TEM Mode

10 Key excerpts on "TEM Mode"

• 

  eBook - PDF

  Theory and Synthesis of Linear Passive Time-Invariant Networks

  • Dante C. Youla(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  14 Multiconductor TEM transmission lines 14.1 Introduction 1 The validity of lumped circuit analysis hinges critically on the assumption that all electrical signals, such as voltage and current, although time-dependent, do not vary continuously with position. Such will indeed be the case if the physical size of any component in the network is much less than the wavelength of the excitation produced in the component by external sources. In the microwave band, however, frequencies range from 1 to 10 GHz and corresponding wavelengths vary from 0.3 to 0.03 meters. It is often impossible, or prohibitively expensive to accurately design discrete components whose dimensions are negligibly small compared with 3 centimeters. Our study of the 1-dimensional TEM line with the aid of field theory will reveal that low-frequency techniques are sometimes applicable, in the small. 14.2 Analytical formulation 2 A transverse electromagnetic or TEM wave is one in which the electric and magnetic field vectors lie in a plane perpendicular to the direction of propagation (which we shall assume to be along the longitudinal z-axis). This is the principal type of wave supported by the structures in Figs. 14.1 (a, b, and c) formed by embedding n + 1 parallel ideal cylindrical conductors in a homogeneous isotropic dielectric. Naturally, even excellent conductors such as gold and platinum are not perfect and have finite conductivities σ (ω). 3 And, of course, neither are they infinitely long. This combination of loss and end-effects inevitably leads to the appearance of longitudinal field components and propagation is not truly TEM. Nevertheless, these z-components attenuate very rapidly as they progress into the interior of such metals and in general contribute negligibly to the field in the bulk of the dielectric. 1 This chapter presupposes at most a first year graduate knowledge of Maxwell’s equations and the relevant Gibbs vector analysis [1].

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Principles of Optics for Engineers

  Diffraction and Modal Analysis

  • William S. C. Chang(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  Since the direction of propagation is z, the solutions of the first group of equations are called the TE, transverse electric, modes. The solutions of the second group of equations are the TM, transverse magnetic, modes. In other words, all planar waveguide modes can be divided into TE and TM types. Since ε is only a function of x, the z variation of the fields must be the same in all layers. This is the consequence of the continuity of E y or H y for all z. For TE modes, the transverse electric E y in Eq. (5.1) can now be written as a product of a function in y and a function in z, i.e. E y ðx; zÞ ¼ E y ðxÞE y ðzÞ. When all these considerations are taken into account, we obtain: ∂ 2 ∂ x 2 þ  ω 2 με ðxÞ  β 2   ! E y ðxÞ ¼ 0 ð5:2aÞ ∂ 2 ∂z 2 þ β 2  ! E y ðzÞ ¼ 0 ð5:2bÞ Similar equations exist for TM modes. Mathematically, Eq. (5.2) and its equivalent for TM modes are second-order differ- ential equations. All the TE modes form a complete set of TE eigenfunctions, meaning that any arbitrary electric field polarized in the y direction with ∂=∂y ¼ 0 can be represented as a summation of TE modes. Similarly, all the TM modes form a complete set of TM eigenfunctions, meaning at any arbitrary electric field polarized in the x direction with ∂=∂y ¼ 0 can be represented as a summation of TM modes. Any radiation field with arbitrary polarization needs to be decomposed first into TE and TM compo- nents, and then analyzed. 5.3 TE modes of planar waveguides The planar TE modes (i.e. modes with ∂=∂y ¼ 0) in the planar waveguides are eigen- solutions of the equation ∂ 2 ∂ x 2 þ ∂ 2 ∂ z 2 þ ω 2 με ðxÞ 2 4 3 5 E y ðxÞE y ðzÞ ¼ 0 ε ðxÞ ¼ n c 2 ε o x ≥ t ¼ n 2 2 ε o t > x > 0 ð5:3Þ ¼ n s 2 ε o 0 ≥ x H x ¼  j ωμ ∂ E y ∂ z ; H z ¼ j ωμ ∂ E y ∂ x 5.3 TE modes of planar waveguides 113 Here, ε o is the free space electric permittivity. The boundary conditions are the continuity of the tangential electric and magnetic fields, E y and H z , at x = 0 and at x = t.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Transmission Lines

