At 10 MHz? When the wavelength greatly exceeds d and other local dimensions of interest, i.e. Summarizing: The Friis transmission equation (Equation 10.14.2) gives the power delivered to a conjugate-matched receiver in response to a distant transmitter, assuming co-polarized antennas and free space conditions. = = 2 0 obj 4.6 Problem-Solving Strategies; 4.7 Further Applications of Newton's Laws of Motion; 4.8 Extended Topic: The Four Basic ForcesAn Introduction; Glossary; Section Summary; . The Problem of Evil Thomas Aquinas wrote his great Summa Theologica that he . If the dipoles are 45 to each other, the receiving cross section is reduced by a factor of \(\sin ^{2} 45^{\circ}=0.5 \Rightarrow P_{\mathrm{A}} \cong 6.4 \times 10^{-6}\ [\mathrm{W}] \). stream This page titled 10.3: Antenna gain, effective area, and circuit properties is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David H. Staelin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Thus, Equation 10.14.1 may be written in the following form: PR, max = PTGT( 4R)2GR. 53 This expression requires that all media near the antenna be reciprocal, which means that no magnetized plasmas or ferrites should be present so that the permittivity and permeabiliy matrices and everywhere equal their own transposes. May 22, 2022. Antenna Theory: Analysis and Design - 4th Edition - Solutions and Answers | Quizlet Science Engineering Antenna Theory: Analysis and Design 4th Edition ISBN: 9781118642061 Constantine A. Balanis Textbook solutions Verified Chapter 2: Fundamental Parameters and Figures-of-Merit of Antennas Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 endstream endobj 138 0 obj <>stream We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. If the critical frequency is 10 MHz in sky wave propagation, what is the best frequency to use assuming 30 0 of . The gain of an antenna takes the directivity into account. If you have an antenna tuner, check to see that it's tuned to the correct band. Problem 2 a) There was a typo in the exam where Eq. 30 Solved Problems on Wave Propagation 1. Gain = 4 . But the ratio of the same equations in terms of \(\underline{\mathrm{Z}}_{\mathrm{ij}}\) also yields: \[\mathrm{\frac{P_{\mathrm{r} 1}}{P_{\mathrm{r} 2}}=\frac{\left|\underline{Z}_{12} \underline I_{2}\right|^{2} R_{\mathrm{r} 2}}{\left|\underline Z_{21} \underline I_{1}\right|^{2}}=\frac{\left|\underline{Z}_{12}\right|^{2} P_{\mathrm{t} 2}}{\left|\underline Z_{2}\right|^{2} P_{\mathrm{t} 1}}}\]. kIj^6~\ @ (e :/sAQX:q ^"o z. The line is a lossless Z0 =50 Z 0 = 50 , = 0.595rad/m = 0.595 r a d / m. The antenna impedance at 20MHz measures ZL = 36+j20 Z L = 36 . 0000004407 00000 n Andrea M. Mitofsky. X|X%=0 V% 7D&4@q^/) fQ ` RDDH@D The maximum radiation intensity of a 90% efficiency antenna is 200 mW/ unit solid angle. Sorry, preview is currently unavailable. Some of these problems will be solved on the blackboard during the tutorials and solutions will also be provided to other problems. Their gain increases as f 2. 588 0 obj <> endobj $'201210 3r0 hi endstream endobj startxref Gain is the radiation intensity of the antenna in a given direction over that of an isotropic (uniformly radiating) source. (z3y[rI#! ( .1** ;{; An AM wave is represented by the expression : v = 5 (1 + 0.6 cos 6280 t) sin 211 104 t volts. \(\mathrm{P_{A}=A\left(P_{t} G_{t} / 4 \pi r^{2}\right)}\) where \(\mathrm{A=G_{r} \lambda^{2} / 4 \pi} \) and Gt 1.5; Gr 1.5. Equation (10.3.3) says that if the directivity or gain is large in one direction, it must be correspondingly diminished elsewhere, as suggested in Figure 10.2.4, where the pattern is plotted relative to an isotropic radiator and exhibits its main lobe in the direction = 90. ^^PzJJM37X,5l001] nqmxe%|h$lOG]"di67( A small wire structure (<< /3) can capture energy from this much larger area if it has a conjugate match, which generally requires a high-Q resonance, large field strengths, and high losses. A C-band earth station has an antenna with a transmit gain of 54 dB. 19 0 obj Typical exceptions to the rule \(\eta_{\mathrm{R}} \cong 1 \) include most short dipoles and antennas that are used over bandwidths much greater than an octave; their impedances are difficult to match. hT[k0+z2/PItBmBhc[OGkh#K9g@Hfp@I &a,5Uf{Y(t3vb=Xu1;eMM0ymL%=\4]FZoU bgAxITvjmneBZ0$\m@_1w+y^nCw$Agr]&1Kr t^MGno>6( frWV_&m)6r-wZ*LJ[z;YUYLMLO$X^7-}LwaE]4]9l+[TK!