Download EXCITATION OF WAVEGUIDES

EXCITATION OF WAVEGUIDES-ELECTRIC AND MAGNETIC
CURRENTS
So far we have considered the propagation, reflection, and transmission of guided waves
in the absence of sources, but obviously the waveguide or transmission line must be
coupled to a generator or some other source of power. For TEM or quasi-TEM lines,
there is usually only one propagating mode that can be excited by a given source,
although there may be reactance (stored energy) associated with a given feed. In the
waveguide case, it may be possible for several propagating modes to be excited, along
with evanescent modes that store energy. In this section we will develop a formalism for
determining the excitation of a given waveguide mode due to an arbitrary electric or
magnetic current source. This theory can then be used impedance of probe and loop feeds
and, in the next , to determine the excitation of waveguides by apertures.
Current Sheets That Excite Only One Waveguide Mode
Consider an infinitely long rectangular waveguid with a transverse sheet of electric
surface current density at z = 0, as shown in Figure 5.28. First assume that this current
has x̂ and ŷ components given as
J sTE ( x, y)   x̂
2A mn n
2A mn m
mx
nx
mx
nx
cos
sin
 ŷ
sin
cos
b
a
b
a
a
b
We will show that such a current excites a TEmn waveguide mode traveling away from
the current source in both the +z and -z directions.
From Table 4.2, the transverse fields for positive and negative traveling TEmn waveguide
modes can be written as
mx
nx  jz
 n 
E x  Z TE  A mn cos
sin
e
a
b
 b 
mx
nx  jz
 m  
E y   Z TE 
cos
e
A mn sin
a
b
 a 
mx
ny  jz
 m  
H x  
cos
e
A mn sin
a
b
 a 
mx
nx  jz
 n 
H y   A mn cos
sin
e
a
b
 b 
where the  notation refers to waves traveling in the z+ direction or –z direction with
amplitude coefficients A mn and A mn respectively.
From (2.36) and (2.37), the following boundary conditions must be satisfied at z = 0
E



 E   ẑ  0

ẑ  H   H   J s
Equation (5.120a) states that the transverse components of the electric field must be
continuous at z = 0, which when applied to (5.119a) and (5.119b) gives
A mn = A mn
Equation (5.120b) states that the discontinuity in the transverse magnetic field is equal to
the electric surface current density. Thus, the surface current density at z=0 must be

 
J S  ŷ H x  H x  x̂ H y  H y
=  x̂

2A mn n
2A mn m
mx
nx
mx
ny
cos
sin
 ŷ
sin
cos
,
b
a
b
a
a
b
where (5.121) was used. This current is seen to be the same as the current of (5.118)
which shows, by the uniqueness theorem, that such a current will excite only the TEmn
FIGURE 5.28 An infinitely long rectangular waveguide with surface current densities at
z = 0.
mode propagating in each direction, since Maxwell's equations and all boundary
conditions are satisfied.
The analogous electric current that excites only the. TMmn mode can be shown to be
J
TM
s
2B mn m
2B mn n
mx
ny
mx
ny
( x, y)   x̂
cos
sin
 ŷ
sin
cos
a
a
b
b
a
b
It is left as a problem to verify that this current excites TMmn modes that satisfy the
appropriate boundary conditions.
Similar results can be derived for magnetic surface current sheets. From (2.36) and (2.37)
the appropriate boundary conditions are
E


 E   ẑ  M S


ẑ  H   H   0
For a magnetic current sheet at z = 0, the TEmn waveguide mode fields of (5.119) must
now have continuous Hx and Hy field components, due to (5. 124b). This results in the
condition that
A mn = -A mn
Then applying (5.124a) gives the source current as
M
TE
s
 x̂ 2Z TE A mn n
2Z TE A mn n
mx
ny
mx
ny

sin
cos
 ŷ
cos
sin
a
a
b
b
a
b
The corresponding magnetic surface current that excites only the TMmn mode can be
shown to be
M sTM 
x̂ 2B mn n
mx
ny ŷ2B mn n
mx
ny
sin
cos

cos
sin
b
a
b
a
a
b
These results show that a single waveguide mode can be selectively excited, to the
exclusion of all other modes, by either an electric or magnetic current sheet of the
appropriate form. In practice, however, such currents are very difficult to generate, and
are usually only approximated with one or two probes or loops. In this case many modes
may be excited, but usually most of these modes are evanescent.
Mode Excitation from an Arbitrary Electric or Magnetic Current Source
We now consider the excitation of waveguide modes by an arbitrary electric or magnetic
current source [3]. With reference to Figure 5.29, first consider an electric current source
J located between two transverse planes at zl and z2, which generates the fields E  , H 
traveling in the +z direction, and the fields E  , H  traveling in the –z direction. These
fields can be expressed in terms of the waveguide modes as follows:
E    A n E n   A n e n  ẑe zn e  jn z ,
n
H    A n H n   A n h n  ẑh zn e  jn z ,
n
z  z2
n
E    A n E n   A n e n  ẑe zn e  jn z ,
n
z  z1
n
H    A n H n   A n  h n  ẑh zn e  jn z ,
n
z  z2
n
n
z  z1
where the single index n is used to represent any possible TE or TM mode. For a given
current 1, we can determine the unknown amplitude A n by using the Lorentz reciprocity
theorem of (2.173) with M1  M 2  0 (since here we are only considering an electric
current source),
 E
s
1
 H 2  E 2  H1  ds   E 2  J1  E1  J 2 dv,
V
where S is a closed surface enclosing the volume V, and * are the fields due to the current
source J i (for i = l or 2).
To apply the reciprocity theorem to the present problem, we let the volume V be the
region between the waveguide walls and the transverse cross-section planes at zl and z2.
Then let E1  E  and H1  H  depending on whether z  z2 or z  z1 and let E 2 , H 2 be
the nth waveguide mode traveling in the negative z direction:
E 2  E n  en  ẑe zn e jn z
H 2  H n   h n  ẑh zn e jn z
FIGURE 5.29 An arbitrary electric or magnetic current source in an infinitely long
waveguide.
Substitution into the above fonn of the reciprocity theorem gives, with * and
 E

s

 H n  E n  H   ds   E n  Jdv,
V
The portion of the surface integral over the waveguide walls vanishes because the
tangential electric field is zero there; that is, E  H  ẑ  H  ẑ  E   0 on the waveguide
walls. This reduces the integration to the guide cross-section, So, at the planes Zl and Z2.
In addition, the waveguide modes are orthogonal over the guide cross-section:

