1
S Parameters and Power gainsS Parameters and Power gains
Training in 1 dayTraining in 1 day
Roberto Antonicelli ST Belgium, Network Division
2
S ParametersS Parameters
Network theory
ZS
ES
ZLSM1 M2
i1
v1
i2
v2
v z i z i
v z i z i1 11 1 12 2
2 21 1 22 2
i y v y v
i y v y v1 11 1 12 2
2 21 1 22 2
Standard amplifier network
Bi-port description
3
S ParametersS Parameters
Two-terminal element
Conjugate power matching
Incident and reflected power
v
i
Zs
Es
E t E ts s 2 cos
Zi
vZ E
Z Zi s
s i
iE
Z Zs
s i
Z Zi s *
v VZ E
Z Z
Z E
Zs s
s s
s s
s
*
*
*
Re2
i IE
Z Z
E
Zs
s s
s
s
* Re2
V Z Is *
P V IE
Zs
sinc Re
Re*
2
4
Incident power
4
S ParametersS Parameters
Voltages and currents
Incident and reflected waves
Reflection coefficients
Incident and reflected power
I
VZs
Es
V+
V
I +I
Zi
vs i s
s i s
V
V
Z Z Z
Z Z Z
*
*
i i s
i s
I
I
Z Z
Z Z
*
bV
ZZ I Z
ss s
Re Re
aV
ZZ I Z
ss s
* Re Re
III
VVV
inc
2
2*
2
2ReRe PZIZ
Z
Va ss
s
rifl
2
2
2
2ReRe PZIZ
Z
Vb ss
s
5
S ParametersS Parameters
Definition
The scatter matrix
Scatter matrix
ba
The incident wave a depends only on the reference impedance and the source Es, while the reflected wave b depends also on the load, being zero when this is matched.
The reflection coefficient depends on both the circuit impedance Zi and the source impedance ZS.
b s a s a
b s a s a1 11 1 12 2
2 21 1 22 2
aSb
a
a
ss
ss
b
b
2
1
2221
1211
2
1
The S parameters depend on both the circuit impedances and a reference impedance Z0
Z 0
E 1
Z 0
I 1 I 2
V 1 V 2
a 1
b 1
a 2
b 2
S
6
S ParametersS Parameters
Analysis
Measurements
Scatter matrix
|a1|² and |a2|² are the incident powers at the ports 1 and 2, while |b1|² and |b2|² are the reflected powers at the two ports
0
2022
0
1011
Re2
Re2
Z
IZVa
Z
IZVa
0
2*02
2
0
1*01
1
Re2
Re2
Z
IZVb
Z
IZVb
sb
a a11
1
12 0
sb
a a12
1
21 0
s11 = input reflection coefficient with matched outputs12 = inverse transmission coefficient with matched outputs21 = forward transmission coefficient with matched outputs22 = output reflection coefficient with matched input
Ex.: s11 is the reflection coefficient at port 1, when a2 = 0, i.e. when the port 2 is terminated over the reference impedance
The S parameters depend on both the device and a reference impedance Z0
7
S ParametersS Parameters
Transmission line’s scatter matrix
De-embedding
Let li be the line length, Zi be the line impedance, the phase constant, i the electrical length
Se
e
j
j
0
0
Se
eS
e
e
j
j
j
j
1
1
1 2
2
2
0
0
0
0
121
211
22221
122
11
jj
jj
eses
esesS
S
Terminal plane 1 Terminal plane 2
Actual matrix S
Meas.d matrix S
Meas.t plane 1 Meas.t plane 2
l 1
1 1 l 2
2 l
Z 2 = 50
l 2
Z 1 = 50
a 1 a 2
b 1 b 2
a 1 a 2
b 1 b 2
8
S ParametersS Parameters
Smith chart
0 1.0
1.0
-1.0
10.0
10.0
-10.0
5.0
5.0
-5.0
2.0
2.0
-2.0
3.0
3.0
-3.0
4.0
4.0
-4.0
0.2
0.2
-0.2
0.4
0.4
-0.4
0.6
0.6
-0.6
0.8
0.8
-0.8
S11 Port 1 ReflectionSwp Max
3GHz
Swp Min0.01GHz
S[1,1]Dev001
0
15
30
45
60
75901
05
120
135
150
165
-180
-165
-150
-135
-120
-105 -
90
-75
-60
-45
-30
-15