|/aaJGvK,4"| `hJ!CIhl13M-~ML[6=> m38Qmbap"Xqd3O%y4mOPIbWAbz Ok_qOHiXI6GE$w4Orc '0 L F `,&tp: `hi%T -[Rp @( *`P> ILqj*\MWwbgJ+reGOsBGZ-. Media characterized by matrices are discussed in Section 9.5.1. . n0%f|U9l 7?j`l7"tiNf]?uhgM Z4i[&LY_x Parabolic Antenna Beamwidth: Where: BW = antenna beamwidth; 8 = wavelength; d = antenna diameter. Antenna with a 20 degree beamwidth has a 20 dB gain. Antenna Theory And Designs Solved Problems Engineering Materials and Design - Sep 15 2020 From Problem Solving to Solution Design - Mar 14 2023 . xZoF~7aR\( You can check your reasoning as you tackle a problem using our interactive solutions viewer. ]&5ZBNFN2>&RA22X."`sw9G6 'T_GOG?R&_M/$Q7w^$7Z! If =90% find directivity. The microstrip antenna (MSA) is a resonant structure that consists of a dielectric substrate sandwiched between a metallic conducting patch and a ground plane. download 1 file . How do I view solution manuals on my smartphone. NW$g^I}}hx xref BVU1t pj.wO2W=dLH xz~PVjI4t J LheR5K{jLh*Ma#P6bh'00/t D = Directivity of the antenna. What is the gain of the antenna? The power received by an antenna with effective area A(,\(\phi\)) in the direction ,\(\phi\) from which the signal arrives is: \[\mathrm{P}_{\mathrm{r}}=\mathrm{I}(\theta, \phi) \mathrm{A}(\theta, \phi) \ [\mathrm{W}] \qquad \qquad \qquad \text{(received power)}\]. A / . A radio link has a 15-W Transmitter connected to an antenna of 2.5m2 effective aperture at 5 GHz. The study of antennas and electromagnetic wave propagation is essential to a complete understanding of radio communications, radar, cell phones, and other electronic communication systems. To maximize this transfer it is first necessary to add an external load reactance, -jXL, in series to cancel the antenna reactance +jX (X is negative for a short dipole antenna because it is capacitive). Bookmark it to easily review again before an exam. Solved Problems Holding Company; Trial Balance; System OF Governance During Vedic Period; Trending. Antennas Question 1: Which of the following is not correct? Value Education.docx question bank (6).docx 5TH April Downloads 2019-converted.pdf Online 2021 April; Sericulture Complete - Lecture notes 1-3; CRPC - Cr.P.C Lecture Notes; OB32 FF ID Unban LIST - Good essay; BSC/BCA English textbook Since Maxwells equations are linear, \(\underline{\mathrm V} \) is linearly related to \( \underline{\mathrm I}\), and we can define an antenna impedance \( \underline{\mathrm Z}_{11}\) consisting of a real part (10.3.14), typically dominated by the radiation resistance Rr (10.3.12), and a reactive part jX (10.3.15). Tb#Yi Jiv%h,C;! A certain matched antenna radiates one watt (P r) when driven with voltage V _ 0 = 10 volts. H10BB&jh488w 7C8'aAvLu!+p.V _yU6CLH>Q0A?B!A' Q!Y 7vFBa0V DEe=E[] &XO{5aJ1s("GjFiS(IA!#l! A transmitter operated at 20MHz, Vg=100V with Zg =50 Z g = 50 internal impedance is connected to an antenna load through l=6.33m of the line. kV)see 9)J 6bVSKlG ^l9s/-JU}^t|A~EPkID(zR!u <> 6 0 obj 0 For an antenna radiating in free space, the electric field at a distance of 1 km is found to be 12 m V/m. R = distance from radar. 'k,l*,6utT,!|oNNi3n!8^Ez'k=KZA59`C"q}QHj K6X.yM ;O.~)X!