宽带匹配网络.

S12S33 − S13S32 ⎤
S 22 S33
−
S 23 S32
⎥ ⎦
C. The relations of scattering parameters of Na and N
|S11a
(
jω )|
=
|S11(
jω )S33(
jω ) − S13( |S33( jω )|
jω )S31(
jω
)|
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) .
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
S13 S23 S33
⎤ ⎥ ⎥ ⎥⎦
=
⎡ S11
⎢ ⎣
S21
S12 ⎤
S22
⎥ ⎦
Then,
[ ] Sa
=
⎡ S11
⎢ ⎣
S21
S12 S22
⎤ ⎥ ⎦
+
⎡ ⎢ ⎣
S13 S23
⎤ ⎥ ⎦
(1
−
ρ
S33
)−1
ρ
S31
S32
=
1 S33
⎡ S11S33
⎢ ⎣
S21S33
Fra Baidu bibliotek− −
S13S31 S23S31
multi-port networks in any way.
S11
=
S11a
+
S12a (Uk
−
S11b
S22a
)
S −1 11b
S 21a
S12 = S12a (Uk − S11b S22a )−1S12b
S21 = S21b (Uk − S22a S11b )−1S21a
S22 = S22b + S21b (Uk − S22a S11b )−1S22a S12b
jω )|
=
|S11( |S33 (
jω )| jω )|
=
S11( jω ) S33( jω )
Therefore, the scattering parameters of negative-resistance amplifier can be computed if
the scattering parameters of lossless three-port N , normalizing to the z1(s) , z2 (s) and z3 (s) = −zl ( − s) , are known.
to
R1 ,
Rˆ 2
and
Rˆ3 .
Sα : normalizing to Rˆ3 and z3 (s) = −zl ( − s) . Sβ : normalizing to Rˆ 2 and R2 .
Nd is combined by Nα and Nβ . Its scattering matrix is
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)
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
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 two-port Na is formed by lossless three-port N interconnected with one-port
active impedance Nb .
Since scattering matrix is defined with strictly passive reference impedance, we should
S12 ( jω)| S( jω)| = S23( jω)S31( jω) − S21( jω)S33( jω)
S11( jω)| S( jω)| = S22 ( jω)S33( jω) − S23( jω )S32 ( jω)
B. The scattering matrix of two-port Na
h∗−1 (s)
=
h(s)
zl
(s)
+ zl 0
(
−
s)
h∗−1 (s)
=
∞
64
is the reflection coefficient of Nb normalizing to z3 ( − s) . For three-port N ,
⎡ S11
S
=
⎢ ⎢
S
21
⎢⎣ S31
S12 S22 S32
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)
=
|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
parameters of network.
63
z1 (s)
+
①
Vg
−
Lossless
three-port
③
zl (s)
N
z2 (s)
②
Nb
Two-port Na
A. The scattering matrix of lossless three-port is unitary, that is, S H ( jω ) = S −1( jω )
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) .
(
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 (
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 .
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) −
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
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 )| .
Here, Nb is a one-port network, S12b = S21b = S22b = 0 . Then, the scattering matrix of
Na is
Sa = S11 + S12 (1− Sb S22 )−1Sb S21
where
Sb
=
ρ
=
h(s)
zl (s) − z3 (s) zl (s) + z3 ( − s)
Chapter 4 Design of Broadband Matching Network: The Active Load
4.1 Special class of active impedances 1) The special class of active impedances considered here is the class of impedances zl (s)
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
+
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