Chapter 8 Waveguides
*Note: The information below can be referenced to: Carr, J., Practical Antenna Handbook, Tab Books, Blue Ridge Summit, PA, 1989, ISBN: 0-8306-9270-3. Edminister, J., Electromagnetics (Schaum’s Outline), McGraw-Hill, New York, NY, 1993, ISBN: 0-07-018993-5.
Chapter 8 Waveguides
The microwave portion of the radio spectrum covers frequencies from about 900 MHz to 300 GHz, with wavelengths in free-space ranging from 33 cm down to 1 mm. Transmission lines are used at frequencies from dc to about 50 or 60 GHz, but anything above 5 GHz only short runs are practical, because attenuation increases dramatically as frequency increases. There are three types of losses in conventional transmission lines: ohmic, dielectric, and radiation. The ohmic losses are caused by the current flowing in the resistance of the conductors making up the transmission lines. The skin effect will increase the resistance at higher frequencies; therefore the losses tend to increase in the microwave region. Dielectric losses are caused by the electric field acting on the molecules of the insulatorand thus, will cause heating through molecular agitation. Radiation losses are the loss of energy as the electromagnetic wave propagates away from the surface of the transmission line conductor.
Losses on long runs of commonly used coaxial transmission line causes concern as low as 400 MHz. Because of the increased losses the power handling capability decreases at higher frequencies, therefore, at higher microwave frequencies, or where long runs make coax attenuation losses unacceptable, or where high power levels causes the coax to overheat, waveguides are used instead of the transmission lines.
This chapter will describe the propagation characteristics in a single conductor transmission lines referred to as waveguides. What is a waveguide? Consider the “light pipe analogy” illustrated in Figure 8.0A. A flashlight serves as our “r-f source,” which given that light is also an electromagnetic wave is not all that unreasonable. The source radiates into free-space, and spreads out as a function of distance. The intensity per unit area at the destination – a wall falls off as a function of distance (D) according to the inverse square law (1/D2). Now consider the transmission scheme in Figure 8.0B, the light wave still propagates over a distance D, but is now confined to the interior of a mirrored pipe. Almost all of the energy coupled to the input end is delivered to the output end, where the intensity is practically undiminished. The light pipe analogy may not be the best way to explain the operation of waveguides, but rather a neat summary on a simple level.
The internal walls of the waveguide are not mirrored surfaces, but instead electrical conductors. Most waveguides are made of aluminum, brass, or copper. Some waveguides internal surfaces are electroplated with either gold or silver to reduce ohmic losses. The gold or silver have lower resistivities than most other metals.
Waveguides are hollow pipes, and may have either circular or rectangular cross sections. Rectangular are, by far, the most common. These waveguides are used for high frequency transmission in the gigahertz (microwave) range. The TEM mode cannot propagate in these single conductor transmission lines. Only higher modes in the form of transverse electric (TE) and transverse magnetic (TM) modes can propagate in the waveguide.
Transverse and Axial Fields
The waveguide is positioned with the longitudinal direction along the z axis.
Waveguide characteristics:
- guide walls have (perfect conductor)
- dielectric-filled hollow has:
- (perfect conductor)
- assumed (no free charge)
The dimensions for the cross section are inside dimensions. Figure 8.1(a) is a rectangular waveguide shown in Cartesian coordinate system; Figure 8.1(b) shows a circular or cylindrical waveguide of radius a in a cylindrical coordinate system.
The time dependence will be assumed for the electromagnetic field in the dielectric core. The following expressions for the field vector F (which stands for either E or H), assuming the wave is propagating in the +z direction.
Rectangular coordinates F = F(x, y) e-jkz where:
Cylindrical coordinates where:
The wave will propagate without attenuation, because the dielectric is lossless (σ = 0). Let (in rad/m) be the wave number and is constrained to be real and positive.
The reason for separating the field vector into a transverse vector component FT and an axial vector component Fzaz is two-fold. The complete EH fields in the waveguide are known once either cartesian component Dz or Hz is known.
Transverse Components from Axial Components
Assume a rectangular coordinate system. Maxwell’s equation yields three scalar equations:
(1a)
(1b)
(1c)
Maxwell’s equation yields three additional scalar equations with σ = 0:
(2a)
(2b)
(2c)
Eliminate Hx between (1a) and (2b) and Hy between (1b) and (2a):
(3a)
(3b)
*() The parameter kc (also rad/m) functions as a critical wave number.
Example for kc:
What is “critical” about the number kc?
For propagation through a lossless dielectric, the wave number k must be real, but
The wave number ko is of a uniform plane wave in the unbounded dielectric at the given ω. Thus kc is a critical wave number in the sense that a guided wave’s same –frequency “twin” must have a wave number exceeding kc. Stated otherwise, the frequency f of the guided wave must exceed the quantity is the wave velocity in the unbounded dielectric.