S12 Inverse TransmissionSwp Max
3 GHz
Swp Min0.01 GHz
Mag Max0.1
0.01Per Div
S[1,2]Dev001
0
15
30
45
60
75901
05
120
135
150
165
-180
-165
-150
-135
-120
-105 -
90
-75
-60
-45
-30
-15
S21 Forward TransmissionSwp Max
3 GHz
Swp Min0.01 GHz
Mag Max6
2Per Div
S[2,1]Dev001
Freq.[GHz]
s11 s21 s12 s22
Mag Angle Mag
Angle Mag Angle Mag Angle
0.01000 0.92681 -0.45872 5.73254 179.257 0.00044 76.0557 0.73428 -0.19327
0.10000 0.92264 -2.90332 5.65939 173.598 0.00313 90.9401 0.73144 -1.71104
0.30000 0.89887 -9.25535 5.48504 160.480 0.00897 88.3476 0.72161 -5.00982
0.70000 0.79630 -17.8064 4.63514 137.791 0.01880 86.4446 0.68115 -9.46235
0.90000 0.74581 -19.7145 4.21554 128.920 0.02351 89.1533 0.65988 -10.6804
1.10000 0.70000 -21.5326 3.83450 121.595 0.02741 91.5389 0.64526 -11.4538
1.50000 0.63412 -22.7401 3.20471 109.782 0.03841 96.7770 0.62723 -11.6775
1.90000 0.58184 -23.5762 2.87512 101.194 0.05373 101.867 0.69004 -9.52874
2.30000 0.55180 -22.6956 2.39804 89.9190 0.06715 101.012 0.62656 -21.3144
2.50000 0.54108 -22.7904 2.24872 87.1711 0.07530 102.366 0.61271 -20.7434
2.70000 0.53229 -22.8477 2.12969 84.3875 0.08419 103.207 0.60842 -20.9636
2.75000 0.53082 -22.8175 2.10360 83.6995 0.08659 103.584 0.60799 -21.1857
3.00000 0.51963 -23.3829 1.98392 80.2487 0.09806 103.250 0.60700 -21.7178
01.0
1.0
-1.0
10.0
10.0
-10.0
5.0
5.0
-5.0
2.0
2.0
-2.0
3.0
3.0
-3.0
4.0
4.0
-4.0
0.2
0.2
-0.2
0.4
0.4
-0.4
0.6
0.6
-0.6
0.8
0.8
-0.8
S22 Port 2 ReflectionSwp Max
3GHz
Swp Min0.01GHz
S[2,2]Dev001
9
S ParametersS Parameters
Input/output reflection coefficients
ZS
ES
ZL
a1
b1
a2
b2
S
S L1 2
0
0
0
0
2
22
1
11
ZZ
ZZ
ZZ
ZZ
a
b
a
b
L
LL
S
SS
S
S
L
L
s
sss
a
b
s
sss
a
b
11
211222
2
22
22
211211
1
11
1
1
1 (2) is the input (output) reflection coefficient that is visible at port 1 (2) when port 2 (1) is terminated on a generically unmatched impedance ZL (ZS ). It is always referred to the reference impedance Z0.
L (S) is the reflection coefficient at the load (source) referred to the reference impedance Z0.
10
S ParametersS Parameters
Power definitions
The available power is the maximum power transferable from the source to the load (conjugate power matching). It depends only on the generator.
ZS
ES
SS
S
Z Z
Z Z
0
0
Zin
inin
in
Z Z
Z Z
0
0
I
V
Source available power
S
SAZZin Z
EPP
inS Re4
2
*
P VI Z I ZE
Z Zin in inS
S in
Re Re Re* 22
11
S ParametersS Parameters
Power definitions
If the source impedance ZS is equal to the reference impedance Z0, the squared magnitude of the incident wave |a|2 gives the source available power. Since the available power only depends on the generator, it is often regarded as a source reflected wave, bS:
ZS
ES
Zin
a
b
P bA S 2
s
s
Z
IZVa
Re2
s
s
Z
IZVb
Re2
*
2
222
Re4Re4Re4
a
Z
IZV
Z
IZIZE
Z
EP
S
S
S
SSS
S
SA
bE
ZSS
S
22
4
Re
12
S ParametersS Parameters
Power definitions
Intuitively, the net power that is transferred to the load is equal to the available power (referred to a reference impedance Z0) minus the reflected power (referred to the same reference impedance).