`VKk+=`Yf'yQ:q+Y#^&&gp27uic+c38/L8JMDQ[CRPaG+eB!x4r,>7v6f`eh$&HY|9/ D %%EOF $f ]p @7Jj~ $IFpO"g $] $V/( Dp! qb[ mq`B pX,8L8AfBF @n This impedance representation easily introduces the reciprocity constraint to the relation between G(,\(\phi\)) and A(,\(\phi\)). startxref The far-field intensity \( \overline{\mathrm{P}}(\mathrm{r}, \theta)\) [W m-2] radiated by any antenna is a function of direction, as given for a short dipole antenna by (10.2.27) and illustrated in Figure 10.2.4. A six by four antenna array was chosen for investigation with uniform element spacing between the elements and the results were presented for each selected geometry, Array elements of dipole and patch antenna were chosen for the investigation and analysis. The ideal length for the cross wires is one-half the . Write the expression for the maximum directive of an antenna. Hence the Directivity of an isotropic antenna is 1. 146 0 obj <>stream The approximate solution for solid beam angle of the patch is calculated as . A Ik_~V{}%=u@t+VV0$H]G,^ Ly&AcsC&tk=!wPB'g03BkVqT-C/-KW%ENB$'A]! For most antennas Rd << Rr. Arktruss h[O8AtT[9hw9R@VmS%9ZwNPzxbQBcsM'\J endobj 4.3: Antenna Components and Definitions. Chegg Study Expert Q&A is a great place to find help on problem sets and Electromagnetic Theory study guides. To obtain more marks in Physics, Download CBSE NCERT . Analysis and design techniques are presented using the transmission line model - uniquely. Can I get help with questions outside of textbook solution manuals? What is the antenna radiation resistance Rr? This matrix \( \overline{\mathrm{\overline Z}}\) does not depend on the network to which the 2-port is connected. Problems of radiation and scattering of water . 01 1 /2 00 1 1 11 /2 1 cos sin 4 L jtkr zL IzL E dE e dz r The net field is just the sum of the fields from all the dipoles: Beyond the Hertzian dipole: longer antennas We now can combine the transmitting and receiving properties of antennas to yield the power that can be transmitted from one place to another. Because Maxwells equations are linear in field strength, antennas have equivalent circuits consisting of a Thevenin equivalent impedance \(\underline{\mathrm{Z}}_{\mathrm{A}}(\omega) \), given by (10.3.13), in series with a Thevenin voltage source \( \underline{\mathrm{V}}_{\mathrm{Th}}(\omega)\) that we can now evaluate. Hb``$WR~|@T#2S/`M. Maximum power transfer occurs when the impedances match so incident waves are not reflected. Solved Problems Chapter 9 Radiation and Antennas Problem 9.3: Parameters of antenna pattern Problem 9.16: Dipole antenna pattern Problem 9.27(a): Automobile antenna beamwidths Fawwaz T. Ulaby, Fundamentals of Applied Electromagnetics. patches or slots are examples of aperture antennas. jypIrL%Y N9dFYY9[0 z N-~(0!.F`%)@m PRH20 &+)b20e @Z"F!LLL;3f`zd``&K/9|~a*@siF buf>#'@ =jZ4 where I(,) is the power intensity [Wm-2] of the plane wave arriving from direction (,), G(,) = D(,) = 1.5 sin2 is the antenna gain of a lossless short-dipole antenna (10.3.7), and A(,) is the antenna effective area as defined by the equation Pr I(,) A(,) [W] for the power received. The cost of an array to meet a given set of performance specifications can vary considerably and depends on many factors, including the amount of preplanning, the experience of the designer, and the amount of money available. Trine University. !G/&$Ld p56~:LR9-*h7& =$l6l`x?c=P,El8RM$ S . This relation can be proven for any arbitrary antenna provided all media in and near the antenna are reciprocal media, i.e., their complex permittivity, permeability, and conductivity matrices \(\underline{\varepsilon}\), \(\underline{\mu}\), and \(\underline{\sigma}\) are all symmetric: \[\underline{\varepsilon}=\underline{\varepsilon}^{\mathrm{t}}, \ \ \underline{\mu}=\underline{\mu}^{\mathrm{t}}, \ \ \underline{\sigma}=\underline{\sigma}^{\mathrm{t}}\]. 