  Equivalent Circuits, Electromagnetic Theory, and Photons

  • Richard Collier(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  The metallic walls cause the tangential or surface electric fields to go to zero. Clearly there is no way a static electric field can exist within this guide, and so a TEM wave is not possible. The waves are in two groups. The first assumes that there are only electric fields transverse to the direction of propagation and this group is called the TE or transverse electric group of waves. The second group is called the TM or transverse magnetic group because that has magnetic fields only in the transverse direction. Starting with the TE waves with variation in just the x direction and referring to the coordinate axes shown in Figure 5.1, the electric field can have a solution of the form 158 Guided electromagnetic waves E y ¼ E 0 sin mπx a   exp j ωt  k z z ð Þ, where m ¼ 1, 2, 3, : : : ð5:31Þ Substituting this into the wave Equation (5.7) gives r 2 E y ¼  mπ a   2 E y  k 2 z E y ¼ ω 2 μ 0 ε 0 E y ¼ k 2 0 E y ; ð5:32Þ this is assuming the waveguide is filled with air. Rearranging Equation (5.32) gives k 2 z ¼ k 2 0  mπ a   2 , ð5:33Þ which is the same as Equation (5.17), with k c ¼ (mπ/a). Now Equation (5.33) shows the major difference between the waves on two wire transmission lines and these waves that are inside a single conductor. As the frequency is reduced there comes a point where the right-hand side of Equation (5.33) goes to zero. In other words, the wavenumber in the direction of propagation is zero. Below this frequency, the wavenumber becomes imaginary and this represents an attenuation of the wave rather than any propagation. Another difference is that the phase velocity in the z direction, which is given by v z ¼ ω k z ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 0  mπ=a ð Þ 2 q , ð5:34Þ approaches infinity as the wavenumber tends to zero.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Electromagnetic Field Theory Fundamentals

  • Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  These various reflected waves interact with each other to produce an infinite number of discrete characteristic patterns called modes . The existence of a discrete mode depends upon (a) the shape and size of the waveguide, (b) the medium within the waveguide, and (c) the operating frequency. Unlike the TEM Mode, which can be excited at any frequency, the TM or TE modes can only propagate when the wave frequency is higher than a certain frequency, called the cutoff frequency . The cutoff frequency is different for each mode. When the operating frequency of the wave is 502 503 10.1 Introduction Figure 10.1 A rectangular waveguide assembly (courtesy of Space-machine and Engineering Corp.) lower than the cutoff frequency of the lowest mode, the wave experiences attenuation and disappears after traveling a very short distance. On the other hand, all those modes with cutoff frequencies lower than the oper-ating frequency can exist simultaneously inside the waveguide. In order to avoid the existence of multiple modes, the waveguide is operated at a frequency that lies between the cutoff frequencies of the lowest and the next lowest modes. Thus, the lowest-order mode can propagate, and all other modes are attenuated. Two types of waveguides are commonly used for guiding signals along their lengths at microwave frequencies. One has a rectangular cross section and is aptly called a rectangular waveguide . The other, of circular cross section, is appropriately referred to as a cylindrical waveguide . The rectangular waveguide is the one most commonly used and is also comparatively easy to analyze. For this reason, we will discuss only rectangular waveguides in this chapter. In our discussion of waveguides, we will presume that (a) the four sides of the waveguide are bounded by perfectly conducting walls, and (b) the medium enclosed by these perfectly conducting walls is a perfect dielectric.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Transmission Lines and Wave Propagation

  • Philip C. Magnusson, Andreas Weisshaar, Vijai K. Tripathi, Gerald C. Alexander(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  For the sake of mathematical simplicity, a particular traveling-wave mode for the waveguide with a rectangular cross section will be studied first. The same general technique employed for the coaxial cable, that of postulating a solution and testing it by substituting in Maxwell's equations, will be used initially. Afterward a more general mathematical approach will be tried. Figure 13-1 shows the geometry of the rectangular waveguide.

  FIGURE 13-1 Dimensions and coordinates for rectangular waveguide.

