In this section we focus on the evolution of modes along the waveguide: in other words, we study the modal voltages and currents. It was said above that each propagation mode is characterized by a number of modal functions and a constant k0ti, kti00 called transverse wavenumber. This wavenumber determines the longitudinal evolution of the mode, since it can be shown that the longitudinal propagation constant kzi is given by
kzi= q
k2− kti2 (5.9)
for both T M and T E modes, where k2 = ω2εµ is the wave number in the medium that fills the waveguide. Moreover, the modal impedances can be shown to be
Zti0 =kzi
ωε for T M modes Zti00 = ωµ
kzi for T E modes
Depending on the frequency of operation, the evolution of modes can be totally different. Consider a specific mode i. It is clear from (5.9) that at very high frequency kzi is real, whereas a very low frequencies it is imaginary. In particular, we define the critical frequency (or cut-off frequency) of each mode
fci= kti
2π√
εµ (5.10)
In terms of it, for both T M and T E modes
• if f ≥ fcithe mode is above cut-off : kzi and Ztiare real
• if f ≤ fcithe mode is below cut-off : kzi and Ztiare pure imaginary At the cut-off frequency, kzi= 0 and Zti0 = 0 and Zti00 → ∞.
The mode with the smallest critical frequency is called fundamental mode.
Let fc0ibe the critical frequency of mode i in an empty waveguide. As it is clear from the definition, the presence of the dielectric makes the critical frequency smaller:
fci= fc0i
√εrµr
ĐŽĂdž 0 z
Figure 5.3. Infinitely long waveguide with a source that excites a mode above cut-off
In order to clearly understand the characteristics of the two regimes, suppose that the waveguide is infinitely long, as shown in Fig. 5.3, so that only the forward wave is excited by the source:
Vi(z) = Vi0+e−jkziz Ii(z) = YtiVi0+e−jkziz for z ≥ 0
1. Suppose f > fci. Let us compute the time evolution of this wave.
vi(z,t) = R{Vi(z)ejωt} = |Vi0+| cos(ωt − kziz + arg(Vi0+)) ii(z,t) = R{Ii(z)ejωt} = Yti|Vi0+| cos(ωt − kziz + arg(Vi0+)) Since kzi is real, the mode is propagating with phase velocity
vph= ω
where we have used (5.10) and the fact that ω
Note that this last quantity is the phase velocity v of a plane wave in the medium that fills the waveguide. Note also that if the waveguide is empty, εr= 1 and the phase velocity of the mode is greater than the speed of light c. However, the theory of relativity is not violated:
indeed, only geometrical points (wave nodes or wave crests) move at the velocity vph and no mass or energy or information. See the analogous discussion concerning plane waves (2.12).
The previous formula can also be written in a slightly different for in terms of the critical frequency of the mode in an empty waveguide:
vph= c/√
We remark that the mode phase velocity depends on frequency, hence propagation in a waveguide is dispersive. When the field in the waveguide is not monochromatic, the various frequency components move at different velocity and the signal suffers distortions. If however the signal is narrow band, as in the case of an amplitude modulated pulse of much longer duration than the period of the carrier, the distortion manifests itself in the fact that the envelope appears to travel at a different velocity than the carrier: the former travels at the group velocity (vg), the latter at the phase velocity (vph). The group velocity can be shown to be given by
vg= 1 d
dωR{kzi} Carrying out the derivative, we find
vg= c
Note that since the envelope does not changes shape, the difference in velocities is conven-tionally not considered a real distortion.
It can also be shown that the energy of the mode travels at the group velocity. We see that the group velocity is always smaller than c and this is very important for the requirements of the theory of relativity.
Finally, it is to be remarked that a simple relationship exists between phase and group velocity:
vgvph= c2 εrµr
The wavelength on the equivalent transmission line is called guided wavelength and is defined as
is the plane wave wavelength in the dielectric that fills the waveguide and λ0the corresponding one in empty space. Obviously λgi> λ. Also this formula can be written in terms of fc0i:
λgi= λ
The modal impedance of T E modes is given by
Zti00 = ωµ
It is useful to recall that
Z = rµ
ε = Z0
rµr
εr
is the plane wave impedance in the dielectric that fills the waveguide.