142 0 obj <>/Filter/FlateDecode/ID[<53E3EDADB1163341820A49FCBE4EE3BE>]/Index[134 13]/Info 133 0 R/Length 56/Prev 100137/Root 135 0 R/Size 147/Type/XRef/W[1 2 1]>>stream web pages U pl@BOz =Y%4X+GFiEQ'%dP52ku~lE%T^V%Ixx'/1'G9"irB;!B]@ eF6N endobj %PDF-1.3 Apply the value of limits as and in equation (2). stream The field lines \( \overline{\mathrm{E}}\) are sketched with solid lines locally perpendicular to the equipotentials. power. Solution Pr = |V _ 0|2 / 2Rr Rr = |V _ 0|2 / 2Pr = 102 / (2 1) = 50 Receiving properties of antennas The effective length of the dipole, \(\overline{\mathrm{d}}_{\mathrm{eff}} \), is defined by (10.3.19), and is the same as the effective length defined in terms of the current distribution (10.2.25) for infinitesimally thin straight wires of length d << . 0000004714 00000 n Signal transmission power = 27.78 dBW. If this doesn't reveal the problem, check all your cables and connectors to make sure they're hooked up correctly. However, the presence The maximum power an antenna can deliver to an external circuit of impedance \( \underline{\mathrm{Z}}_{\mathrm{L}}\) is easily computed once the antenna equivalent circuit is known. Solution The effective area of an antenna is given as, where D is the directivity of the antenna. 8rdWT'eL~.u"A=9]>313X3-$e}u,gmg664$EzL*LZ_j_]Xy[?Xs N/]|msk_WfA2)oz-di2|mj|5ej8eE7[Q|IM%xf)|6\ k`. on the Internet. The receiving antenna has an effective aperture of 0.5 m2the transmitting and is located at a 15-Km Line-of-sight distance from the transmitting antenna. In practice the real part of the j term in (10.3.14) is usually zero, as is the imaginary part of the \(\mathrm{\overline{\underline E } \bullet \overline{\underline J}^{*}}\) term in (10.3.15), but there can be exceptions. To prove (10.3.24) we characterize a general linear 2-port network by its impedance matrix: \[\overline{\underline{\mathrm{Z}}}=\left[\begin{array}{ll} \underline{\mathrm{Z}}_{11} & \underline{\mathrm{Z}}_{12} \\ \underline{\mathrm{Z}}_{21} & \underline{\mathrm{Z}}_{22} \end{array}\right] \qquad\qquad\qquad \text{(impedance matrix)}\], \[\overline{\mathrm{\underline V}}=\overline{\overline{\mathrm{\underline Z}}} \bar{\mathrm{\underline I}}\]. What is PA at 1 MHz if the two dipoles are 45 to each other? Mathematically, the gain is defined as: G = e D. e = Antennas efficiency. This is the Friis transmission equation. 0000006063 00000 n An analysis of the effect of the chosen antenna array was done by investigating its response when an incident emw (electromagnetic waves) from a mobile phone impinge on the array antenna where the tapering and beam-forming techniques were used for analysis and results presented. We can use the following modified form of Radar range equation in order to calculate the maximum range of Radar for given specifications. This paper intends to investigate the effects of different geometries of array antenna on directivity and side lobe levels. directive antenna. No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. hb```bf6~g`C]b,&t`X*AykvG)QSVjMV@\bFjFL }M K1040B -XN^M . Aperture antennas are commonly used at UHF and above where their sizes are relatively small. 599 0 obj <>stream \(\mathrm{P_{r}=\left|\underline{V}_{0}\right|^{2} / 2 R_{r} \Rightarrow R_{r}=\left|\underline{V}_{0}\right|^{2} / 2 P_{r}=10^{2} /(2 \times 1)=50 \Omega}\). The R and X of antennas are seldom computed analytically, but are usually determined by experiment or computational tools. Notations: lers = effective length (m) c = speed of light (m/s) 4 = wavelength (m) f = frequency (Hz) Solution: leg =a aL Re 3x108 1, = 32108 eT ge 1.83 m less . What is the antenna radiation resistance R r? Then the resistive part of the load RL must match that of the antenna, i.e., RL = Rr. stream First, verify that the antenna is connected. Legal. Solutions Manuals are available for