S-parameters
MUSE
Material developed by Prof. L. Dunleavy, USF
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1
2-Port Parameters
z
Recall Z-Parameters:
V =Z I +Z I
1
11 1
12 2
V =Z I +Z I
2
21 1
22 2
⎡ V1 ⎤ ⎡Z 11
==> ⎢ ⎥ = ⎢
⎣ V2 ⎦ ⎣Z 21
I1
V1
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Z 12 ⎤ ⎡ I1 ⎤
Z 22 ⎥⎦ ⎢⎣ I 2 ⎥⎦
I2
2-port
V2
2
2-Port Parameters
z
Recall Z-Parameters:
V =Z I +Z I
1
11 1
12 2
V =Z I +Z I
2
21 1
22 2
⎡ V1 ⎤ ⎡Z 11
==> ⎢ ⎥ = ⎢
⎣ V2 ⎦ ⎣Z 21
I1
V1
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Z 12 ⎤ ⎡ I1 ⎤
Z 22 ⎥⎦ ⎢⎣ I 2 ⎥⎦
I2
2-port
network
V2
3
2-PORT Parameters (cont’d)
z
Y-Parameters:
I1 = Y11V1 + Y12 V2
I 2 = Y21V1 + Y22 V2
z
⎡ I1 ⎤
==> ⎢ ⎥ =
⎣I 2 ⎦
⎡ Y11
⎢
⎣Y21
Y12 ⎤ ⎡ V1 ⎤
⎥⎢ ⎥
Y22 ⎦ ⎣ V2 ⎦
h-Parameters:
V1 = h11I1 + h12 V2
⎡ V1 ⎤ ⎡ h11 h12 ⎤ ⎡ I1 ⎤
==> ⎢ ⎥ = ⎢
⎥ ⎢V ⎥
I
h
h
I 2 = h 21I1 + h 22 V2
⎣ 2 ⎦ ⎣ 21
22 ⎦ ⎣ 2 ⎦
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4
2-Port Parameters (cont’d)
z
2-port Parameter Determination:
h 11 =
v1
|v = 0
I1 2
I2
h21 = |V = 0
I1 2
V
h = 1|
12 V I =0
2 1
I
h22 = 2 |I =0
V 1
2
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(Put a Short Circuit at Port #2)
(Put an Open Circuit at Port # 1)
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S-Parameters
At high RF and Microwave frequencies direct
measurement of Y- , Z-, or H- parameters is
difficult due to:
• Unavailability of equipment to measure
RF/MW total current and voltage.
• Difficulty of obtaining perfect opens/shorts
• Active devices may be unstable under
open/short conditions.
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6
S-Parameters
For a two-port device there are four Sparameters S11, S21, S12, and S22
z S11, and S22 are simply the forward and
reverse reflection coefficients, with the
opposite port terminated in Z0 (usually 50
ohms.)
z S21 and S12 are simply the forward and
reverse gains assuming a Z0 source and load
(again usually 50 ohms).
z
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7
S-Parameters (cont’d)
z
S-Parameters:
b =S a +s a
1
11 1 12 2
⎡ b 1 ⎤ ⎡S 11 S 12 ⎤ ⎡ a 1 ⎤
⎥ ⎢a ⎥
b = S a + S a ==> ⎢b ⎥ = ⎢S
S
⎣ 2 ⎦ ⎣ 21
2
21 1
22 2
22 ⎦ ⎣ 2 ⎦
a1
b1
b2
a2
2-port
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8
S-Parameters (cont’d)
z
Q. So what’s the deal with the a’s and b’s?
a1
b1
b2
a2
Zs
Vg
z
V1
2-port
V2
ZL
A. a1 and a2 are incident waves; b1 and b2 are
reflected waves
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9
Incident & Reflected Waves:
Simplified Case: Z1=Zs=Z2=ZL=Zo (real)
a1 =
V1 + Z 0 I 1
=
Incident port 1 voltage
2 Z0
E i1
=
Z0
Z0
V2 + Z 0 I 2
E i2
Incident port 2 voltage
=
=
a2 =
2 Z0
Z0
Z0
b1 =
b2 =
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V1 − Z 0 I 1
=
reflected port 1 voltage
2 Z0
V2 − Z 0 I 2
2 Z0
=
E r1
Z0
=
reflected port 2 voltage
Z0
Z0
=
E r2
Z0
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S-Parameter Determination
b1
s11 = |
a 1 a2 = 0
b2
s21 = |
a1 a2 = 0
b1
s12 = |
a2 a1 = 0
b2
s22 = |
a2 a1 = 0
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= Input reflection coefficient Γin
for case of ZL=Z0
= Forward transmission (insertion) gain
for case of ZL=Z0
= Reverse transmission (insertion) gain
for case of Zs=Z0
= Output reflection coefficient Γout
for case of Zs=Z0
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Desired Measurement Conditions
Zs = Z0
a1
b1
Vg
b2
Z0
2-port
Z0
a2 = 0
V2
ZL =
Z0
Note…the input and output are terminated in Z0
a1
b1
Zs = Z0
Vg
Z0
Γ
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1-port
S11=Γ
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GRAPHICAL VIEW OF SPARAMETERS
S12
S11,
Input
Refl.
Coeff. Γin,
Return
Loss,
VSWR
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Reverse Gain
Insertion Loss,
Transmission Phase
Device
Under Test
S21, Forward Gain
Insertion Loss,
Transmission Phase
S22,
Output
Refl.
Coeff. Γout,
Return
Loss,
VSWR
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S-Parameters in Decibels
dB
Meaning or interpretation
S11
20 log10 |S11| Corresponds to the algebraic negative of the input
S12
20 log10 |S12| Reverse isolation (active device or amplifier), or
S21
20 log10 |S21| Power gain (active device or amplifier), or algebraic
S22
20 log10 |S22| Corresponds to the algebraic negative of the
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return loss of a 2-port with an R0 termination on
the opposite port.
algebraic negative of the insertion loss (I.L.) for a
passive device, with R0 at ports 1 and 2.
negative of the insertion loss (I.L.) for a passive
device, under matched R0 at ports 1 and 2.
output return loss of a 2-port with an R0
termination on the opposite port.
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WHAT TO EXPECT
Ideal Lossless T-line
θ = βd
Zo, εr
Ideal “X” dB Attenuator
PAD
S11=S22=0
S21=S12=1e-jθ
S21DB=0
S11=S22=0
S21=S12=xe-jθpad
S21DB= X=20log(|S21|)
x =10-X/20
Ideal “G” dB Gain Amp
S11 = S22 = 0 = S12
S21 = gve-jθamp
S21DB= G=20log(S21)
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gv
G
= 10 20
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WHAT TO EXPECT: Ideal Filters
Ideal Band Pass
S21
Ideal Low Pass
S21
fc1
fc2
Ideal High Pass
0dB
S21
fc
fc
GENERAL “IN-BAND”
GENERAL “OUT-OF-BAND”
S11=S22=0
S21=S12=1e-jθ(f)
S21(dB)=S12(dB)=0dB
|S11|=|S22|=1
S21=S12=0
S11(dB)=S22(dB)=0dB
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