Finally, take (3b) and (3a) substitute into (2a) and (2b):
(3c)
(3d)
It is possible to force either Ez or Hz (but not both) to vanish identically. The non-vanishing axial component will determine all other components via equations (3).
Example 8.1:
Express Maxwell’s equations (1) and (2) in scalar form in cylindrical coordinate system.
(1)
(2)
*Note: For the curl in cylindrical coordinates refer to –
Equation (1) yields (σ = 0):
(i)
(ii)
(iii)
Equation (2) yields:
(iv)
(v)
(vi)
Example 8.2:
Using the equations of example 8.1, find all cylindrical field components in terms of Ez and Hz.
From (i) and (v), with kc as previously defined,
(1)
From (ii) and (iv),
(2)
From (1) and (i),
(3)
From (1) and (ii),
(4)
Propagation Modes in Waveguide
In a waveguide a signal will propagate as an electromagnetic wave. Even in a transmission line the signal propagates as a wave because the current in motion down the line gives rise to the electric and magnetic fields that behaves as an electromagnetic field. The transverse electromagnetic (TEM) field is the specific type of field found in transmission lines. We also know that the term “transverse” implies to things at right angles to each other, so the electric and magnetic fields are perpendicular to the direction of travel. These right angle waves are said to be “normal” or “orthogonal “to the direction of travel.
The boundary conditions that apply to waveguides will not allow a TEM wave to propagate. However, the wave in the waveguide will propagate through air or inert gas dielectric in a manner similar to free space propagation, the phenomena is bounded by the walls of the waveguide and that implies certain conditions that must be met. The boundary conditions for waveguides are:
- The electric field must be orthogonal to the conductor in order to exist at the surface of that conductor.
- The magnetic field must not be orthogonal to the surface of the waveguide.
The waveguide has two different types of propagation modes to satisfy these boundary conditions:
- TE – transverse electric (Ez = 0)
- TM – transverse magnetic (Hz = 0)
The transverse electric field requirement means that the E-field must be perpendicular to the conductor wall of the waveguide. This requirement can be met with proper coupling at the input end of the waveguide. A vertically polarized coupling radiator will provide the necessary transverse field.
One boundary condition will require the magnetic (H) field not to be orthogonal to the conductor surface. Since it is at right angles to the E-field, the requirement will be met. The planes that are formed by the H-field will be parallel to the direction of propagation and to the surface.
Waveguide Impedances
For any transverse electromagnetic wave , the wave impedance (in ohms) is defined as being approximately equal to the ratio of the electric and magnetic fields, and converges as a function of frequency to the intrinsic impedance of the dielectric:
(4)
For a TE mode waveguide, (1a) & (1b) imply:
Or(5)
Equation (4) involves only lengths of two-dimensional vectors, so η must be independent of the coordinate system. Example 8.3 will confirm the value of ηTE by recalculating it in cylindrical coordinates. Example 8.4 shows (using rectangular coordinates) that:
(6)
Example 8.3:
Calculate ηTEfrom the field components in cylindrical coordinates.
Ez ≡ 0, (iv) and (v) of Example 8.1 gives:
(iv)
(v)
Example 8.4
Calculate ηTM from the field of components in rectangular coordinates.
Hz≡ 0, (2a) and (2b):
(2a)
(2b)
Solution:
Determination of the Axial Fields
All that remains for a complete description of TE and TM modes is the determination of the respective axial fields:
Fz = Hz TE
Fz = Ez TM
The cartesian coordinate of F (in either rectangular or cylindrical coordinates), must satisfy the scalar wave equation,
(7)
And the appropriate boundary conditions which are inferred from the boundary conditions on the components of FT. *Note: Transverse components such as are not cartesian components and do not obey a scalar wave equation.
Explicit Solutions for TE Modes of a Rectangular Guide
The wave equation (7) becomes:
This was previously defined as. Solve by using separation of variables:
(8)
. The separation constants kx and ky are determined by the boundary conditions. Consider first the x-conditions; in view of (3a) - and Ez ≡ 0these translate into:
Apply these conditions to: (8) -
This will result in Bx = 0 and and by symmetry, the boundary conditions in y force By = 0 and
Each pair of nonnegative integers (m, n) –with the exception of (0, 0) which will result in a trivial solution-identifies a distinct TE mode, indicated as TEmn. This mode has the axial field
(9)
And the transverse field is obtained through (3) –(refer to pages 2-3 of these notes). The critical wave number for TEmnis:
This is in terms of which the wave number and the wave impedance for TEmn are:
(10)
(11)
*Note: m, n are integers that define the number of half wavelengths that will fit in the (a) and (b) dimensions, respectively; a, b are the waveguide dimensions. (see Figure 8.2)
Example 8.5:
This example will show for the TMmn modes of a rectangular waveguide and it will show that kcTMmn = kcTEmn. The subscripts TE and TM can be dropped from all modal parameters of rectangular guides save the wave impedance.