thousands of the most popular college and high school textbooks in subjects such as Math, Science (. Parameter. endobj endobj (*(%8H8c- fd9@6_IjH9(3=DR1%? << /Length 5 0 R /Filter /FlateDecode >> SHOW ALL. The trapezoidal patch on the sphere will resemble like Figure 1. The field lines terminate at charges on the surface of the conductors and possibly at infinity, as governed by Gausss law: \(\hat{n} \bullet \overline{\mathrm{D}}=\sigma_{\mathrm{S}} \). X() is the antenna reactance, and the integral in (10.3.14) is the dissipative component Rd() of antenna resistance R(). 16.2 Smart-Antenna Analogy 931. The transmitted power is less than the available power if the antenna is mismatched or lossy. This Instructors' Manual provides solutions to most of the problems in ANTENNAS: FOR ALL APPLICATIONS, THIRD EDITION. G(,) is often called gain over isotropic where: \[\mathrm{G}(\theta, \phi) \equiv \frac{\mathrm{P}(\mathrm{r}, \theta, \phi)}{\left(\mathrm{P}_{\mathrm{A}} / 4 \pi \mathrm{r}^{2}\right)} \qquad \qquad \qquad \text{(antenna gain definition) }\]. Uploaded By ZvnaS. You can also find solutions immediately by searching the millions of fully answered study questions in our archive. Answers are given for most of the problems. Of[/j3(/J,F1L:"u9Y,QkGw%(:DXJYdts1;a%twT?}> I?o[qYB"$V5PF# gS8B/GZ^nbD,7kqv6t!'+ Just post a question you need help with, and one of our experts will provide a custom solution. What is required is an antenna with a known gain, preferably gain of the same order of magnitude as the antennas to be measured. %PDF-1.4 Pages 30. %PDF-1.5 Equation (10.3.33) says: \[\mathrm{A}(\theta, \phi)=\frac{\lambda^{2}}{4 \pi} \mathrm{G}_{\mathrm{r}}(\theta, \phi)\]. p/`jxAc=B:GbX4E1I`q8}JE JJK `~DD&gMcL!XqjI!p` NR{"E:q%F?rYB?U+PbgH-8ocOB?"Y3tIF@`N_pB-vk\#tu)oU#mo(:95_L%rA 2:4! [e q++VWHaK%lp`cf2;GfKxviTAQW-Tf_]It_mc$[(z# 3]3S! Figure 10.3.1(a) illustrates the Thevenin equivalent circuit for any antenna, and Figure 10.3.1(b) illustrates the electric fields and equipotentials associated with a short dipole antenna intercepting a uniform plane wave polarized parallel to the dipole axis. This paper will review the key processing technologies which can be potentially integrated into 22 nm and beyond technology nodes, including double patterning technology with high NA water immersion lithography and EUV lithography, new device architectures, high K/metal gate (HK/MG) stack and integration technology, mobility enhancement Therefore, the exact solution for equivalent solid beam angle of the patch is calculated as. Consider the approximate solution for finding solid beam angle can be evaluated as follows: Substitute for , for , for , and for in the equation (4). \(\mathrm{G}_{\mathrm{o}} \mathrm{P}_{\mathrm{T}} / 4 \pi \mathrm{r}^{2}=\left|\mathrm{E}_{\mathrm{o}}\right|^{2} / 2 \eta_{\mathrm{o}} \Rightarrow \mathrm{P}_{\mathrm{T}}=4 \pi \mathrm{r}^{2}\left|\mathrm{\underline E}_{0}\right|^{2} / 2 \eta_{0} \mathrm{G}_{0}=4 \pi\left(10^{4}\right)^{2} \times 1^{2} /\left(2 \times 377 \times 10^{4}\right) \cong 166 \ [\mathrm{W}] \). Isotropic Antenna: It radiates equally in all directions. hbbd``b`6 @H06 Problems 867. To find \(\underline{\mathrm{Z}}_{\mathrm{A}}(\omega) \) we can use the integral form of Poyntings theorem (2.7.23) for a volume V bounded by surface area A to relate the terminal voltage \( \underline{\mathrm V}\) and current \( \underline{\mathrm I}\) to the near and far fields of any antenna: \[\oiint_{\mathrm{A}}\left(\overline{\mathrm{\underline E}} \times \overline{\mathrm{\underline H}}^{*}\right) \bullet \hat{n} \mathrm{d} \mathrm{a}=-\int \int \int_{\mathrm{V}}\left\{\overline{\mathrm{\underline E}} \bullet \overline{\mathrm{\underline J}}^{*}+\mathrm{j} \omega\left(\overline{\mathrm{\underline H}}^{*} \bullet \overline{\mathrm{\underline B}}-\overline{\mathrm{\underline E}} \bullet \overline{\mathrm{\underline D}}^{*}\right)\right\} \mathrm{d} \mathrm{v}\]. Uploaded by W`Hxgd'+ aoX\!z5QfD\iJ47!8~&8-,p#y4B5UL7iI?a S#A&LKu |>2$e-Wp)t?G7tOX}? lzYj!dohE@E?