Introduction to TEM Waves

13.1.1^*With a short at y = 0, it is possible to find the fields for Example 13.1.1 by recognizing at the outset that standing wave solutions meeting the homogeneous boundary condition of (12) are of the form E_x = Re A sin ( y) \exp (j t).

H_z and the dispersion equation (relation between and ).

(a) Use (13.1.2) and (13.1.3) to determine the associated

(b) Now use the boundary condition at y = -b to show that the fields are as given by (13.1.16) and (13.1.17).

13.1.2^*Take the approach outlined in Prob. 13.1.1 for finding the fields [(13.1.28) and (13.1.29)] in Example 13.1.2.

13.1.3Assume that \hat K_o is real and express the standing wave of (13.1.17) so as to make it evident that it is the sum of equal-amplitude waves traveling in the y directions, each with a magnitude of phase velocity / = c and wavelength 2 /.

13.1.4^*Coaxial perfectly conducting circular cylinders having outer and inner radii a and b, respectively, form the transmission line shown in Fig. P13.1.4. driven by a voltage source V at z = -l, show that the EQS electric field is radial and given by V/[r ln(a/b)].

(a) If the conductors were ``open circuit'' at z = 0 and

(b) If the conductors were ``shorted'' at z = 0 and driven by a current source I at z = -l, show that the MQS magnetic field intensity is directed and given by I/2 r.

(c) With the motivation provided by these limiting solutions, show that solutions to all of Maxwell's equations (in the region between the conductors) that satisfy the boundary conditions on the surfaces of the coaxial conductors are equation GIF #13.131 provided that V and I are now functions not only of t but of z as well that satisfy equations taking the same form as (13.1.2) and (13.1.3). equation GIF #13.132 equation GIF #13.133

13.1.5For the coaxial configuration of Prob. 13.1.4, there is a perfectly conducting ``short'' at z = 0, and the conductors are driven by a current source I = Re [I_o e^{j t}] at z = -l. H.

(a)	Find I(z, t) and V(z, t) and hence E and
(b)	Take the low frequency limit where l \ll 1 and show that E and H are the same as for a coaxial inductor.
(c)	Find E and H directly from the MQS laws and show that they agree with the results of part (b).

13.1.6For the coaxial configuration of Prob. 13.1.4, the conductors are ``open circuited'' at z = 0 and driven by a voltage source V = Re [V_o \exp (j t] at x = -l.

(a)	Find I(z, t) and V(z, t) and hence E and H.
(b)	Take the low-frequency limit where l \ll 1 and show that E and H are the same as for a coaxial capacitor.

\item (c) Find E and H directly from the EQS laws and show that they agree with the results of (b).

Two-Dimensional Modes Between Parallel Plates

13.2.1^*Show that each of the higher-order modes propagating in the +y direction, represented by A⁺_n and C⁺_n in (13.2.19) and (13.2.20), respectively, can be regarded as the sum of plane waves propagating in the directions represented by the vector wave number equation GIF #13.134

and interfering in the planes x = 0 and x = a so as to satisfy the boundary conditions.

13.2.2The TM and TE modes can themselves be classified into odd or even modes that, respectively, have \hat h_z or \hat e_z odd or even functions of x. With this in mind, the origin of the coordinate system is moved so that it is midway between the perfectly conducting plates, as shown in Fig. P13.2.2. when the boundary condition is met at x = d \equiv a/2 for these functions, it is automatically met at x = -d.

(a) Find the odd TM and TE solutions. Note that

(b) Find the even TM and TE solutions, again noting that if the conditions are met at x = d, then they are at x = -d as well.

TE and TM Standing Waves between Parallel Plates

13.3.1^*Starting with (13.3.1) (for TM modes) and (13.3.2) (for TE modes) use steps similar to those illustrated by (5.5.20)-(5.5.26) to obtain the orthogonality conditions of (13.3.3) and (13.3.4), respectively. 13.3.2In the system of Example 13.3.1, the wall at y = 0 is replaced by that shown in Fig. P13.3.2. A strip electrode is embedded in, but insulated from, the wall at y = 0. The resistance R is low enough so that E tangential to the boundary at y = 0, even at the insulating gaps between the strip electrode and the surrounding wall, is negligible.

(a)	Determine the output voltage v_o in terms of v.
(b)	For b/a = 2, describe the dependence of \|v_o\| on frequency over the range a = 0 5/4, specifying the low-frequency range where the response has a linear dependence on frequency and the resonance frequencies.
(c)	What is the distribution of H_z(x, y) at the resonance frequencies?

13.3.3^*In the two-dimensional system of Fig. P13.3.3, each driven electrode has the same nature as the one in Fig. 13.3.1. The origin of the y axis has been chosen to be in the plane of symmetry.

(a)	Use the symmetry to argue that H_z(y = 0) =0.
(b)	Show that in the interior region,

13.3.4The one-turn loop of Fig. P13.3.4 has dimensions that are small compared to a, b, or wavelengths of interest and has area A in the x - y plane. bottom electrode in Fig. 13.3.1. Assume that the resistance is large enough so that the current induced in this loop gives rise to a magnetic field that is negligible compared to that already found. In terms of H_z, what is v_o?

(a)	It is used to detect the TM H field at the middle of the
(b)	At what locations x = X of the loop is \|v_o\| a maximum?
(c)	If the same loop were in the plate at y = 0 in the configuration of Fig. 13.1.3 and used to detect H_z at y = 0 for the TEM fields of Example 13.1.1, what would be the dependence of \|v_o\| on the location x = X of the loop?
(d)	If the loop were located in the plate at y = 0 in the TE configuration of Fig. 13.3.4, how should the loop be oriented to detect H?