S0
E m H n .d s   e m  ẑe zn    h n  ẑh zn   ẑds
S0
=   e m  h n  ẑds  0,
for m  n
S0
Using (5.128) and (5.130) then reduces (5.129) to




A n  E n  H n  E n  H n  ds A n  E n  H n  E n  H n  ds,
z2
z1
  E n  Jdv
V
Since the second integral vanishes, this further reduces to
A n 
z2
e
n

 ẑe zn    h n  ẑh zn   e n  ẑe zn   h n  ẑh zn   ẑds
 2A n  e n  h n  ẑds   E n  Jdv,
z2
or
A n 
1
1
e n  ẑe zn   Je jn z dv,
E n  Jdv 


V
V
Pn
Pn
where
Pn  2 e n  h n  ẑds.
S0
is a normalization constant proportional to the power flow of the nth mode.
V
By repeating the above procedure with E 2  E n and H 2  H n , the amplitude of the
negatively traveling waves can be derived as
A n 
1
1
e n  ẑe zn   Je  jn z dv,
E n  Jdv 


V
V
Pn
Pn
The above results are quite general, being applicable to any type of waveguide (including
planar lines such as stripline and microstrip), where modal fields can be defined.
Example 5.11 applies this theory to the problem of a probe-fed rectangular waveguide.
EXAMPLE 5.11
For the probe-fed rectangular waveguide shown in Figure 5.30, determine the amplitudes
of the forward and backward traveling TE10 modes, and the input resistance seen by the
probe. Assume that the TE10 mode is the only propagating mode.
Solution
If the current probe is assumed to have an infinitesimal diameter, the source
volume current density J can be written as
a

J x, y, z   I 0  x  (z) ŷ,
2

for 0  y  b
From Chapter 4 the TE10 modal fields can be written as
x
,
a
 x̂
x
h1 
sin
Z1
a
e1  ŷ sin
where Z1  k 0 0 / 1 is the TE10 wave impedance. From (5.132) the normalization
constant P1 is,
P1 
2
Z1
a
 
b
x 0 y 0
sin 2
x
ab
dxdy 
a
Z1
Then from (5.131) the amplitude A 1 is
A1 
 I 0 b  Z1I 0
1
x j1z 
a
sin
e
I

x


(
z
)
dxdydz




0
P1 V
a
2
P1
a

Similarly,
A1 
 Z1 I 0
a
FIGURE 5.30 A uniform current probe in a rectangular waveguide.
If the TE10 mode is the only propagating mode in the waveguide, then this mode carries
all of the average power, which can be calculated for real Z1 as
P
1
1
E   H *  d s   E   H * .d s

S
2 0
2 S0
=  E   H *  d s
S0
If the input resistance seen looking into the probe is Rin, and the terminal current is I0,
then P  I 02 R in / 2 , so that the input resistance is

2P ab A1
R in  2  2
I0
I 0 Z1
2

bZ1
a
which is real for real Z1 (corresponding to a propagating TE10 mode).
A similar derivation can be carried out for a magnetic current source M .This source will
also generate positively and negatively traveling waves which can be expressed as a
superposition of waveguide modes, as in (5.128). For J1  J 2  0 , the reciprocity
theorem of (2.173) reduces to
 E
s
1
 H 2  E 2  H1  ds   H1  M 2  H 2  M1 dv,
V
By following the same procedure as for the electric current case, the excitation
coefficients of the nth waveguide mode can be derived as
A n 
1
1
H n  Mdv 

Pn V
Pn
  h
A n 
1
Pn
 h

V
H n  Mdv 
1
Pn
V
V
n
n
 ẑh zn   Me jn z dv,
 ẑh zn   Me  jn z dv,
where Pn is defined in (5.132).
EXAMPLE 5.12
Find the excitation coefficient of the forward traveling TE10 mode generated by the loop
in the end wall of the waveguide shown in Figure 5.31a.
Solution
By image theory, the half-loop of current 10 on the end wall of the waveguide can be
replaced by a full loop of current 10, without the end wall, as shown in
FIGURE 5.31 Application of image theory to a loop in the end wall of a
rectangularwaveguide. (a) Original geometry. (b) Using image theory to replace the
end wall with the image of the half-loop.
Figure 5.31b. Assuming that the current loop is very small, it is equivalent to a magnetic
dipole moment,
a 
b

Pm  x̂I 0 r02  x   y  z 
2 
2

Now since   E   jB  M   j 0 H  j 0 Pm  M , a magnetic polarization
current P m can be related to an equivalent magnetic current density M as
M  j 0 Pm
Thus, the loop can be represented as a magnetic current density:
a 
b

M  x̂j 0 I 0 r02  x   y  z V / m 2
2 
2

If we define the modal h 1 field as
h1 
 x̂
x
sin
,
Z1
a
then (5.135) gives the forward wave excitation coefficient A 1 as
1
A 
P1

1
jk 0 0 I 0 r02
V  h1  Mdv  ab