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- MicrowaveMicrowave EngineeringEngineering
- OutlineOutline Introduction to Microwaves Microwave Frequency Bands Advantages of Microwaves Applications of Microwaves Comparison of Transmission Line and Waveguide Rectangular Waveguide Analysis of TM modes for Rectangular Waveguide Analysis of TE modes for Rectangular Waveguide Waveguide Parameters Rectangular waveguide Cavity Resonators. Examples
- Electromagnetic SpectrumElectromagnetic Spectrum
- Microwave PropertiesMicrowave Properties Microwaves frequency range 1 GHz – 300 GHz Microwave is an electromagnetic radiation of short wavelength. Reflected by conducting surfaces Microwave currents flow through a thin outer layer of an ordinary cable. They are not reflected by ionosphere
- Microwave Bands DesignationMicrowave Bands Designation Band Frequency (GHz) Wavelength(cm) L 1 to 2 30.0 to 15.0 S 2 to 4 15 to 7.5 C 4 to 8 7.5 to 3.8 X 8 to 12 3.8 to 2.5 Ku 12 to 18 2.5 to 1.7 K 18 to 27 1.7 to 1.1 Ka 27 to 40 1.1 to 0.75 Millimeter 40-300 0.75-0.1
- Advantages of Microwaves Increased Bandwidth Availability Improved Directive properties Less fading effect and more reliable Lower power requirement Transparency Property of Microwaves
- Applications Telecommunication: Intercontinental Telephone and TV Space communication Telemetry Radars: Detect aircraft Track / guide supersonic missiles Observe and track weather patterns Air Traffic Control (ATC)
- Applications Commercial Applications Microwave ovens Drying machines Rubber / Plastics / Chemical /Food process industry Biomedical Applications Electronic Warfare and Counter Warfare
- Waveguides A hollow metallic tube of uniform cross section for transmitting electromagnetic waves by successive reflections from the inner walls of the tube is called waveguide.
- WaveguidesWaveguides Waveguides, like transmission lines, are structures used to guide electromagnetic waves from point to point. However, the fundamental characteristics of waveguide and transmission line waves (modes) are quite different. The differences in these modes result from the basic differences in geometry for a transmission line and a waveguide. Waveguides can be generally classified as either metal waveguides or dielectric waveguides. Metal waveguides normally take the form of an enclosed conducting metal pipe. The waves propagating inside the metal waveguide may be characterized by reflections from the conducting walls.
- Transmission line and Wave guide -1Transmission line and Wave guide -1 Transmission line Wave guide Two or more conductors separated by some insulating medium (two- wire, coaxial, micro strip, etc.) Metal waveguides are typically one enclosed conductor filled with an insulating medium. Normal operating mode is the TEM or quasi-TEM mode (can support TE and TM modes but these modes are typically undesirable). Operating modes are TE or TM modes (cannot support a TEM mode).
- Transmission line and Wave guide -2Transmission line and Wave guide -2 Transmission line Wave guide No cutoff frequency for the TEM mode. Transmission lines can transmit signals from DC up to high frequency. Must operate the waveguide at a frequency above the respective TE or TM mode cutoff frequency for that mode to propagate. Significant signal attenuation at high frequencies due to conductor and dielectric losses. Lower signal attenuation at high frequencies than transmission lines.
- Transmission line and Wave guide -3Transmission line and Wave guide -3 Transmission line Wave guide Small cross-section transmission lines (like coaxial cables) can only transmit low power levels due to the relatively high fields concentrated at specific locations within the device Metal waveguides can transmit high power levels. The fields of the propagating wave are spread more uniformly over a larger cross-sectional area than the small cross-section transmission line. Large cross-section transmission lines can transmit high power levels. Large cross-section (low frequency) waveguides are impractical due to large size and high cost.
- Rectangular WaveguideRectangular Waveguide The walls of the waveguide are made of ideal conductor and the medium filling the waveguide is ideal dielectric.
- AssumptionsAssumptions The waveguide is infinitely long, oriented along the z-axis, and uniform along its length. The waveguide is constructed from ideal materials. [perfectly conducting pipe (PEC) is filled with a perfect insulator (lossless dielectric)]. Fields are time-harmonic.