     

  13-1 TE

  m 0

  Propagating Modes

  The simplest and also the most commonly used mode in a rectangular waveguide has these properties: (1) the electric field is confined to planes perpendicular to the direction of propagation, whereas the magnetic field has both transverse and longitudinal components, and (2) the electric field is directed from one of the guide surfaces straight across to the opposite surface. Actually an infinite number of modes possess these properties in common; they are referred to as TE

  m 0

  modes by which the following is meant: (1) TE designates transverse electric (property 1 above), (2) the second subscript, 0, indicates an absence of variation in field configuration as a function of the shorter transverse direction (y direction in Fig. 13-1 ), and (3) the first subscript, m, which is an integer, designates the particular mode of this class. This third item will be described specifically in Sec. 13-la(2), after the TE

  m 0

  field patterns have been derived.

  a. Electric and Magnetic Fields

  For the ordinarily used approximation of the electric and magnetic fields, the waveguide walls may be assumed to be of infinite conductivity, in which case a traveling wave should propagate without attenuation.

  (1) Traveling-Wave Solution

  In view of the success experienced in finding traveling-wave solutions in unbounded space and in a coaxial cable, such a solution will be postulated here. The phase function β will be regarded as an unknown and may well differ from that for the TEM Mode. In accordance with the properties just listed for the TE

  m 0

  mode, the E field will be assumed to consist solely of a y -directed component,

  Ey

  . However, because the component of E tangential to a perfectly conducting surface must vanish (Sec. 11-2b) and, as shown in Fig. 13-1 , such surfaces exist at x = 0 and at

  x —a, Ey

  will be assumed to be a function of x. The requirement that the divergence of D should vanish (Eq. 10-19 , with p = 0) will be met if the E thus chosen is not a function of y

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Electromagnetic Waves with Maxwell's Equations

  • Ozgur Ergul(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  It should be emphasized that propagating modes propagate with different velocities: i.e. lower-order modes propagate slower than higher-order modes. On the other hand, since most electromagnetic energy is car-ried by lower-order modes, the mode with the smallest mode numbers is called the dominant (fundamental) mode. In a rectangular waveguide, the cutoff frequencies for TE 01 and TE 10 modes can be written f 01 c = 1 2 π √ με π b = 1 2 b √ με (7.307) 478 Introduction to electromagnetic waves with Maxwell’s equations y = 0.5 m y x x z 200 MHz 100 MHz Cutoff z = 0.2 m – 0.5 0.5 – 0.5 0.5 – 0.5 0.5 V/m Figure 7.76 The real part of the z component of the electric field intensity for the TM 11 mode in a 2 × 1 m rectangular waveguide (a vacuum inside) at three different frequencies: i.e. at 200 MHz (propagating), at approximately 168 MHz (cutoff), and at 100 MHz (evanescent). The values are shown with respect to position on the y = 0 . 5 m plane (with respect to z ∈ [0 , 6] m and x ∈ [0 , 2] m) and on the z = 0 . 2 m plane (with respect to x ∈ [0 , 2] m and y ∈ [0 , 1] m). and f 10 c = 1 2 π √ με π a = 1 2 a √ με , (7.308) respectively. Therefore, depending on the values of a and b , either TE 01 or TE 10 is the dominant mode of a rectangular waveguide. 70 70 We can write field expressions for TE 10 as H TE x (¯ r ) = γB 10 a π sin πx a exp( − γ 10 z ) E TE y (¯ r ) = − jωμB 10 a π sin πx a exp( − γ 10 z ) H TE z (¯ r ) = B 10 cos πx a exp( − γ 10 z ) E TE x (¯ r ) = 0 , H TE y (¯ r ) = 0 , ¯ E TE z = 0 . Similarly, for TE 01 , we have E TE x (¯ r ) = jωμB 01 b π sin nπy b exp( − γ 01 z ) H TE y (¯ r ) = γ 01 B 01 b π sin nπy b exp( − γ 01 z ) H TE z (¯ r ) = B 01 cos πy b exp( − γ 01 z ) H TE x (¯ r ) = 0 , E TE y (¯ r ) = 0 , ¯ E TE z = 0 . Similar to the wave impedance in a homogeneous space, one can define a wave impedance inside a waveguide.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Principles of Lasers and Optics