Obtain the analogues of (9) – (12) for TMmn.
Analogous to (8),
But now the boundary conditions are:
This will require that:
Note that neither m nor n is zero in a TM mode.
The required formulas are:
(1)
(2)
(3)
(4)
Velocity and Wavelength in Waveguides:
Figure 8.3 illustrates the geometry for two wave components simplified for sake of illustration. There are three different wave velocities to consider with respect to waveguides: free space velocity (c), group velocity (Vg), and phase velocity (Vp).
The space velocity of propagation in unbounded free-space, i.e., the speed of light
(c = 3 * 108 m/s).
The group velocity is the straight line velocity of propagation of the wave down the center-line (z-axis) of the waveguides. The value of Vg is always less than c, because the actual path length taken as the wave bounces back and forth is longer than the straight line path (i.e., path ABC is longer than path AC). The relationship between c and Vg is:
Vg= c sin a
*Note: Vg is the group velocity in (m/s), c is the free space velocity (3 * 108 m/s), and ais the angle of incidence in the waveguide.
The phase velocity is the velocity of propagation of the spot on the waveguide wall where the wave impinges (e.g., point “B” in Figure 8.4). This velocity is actually faster than both the group velocity and the speed of light. The relationship between the phase and group velocities can be seen in the “Beach analogy.” If we consider an ocean beach that waves will arrive from offshore at an angle other than 90°, meaning the arriving wave fronts will not be parallel to the shore. The arriving waves at Vgas it hits the shore will strike a point down the beach first, and the “point of strike” races up the beach at a faster phase velocity, Vp, that is faster than Vg. In a microwave waveguide the phase velocity can be greater than c.
Mode Cutoff Frequencies
The propagation of signals in a waveguide depends in part upon the operating frequency of the applied signal. The angle of incidence made by the plane wave to the waveguide wall is a function of frequency. As the frequency drops, the angle of incidence increases towards 90°.
In practice one may deal with frequencies and not wave numbers. It is desirable to replace the concept of the critical wave number (kc) by one of the cutoff frequency (fc). This was accomplished in the example for (kc) (refer to page 3 of these notes):
(13)
In terms of the cutoff frequency fc and the operating frequency
(10), (11), and (12) will become:
(Rectangular waveguide) (10bis)
(11bis)
(12bis)
is the wavelength of an imaginary uniform plane wave at the operating frequency and where is the plane wave impedance of the lossless dielectric. The second form of (11 bis) exhibits the relation between the operating wavelength λo and the actual guide wavelength λmn. For TMmn waves, (12 bis) is replaced by [see (6)]
(14)
The phase velocity of a TEmn or TMmn wave is given by:
(15)
The meaning of cutoff is made particularly clear in (15). As the operating frequency drops to the cutoff frequency, the velocity becomes infinite. This is a characteristic, not of wave propagation, but of diffusion (instantaneous spread of exponentially small disturbances).
Example 8.6
Define the notion of cutoff wavelength.
The cutoff wavelength λc is the wavelength of an unguided plane wave whose frequency is the cutoff frequency; i.e., λc* fc = uo
Is the cutoff wavelength an upper limit on the guide wavelength, just as the cutoff frequency is a lower limit on the guide frequency?
No; in fact, the formula shows that an (m, n) mode can propagate with any guide wavelength greater than λ.
Dominant Mode
The dominant mode of any waveguide is that of the lowest cutoff frequency. Now, for a rectangular guide, the coordinate system may always be oriented to make a ≥ b.
Sincefor either TE or TM, but neither m nor n can vanish in TM, the dominant mode of a rectangular guide is invariably TE10, with
From (9), Ez10 ≡0, and the equations (Transverse Components from Axial Components Section):
(16)
For H10 real, the three nonzero field components have the time-domain expressions:
(17)
Plots of the dominant-mode fields (17) at t = 0 are given in Figs. 8.5 and 8.6. Both vary as. This is indicated in Figure 8.5 by drawing the lines of E close together near x = a/2 and far apart near x = 0 and x = a. The lines of H are shown evenly spaced because there is no variation with y. This same line-density convention is used to indicated the local value of in Figure 8.6(a) and of
in Figure 8.6(b). Notice that the lines of H are closed curves (div H = 0); the H field may be considered as circulating about the perpendicular displacement current density JD.
1
Notes by: Debbie Prestridge