TWO-PORT NETWORKS INCLUDING
SCATTERING P ARAMETERS
• Conventional Two-Port Network Parameters.
Several two ports, sources, and loads in different parameter systems with
corresponding quantities indicated.
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To demonstrate the expression forms independent of the particular two-port
parameters, two derivations are carried out.
E1 E1
Zin = Zin =
I1 I1
E1 = z11I1 + z12I 2 E1 = h11I1 + h12 E2
E1 I E1 E
= z11 + 2 z12 = h11 + 2 h12
I1 I1 I1 I1
E2 = − I 2 Z L = I1z12 + I 2 z22 I 2 = − E2YL = I1h21 + E2 h22
I2 − z21 E2 − h21
= =
I1 z22 + Z L I1 h22 + YL
z12 z21 h12h21
Zin = z11 − Zin = h11 −
z22 + Z L h22 + YL
A similar derivation using the generalized k parameters gives
k12k 21
M in = k11 −
k 22 + M L
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Transducer (Power) Gain GT.
power delivered to the load P
GT = = o
power available from the source Pavs
Available (Power) Gain, GA.
power available at the output Pavo
GA = =
power available from the source Pavs
Po
Pav s Pavo Power gain = G =
Pi
Z s or Ys
Two- ZL
Port or Po
Es E1 E2
YL Transducer gain = GT =
Network Pavs
Pi Po
Pavo
Power gains of a two port. Available gain = GA =
Pavs
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Tabulation of Functions in Terms of General Parameters
And Variables
Independent input variable . . . . . . . . . . . . . . . . . . . Uii
Independent output variable . . . . . . . . . . . . . . . . . . Uoi
Source Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . Us
Source immittance . . . . . . . . . . . . . . . . . . . . . . . . . M s
Input immittance . . . . . . . . . . . . . . . . . . . . . . . . . . M in
Dependent input variable . . . . . . . . . . . . . . . . . . . Uid
Dependent output variable . . . . . . . . . . . . . . . . . . Uod
Load variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . UL
Load immittance . . . . . . . . . . . . . . . . . . . . . . . . . ML
Output immittance . . . . . . . . . . . . . . . . . . . . . . . . Mo
Defining equations: U id = Uii k11 + U oi k12
U od = Uii k21 + U oik 22
Two-port parameters: k11k12
k21k22
k12k21 k12k 21
M in = k11 − M o = k22 −
k22 + M L k11 + M s
Amplification 1
U od k21M L k M
= = 21 L
U s (k11 + M s )(k22 + M L ) − k12k 21 D
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General Parameters (continues)
Amplificat ion 2 :
U oi − k21 − k21
= =
U s (k11 + M s )(k 22 + M L ) − k12k21 D
Power gains :
2
M Lr k21
G=
Re[k11 − k12k21 (k 22 + M L )] k22 + M L
2
2
k21 M sr
GA =
Re[(k11k22 − k12k 21 + k22 M s )(k11 + M s )]
2
4 k21 M Lr M sr
GT =
(k11 + M s )(k22 + M L ) − k12k 21 2
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Sensitivit ies of U oi / U s or U od / U s to changes in k ' s :
− k11 (k22 + M L )
Sk11 =
D
k k
Sk12 = 12 21
D
(k + M s )(k22 + M L )
Sk 21 = 11
D
− k22 (k11 + M s )
Sk22 =
D
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Stability Conditions
For a two-port that is not potentially unstable the following inequalities must be
satisfied:
Po P
i) = GT ≤ G = o
Pavs P1
ii ) GT = Gmax
Po Pav
= GT ≤ GA = o
Pavs Pavs
Transducer gain only equals the available gain when the output port is conjugate-
matched. It can be proved that the condition for potentially instability of the ports
with real parameters is
h12h21 ≥ h11h22
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Stability Summary of Results.
Po o
is within 3dB of the maximum available gain unless the port is potentially unstable
Pi o
at the frequency in question.
Poo
The port is potentially unstable if is negative or if the critical factor C exceed unity
Pio
2 P o o h12
C=
Pi o h21
When C is less than 1, the maximum available gain is
(
KG P o o Po o 1 − 1 − C 2
=
)
Pi o Pi o C2
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Where
2
Poo h21
=
Pi o 4h11r h22r − 2 Re(h12h21 )
θ = arg(− h12h21 ) = − arg(− h12h21 )
*
*
h h CK G exp jθ
Z s,opt = Zin
*
= h11 − 12 21 1 −
2h22r 2
2 h22r
YL ,opt = Yo* = − h22 +
1 − CK G exp jθ / 2
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Scattering Parameters.
For very high-frequencies where short- or open-circuit conditions are not possible,
an alternative method of representing a linear time-invariant network is by means
of its scattering parameters, also known as S-Parameters.
