(
jω
)|
=
|S21(
jω
)S33
(
jω ) − S23( |S33( jω )|
jω
)S31(
jω
)|
=
|S12 ( |S33 (
jω jω
)| )|
=
S12 ( jω ) S33( jω )
|S22a
(
jω
)|
=
|S22 (
jω )S33(
jω ) − S23( |S33( jω )|
jω
)S32 (
1) General configuration of the negative-resistance amplifiers Our task is to obtain the transducer power-gain from port 1 to port 2 in terms of scattering
Since Z11( jω ) , Z22 ( jω ) and Z12 ( jω ) are images, Z (− jω ) = Z ( jω ) = −Z ( jω ) .
Using the theory of analytic continuation, Z ( − s) = −Z (s) .
65
R2
2 2
Nβ
②
R1
+
1 1①
Vg
−
Nc
③
Na N
Nα
3
zl (s)
②
R1
+
1①
Nc
Vg
−
③
1
①
Nβ ② 3 2
R2
Nd
2
① Nα ② 4 3
zl (s)
N
⎡0 1 0⎤
Sc = ⎢⎢0 ⎢⎣1
0 0
1⎥⎥ 0⎥⎦
=
⎡ S11c
⎢ ⎣
S21c
S12c S22c
⎤ ⎥ ⎦
:
normalizing
The transducer power-gain of amplifier is
G(ω2 ) = |S21a ( jω )|2
=
S12 ( jω ) 2 S33( jω )
i)
Since
|S12 ( jω )| ≤ 1,
G(ω2 )
≤
1 |S33( jω )|2
.
ii) the optimum amplifier should have a maximum |S12 ( jω )| and a minimum |S33( jω )| .
output port by a negative resistor with resistance −RΩ belongs to the special class.
zl (s)
Lossless two-port
−R
Proof:
A. Since
zl (s)
=
Z11 (s )
−
Z122 (s) Z22 (s) −
compute the scattering matrix of two-port Na formed by three-port N and one-port Nb
using following formulas which are derived in chapter 1 from the interconnection of two
+
R RCs +1
(b)
C
−R
−zl (
−
s)
=
R RCs +1
N
(c)
−R
−zl ( − s) = R
N
(d)
− R1
zl
(s)
=
− R1
+
R RCs
−1
C N
−R
− zl
(
−
s)
=
R1
+
R RCs +1
(e)
4.2 The design of nonreciprocal negative resistance amplifier
R
=
Z11(s)Z22 (s) − Z122 (s) Z22 (s) − R
−
Z11 (s )R
Y22 (s)
=
Z11 (s) Z11(s)Z22 (s) − Z122 (s)
so
1 −R
zl
(s)
=
Z11 (s)
Y22 (s) Z22 (s)
−
R
.
B. Obtain associated impedance −zl ( − s)
2) The design of nonreciprocal negative resistance amplifier A. Circuit of nonreciprocal negative resistance amplifier
The lossless three-port network N consists of three parts: i) Lossless two-port network Nα ; ii) Lossless two-port network Nβ ; iii) Circulator Nc . B. The relations between S and Sα , Sβ , Sc .
S12S23 − S13S22 ⎤
S13S21
−
S11S23
⎥ ⎥
S11S22 − S12S21 ⎥⎦
Then, there exist following relations:
S22 ( jω )| S( jω )| = S11( jω )S33( jω ) − S13( jω )S31( jω ) S21( jω )| S( jω )| = S13( jω )S32 ( jω ) − S12 ( jω )S33( jω)
=
|S22 ( |S33 (
jω )| jω )|
=
S22 ( jω ) S33( jω )
|S12a
(
jω
)|
=
|S12
(
jω
)S33
(
jω ) − S13( |S33( jω )|
jω
)S32
(
jω
)|
=
|S21( |S33 (
jω jω
)| )|
=
S21( jω ) S33( jω )
|S21a
that are active over a frequency band of interest and such that the function defined by the relation
z3 (s) = −zl ( − s)
is a strictly passive impedance function. 2) Any active impedance which is formed by a lossless two-port network terminated at the
3) Examples of special class of active impedance.
Rs
Ls
zl
(s)
=
Rs
+
Ls s
+
R RCs −1
Rs C N
−R
− zl
(
−
s)
=
− Rs
+
Ls s
+
R RCs
+1
(a)
Ls Rs C
−R
zl
(s)
=
Ls s
+
R RCs
−1
N
− zl
(
−
s)
=
Ls s
⎡S11 ⎢⎢S21 ⎢⎣S31
S12 S22 S32
S13 ⎤H
S23
⎥ ⎥
=
⎢⎡⎢SS1121
S33 ⎥⎦ ⎢⎣S13
S21 S22 S23
S31 S32 S33
⎤ ⎥ ⎥ ⎥⎦
=
|
1 S
|
⎡S22S33
⎢ ⎢
S23S31
⎣⎢S21S32
− − −
S23S32 S21S33 S22S31
S13S32 − S12S33 S11S33 − S13S31 S12S31 − S11S32
1 −R
1 +R
z3
(s)
=
−
zl
(
−
s)
=
−[Z11
(
−
s)
Y22 ( Z22 (
− −
s) s)
−
R
]
=
Z11
(s
)
Y22 (s) Z22 (s)
+
R
z3 (s)
Lossless two-port
R
62
−zl ( − s) is the driving-point impedance of the same two-port terminated at the output port in a passive resistor of resistance RΩ . Thus, z3 (s) = −zl ( − s) is a strictly passive impedance and called associated passive impedance of zl (s) .
jω )|
=
|S11( |S33 (