- General Wave Characteristics as DefinedGeneral Wave Characteristics as Defined by Maxwell’s Equationsby Maxwell’s Equations Given any time-harmonic source of electromagnetic radiation, the phasor electric and magnetic fields associated with the electromagnetic waves that propagate away from the source through a medium characterized by (μ,ε) must satisfy the source-free Maxwell’s equations given by
- General Wave Characteristics as DefinedGeneral Wave Characteristics as Defined by Maxwell’s Equationsby Maxwell’s Equations The source-free Maxwell’s equations can be manipulated into wave equations for the electric and magnetic fields. These wave equations are where the wave number k is real-valued for lossless media and complex valued for lossy media.
- Types of modes in WaveguideTypes of modes in Waveguide Transverse Magnetic (TM) wave: Here only magnetic field is transverse to the direction of propagation and the electric field is not purely transverse. (i.e.) Ez ≠ 0, Hz = 0. Transverse Electric (TE) wave: Here only the electric field is purely transverse to the direction of propagation and the magnetic field is not purely transverse. (i.e.) Ez = 0, Hz ≠ 0.
- Analysis of TM modesAnalysis of TM modes The wave equation is to be solved for Ez with appropriate boundary conditions. In Cartesian co- ordinates the wave equation for Ez can be written as: The equation can be solved by the separation of variables i.e. by assuming that Ezis given as: The fields are assumed to be of sinusoidal nature with an angular frequency ω.
- Analysis of TM modesAnalysis of TM modes Substituting this solution of Ez and re-arranging the terms, the wave equation becomes- The first term is a function of x only, the second term is a function of y only, the third term is a function of z only and fourth term is a constant. Since the equation is to be satisfied for every value of x, y, z in each term in equation must be constant.
- where, A, B and β are real constants. The parameter β is commonly referred as modal phase constant. Analysis of TM mode
- Analysis of TM modesAnalysis of TM modes From the physical understanding of reflection of waves from parallel conducting boundaries we expect a standing wave kind of behavior in x and y directions and a travelling wave kind of behavior in z direction. In any case, we expect a wave phenomenon in x,y,z direction which can be properly represented by putting a negative sign in front of the constants A2 , B2 and β2 . Instead of negative sign if the positive sign was used the solutions will have real exponential functions which would not represent the wave phenomenon.
- Analysis of TM mode These equations can be re-written as: These equations are identical to the transmission line equations.
- Analysis of TM mode The solution to these equations can be appropriately written as: If we assume that waveguide is of infinite length, we can take only one travelling wave in +z-direction.
- Analysis of TM mode Boundary condition 1: Ez= 0 @ x = 0 leads to C1 = 0 Boundary condition 2: Ez= 0 @ y = 0 leads to C3 = 0 The equation of Ez now reduces to Ez = C * sin(Ax) * sin(By) * e-jβz
- Analysis of TM mode The equation of Ez Ez = C * sin(Ax) * sin(By) * e-jβz Boundary condition 3: Ez = 0 @ x = a leads to Boundary condition 4: Ez = 0 @ y = b leads to where, m & n are integers.
- Analysis of TM mode Substituting for A and B, we get the solution for Ez as: One can get, the values of transverse components Ex, Ey, Hx, Hy from longitudinal components Ez and Hz. In case of TM mode, as Hz = 0, the all 4 transverse components can be obtained from Ez only.
- Analysis of TM mode where, h2 is transverse propagation constant given by-
- Observations for TMmn Modes The fields existing in the discrete electric and magnetic field pattern called modes of waveguide. All field components vary sinusoidal in x and y directions. All transverse fields go to zero if either m or n is zero. Both the indices m and n have to be non-zero for existence of the TM mode. TMm0 and TM0n modes can not exist. Consequently, the lowest order mode which can exist is mode TM11 mode.
