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Microwave Engineering Course 4th Stage Electrical Engineering Department College of Engineering Misan University 2016-2017 Prof. Dr. Ahmad H. Abood Part I Electromagnetic Theory Outline course Maxwell’s Plane Filed Equations wave solution component of plane wave General development of the wave equation Plane wave propagation in lossy medium Plane wave propagation in conducting medium Plane wave propagation in lossy dielectric medium Text Book D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. Solution Manual of Microwave Engineering, 4th ed., Wiley, 2012. References 1. D. M. Pozar, Microwave Engineering, 3rd ed., Wiley, 2005. 2. S. C. Harsany, Principles of Microwave Technology, Prentice-Hall, 1997. (undergraduate text) 3. R. E. Collin, Foundations for Microwave Engineering, 2nd ed., IEEE Press, 2001. (advanced text) Outline course Maxwell’s Equations Plane Filed wave solution component of plane wave General development of the wave equation Plane wave propagation in lossy medium Plane wave propagation in conducting medium Plane wave propagation in lossy dielectric medium A BRIEF HISTORY OF ELECTROMAGNETICS Magnetism was known to the ancient Greeks (Plato and Socrates). Hans Christian Oersted (1777-1851) discovers the relation between current carrying wire and the magnetic field. André Ampère (1775-1836) discovers the force between two current carrying wires. Jean-Baptiste Biot (1774-1862) and Félix Savart (1791-1841) formulate their law quantifying the force between current element. Benjamin Franklin (1706-1790) and Joseph Priestly (1733-1804) postulate the inverse square law of electrostatics. A BRIEF HISTORY OF ELECTROMAGNETICS, cont. Coulomb (in 1785) proves experimentally the inverse square law for stationary electric charges. Alessandro Volta (1745-1827) investigated reactions between dissimilar metals, and developed the first electric battery (1800). Karl Friedrich Gauss (1777-1855) discovers the divergence theorem of electricity. A BRIEF HISTORY OF ELECTROMAGNETICS, cont. A BRIEF HISTORY OF ELECTROMAGNETICS, cont. Four Time-domain Maxwell’s Equations Definitions: Faraday's law states: The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit. B B dS S B B ds cos S B Wb 1Wb 1T m 2 d m Therefore, E dt Definitions: Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop. Ampere’s Where law, B.ds 0ienc the integral is a line integral. B.ds is integrated around a closed loop called an Amperian loop. The current ienc is net current enclosed by the loop. Definitions: Gauss' Law of electric field is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density Definitions: Gauss' law of magnetic filed, the electric flux through any closed surface is directly proportional to the net electric charge enclosed by that surface. Gauss’ Law in magnetism: the net magnetic flux through a closed surface is zero B d A 0 (1) (2) (3) (4) Outline course Maxwell’s Equations Plane wave solution Filed component of plane wave General development of the wave equation Plane wave propagation in lossy medium Plane wave propagation in conducting medium Plane wave propagation in lossy dielectric medium A B Outline course Maxwell’s Plane Equations wave solution Filed