  • William S. C. Chang(Author)
  • 2005(Publication Date)
  • Cambridge University Press

    (Publisher)

  In Sections 3.2 to 3.4 , we will solve rigorously Maxwell’s equations and obtain all the modes for the asymmetric planar waveguide shown in Fig. 3.1 . We will identify the solutions thus obtained with the three types of waves discussed above. 3.1.1 TE and TM modes in planar waveguides The variation of the refractive index in the transverse direction is independent of z in Fig. 3.1 . From discussions of electromagnetic theory in classical electri-cal engineering textbooks, we know that modes for structures that have constant 76 Guided wave modes and their propagation transverse cross-section in the direction of propagation can often be divided into TE (transverse electric) type and TM (transverse magnetic) type. TE means no electric field component in the direction of propagation. TM means no magnetic field component in the direction of propagation. For the planar waveguide case, if we substitute ∂/∂ y = 0 into the ∇× e and ∇× h Maxwell equations, we obtain two separate groups of equations: ∂ E y ∂ z = µ∂ H x /∂ t , ∂ E y ∂ x = − µ∂ H z /∂ t , ∂ H z ∂ x − ∂ H x ∂ z = − ε∂ E y /∂ t ;                (3.1a) and ∂ H y ∂ z = − ε∂ E x /∂ t , ∂ H y ∂ x = ε∂ E z /∂ t , ∂ E z ∂ x − ∂ E x ∂ z = µ∂ H y /∂ t .                (3.1b) Clearly, E y , H x and H z are related only to each other, and H y , E x and E z are related only to each other. Since the direction of propagation is z , the solutions of the first group of equations ( 3.1a ) are the TE modes. The solutions of the second group of equations ( 3.1b ) are the TM modes. Thus we have shown the separation of all planar waveguide modes into TE and TM types. Since ε is only a function of x , the z variation of the solution must be the same in all layers. This is a consequence of the requirement for the continuity of E y or H y for all z .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Electromagnetic Waves

  Propagation in Complex Matter

  • Ahmed Kishk(Author)
  • 2011(Publication Date)
  • IntechOpen

    (Publisher)

  We stress again the use of the coefficient of potential is essential for the formulation of the three-line system. Hence, the ordinary TEM Mode propagation of the EM wave is achieved only when the symmetrization is introduced for the electric circuit in the circumstance. We shall calculate the electromagnetic power of the three-conductor transmission-line system. P ( x ) = 1 4 ( V ∗ 1 I 1 + V ∗ 2 I 2 + V ∗ 3 I 3 + V 1 I ∗ 1 + V 2 I ∗ 2 + V 3 I ∗ 3 ) (76) = 1 4 ( V ∗ n I n + V ∗ c I c + V ∗ a I a + V n I ∗ n + V c I ∗ c + V a I ∗ a ) It is interesting to calculate the change of the power with distance so that we can pick up only the terms which change with distance. We take the time dependence of all the modes as exp ( − j ω t ) . We can work out the change rate dP ( x ) dx on the basis of Eqs. (65) and (69) in exactly the same way as the case of the one-conductor transmission line. We write only the final result. dP ( x ) dx = − 1 2 ( R n | I n ( x ) | 2 + R c | I c ( x ) | 2 + R a | I a ( x ) | 2 (77) + R nc ( I n ( x ) I ∗ c ( x ) + I ∗ n ( x ) I c ( x )) + R na ( I n ( x ) I ∗ a ( x ) + I ∗ n ( x ) I a ( x )) + R ca ( I c ( x ) I ∗ a ( x ) + I ∗ c ( x ) I a ( x ))) − 1 2 M ω c ( I I ∗ t ( l , x ) I a ( x ) + I I , x t ( l ) I ∗ a ( x )) − j 1 2 Mc parenleftbigg Q I ∗ t ( l , x ) dI a ( x ) dx − Q I t ( l . x ) dI ∗ a ( x ) dx parenrightbigg This expression agrees with the one of the line-antenna (62), when the change rate is expressed with the total current by using the relation I a ( x ) = 1 2 I t ( x ) . It is interesting to point out that the change of the electric power is made by the resistance terms and the antenna mode terms. 250 Electromagnetic Waves Propagation in Complex Matter Electrodynamics of Multiconductor Transmission-line Theory with Antenna Mode 19 There are two effects in the antenna mode terms. One is a term associated with emission and the other is a term associated with absorption.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Advanced Electromagnetic Waves