&;y`2\x 8Psxsk_MmR&Yx K8@{5|PMkd!;Z *KnW i@*0_\FE!^j3qAm*8]$,Eg/otfqE$":: %*J9Fnq"A;HMx^<9lIO'ieS6l\rY,,{D i?X}7C%Iv@+?+`rj. To identify the Problems and challenges affecting the performance of radar systems II. The open-circuit voltage \( \mathrm{\underline{V}_{T h}}\) induced at the terminals of a small wire loop (D << ) follows from Amperes law: \(\underline{\mathrm{V}}_{\mathrm{Th}}=\int_{\mathrm{C}} \overline{\mathrm{\underline E}} \bullet \mathrm{d} \overline{\mathrm{s}}=-\mathrm{N} \int \int \mathrm{j} \omega \mu_{\mathrm{o}} \overline{\mathrm{\underline H}} \bullet \mathrm{d} \overline{\mathrm{a}}=-\mathrm{Nj} \omega \mu_{\mathrm{o}} \underline{\mathrm{H}} \pi \mathrm{D}^{2} / 4=-\mathrm{Nj} \omega \mu_{\mathrm{o}} \mathrm{\underline E} \pi \mathrm{D}^{2} / 4 \eta_{\mathrm{o}} \). 2.1.2 Free space, moving antenna Next consider the fixed antenna and free space model above with a receive antenna that is moving with speed v in the direction of increasing distance from the transmit antenna. Hit a particularly tricky question? << /Length 11 0 R /N 3 /Alternate /DeviceRGB /Filter /FlateDecode >> This. The electric lines of force start from a positive charge and end at a negative charge of same polarity repel each other and opposite polarity attract each other always enter or leave a conducting body at right angle always intersect each other Answer (Detailed Solution Below) 217 0 obj <>stream improving the gain of an antenna: right dimensioning (e.g., half the wavelength), multiple antennas plus a signal processor combining the signals, active and passive components attached to the antenna (compare with traditional TV antennas, satellite dishes etc.). 1657 The linearity of Maxwells equations applies to antennas, so they can therefore be modeled by a Thevenin equivalent circuit consisting of a Thevenin equivalent impedance \( \underline{\mathrm{Z}}_ \mathrm{A}\) in series with a Thevenin voltage source \( \underline{\mathrm{V}}_{\mathrm{Th}}\). Fig.1. Q2. 5 0 obj NCERT Solutions for Wave Optics Class 12 are now available for students who want to get the best preparation for the exam. Our interactive player makes it easy to find solutions to Antenna Theory 4th Edition problems you're working on - just go to the chapter for your book. The canonical forms of the MSA are the rectangular and circular patch MSAs. Fig. hb```f``c`e` ,@Q So, the approximate solution deviates 43.05% from the exact solution. stream By combining the expression for \( \underline{\mathrm Z}(\omega)\) in (10.3.10) with equations (10.3.912) we obtain: \[\mathrm{\underline{Z}(\omega)=R+j X=R_{r}+\int \int \int_{V}\left\{\left[\overline{\underline E} \bullet \overline{\underline J}^{*}+j \omega\left(\overline {\underline H}^{*} \bullet \overline{\underline B}-\overline{\underline E} \bullet {\overline{\underline D}}^{*}\right)\right] \Big/\left|\underline{I}_{0}\right|^{2}\right\} d v}\], \[\mathrm{R(\omega)=R_{r}+\int \int \int_{V} j R_{e}\left\{\left[\overline{\underline E } \bullet \overline{\underline J}^{*}+\omega\left(\overline {\underline H}^{*} \bullet \overline{\underline B}-\overline {\underline E} \bullet \overline {\underline D}^{*}\right)\right] \Big/\left|\underline I_{0}\right|^{2}\right\} d v=R_{r}+R_{d}}\], \[\mathrm{X(\omega)=\int \int \int_{V} I_m\left\{\left[\overline{\underline E } \bullet \overline{\underline J}^{*}+j\omega\left(\overline {\underline H}^{*} \bullet \overline{\underline B}-\overline {\underline E} \bullet \overline {\underline D}^{*}\right)\right] \Big/\left|\underline I_{0}\right|^{2}\right\} d v}\]. 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