宽带匹配网络

Rs Ls Rs N C
−R
zl (s ) = Rs + Ls s +
R RCs − 1
R RCs + 1
− zl ( − s ) = − Rs + Ls s +
(a)
Ls Rs N C
−R
R RCs − 1 R − zl ( − s ) = Ls s + RCs + 1 zl (s ) = Ls s +
⎡ S11 S=⎢ ⎢ S 21 ⎢ ⎣ S31
Then,
S12 S 22 S32
S13 ⎤ ⎡ S11 = S 23 ⎥ ⎥ ⎢S ⎣ 21 S33 ⎥ ⎦
S12 ⎤ S22 ⎥ ⎦
⎡S Sa = ⎢ 11 ⎣ S 21 =
Hale Waihona Puke S12 ⎤ ⎡ S13 ⎤ + ⎢ ⎥ (1 − ρ S33 ) −1 ρ [ S31 ⎥ S 22 ⎦ ⎣ S23 ⎦ S12 S33 − S13 S32 ⎤ S 22 S33 − S 23 S32 ⎥ ⎦
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 ) that are active over a frequency band of interest and such that the function defined by the relation
S11 = S11a + S12 a (U k − S11b S22 a ) −1 S11b S21a S12 = S12 a (U k − S11b S22 a ) −1 S12b S21 = S 21b (U k − S22 a S11b ) −1 S21a S22 = S22b + S 21b (U k − S22 a S11b ) −1 S22 a S12b
z3 (s ) = − zl ( − s )
2) is a strictly passive impedance function. Any active impedance which is formed by a lossless two-port network terminated at the output port by a negative resistor with resistance − RΩ belongs to the special class.
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 parameters of network.
zl (s ) − z3 (s ) −1 z (s ) + zl ( − s ) −1 h∗ (s ) = h(s ) l h∗ (s ) = ∞ zl (s ) + z3 ( − s ) 0
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is the reflection coefficient of N b normalizing to z3 ( − s ) . For three-port N ,
1 1 −R +R Y22 ( − s ) Y22 (s ) z3 (s ) = − zl ( − s ) = −[Z11 ( − s ) ] = Z11 (s ) Z 22 ( − s ) − R Z 22 (s ) + R
z3 (s)
Lossless two-port
R
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− zl ( − s ) is the driving-point impedance of the same two-port terminated at the output
Here, N b is a one-port network, S12b = S 21b = S 22b = 0 . Then, the scattering matrix of
N a is Sa = S11 + S12 (1 − Sb S 22 ) −1 Sb S21
where
Sb = ρ = h(s )
H
−1
⎡ S11 S12 S13 ⎤ ⎡ S11 S21 S31 ⎤ ⎡S22 S33 − S23S32 S13S32 − S12 S33 S12 S23 − S13S22 ⎤ ⎢S S S ⎥ = ⎢S S S ⎥ = 1 ⎢ S S − S S ⎥ ⎢ 21 22 23 ⎥ ⎢ 12 22 32 ⎥ | S | ⎢ 23 31 21 33 S11S33 − S13S31 S13S21 − S11S23 ⎥ ⎥ ⎢ ⎢ S21S32 − S22 S31 S12 S31 − S11S32 S11S22 − S12 S21 ⎥ ⎣S31 S32 S33 ⎥ ⎦ ⎢ ⎣ ⎦ ⎣S13 S23 S33 ⎦
|S12 a ( jω )| = |S 21a ( jω )| = |S 22 a ( jω )| =
|S12 ( jω )S33 ( jω ) − S13 ( jω )S32 ( jω )| |S 21 ( jω )| S 21 ( jω ) = = |S33 ( jω )| |S33 ( jω )| S33 ( jω ) |S 21 ( jω )S33 ( jω ) − S23 ( jω )S31 ( jω )| |S12 ( jω )| S12 ( jω ) = = |S33 ( jω )| |S33 ( jω )| S33 ( jω ) |S 22 ( jω )S33 ( jω ) − S23 ( jω )S32 ( jω )| |S11 ( jω )| S11 ( jω ) = = |S33 ( jω )| |S33 ( jω )| S33 ( jω )
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z1 (s ) Vg + −
①
Lossless three-port N
③
zl (s)
z2 (s )
②
Nb
Two-port N a
A. The scattering matrix of lossless three-port is unitary, that is, S ( jω ) = S ( 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
Then, there exist following relations:
H
S 22 ( jω )| S ( jω )| = S11 ( jω )S33 ( jω ) − S13 ( jω )S31 ( jω )
S 21 ( jω )| S ( jω )| = S13 ( jω )S32 ( jω ) − S12 ( jω )S33 ( jω ) S12 ( jω )| S ( jω )| = S 23 ( jω )S31 ( jω ) − S21 ( jω )S33 ( jω ) S11 ( jω )| S ( jω )| = S22 ( jω )S33 ( jω ) − S23 ( jω )S32 ( jω )
2
1 . |S33 ( jω )|2
the optimum amplifier should have a maximum |S12 ( jω )| and a minimum |S33 ( jω )| .
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) ii) Lossless two-port network Nα ; Lossless two-port network N β ;
(b)
C N
−R
− zl ( − s ) =
R RCs + 1
(c)
−R
− zl ( − s ) = R
N
(d)
− R1
C
−R
N
R RCs − 1 R − zl ( − s ) = R1 + RCs + 1 zl (s ) = − R1 +
(e)
4.2 The design of nonreciprocal negative resistance amplifier
zl (s )
Lossless two-port
−R
Proof: A. Since
2 2 Z12 (s ) Z11 (s )Z 22 (s ) − Z12 (s ) − Z11 (s )R zl (s ) = Z11 (s ) − = Z 22 (s ) − R Z 22 (s ) − R
Y22 (s ) =
B. The scattering matrix of two-port N a The two-port N a is formed by lossless three-port N interconnected with one-port active impedance N b . Since scattering matrix is defined with strictly passive reference impedance, we should compute the scattering matrix of two-port N a formed by three-port N and one-port N b using following formulas which are derived in chapter 1 from the interconnection of two multi-port networks in any way.