13.3.5In the system shown in Fig. P13.3.5, \ll d and the driving sources v = Re [\hat v \exp (j t)] are uniformly distributed in the z direction so that the fields are two dimensional. Thus, the driving electrode is like that of Fig. 13.3.1 except that it spans the width d rather than the full width a. Find H and E in terms of v.

13.3.6In the system shown in Fig. P13.3.6, the excitation electrode is like that for Fig. 13.3.4 except that it has a width d rather than a. Find H and E in terms of \hat\Lambda.

Rectangular Waveguide Modes

13.4.1^*Show that an alternative method of exciting and detecting the TE_10 mode in Demonstration 13.4.1 is to introduce one-turn loops as shown in Fig. P13.4.1. The excitation loop is inserted through a hole in the conducting wall while the detection loop passes through a slot, so that it can be moved in the y direction. The loops are each in the y - z plane. To minimize disturbance of the field, the detection loop is terminated in a high enough impedance so that the field from the current in the loop is negligible. Compare the y dependence of the detected signal to that measured using the electric probe.

13.4.2A rectangular waveguide has w/a = 0.75. Presuming that all TE and TM modes are excited in the guide, in what order do the lowest six modes begin to propagate in the y direction as the frequency is raised? 13.4.3^*The rectangular waveguide shown in Fig. P13.4.3 is terminated in a perfectly conducting plate at y = 0 that makes contact with the guide walls. An electrode at y = b has a gap of width \ll a and \ll w around its edges. Distributed around this gap are sources that constrain the field from the edges of the plate to the guide walls to v(t)/ = Re (\hat v/ ) \exp(j t). condition at y = 0 to show that equation GIF #13.136

[Hint: If (13.4.9) and (13.4.10) are used, remember that k_y = + _mn for the A⁺_mn mode but k_y = -_mn for the A^-_mn mode. ]

(a)	Argue that the fields should be TM and use the boundary
(b)	Show that, for m and n both odd, A⁺_mn = 8\hat v (² - _mn²)/nm ² _mn sin (k_mn b), while for either m or n even, A⁺_mn = 0.
(c)	Show that for these modes the resonance frequencies (normalized to 1/ a) are where m, n, and p are integers, m and n odd.
(d)	Show that under quasistatic conditions, the field which has been found is consistent with that implied by the EQS potential given by (5.10.10) and (5.10.15).

13.4.4The rectangular waveguide shown in Fig. P13.4.3 is terminated in a perfectly conducting plate at y = 0 that makes contact with the guide walls. However, instead of the excitation electrode shown, at y = b there is the perfectly conducting plate with a square hole cut in its center, shown in Fig. P13.4.4. In this hole, the pole faces of a magnetic circuit are flush with the plate and are used to excite fields within the guide. Approximate the normal fields over the surface of the pole faces as equation GIF #13.138

where \hat H_o is a complex constant. (Note that, if the magnetic circuit is driven by a one turn coil, the terminal voltage v = j (² /2) \hat H_o.) Determine H_y, and hence E and H, inside the guide.

Dielectric Waveguides: Optical Fibers

13.5.1^*For the dielectric slab waveguide of Fig. 13.5.1, consider the TE modes that have E_z an odd function of z. k_y is again found from (13.5.10), but with (k_x d) found by simultaneously solving (13.5.8) and equation GIF #13.139

(a)	Show that the dispersion relation between and
(b)	Sketch the graphical solution for k_x d \equiv _m (m odd) and show that the cutoff frequency is again given by (13.5.9), but with m odd rather than even.
(c)	Show that these odd modes also have the asymptote of unity slope shown in Fig. 13.5.3.
(d)	Sketch the odd mode dispersion relation on that for the even modes (Fig. 13.5.3).

13.5.2For the dielectric slab waveguide shown in Fig. 13.5.1, _i / = 2.5, = _o, and d = 1 cm. In Hz, what is the highest frequency that can be used to guide only one TE mode. (Note the result of Prob. 13.5.1.) 13.5.3^*The dielectric slab waveguide of Fig. 13.5.1 is the same as that considered in this problem except that it now has a permeability _i that differs from that outside, where it is . by equation GIF #13.140

(a)	Show that (13.5.7) and (13.5.8), respectively, are replaced
(b)	Show that making _i > lowers the cutoff frequency.
(c)	For a given frequency, does making _i / > 1 increase or decrease the wavelength \equiv 2 /k_y?

13.5.4The dielectric slab of Fig. 13.5.1 has permittivity _i and permeability _i, while in the surrounding regions these are and , respectively. Consider the TM modes. (13.5.10) that can be used to determine the dispersion relation = (k_y) for modes that have H_z even and odd functions of x.

(a)	Determine expressions analogous to (13.5.7), (13.5.8), and
(b)	What are the cutoff frequencies?
(c)	For _i = and _i = = 2.5, draw the dispersion plot for the lowest three modes that is analogous to that of Fig. 13.5.3.

(a)	Use (13.1.2) and (13.1.3) to determine the associated
(b)	Now use the boundary condition at y = -b to show that the fields are as given by (13.1.16) and (13.1.17).

(a)	If the conductors were ``open circuit'' at z = 0 and
(b)	If the conductors were ``shorted'' at z = 0 and driven by a current source I at z = -l, show that the MQS magnetic field intensity is directed and given by I/2 r.
(c)	With the motivation provided by these limiting solutions, show that solutions to all of Maxwell's equations (in the region between the conductors) that satisfy the boundary conditions on the surfaces of the coaxial conductors are provided that V and I are now functions not only of t but of z as well that satisfy equations taking the same form as (13.1.2) and (13.1.3).

(a)	Find the odd TM and TE solutions. Note that
(b)	Find the even TM and TE solutions, again noting that if the conditions are met at x = d, then they are at x = -d as well.