XV.
MICROWAVE
THEORY
E. F. Bolinder
A.
GEOMETRIC-ANALYTIC
THEORY OF NOISY TWO-PORTS
Sections XV-A, B, and C are extensions of the geometric-analytic theory of noisy
two-ports published in the Quarterly Progress Report of July 15, 1957, pages 163-169.
This report will use the notation that was given in the previous report.
tation of the geometric-analytic theory voltages and currents were used.
In the presenIn many micro-
wave applications, however, it is more convenient to use a wave representation.
We split the reflected wave at the output of the two-port network into two components
a2
a2n + IFcor
1
and Fco r is a complex correlation reflection coeffi-
where a 1 is the transmitted wave,
cient, defined by
a
2
aI
(2)
cor
a1
al
In Eq. 1, a2n and
co
a
are uncorrelated and thus
a2n al = 0
(3)
If we set
a
2
a
2
= kzf T 2
a2n a2n = kAf T2n
a1 a 1 = kAf T
where T 2 , T 2 n, and T
T2 = T 2 n +
1
cno
are noise temperatures,
2
T
Eq.
1 yields
(5)
1
The four-vector Q can now be written in the wave representation as
wl
1
Qz
w
Qw2
a
2
a
a
2
a1
w
w3
al a 2
w4
a
T2 n
T
+ ircor 2
cor
1
al
123
(XV.
MICROWAVE
THEORY)
The reflected and transmitted waves at the input of a noise-free two-port network
are expressed in the transmitted and reflected waves at the output by the equation
a'
A
B
a2
D
a
= TW
w
a1
w ww
(7)
; AD - BC = 1
By using the chain matrix, w *e obtain
b
T
w
21
11
T S
(8)
1
c
-1
d
1
For a noise process, we hav,
AA*
AB
BA
BB
a' a'
AC*
AD
BC
BD
a' a'
1 2
CA
CB
DA
DB
/
a a2
a 2 a1
1
Q21
w
= LwQ w
/
Saa
ala
CC
CD
DC
DD
"1"2
(9)
where
1
L
w
4
-1
1
1
-1
1
-1
1
aa
ab
ba
bb
1
-1
-1
ac
ad
bc
bd
1
1
-1
ca
cb
da
db*
1
1
cc
cd
dc
dd
/
1
1
-1
1
-1
1
1
-1
-!
1/
(10)
L
*
w
is the product of the three Kronecker products S X S, T X T , and S
-1
xS
-1
If we set
V
-
(a
+ a2
)
a2
(V-I)
(11)
I
V2 (a
a)
a,
v-a22(V+I)
we obtain
124
(XV.
PZ1
-
Z
P
PZ3 PZ4
aa
I) =-a
-V
(VI
PZ2 -
a 2 a 2 -a
+ VI) = -
2(VI
2
l
-
II)
2(VV
+ II*)
a2 a
=
P
- ala2
2(VVI
2a
(VV
2(2
MICROWAVE THEORY)
12
+ aa
=
2 a a2 + ala
Pr
12
(12)
1
1
Thus, for bilateral, two-port networks, Eq. 9 can be interpreted geometrically as a
movement in a Poincare model of the three-dimensional hyperbolic space that has the
complex reflection-coefficient
plane for its absolute surface.
The corresponding
Cayley-Klein model, which has the unit sphere as the absolute surface,
is obtained from
the Cayley-Klein model of the last report by a 90' rotation around the y-axis.
E. F.
B.
Bolinder
CASCADING OF NOISY TWO-PORT NETWORKS
With the notation of Fig. XV-1,on a voltage-current basis, we write
S"t = o
qi0
+
+ T
(1)
Similarly, on a power basis, we obtain
=O + Tx T Q
Q," = QQ'
If we insert in Eq. 2 the Q-vectors expressed in the equivalent noise resistance
rn, the equivalent noise conductance gn, and the complex correlation impedance
Z cor
o r (1),
r
n
+
cor
n
gn
cor
gn
cor
Q = 4kAf
gn
VO
Fig. XV-1.
125
Noisy two-port network.
(XV.
MICROWAVE
THEORY)
we obtain
g,
o
,
(4)
o Z 0cor + g n Z' cor
Zn
Z"
cor
0
n
From Eqs.
g'n
Z'cor - Zoco
o
n'
go
gn
o
r" = gr + r' +
n
n
n
n
2 and 3 we also obtain
cc
r n + Ic Z cor + d2g n
ad - bc
2
Z'
g
r
n
n
cc
r
ac
r
n
+ Ic Z
cor
(5)
n
+ d2
g
+ (a Z
n
cor + b) (c
cc rn + c Zo
n
Z cor + d)
+ d
2
gn
g
Equations 4 and 5 are formulas obtained by Dahlke (2).