One-Port Example
Rs i Incident voltage vi
+ Transmission
vs +- v line with Reflected voltage vr
ZL
- Zo characterist
impedance.
v = vi + vr
i = ii − ir
i = (vi − vr )
1
Zo
Thus
vr = (v − Z oi )
1
2
1
vi = (v + Zoi )
2
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The reflection coefficient is defined as :
v v − Z oi
ρ= r =
vi v + Z oi
Instant power p( t ) is given by
p( t ) = v( t )i( t ) =
1
Zo
(
(vi + vr )(vi − vr ) = 1 vi2 − vr2
Zo
)
maximizing p( t ) implies minimizing vr and ρ
The scattering parameter of the one - port is defined as :
vr ir v − Ri
S= = =
v i ii v + Ri
for the (lumped) case of Zo becoming equal to R. Furthermore, the open circuit
impedance is related as
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Z oc − R 1 − Rysc
S= =
Z oc + R 1 + Rysc
Example
Rs=1Ω C
1 1
Zoc = +
sC 1 1
+- vc +
R sL
L R
s2 RLC + sL + R
Zoc Zoc = 2
s LC + sRC
Z oc − Rs s 2 LC (R − Rs ) + s( L − RRsC ) + R
S= =
Z oc + Rs s2 LC (R + Rs ) + s( L + RRs C ) + R
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Let us consider a two-port network
a1 b2
Zs a1 = Vi1 Zo
Two-Port
+ V
- s b1
1
Network
2
a2
ZL
b1 = Vr1 Zo
a 2 = Vi 2 Zo
b2 = Vr 2 Zo
b1 S11 S12 a1
=
b2 S 21 S 22 a2
i.e.,
b2 = S 21a1 + S 22a2
b2 b2
S 21 ≡ ; S 22 =
a1 a =0
a2 a1 =0
2
If Z L = Zo , a2 = 0
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b1, b2, a1, a2 have the dimensions of power. Next signal flow graph
representations are introduced.
2
bs 1 b bs
Pavs = =b −a
2 2
Zs
1 − Γs
2
+- Vs b Γs
Vs Z o Z s .L − Z o
a bs = ; Γs ,L =
Zs + Zo Z s ,L − Z o
a
Vs
bs =
A two-port with a source and a load. Zs Zo + Z o
bs 1 a1 s 21 b2 Using Mason' s Rule
Γs
s 11 s
22 ΓL
b1 S11(1 − S22Γ L ) + S 21Γ L S12
b1 s 12 a2 =
bs 1 − ( S11Γs + S 22Γ L + S 21Γ L S12Γ s ) + S11Γ s S 22ΓL
P b 1 − ΓL
2
( 2
)
GT = del = 22
Pavs bs 1 − Γ s 2 ( ) b2
=
S21
bs 1 − (S11Γ s + S22Γ L + S 21ΓL S12Γ s ) + S11Γ s S 22Γ L
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GT =
(
S 21 1 − Γs
2 2
)(1 − Γ )
L
2
(1 − S11Γs )(1 − S 22ΓL ) − S 21S12ΓLΓs 2
For ideal amplifiers S12 = 0, then
GT ≅
(
S 21 1 − Γs
2 2
)(1 − Γ )
L
2
; GT = S21
2
(1 − S11Γs )(1 − S 22ΓL ) 2 ΓL =Γs = 0
Also the total voltage gain is given by
a2 b2
+
a2 + b2 bs bs
AV = =
a1 + b1 a1 + b1
bs bs
S21ΓL + S 21 S 21
AV = =
(1 − S 22ΓL ) + S11 (1 − S 22ΓL ) +S 21S12 ΓL ΓL = 0
1 + S11
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Stability Considerations using S-Parameters
• Max GT occurs when Γ s = Γin* and Γ L = Γout
*
• Conditionally Stable if Re{Zin } and Re{Zout } > 0 for some specific positive real load
and source impedance at a specific frequency. If this condition is satisfied for all
positive real oad and source impedances then the network is Unconditionally
Stable. Γs ≤ 1 , ΓL ≤ 1
• For conjugately match the input and output, the network wi ll be stable if
K >1
2 2 2
1 + S11S 22 − S12 S 21 − S11 − S22
K=
2 S12 S 21
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REFLECTION COEFFICIENTS
b1 S S Γ
ΓIN = = S11 + 21 12 L
a1 1 − S 22ΓL
b S S Γ
Γout = 2 = S 22 + 21 12 s
a2 1 − S11Γs
If one sets Γin = 1, the solutions of ΓL lie on a circle. The radius, and center are given by
S12 S 21
radius = rL = 2 2
S 22 − ∆
∆ = S11S22 − S12S 21
( S22 − ∆S11 )*
center = CL = 2 2
S22 − ∆
This circle can be plotted on a Smith Chart to determine all values of ΓL that make Γin = 1.
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