The modal impedance of T M modes is given by
Zti0 =kzi
Since the characteristic impedance Zti is real, the wave is characterized by the power flow P = 1
2
|Vi0+|2 Zti
2. Suppose f < fci. In this case kzi = −j|kzi| is imaginary. The negative sign of the propagation constant is related, as always, to our time convention exp(jω0t) for phasors. The time evolution of the modal voltage and current wave is
vi(z,t) = R{Vi(z)ejωt} = |Vi0+|e−|kziz| cos(ωt + arg(Vi0+))
ii(z,t) = R{Ii(z)ejωt} = |Yti||Vi0+||e−|kziz| cos(ωt + arg(Vi0+) ± 90◦)
ĐŽĂdž 0 z
Figure 5.4. Infinitely long waveguide with a source that excites a mode below cut-off
Also the modal admittance is imaginary, but its sign depends on the polarization: negative for T E modes and positive for T M . Fig. 5.4 shows clearly that the voltage and current waves are evanescent. Since the phase of the wave does not depend on z, phase velocity, group velocity and guided wavelength are not defined. The decay constant, i.e. the length necessary for the amplitude to reach the level 1/e of the starting value, is
0 2 4 6 8 10 12 14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Li 1/e
Figure 5.5. Decay constant of a mode below cut-off
Li = 1
|kzi|
and is shown in Fig. 5.5. We see also that voltage and current are in phase quadrature, hence no active power flow is associated to a purely forward evanescent wave. The situation is different when both forward and backward wave are present in the same region, as we will explain later. Nevertheless, an evanescent wave plays an important role in the energy budget, since it stores energy. Since the wave does not move, but remains attached to the source, it does not give rise to any power flow. The wave oscillates in time and exchanges its energy with the source twice per period.
If we look at the equations derived above, we see that the various modal parameters have two different frequency behaviors. In particular
kzi
k = vg
v =Zti0 Z =
s 1 −
µfci
f
¶2
and λg
λ = vph
v =Zti00
Z = 1
s 1 −
µfci f
¶2
The symbols λ, v, Z, denote the values of wavelength, phase velocity and wave impedance in the dielectric that fills the waveguide. The two behaviors are plotted in Fig. 5.6 It is evident that the
0 1 2 3 4 5
0 0.5 1 1.5 2
f/fc
kz/k=v
g/v=Z’
t/Z λg/λ=v
ph/v=Z’’
t/Z
Figure 5.6. Dispersion curves of various parameters of a mode above cut-off. The two vertical dash–
dotted lines indicate the standard operating band of a rectangular waveguide in the fundamental mode. Note that the first higher order mode goes above cut-off at f /fc= 2.
waveguide is very dispersive close to cut-off. However, it is not possible to use it in the frequency band where the curves are almost flat, since the guide is not single mode there. Actually, assuming that the curves refer to the fundamental mode of a rectangular waveguide, the vertical lines are the limits of the standard operating range. Indeed, the first higher order mode goes above cut-off at f /fc= 2.
In general a source excites all the modes of a waveguide, the amount of excitation depending on the geometry of the source. At a given frequency, a finite number of excited modes are above cut-off and, by carrying energy away from the source, is responsible for the radiation phenomenon.
The value of the radiation resistance of the dipole is related to them. At the same time, an infinite number of excited modes are below cut-off: their effect is important only in the neighborhood of the source, where they describe the reactive field. Moreover, evanescent modes are responsible for the imaginary part of the input impedance of the dipole.
A waveguide is said to be single mode if at the operation frequency only the fundamental mode is above cut-off. At microwave frequency, waveguides are essentially always single mode, in order to avoid interference effects between the various modes above cut-off.
The sources present in the waveguide can be represented in circuit terms by means of voltage and current generators to be inserted on the modal equivalent lines, as shown in Fig. 5.7. If the source is described by the distributions J(r), M(r), it can be shown that the corresponding generators are
vi(z) =< Mt,hi> + < Jz,Ztiezi >
ii(z) =< Jt,ei> + < Mz,Ytihzi> (5.11) The inner products, defined in (5.8), are integrals over the waveguide cross section, hence, the generators are distributed on the line since their strength is a function of z.