In Dahlke' s formulas all
=
d,
a12 = c, a21 = b, and a 2 2 = a.
E.
F.
Bolinder
References
1.
H. Rothe, W. Dahlke, Proc. IRE 44, 811-818 (1956); AEU 9, 117-121 (1955);
cf. E. F. Bolinder, Quarterly Progress Report, July 15, 1957, p. 163.
2.
W.
C.
Dahlke, AEU 9,
391-401 (1955).
GEOMETRIC-ANALYTIC
THEORY OF PARTLY POLARIZED ELECTRO-
MAGNETIC WAVES
The geometric-analytic
theory of noisy two-ports is
partly polarized electromagnetic
waves,
converted into a theory of
if the voltages and currents
are exchanged
for two complex electric field-strength quantities E
and E x, perpendicular in space
y
to each other and to the direction of propagation of the wave.
The complex correlation impedance
Zco r
is replaced by a complex correlation polarization ratio Pcor,
where
126
(XV.
MICROWAVE
THEORY)
EE
cor
EE
x
x
For a polarized wave (beam),
E
y
p =E
x
(Some authors (1) use -jp instead of p.)
The mapping of the complex polarization ratio
p on the Riemann unit sphere was introduced into optics by Poincare (2),
the "Poincare
sphere," and into antenna theory by Deschamps (3).
The transformations of elliptically polarized waves through lossless transducers
form a unitary group.
E'
Therefore,
a-c
' =
E
2
=
le
E'
x
= Te
ee
c
a
E
;
a
2
+
c
= 1
(3)
x
In the p-plane
p? = ap - c
(4)
cp + a
A simple example of a lossless p-transformation (a = (/6/3)e j
p = 0,
and p'
= (V/2)e
j2 10
60
, c = (3/3)e
) is performed by the isometric circle method (4) and shown
r
Fig. XV-2.
j 30 °
Example of a lossless polarization-ratio transformation.
127
(XV.
MICROWAVE
in Fig. XV-2.
THEORY)
The isometric circles intersect, so that the transformation is elliptic.
The fixed points of the transformation (marked by crosses) fall, by a stereographic
mapping, on a diameter of the unit sphere.
of the sphere through an angle 24
eigenvalues of the matrix T
.
The transformation consists of a rotation
around the diameter, where
4
is given by one of the
Thus we have
e
e
a + a') ±
=-
a + a2
(5)
- 4
An elegant and convenient tool to use in connection with unitary groups is Hamilton' s
Partly polarized waves can be represented by the Qe- or
theory of quaternions (5).
Pe-power four-vectors
Points,
(Stokes vectors).
representing power ratios,
are
obtained both in a three-dimensional space that is formed by an extension of the
polarization-ratio
plane,
and inside the unit sphere.
An unpolarized wave (natural
light) is represented by the center of the sphere or by the point (0, 0, 1) in the Qe-space.
Example of a lossless power-ratio
transformation in the Qe -space.
Fig. XV-3.
A--t
Figure XV-3 shows a cross section through A-A of Fig. XV-2 extended to three
dimensions.
3
(6-/4,
The point (
, Te,
e) = (/-,
0,
2) is transformed into ( e' r',
/2-/4, 1/2) by a rotation through the angle
ered as a non-Euclidean rotation around the point F,
plane and a semicircle through the fixed points.
e -e
-
0e)
4. This rotation may be considthe crossover point between the
The semicircle is orthogonal to the
For a unitary transformation, the semicircle passes through (0, 0, 1).
E. F. Bolinder
plane.
References
i. V. H. Rumsey, Proc. IRE 39, 535-540 (1951).
2.
H. Poincare, Theorie Mathematique de la Lumiere (G.
Vol. II, 1892).
3.
G. A. Deschamps,
4.
E. F. Bolinder, Quarterly Progress Report, Research Laboratory of Electronics,
M.I.T., April 15, 1956, pp. 123-126.
5.
W.
R. Hamilton,
Proc. IRE 39,
Carre,
Paris, Vol. I,
1889;
540-544 (1951).
Lectures on Quaternions (Hodges and Smith, Dublin,
128
1853).