Franek, Ondrej

English

Two versions of phasor transmission equations derived using power wave theory are presented. These equations aim at having similar structure as the well-known Friis’ transmission equation and thus being easy to apply whenever the phase shift in the wireless link cannot be omitted. All versions are arranged so that they intuitively follow the flow of electromagnetic energy from the transmitter all the way to the receiver. In addition, two versions of phasor scattering equations using the same path loss and antenna quantities are introduced. These are intended to help with simplified analysis of communication channels with unintentional or intentional scatterers such as the reconfigurable intelligent surfaces

    Aalborg UniversitetStructured and Intuitive Phasor Transmission and Scattering EquationsFranek, OndrejPublished in:Proceedings of the 2023 International Conference on Electromagnetics in Advanced Applications, ICEAA 2023Publication date:2023Document VersionAccepted author manuscript, peer reviewed versionLink to publication from Aalborg UniversityCitation for published version (APA):Franek, O. (2023). Structured and Intuitive Phasor Transmission and Scattering Equations. In Proceedings ofthe 2023 International Conference on Electromagnetics in Advanced Applications, ICEAA 2023 (pp. 581-583)General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            - Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            - You may not further distribute the material or use it for any profit-making activity or commercial gain            - You may freely distribute the URL identifying the publication in the public portal -Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: October 29, 2023Structured and Intuitive PhasorTransmission and Scattering EquationsOndřej FranekDepartment of Electronic Systems, APMS SectionAalborg UniversityAalborg, Denmarkhttps://orcid.org/0000-0003-4928-2208Abstract—Two versions of phasor transmission equations de-rived using power wave theory are presented. These equationsaim at having similar structure as the well-known Friis’ trans-mission equation and thus being easy to apply whenever thephase shift in the wireless link cannot be omitted. All versionsare arranged so that they intuitively follow the flow of electro-magnetic energy from the transmitter all the way to the receiver.In addition, two versions of phasor scattering equations usingthe same path loss and antenna quantities are introduced. Theseare intended to help with simplified analysis of communicationchannels with unintentional or intentional scatterers such as thereconfigurable intelligent surfaces.Index Terms—Antenna theory, Radiowave propagationI. INTRODUCTIONFriis’ transmission equation [1] is without doubt one of themost essential pieces of knowledge in wireless communicationengineering. Its popularity is largely due to its structure, whichnaturally follows the flow of electromagnetic energy alongthe communication channel. Factors of the formula may beidentified with distinct parts of the wireless link, and severaluseful quantities, such as the total radiated power, equivalentisotropic radiated power, or power density can be obtained assub-expressions of the equation.The drawback of the original Friis’ equation is that itoperates with power quantities, which do not satisfy the super-position principle. Therefore, it is not suitable for analyzingwireless links with multiple transmitters or receivers (includingelectrically large arrays for massive MIMO), or with multipathpropagation, where different paths contribute with differentphase shifts. Moreover, it is not applicable to wireless linkswith ultrawideband pulses that require time domain analysis,and for which various versions of time-domain transmissionequations have been proposed. The need for phase in the trans-mission equation has been addressed by various alternativesto Friis’ equation using phasors, with antennas described byeither a transfer function or a complex effective length [2].An interesting finding followed, namely that transmitting andreceiving antennas contribute differently to the wireless link,as a consequence of the Lorentz’ reciprocity theorem, resultingin an asymmetric transmission equation [3].This work has been supported by the EU Horizon 2020 project Reconfig-urable Intelligent Sustainable Environments for 6G Wireless Networks (RISE-6G).Fig. 1. Phasor transmission equation (top) and Friis’ equation (bottom) whereantennas are characterized by complex transfer functions and realized effectiveareas, respectively.In our recent paper, we have proposed phasor alternativesto Friis’ transmission equation that preserve the symmetry andmodularity of the original [4]. The equations employ complexeffective length vectors, both