1 S Parameters and Power gains Training in 1 day Roberto Antonicelli ST Belgium, Network Division.

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


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


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


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


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


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


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


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


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


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


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


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



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



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



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



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



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



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

1 S Parameters and Power gains Training in 1 day Roberto Antonicelli ST Belgium, Network Division.

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Transcript of 1 S Parameters and Power gains Training in 1 day Roberto Antonicelli ST Belgium, Network Division.