  • Saad Osman Bashir(Author)
  • 2015(Publication Date)
  • IntechOpen

    (Publisher)

  In the case of a good conductor, an approximation by a real number for Γ is presented in [28]. However, Γ is in general complex and would alter the shape of the excitation pulses, which cannot be accounted for in the TLM method [7]. 3.2. Application to confined environments modeled by arbitrary cross-section oversized waveguides: 2.5D TLM approach The analysis of electrically large structures like tunnels involves a high computation time by using full-wave methods like TLM. However, some astutenesses can be considered to efficiently model them, which translates in finding an approach to simplify the analysis of oversized waveguides. From the geometrical point of view, tunnels can have any arbitrary shape, and this constraint has to be considered. From the electromagnetic point of view, waves propagating in tunnels can be classified in modes. This problem is well known in electromagnetic theory. As we shall see, a full-wave model based on a modal decomposition of the waves for the analysis of the radio propagation in tunnels at high frequencies is presented. Besides, the concept is used to further reduce the complexity of this electromagnetic problem. 3.2.1. The TLM modal approach Modal approaches in frequency domain are based on the expansion of fields in terms of modes. In general, total fields can be represented as the sum of modes as shown in (5): Advanced Electromagnetic Waves 244 F ( x , y , z ) = N ∑ i = 1 A i ( x , z ) e -( α i + j β i ) y , (5) where N is the total number of modes, A i ( x , z ) the amplitude of the i -th mode at the coordinate ( x , z ) in the plane perpendicular to the propagation direction y , α i its attenuation, and β i its phase constant. The mode amplitude A i ( x , z ) gives the electric or magnetic field distribution of the mode. The phase constant β represents the change in phase along the path traveled by the wave. Lastly, the attenuation constant is a measure of losses in the structure.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Microwave Electronics

  Measurement and Materials Characterization

  • L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan, Vijay K. Varadan(Authors)
  • 2004(Publication Date)
  • Wiley

    (Publisher)

  So the wave is in a cutoff state. There is a critical state between the transmis- sion state and cutoff state: k = k c and so γ = 0. The frequency corresponding to the critical state is called the cutoff frequency f c , and its correspond- ing wavelength is called the cutoff wavelength λ c . The relationship between k c and λ c is k c = 2π λ c (2.166) Both k c and λ c are related to the transverse field distribution in the waveguide. The transmission requirement can be described as k > k c , λ < λ c , and Eq. (2.160) can be rewritten as β = 2π λ 1 − λ λ c 2 (2.167) For a TEM wave, as β = k, the cutoff wavelength is infinity, so TEM waves with any frequency satisfy the propagation requirement. In the following, we discuss parameters often used in describing the propagation properties of hollow metallic waveguides, including phase velocity, group velocity, and the wave impedances for TE and TM waves. According to the definition, phase velocity is the velocity of the movement of phase planes. From Eqs. (2.161) and (2.162), for a certain phase plane moving along the z-axis, following requirement is satisfied: ωt − βz = constant (2.168) From Eq. (2.168), we have d dt (ωt − βz) = ω − β dz dt = 0 (2.169) So the phase velocity is v p = dz dt = ω β (2.170) From (2.167), we have v p = ω β = v 1 − (λ/λ c ) 2 = v 1 − (λ/λ c ) 2 , (2.171) where v is the velocity of electromagnetic wave: v = ω 2π λ = f λ (2.172) Equation (2.171) indicates that the phase velocity of TE or TM wave is frequency-dependent, while 64 Microwave Electronics: Measurement and Materials Characterization for TEM wave, the phase velocity does not change with frequency: v p = ω/k = 1/ √ εµ. (2.173) The waveguide wavelength refers to the wave- length of a wave propagating along a waveguide. For TEM wave, the waveguide wavelength λ g = v/f (2.174) So it is the same as the wavelength when the TEM wave propagates in a free space filled with the same medium.

  Sign up to read

  Learn more about book