original and normalized, and anewly proposed field gain, serving as a phasor counterpartto the commonly used power gain. Voltage and field wavesare used as sub-expressions of the equations and intermediatequantities along the communication channel.In this paper, which is a contribution to the ElectromagneticEducation session, we propose a symmetric phasor transmis-sion equation that is further simplified by using power wavetheory [5]. We demonstrate that the proposed equation isstructured in a similar way as the Friis’ original, i.e. it issymmetric and modular (see Fig. 1), and its application topractical problems is thus very intuitive, which is of greathelp when teaching new generations of wireless engineers.Both transfer function and field gain versions of the equationhave been derived.In addition, two versions of phasor scattering equationsare proposed, following the same symmetric and modular ap-proach. These should help analyzing wireless communicationchannels with scatterers, either unintentional or intentionalsuch as the reconfigurable intelligent surfaces (RIS) [6].II. TRANSMISSION EQUATIONLet us assume that all field and circuit quantities arerms phasors with the ejωt convention, with frequency f inω = 2πf , and wavelength λ. Also, the transmitting (TX) andreceiving (RX) antennas separated by distance r are placed infree space described by propagation constant k, both are fedwith single mode transmission lines (such as coaxial cables),and there is a polarization match between them.Although the original paper of Friis [1] does not display it inthis form, the classical version of Friis’ transmission equationcan be written asP1 ·A1 ·(1λr)2·A2 = P2 (1)The arrangement of terms in (1) has the advantage that itvisually follows the flow of electromagnetic energy throughthe wireless link from left to right. Input power P1 feedsthe TX antenna with effective area A1, which radiates theelectromagnetic energy through free space with path loss(1/λr)2, until it arrives at the RX antenna with effective areaA2, which then produces power P2 at its output. The effectiveareas are considered as realized, i.e. containing any mismatcheffects between the antenna and its feeder. In addition, thesubexpression of (1) to the left of A2 is equal to the powerdensity of the propagating plane wave at distance r from theTX antenna, right before it arrives at the RX antenna.In many situations, however, it is necessary to know also thephase of the involved signals, such as in multipath propagationor with multiple TX and RX. This can be solved by using aphasor transmission equation, of which one possible form canbe [4]P̃+1 · h̃1 ·je−jkrλr· h̃2 = P̃−2 (2)Here, P̃+1 is the power wave incoming (+) from TX to theterminals of the TX antenna and P̃−2 is the power waveoutgoing (−) from the RX antenna terminals to RX. The TXand RX antennas are characterized by complex values h̃1 andh̃2, respectively. This value can be called a transfer functionof an antenna [7], or, alternatively, a normalized effectivelength [4], since its dimension is meters and it is proportionalto the traditional effective length h byh̃ =h2√η0Z0(3)where η0 is the free-space wave impedance and Z0 is thereference impedance for the power waves at the antennaterminals. Just like the effective areas in (1), also the transferfunction contains the mismatch effects of the antenna and itsfeeder.The elegance of (2) lies in its one-to-one relationship withthe Friis’ equation (1) as shown in Fig. 1. Each term of theFriis’ equation can be obtained simply as a square of themagnitude of its counterpart in the phasor equation, and thewireless link is again intuitively traversed from left to rightfollowing the multiplication.Furthermore, similarly to (1), the subexpression of (2) tothe left of h̃2 is equal to the normalized electric field ofthe plane wave incident on the RX antenna Ẽ2 = E2/√η0,where E2 is the incident plane wave electric field. FromFig. 1 it follows that the power wave-like quantity Ẽ2 has thefamiliar simple relation to the power density S2 = |Ẽ2|2. Thementioned subexpression can, for example, be convenientlyFig. 