component of plane wave General development of the wave equation Plane wave propagation in lossy medium Plane wave propagation in conducting medium Plane wave propagation in lossy dielectric medium 𝒙 𝒚 𝒛 𝝏 𝝏 𝝏 𝝏𝑬𝒚 𝝏𝑬𝒙 𝝏𝑬𝒛 𝝏𝑬𝒚 𝝏𝑬𝒙 𝝏𝑬𝒛 𝜵×𝑬= = − 𝒙+ − 𝒚+ − 𝒛 𝝏𝒙 𝝏𝒚 𝝏𝒛 𝝏𝒚 𝝏𝒛 𝝏𝒛 𝝏𝒙 𝝏𝒙 𝝏𝒚 𝑬𝒙 𝑬𝒚 𝑬𝒛 In the same manner 𝒙 𝝏 𝜵×𝑯= 𝝏𝒙 𝑯𝒙 𝒚 𝝏 𝝏𝒚 𝑯𝒚 𝒛 𝝏 𝝏𝑯𝒚 𝝏𝑯𝒙 𝝏𝑯𝒛 𝝏𝑯𝒚 𝝏𝑯𝒙 𝝏𝑯𝒛 = − 𝒙+ − 𝒚+ − 𝒛 𝝏𝒛 𝝏𝒚 𝝏𝒛 𝝏𝒛 𝝏𝒙 𝝏𝒙 𝝏𝒚 𝑯𝒛 So, 𝜕𝐸𝑧 𝜕𝐸𝑦 𝜕𝐻𝑥 − = −𝜇 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝐻𝑦 𝜕𝐸𝑧 𝜕𝐸𝑦 − = −𝜇 𝜕𝑦 𝜕𝑧 𝜕𝑦 𝜕𝐸𝑧 𝜕𝐸𝑦 𝜕𝐻𝑧 − = −𝜇 𝜕𝑦 𝜕𝑧 𝜕𝑧 𝜕𝐻𝑧 𝜕𝐻𝑦 𝜕𝐸𝑥 − =𝜀 𝜕𝑦 𝜕𝑧 𝜕𝑥 and, 𝜕𝐸𝑦 𝜕𝐻𝑧 𝜕𝐻𝑦 − =𝜀 𝜕𝑦 𝜕𝑧 𝜕𝑦 𝜕𝐻𝑧 𝜕𝐻𝑦 𝜕𝐸𝑧 − =𝜀 𝜕𝑦 𝜕𝑧 𝜕𝑧 Outline course Maxwell’s Plane Filed Equations wave solution component of plane wave General development of the wave equation Plane wave propagation in lossy medium Plane wave propagation in conducting medium Plane wave propagation in lossy dielectric medium Now consider the effect of a lossy medium. If the medium is conductive, with a conductivity σ, Maxwell’s curl equations can be written, from (1) and (2) as .................. (1) …………….. (2) The resulting wave equation for 𝐸 then becomes …………….. (3) where we see a similarity with the wave equation for .E in the lossless case. The difference is that the quantity 𝑘 2 = 𝜔2 𝜇𝜀 of lossless case is replaced by 𝑘 2 = 𝜔2 𝜇𝜀 1 − 𝑗 𝜎 𝜔𝜀 in (3). We then define a complex propagation constant for the medium as …………….. (4) where α is the attenuation constant and β is the phase constant. If we again assume an electric field with only an 𝑥 component and uniform in x and y, the wave equation of (3) reduces to …………….. (5) …………….. (6) …………….. (7) …………….. (8) c Outline course Maxwell’s Plane Filed Equations wave solution component of plane wave General Plane development of the wave equation wave propagation in lossy medium Plane wave propagation Plane in conducting medium wave propagation in lossy dielectric medium Many problems of practical interest involve loss or attenuation due to good (but not perfect) conductors. A good conductor is a special case of the preceding analysis, where the conductive current is much greater than the displacement current, which means that σ>>ω. Most metals can be categorized as good conductors. In terms of a complex, rather than conductivity, this condition is equivalent to 𝜖 ′′ ≫ 𝜖 ′ . The propagation constant of (4) can then be adequately approximated by ignoring the displacement current term, to give (1) (2) Thus the amplitude of the fields in the conductor will decay by an amount 1/e, or 36.8%, after traveling a distance of one skin depth, because . At microwave frequencies, for a good conductor, this distance is very small. The practical importance of this result is that only a thin plating of a good conductor (e.g., silver or gold) is necessary for low-loss microwave components. Compute the skin depth of aluminum, copper, gold, and silver at a frequency of 10 GHz? Solution The conductivities for these metals are listed in Appendix F. Equation (2) gives the skin depths as Outline course Maxwell’s Plane Filed Equations wave solution component of plane wave General development of the wave equation Plane wave propagation in lossy medium Plane wave propagation in conducting medium Plane wave propagation in lossy dielectric medium