- Analysis of TM mode We started with ,
- Analysis of TM mode Re-writing this expression, -A2 - B2 - β2 + ω2 µε = 0 Also, we have, which leads to,
- Analysis of TE ModeAnalysis of TE Mode For TE mode, Ez = 0 and only Hz is present, all 4 transverse components of the field can be obtained from Hz only. In the case of TM mode, the wave equation was solved for Ez which was tangential to all the four walls of the waveguides. We therefore had boundary conditions on Ez. In the TE case however the independent component Hz is tangential to the walls of the waveguide which do not impose any boundary conditions on Hz.
- Analysis of TE Mode One can note that for x=0 and x=a , (vertical walls) and for y= 0 and y=b, (horizontal walls) the tangential component of magnetic field is maximum. Proceeding the analysis for TE mode similar to TM mode, with knowledge that Hz is maximum at the boundaries, one can obtain the solution for Hz as:
- Analysis of TE Mode Where, h2 is transverse propagation constant given by-
- Analysis of TE Mode The fields for the TE modes have similar behavior to the fields of the TM modes i.e. they exist in the form of discrete pattern. They have sinusoidal variations in x and y directions, indices m and n represent number of half cycles of the field amplitudes in x and y direction respectively. Unlike TM mode both indices m and n need not be non-zero for the existence of the TE mode. TE00 mode cannot exist but TEm0 and TE0n modes can exist. The lowest order mode for the TE case therefore is TE10 and TE01 .
- Phase constant of TE and TM modePhase constant of TE and TM mode For both TEmn and TMmn modes the modal phase constant β is given by: For the mode to be travelling β has to be a real quantity. If β becomes imaginary then the fields no more remain travelling but become exponentially decaying.
- Cut-off Frequency of TE and TMCut-off Frequency of TE and TM modemode The frequency at which β changes from real to imaginary is called the cut-off frequency of the mode. At cut-off frequency, therefore β = 0 gives,
- Cut-off Frequency of TE and TMCut-off Frequency of TE and TM modemode The cut-off frequencies for lowest TM and TE modes i.e. TM11, TE10and TE01 can be obtained as:
- Cut-off Frequency of TE and TMCut-off Frequency of TE and TM modemode Since by definition we have a > b we get the frequencies as- We can make an important observation that, if at all the electromagnetic energy travels on a rectangular waveguide its frequency has to be more than the lowest cut-off frequency i.e. fc of TE10 mode. As the order of the mode increases the cut-off frequency also increases.
- Cut-offCut-off wavelength ofof TE and TM modeTE and TM mode The very first mode that propagates on the rectangular waveguide is TE10 mode and therefore this mode is called the dominant mode of the rectangular waveguide. The cut-off wavelength is given by For dominant mode , λc=2a. For propagation of wave in the waveguide λ < λc or f > fc 22 2 + = b n a m cλ
- Field PatternField Pattern The visualization of the modal fields is important for identifying regions from where fields can be tapped efficiently by the probes. The field probes are devices which can induce fields inside a waveguide or extract energy from the fields propagating inside the waveguide. One can see from the modal field expression that the fields are periodic over one guided wavelength λg along the length of the waveguide. So essentially one has to develop a three dimensional picture of the fields only over a block of λg.
- Waveguide ParametersWaveguide Parameters Guide Wavelength It is defined as the distance travelled by the wave in order to undergo a phase shift of 2π radians. It is related to phase constant by the relation λg = 2π / β Wave Impedance It is defined as ratio of strength of electric field in one transverse direction to the strength of magnetic field along other transverse direction.
- Waveguide ParametersWaveguide Parameters Phase Velocity The phase velocity is defined as the velocity with which the wave changes phase in terms of the guide wavelength Vp = λg * f Group Velocity The group velocity of a wave is defined as the rate at which the wave propagates through the waveguide. Vg = dω / dβ The product of phase and group velocities is equal to square of the velocity of light. i.e. vp * vg = c2
- Example 1Example 1
- Example 2Example 2
- Example 3Example 3
- Example 4Example 4 Find the resonant frequencies of the dominant mode of an air filled rectangular cavity of dimensions 5cm x 4cm x 2.5cm.The dominant mode for rectangular cavity resonator is TE101 Given a = 5 cm, b = 4 cm, d = 2.5 cm Resonant frequency of TE and TM modes is given by fr = 6.71 GHz
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