-receiver pair. VC 2013 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4755780] I. INTRODUCTION P A A r ¼ r t : (2) 2 2 It is becoming increasingly common for physics and elec- Pt k R trical engineering students to take courses together in optics, Following his derivation of the transmission equation photonics, and topics in applied electromagnetics such as 1 wireless communications, antennas, or remote sensing. For from geometry and antenna impedance, Friis proposed that effective learning with students from different disciplines, it henceforth antennas should be described in terms of their is helpful to identify examples where principles from one effective area rather than antenna gain or radiation resist- subject area can be applied to another. Exactly such an op- ance, and that antenna radiation should be expressed as portunity arises when students encounter principles of radio- power flow per unit area rather than field strength in Volts metry in optics courses or the free-space Friis transmission per meter. In doing so, he effectively proposed moving from equation in antennas or wireless communications courses. a traditional circuit viewpoint, which deals with current and The Friis transmission equation relates the received power voltage, to a radiometric one, which deals with power, power to the transmitted power, antenna-separation distance, and density, etc. However, in textbooks and classroom presenta- antenna gains in a free-space communication link.1 How- tions, the Friis equation is normally derived or explained in ever, students first encountering the usual textbook deriva- terms of antenna impedance, leading to Eq. (1). Unfortu- tion, based on equivalent dipole antenna impedance, tend to nately, without a deeper understanding of how the antenna gain little physical insight into the Friis equation. Even for gain depends on wavelength, Eq. (1) can be mistakenly inter- students familiar with impedance concepts, the physical ba- preted to mean that the received power increases with wave- sis of the Friis equation becomes more apparent with a deri- length, whereas in fact the opposite occurs. Additional vation based on radiometry2–4 and scalar diffraction confusion often arises because the antenna impedance deri- 5–8 vation is based on the effective area of an infinitesimal theory. Radiometry is the field of science devoted to 10,11 quantifying the amount of power carried by a beam of elec- dipole, but applied broadly to antennas of any shape. tromagnetic radiation using geometric optics, areas, and Several authors have discussed alternate derivations of solid angles. Scalar diffraction theory allows us to use simple the Friis equation. Friis himself presented an argument that Fourier transform relationships to account for the beam- resembles a simpler form of the antenna derivation that is common in modern antenna textbooks, but he left the result spreading effects induced by the finite size of antenna or op- 14 tical apertures.5–8 (Vector diffraction theory does the same in terms of antenna areas as seen in Eq. (2). Hogg derived thing more accurately by using Maxwell’s equations without the Friis transmission equation using three different 9 approaches—Fresnel zones, Gaussian beams, and modes— the simplifying scalar-field assumptions. ) 15 Textbooks on antennas and communications10–13 often all of which lead to Eq. (2).Bush outlined a derivation write the Friis equation as based on Fraunhofer diffraction integrals but did not com- bine diffraction with radiometry. Shortly thereafter, Heald16 2 2 pointed out that the most significant benefit of Bush’s paper Pr k k was that it removed the oft-confusing antenna impedance ¼ eret DrDt ¼ GorGot; (1) Pt 4pR 4pR from the derivation. The development presented here builds on this back- where P is power in Watts, e is the antenna efficiency, k is ground to provide an even simpler path to the Friis equation the electromagnetic wavelength, R is the far-field line- based on optical radiometry and a simple result of scalar dif- of-sight distance between the antennas, D is the antenna di- fraction theory. Many undergraduate and graduate students rectivity relative to isotropic, G is the antenna gain (G ¼ eD), have been exposed to enough basic optical principles that and the subscripts r and t denote the receiver and transmitter, this development is intuitively appealing, particularly in a respectively. Friis actually developed the transmission equa- mixed class of physics and engineering students. This paper tion in terms of effective areas Ar and At, of receiving and presents a combined discussion of the Friis transmission transmitting antennas, respectively, resulting in equation and optical radiometry, relying on the geometric 33 Am. J. Phys. 81 (1), January 2013 http://aapt.org/ajp VC 2013 American Association of Physics Teachers 33 Downloaded 07 Jan 2013 to 153.90.120.11. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission optics propagation of radiance (power per unit area per unit solid angle) through free space. Beam spreading is intro- duced via scalar diffraction at the antenna aperture. This approach still relies on aperture areas, but in a much more direct manner and without any reference to antenna imped- ance. In this development, it becomes clear that the source of the wavelength-squared dependence is diffraction at the antenna aperture. This discussion also conveniently merges several topics from antenna theory and radiometry. There- fore, for students familiar with radiometry this approach builds on familiar principles to enhance understanding of the Friis equation, and for students not familiar with radiometry this approach teaches radiometry in the practical context of a free-space communication link. Fig. 1. Geometry for radiometric calculations, showing that a solid angle II. BASIC CONCEPTS OF OPTICAL RADIOMETRY always opens away from the area (A) in which incident power is calculated. The radiometric development of the Friis transmission equation relies on the concepts of radiance and throughput, 2,3 radians is defined as the ratio of (circular) arc length to ra- both central to radiometry. As indicated in Table I, radi- dius (s/r), whereas a solid angle in steradians is defined as ance is the power incident on (emitted from) a surface the ratio of (spherical) surface area to the square of the radius per unit area, from (into) a given solid angle, with units of 2 2 (A /r ). In a more complicated situation, we can find a solid W/(m sr). Suggestive of its physical meaning, radiance of- sph angle x by integrating in spherical coordinates over the ten is called brightness in the microwave radiometry and ra- 17,18 appropriate range of polar and azimuthal angles dio astronomy communities. The concepts of radiance and related radiometric quantities can be introduced and used in courses ranging from antennas to photonics and opti- x ¼ ͜ sin h dh d/: (3) cal design because they provide a consistent framework for calculating power transmitted from or received by an optical Although many antenna and physics textbooks use a capi- detector or antenna. While the five quantities in Table I are tal omega (X) for all solid angles, in radiometry a careful dis- sufficient for power flow calculations, radiance is the most tinction is made between solid angle x and projected solid fundamental because it is invariant in a lossless system. Sim- angle X. For a projected solid angle, we replace the spherical ilarly, conservation of energy requires that the product of surface area with a flat cross-sectional area, such as the pro- area and solid angle is invariant, a quantity referred to by op- jected aperture area of a lens, mirror, or antenna. The integral tical scientists and engineers as throughput (sometimes definition of the projected solid angle X is the same as called etendue ).