2. Phasor transmission equation (top) and Friis’ equation (bottom) whereantennas are characterized by field gains and realized gains, respectively.Fig. 3. Phasor scattering equation (top) and symmetrized radar equation(bottom) where the scatterer is characterized by the complex scattering fieldgain and RCS, respectively.used to determine the transfer function h̃1 from measuredelectric field when the input power wave is known.Another form of Friis’ equation is also common in practice,using realized power gains G1 and G2 instead of realized ef-fective areas [8]. Also this form can have a phasor version withone-by-one counterparts to each of the terms, as demonstratedin Fig. 2. The counterpart to the realized power gain G is thefield gain g̃ defined in [4] and equivalent to another form oftransfer function introduced in [9].III. SCATTERING EQUATIONIf the wireless link contains a scatterer, as in the case of aradar target or RIS, the power received at RX is expressedusing the radar range equation [8], in its simplest formomitting the polarization mismatch and reshuffling the termsto again follow the signal path:P1 ·G1 ·14πr21· σ ·(λ4πr2)2·G2 = P2 (4)Here, r1 and r2 are the distances of TX and RX from thescatterer, respectively, and σ is the radar cross section (RCS)of the scatterer. What is not very satisfying on (4) is theasymmetry of the equation, in that it uses two different types ofpath loss: the spherical power spreading for the first segmentfrom TX to the scatterer and the path loss term from Friis’equation in Fig. 2 for the second segment from the scattererto RX.Figure 3 presents a symmetrized radar equation in the samestyle as Fig. 2 together with the proposed phasor version. Toensure a simple conversion, the scatterer is characterized by acomplex scattering field gain g̃S. This dimensionless quantityFig. 4. Phasor scattering equation (top) and symmetrized radar equation(bottom) where the scatterer is characterized by the complex scattering areaand RCS, respectively.is equivalent to a product of two field gains, as if the scattererconsisted of two interconnected antennas characterized bythese field gains.Another version of the scattering equation is also possible,with path loss and antenna characterization as in Fig. 1. Thisversion is shown in Fig. 4 and features a complex scatteringarea ãS as the phasor counterpart to RCS. This quantity has adimension of square meters and is equivalent to a product oftwo transfer functions (or normalized effective lengths), againas if the scatterer consisted of two interconnected antennascharacterized by the respective transfer functions.IV. DISCUSSION AND CONCLUSIONTwo versions of phasor transmission equations and two ver-sions of phasor scattering equations have been introduced. Theequations are intentionally arranged in a structured way thatallows to visually follow the flow of electromagnetic energythrough distinct parts of the communication channel. Unlikethe recently proposed antenna equation [10], the presentapproach aims at symmetric arrangement with the TX andRX antennas characterized by the same quantity, in the besttradition of the well-established Friis’ transmission equation.Besides, the notation of the involved phasor quantities utilizesa simple accent, the tilde (˜ ), which, thanks to its shape(“wave”), gives a hint that the particular quantity is basedon power wave theory. Overall, these properties should helpwith simplified analysis of wireless links, where phase cannotbe omitted from the calculations, such as in fading scenariosboth with or without scatterers.REFERENCES[1] H. T. Friis, “A note on a simple transmission formula,” Proc. IRE,vol. 34, no. 5, pp. 254–256, May 1946.[2] Y. Duroc, “On the system modeling of antennas,” PIER B, vol. 21, pp.69–85, 2010.[3] J. Kunisch, “Implications of Lorentz reciprocity for ultra-widebandantennas,” in Proc. IEEE International Conference on Ultra-Wideband(ICUWB 2007), 2007, pp. 214–219.[4] O. Franek, “Phasor alternatives to Friis’ transmission equation,” IEEEAntennas and Wireless Propagation Letters, vol. 17, no. 1, pp. 90–93,2017.[5] E. G. Farr, “A power wave theory of antennas,” inForum for Electromagnetic Research Methods and ApplicationTechnologies (FERMAT), 2015. [Online]. Available: http://www.e-fermat.org/files/articles/15420b1e99d097.pdf[6] E. C. Strinati, G. C. Alexandropoulos, V. Sciancalepore, M. Di Renzo,H. Wymeersch, D.-T. Phan-Huy, M. Crozzoli, R. d’Errico, E. De Car-valho, P. Popovski et al., “Wireless environment as a service enabled byreconfigurable intelligent surfaces: The RISE-6G perspective,” in 2021Joint European Conference on Networks and Communications & 6GSummit (EuCNC/6G Summit). IEEE, 2021, pp. 562–567.[7] W. Wiesbeck, G. Adamiuk, and C. Sturm, “Basic properties and designprinciples of UWB antennas,” Proc. IEEE, vol. 97, no. 2, pp. 372–385,Feb. 2009.[8] C. A. Balanis, Antenna Theory: Analysis and Design. John Wiley &Sons, 2016.[9] C. Roblin, S. Bories, and A. Sibille, “Characterization tools of antennasin the time domain,” Proc. IWUWBS, 2003.[10] E. G. Farr, “Ten fundamental antenna-theory puzzles solved by theantenna equation: A remarkable array of solutions,” IEEE Antennas andPropagation Magazine, vol. 64, no. 1, pp. 61–71, 2021.

Structured and Intuitive Phasor Transmission and Scattering Equations

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Structured and Intuitive Phasor Transmission and Scattering Equations

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