Lecture 10: TEM, TE, and TM Modes for …

1 Whites, EE 481/581 Lecture 10 Page 1 of 10 2015 Keith W. Whites Lecture 10: TEM, TE, and TM Modes for Waveguides. rectangular waveguide . We will now generalize our discussion of transmission lines by considering EM waveguides. These are pipes that guide EM waves. Coaxial cables, hollow metal pipes, and fiber optical cables are all examples of waveguides. We will assume that the waveguide is invariant in the z-direction: xyzab , Metal walls and that the wave is propagating in z as jze . (We could also have assumed propagation in z.) Types of EM Waves We will first develop an extremely interesting property of EM waves that propagate in homogeneous waveguides.

2 This will lead to the concept of Modes and their classification as Transverse Electric and Magnetic (TEM), Whites, EE 481/581 Lecture 10 Page 2 of 10 Transverse Electric (TE), or Transverse Magnetic (TM). Proceeding from the Maxwell curl equations: xyzxyzEjHjHxyzEEE or x: yzxEEjHyz y: xzyEEjHxz z: yxzEEjHxy However, the spatial variation in z is known so that jzjzejez Consequently, these curl equations simplify to zyxEjEj Hy ( ),(1) zxyEjEj Hx ( ),(2) yxzEEjHxy ( ),(3) Whites, EE 481/581 Lecture 10 Page 3 of 10 We can perform a similar expansion of Amp re s equation HjE to obtain zyxHjH j Ey ( ),(4) zxyHjHj Ex ( ),(5) yxzHHjExy ( ),(6) Now, (1)-(6)

3 Can be manipulated to produce simple algebraic equations for the transverse (x and y) components of E and H. For example, from (1): zxyjEHjEy Substituting for Ey from (5) we find 2221zzxxzzxjEHHjjHyjxjEj HHyx or, 2zzxcjEHHkyx ( ),(7) where 22 2ckk and 22k . ( ) Similarly, we can show that Whites, EE 481/581 Lecture 10 Page 4 of 10 2zzycjEHHkxy ( ),(8) 2zzxcjEHEkx y ( ),(9) 2zzycjE HEky x ( ),(10) Most important point: From (7)-(10), we can see that all transverse components of E and H can be determined from only the axial components zE and zH.

4 It is this fact that allows the mode designations TEM, TE, and TM. Furthermore, we can use superposition to reduce the complexity of the solution by considering each of these mode types separately, then adding the fields together at the end. TE Modes and rectangular Waveguides A transverse electric (TE) wave has 0zE and 0zH . Consequently, all E components are transverse to the direction of propagation. Hence, in (7)-(10) with 0zE , then all transverse components of E and H are known once we find a solution for only zH. Neat! Whites, EE 481/581 Lecture 10 Page 5 of 10 For a rectangular waveguide , the solutions for xE, yE, xH, yH, and zH are obtained in Section of the text.

5 The solution and the solution process are interesting, but not needed in this course. What is found in that section is that 22,,0,1,(0)cmnmnmnkmnab (11) Therefore, from ( ) 22,mnc mnkk (12) These m and n indices indicate that only discrete solutions for the transverse wavenumber (kc) are allowed. Physically, this occurs because we ve bounded the system in the x and y directions. (A vaguely similar situation occurs in atoms, leading to shell orbitals.) Notice something important. From (11), we find that 0mn means that ,000ck . In (7)-(10), this implies infinite field amplitudes, which is not a physical result.

6 Consequently, the 0mn TE (or TM) Modes are not allowed. One exception might occur if 0zzEH (a transverse electric and magnetic, , TEM, wave) since this leads to indeterminate forms in (7)-(10). However, it can be shown that inside hollow Whites, EE 481/581 Lecture 10 Page 6 of 10 metallic waveguides when both 0mn and 0zzEH , then 0EH . This means there is no TEM mode in these hollow metallic waveguides. Consequently, EM waves will propagate in hollow metallic waveguides only when the frequency is large enough since the TEM mode cannot exist. To understand this concept, consider (12) and using 22k 22222,11cmnmnkab (13) If 0mn is not allowed in the hollow metallic waveguide ( , no TEM mode), then must be large enough so that is a real number as required for a propagating mode.

7 Otherwise will be imaginary (j ), leading to pure attenuation and no propagation of the wave jzzee . This turns out to be a general result. That is, for a hollow conductor waveguide , EM waves will propagate only when the frequency is large enough and exceeds some lower threshold. This minimum frequency for wave propagation is called the cutoff frequency ,cmnf. It can be shown that guided EM waves require at least two distinct conductors in order to support wave propagation all the way down to 0+ Hz. Whites, EE 481/581 Lecture 10 Page 7 of 10 The cutoff frequencies for TE Modes in a rectangular waveguide are determined from (13) with 0 to be 22,,0,1,1(0)2cmnmnmnfmnab ( ),(14) In other words, these are the frequencies where 0mn and wave propagation begins when the frequency slightly exceeds ,cmnf.

8 For an X-band rectangular waveguide , the cross-sectional dimensions are a = cm and b = cm. Using (14): TEm,n Mode Cutoff Frequencies m n fc,mn (GHz) 1 0 2 0 0 1 1 1 In the X-band region ( GHz), only the TE10 mode can propagate in the waveguide regardless of how it is excited. (We ll also see shortly that no TM Modes will propagate either.) This is called single mode operation and is most often the preferred application for hollow waveguides. On the other hand, at GHz any combination of the first three of these Modes could exist and propagate inside a metal, Whites, EE 481/581 Lecture 10 Page 8 of 10 rectangular waveguide .

9 Which combination actually exists will depend on how the waveguide is excited. Note that the TE11 mode (and all higher-ordered TE Modes ) could not propagate. (We ll also see next that no TM Modes will propagate at GHz either.) TM Modes and rectangular Waveguides Conversely to TE Modes , transverse magnetic (TM) Modes have 0zH and 0zE . The expression for the cutoff frequencies of TM Modes in a rectangular waveguide 22,1,1,22cmnmnfmnab (15) is very similar to that for TE Modes given in (14). It can be shown that if either 0m or 0n for TM Modes , then 0EH . This means that no TM Modes with 0m or 0n are allowable in a rectangular waveguide .

10 For an X-band waveguide from (15): Whites, EE 481/581 Lecture 10 Page 9 of 10 TMm,n Mode Cutoff Frequencies m n fc,mn (GHz) 1 1 1 2 2 1 Therefore, no TM Modes can propagate in an X-band rectangular waveguide when f < GHz. Dominant Mode Note that from GHz f GHz in the X-band rectangular waveguide , only the TE10 mode can propagate. This mode is called the dominant mode of the waveguide . See Fig in the text for plots of the electric and magnetic fields associated with this mode.