ZS
ES
Zin
a
b
P a bin 2 2
Power transferred to the load
riflincin
in
PPbaP
Z
IZV
Z
IZV
Z
IZVIZVIZVIZV
Z
IVVIZZ
Z
VIZ
Z
VIZ
VIP
22
0
2*0
0
2
0
0
**0
*0
*00
0
***00
0
*0
0
*0
*
Re4Re4
Re4
Re4
Re4
Re2Re2
Re
ReRe
Re
13
S ParametersS Parameters
Power gain definitions
In the reference impedance domain:
Transducer gain
ZS
ES
ZL
a1
b1
a2
b2
S
PA
PL
1 2
LL
L
Z Z
Z Z
0
0
SS
S
Z Z
Z Z
0
0
2
21122211
222
21
2
2
2
11
222
21
11
11
11
11
,,
LSLS
LS
LS
SL
A
LLSTT
ssss
s
s
s
P
PSGG
P Pb b
AS
S
S
SS
S
1
2
12 1
22
21
1
11
1
*
*
By definition, the transducer power gain (TG) is the ratio btw the power transferred to the load and the source available power:
14
S ParametersS Parameters
Power gain definitions
Maximum available gain
ZS
ES
ZL
a1
b1
a2
b2
S
PA
PL
1 2
LL
L
Z Z
Z Z
0
0
SS
S
Z Z
Z Z
0
0
The highest transducer gain achievable is called Maximum available gain (MAG). It depends only on the device parameters.
S
ss
ssK
KKs
s
ssss
sG
LmSmLmSm
LmSmma
det
2
1
1
11
11
2112
22
22
2
11
2
12
21
2
21122211
222
21
15
S ParametersS Parameters
Power gain definitions
Available gain
By design strategy, the source is in a controlled mismatch. The available gain depends on the device and the (unmatched) source impedance.ZS
ES
ZL
a1
b1
a2
b2
S
PL
1 2
LL
L
Z Z
Z Z
0
0
M2
Z0
PA
PoutSS
S
Z Z
Z Z
0
0
11
22
22
22
21
2
11
2
2
22
21
out
Re21
1
11
1
,
CDs
s
s
s
P
PSGG
SS
S
S
S
ASAA
B s s B s s
C s s C s s
D s D s
E s E s
m s s
1 112
222 2
2 222
112 2
1 11 22 2 22 11
1 112 2
2 222 2
1 222
2 112
12 21
1 1
1 1
* *
16
S ParametersS Parameters
Power gain definitions
Operative gain
By design strategy, the load is in a controlled mismatch. The operative gain depends on the device and the (unmatched) load impedance.
B s s B s s
C s s C s s
D s D s
E s E s
m s s
1 112
222 2
2 222
112 2
1 11 22 2 22 11
1 112 2
2 222 2
1 222
2 112
12 21
1 1
1 1
* *
ZS
ES
ZL
a1
b1
a2
b2
S
PL
1 2
LL
L
Z Z
Z Z
0
0
M1
Z0
PS
SS
S
Z Z
Z Z
0
0
Pin
22
22
11
22
21
2
22
2
1
22
21
in
Re21
1
11
1
,
CDs
s
s
s
P
PSGG
LL
L
L
L
LLWW
17
S ParametersS Parameters
Power gain definitions
Unilateral transducer gain
Usually, the unilateral approximation is used (s12 = 0):
G
s
s sTU
L S
L S
212 2 2
222
112
1 1
1 1
Here, the mismatch effect on both the source and load sections is in. By simultaneously conjugate matching, we have the maximum unilateral transducer gain:
Gs
s sTU ,max
21
2
112
222
1 1
18
S ParametersS Parameters
Power gain definitions
Constant available gain circles
By varying the source reflection coefficients on the input Smith Chart, the resulting available power gain changes Constant gain loci (circles). Centers and radiuses are function of the selected available gain GA.
A
AAA
A
AAAA
GDs
GssGssKsr
GDs
CGvuC
1
2
21
22
2112
3
2112
4
21
1
2
21
*1
2
j
S
GA = 8 dB
1012
14 Sm
Over all those circles, the available gain is constantly equal to GA (provided the output is in conjugate matching).
19
S ParametersS Parameters
Power gain definitions
Constant operative gain circles
By varying the load reflection coefficients on the output Smith Chart, the resulting operative power gain changes Constant gain loci (circles). Centers and radiuses are function of the selected oeprative gain GW.
W
WWW
W
WWWW
GDs
GssGssKsr
GDs
CGvuC
2
2
21
22
2112
3
2112
4
21
2
2
21
*2
2
j
Over all those circles, the operative gain is constantly equal to GW (provided the input is in conjugate matching).
L
GA = 8 dB10
1214
Lm
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