2nd Week Seminar Sunryul Kim 2018.07.18 Antennas & RF Devices Lab.

MICROWAVE ENGINEERING Contents 1. GENERAL SOLUTIONS FOR TEM, TE, AND TM WAVES 2. PARALLEL PLATE WAVEGUIDE MICROWAVE ENGINEERING David M . Pozar Antennas & RF Devices Lab.

General Solutions Antennas & RF Devices Lab. Time-harmonic fields with an 𝒆 𝒋𝝎𝒕 dependence and wave propagation along the z-zxis (3.1a) (3.1b) Transverse components Longitudinal components In terms of 𝑬 𝒛 and 𝑯 𝒛 Assume source free (3.2a) (3.2b) Cutoff wave number Antennas & RF Devices Lab.

General Solutions TEM Waves (Transverse ElectroMagnetic waves) http://www.rfdh.com As a result of the definition of TEM Waves and (3.5) 𝐸 𝑥 , 𝐸 𝑦 , 𝐻 𝑥 , 𝐻 𝑦 =0 𝒌 : wave number 𝒌 𝒄 : cutoff wave number 𝜷 : propagation constant In TEM waves, the cutoff wave number is zero! Antennas & RF Devices Lab.

General Solutions TEM Waves (Transverse ElectroMagnetic waves) Helmholtz equation E field is the gradient of scalar potential ! ( ∵𝐸 𝑥 =𝑓 𝑥 𝑒 −𝑗𝛽𝑧 ) ( 𝛻 2 𝑓=𝛻∙𝛻𝑓 ) Antennas & RF Devices Lab.

General Solutions TEM Waves (Transverse ElectroMagnetic waves) Wave impedance of TEM wave Antennas & RF Devices Lab.

General Solutions TE Waves (Transverse Electric waves) http://www.rfdh.com Propagation constant is a function of frequency and geometry of the line or guide Antennas & RF Devices Lab.

General Solutions TE Waves (Transverse Electric waves) Helmholtz equation Wave impedance of TE wave TEM wave × 𝑘 𝛽 Antennas & RF Devices Lab.

General Solutions TM Waves (Transverse Magnetic waves) http://www.rfdh.com Propagation constant is a function of frequency and geometry of the line or guide Antennas & RF Devices Lab.

General Solutions TM Waves (Transverse Magnetic waves) Helmholtz equation Wave impedance of TE wave TEM wave × 𝛽 𝑘 Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE * The parallel plate waveguide is the simplest type of guide that can support TM and TE modes. * It can be useful modeling the propagation of higher order modes in stripline. * 𝑑≪𝑊 FIGURE 3.2 Geometry of a parallel plate wave guide. Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE TEM Modes Laplace’s equation ( 𝛻 2 𝑓=𝛻∙𝛻𝑓 ) Boundary Conditions Transverse electric field Total electric field Total magnetic field Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE TEM Modes Voltage Current Characteristic impedance Phase velocity Characteristic impedance depends only on the medium and the geometry. Phase velocity is equal to the speed of light in the medium. Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE Wave number 1 𝜆 2 = 1 𝜆 𝑥 2 + 1 𝜆 𝑦 2 + 1 𝜆 𝑧 2 𝑘= 2𝜋 𝜆 , 𝑘 𝑥 = 2𝜋 𝜆 𝑥 , 𝑘 𝑦 = 2𝜋 𝜆 𝑦 , 𝑘 𝑧 = 2𝜋 𝜆 𝑧 𝑘 2 = 𝑘 𝑥 2 + 𝑘 𝑦 2 + 𝑘 𝑧 2 𝝀 Wavelength along the propagation direction 𝝀 𝒙 Wavelength along the x-axis 𝝀 𝒚 Wavelength along the y-axis 𝝀 𝒛 Wavelength along the z-axis 𝑘 2 = 𝑘 𝑥𝑦 2 + 𝑘 𝑧 2 Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE Wave number (example) How many cycles are there in the same distance? 10𝜋 , 6𝜋 , 8𝜋 Wave lengths? 𝜆= 𝑑 5 , 𝜆 𝑐 = 𝑑 3 , 𝜆 𝑔 = 𝑑 4 Wave numbers? 𝑘= 2𝜋 𝜆 = 10𝜋 𝑑 , 𝑘 𝑐 = 2𝜋 𝜆 𝑐 = 6𝜋 𝑑 , 𝛽= 2𝜋 𝜆 𝑔 = 8𝜋 𝑑 Unit distance : dot line Wavefront : gray line 𝝀 𝒄 : cutoff wavelength 𝝀 𝒈 : guide wavelength 𝒌 𝒄 : cutoff wave number 𝜷 : propagation constant Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE TM Modes Cutoff wave number General solution Boundary condition *When, 𝒏=𝟎 𝐓𝐌 𝟎 =𝐓𝐄𝐌 *𝜷 is real number 𝒌> 𝒌 𝒄 Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE TM Modes Cutoff frequency (𝐓 𝐌 𝐧 𝐦𝐨𝐝𝐞) Cutoff modes (Evanescent modes) : At frequencies below the cutoff frequency of a given mode, the propagation constant is purely imaginary, corresponding to a rapid exponential decay of the fields. 𝑓 𝜆 > < Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE TM Modes Phase velocity 𝒗 𝒑 is greater than the speed of light( ) in the medium. How? 𝝀 𝒈 >𝝀 → 𝒗 𝒑 >𝒗 Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE TM Modes Power propagation When the mode is below cutoff, 𝜷 is imaginary, and then 𝑷 𝒐 =𝟎 Antennas & RF Devices Lab.

PARALLEL PLATE WAVEGUIDE TE Modes Cutoff wave number General solution Cutoff frequency Boundary condition Wave impedance Power propagation Antennas & RF Devices Lab.

Thank you