This document discusses traveling waves and scattering parameters for analyzing multi-port networks. It begins by defining traveling waves as voltage and current waves that propagate through transmission lines. It then introduces scattering parameters (S-parameters) which describe the input-output relationship of linear electrical networks with multiple ports. S-parameters are presented as elements of a scattering matrix that relates incoming and outgoing wave amplitudes at each port. Methods for calculating reflection and transmission coefficients from S-parameters are provided for characterizing two-port networks. The analysis is then generalized to n-port networks using scattering matrices. Key parameters like return loss, insertion loss, and available power are defined in terms of S-parameters.

1. Department of Electronic Engineering
National Taipei University of Technology

2. •
•
•
• (Power waves)
• (Traveling waves) −
Department of Electronic Engineering, NTUT2/25

3. (Traveling Waves)
( ) ( ) ( ) j x j x
V x V x V x Ae Beβ β+ − −
= + = +
( ) ( ) ( )
( ) ( )
0 0
V x V x
I x I x I x
Z Z
+ −
+ −
= − = −
( )
( )
( )
V x
x
V x
−
+
Γ =
•
•
• (Normalized traveling waves)
( )
( )
0
V x
a x
Z
+
=
( )
( )
0
V x
b x
Z
−
=
( )
( )
2
2
0
V x
a x
Z
+
  =
( ) ( )
( )
2
10log 10log
20log
aP a x
a x
=
=
( ) ( ) ( ) ( ) ( )0
0
1 1
2 2
b x v x i x V x Z I x
Z
= − = −      
( ) ( ) ( ) ( ) ( )0
0
1 1
2 2
a x v x i x V x Z I x
Z
= + = +      
( ) ( ) ( )
( )
0
V x
v x a x b x
Z
= + =
( ) ( ) ( ) ( )0i x a x b x Z I x= − =
( ) ( ) ( )b x x a x= Γ
slide 16, 30
3/25 Department of Electronic Engineering, NTUT

4. Two-port
Network
( )2 2a l
( )2 2b l
( )2 2a x
( )2 2b x
( )1 1a l
( )1 1b l
( )1 1a x
( )1 1b x
1oZ 2oZ
Input port Output port
Port 1
1 1x = l
Port 2
2 2x = l
• Port 1 ( )
• Port 2 ( )
( )1 1a l ( )1 1b l1 1x = l
( )2 2a l ( )2 2b l2 2x = l
Incident wave
Reflected wave
• (Scattering Matrix)
( ) ( ) ( )1 1 11 1 1 12 2 2b S a S a= +l l l
( ) ( ) ( )2 2 21 1 1 22 2 2b S a S a= +l l l
( )
( )
( )
( )
1 1 1 111 12
2 2 2 221 22
b aS S
b aS S
    
=    
    
l l
l l
xx
4/25 Department of Electronic Engineering, NTUT

5. ( )
( ) ( )2 2
1 1
11
1 1 0a
b
S
a =
=
l
l
l
Input reflection coefficient with output properly terminated
( )
( ) ( )1 1
2 2
22
2 2 0a
b
S
a =
=
l
l
l
Output reflection coefficient with input properly terminated
Forward transmission coefficient with output properly terminated
( )
( ) ( )1 1
1 1
12
2 2 0a
b
S
a =
=
l
l
l
Reverse transmission coefficient with output properly terminated
(measured with port 2 properly terminated)
(measured with port 2 properly terminated)
(measured with port 1 properly terminated)
(measured with port 1 properly terminated)
( )
( ) ( )2 2
2 2
21
1 1 0a
b
S
a =
=
l
l
l
5/25 Department of Electronic Engineering, NTUT

6. ( )
( ) ( )2 2
1 1
11
1 1 0a
b
S
a =
=
l
l
l
• Return Loss (RL)
( )
( ) ( )2 2
2 2
21
1 1 0a
b
S
a =
=
l
l
l
( )
( )
2
1 12 1
11 2
1 1 1
b
a
b P
S
a P
= =
l
l
( )21
11 11
1
10log 10log 20log (dB)b
a
P
S S
P
 
= = 
 
11Return Loss (RL) 10log 20log (dB)in
reft
P
S
P
 
= = −  
 
( )
( )
( )
2
2 22 2
21 2
1 1 1
b
a
b P
S
a P
= =
l
l
( )22
21 21
1
10log 10log 20log (dB)b
a
P
S S
P
 
= = 
 
21Insertion Loss (IL) 10log 20log (dB)transmit
receive
P
S
P
 
= = − 
 
( )• Insertion Loss (IL)
|S11| −12 dB
12 dB
6/25 Department of Electronic Engineering, NTUT

7. S11
Two-port
Network
( )2 2 0a =l
( )2 2b l
( )1 1a l
( )1 1b l
1oZ 2oZ
Port 1
1 1x = l
Port 2
2 2x = l
2 2oZ Z=+
−
1E
1 1oZ Z=
( )
( ) ( )2 2
1 1
11
1 1 0a
b
S
a =
=
l
l
l OUTZ
• Z2=Zo2 (i.e. Zo1=Zo2)
50
( )2 2 0a =l
( )
( ) ( )2 2
1 1
11
1 1 0a
b
S
a =
=
l
l
l
( ) ( ) ( )1 1 11 1 1 12 2 2b S a S a= +l l l
0
port 2 ( )2 2 0a =l
7/25 Department of Electronic Engineering, NTUT

8. •
Zoi (i=1 to n) n port
[ ] [ ][ ]b S a=
n-port
Network
1oZ
Port 1Port 1'
1TZ
( )1 1a l
( )1 1b l
2oZ
Port 2Port 2'
( )2 2a l
( )2 2b l
onZ
Port nPort n'
( )n na l
( )n nb l
[ ]
11 12 1
21 22 2
1 2
n
n
n n nn
S S S
S S S
S
S S S
⋅ ⋅ 
 ⋅ ⋅ 
 = ⋅ ⋅ ⋅ ⋅ ⋅
 
⋅ ⋅ ⋅ ⋅ ⋅ 
 ⋅ ⋅ 
8/25 Department of Electronic Engineering, NTUT

9. •
•
(i=1 for port 1 and i=2 for port 2)
( )0iP+
( )0iP−
+
1oZ 2oZ
Port 1
1 1x = l
Port 2
2 2x = l
Port 1'
1 0x =
( )1 1I x
Port 2'
2 0x =
2l1l
−
( )1 1V x
( )2 2I x
+
−
( )2 2V x
( )0iP+
( )0iP−
( )0iP+
( )0iP−
11 12
21 22
S S
S S
 
 
 
( )
( )
( )
( )
1 1 1 111 12
2 2 2 221 22
b aS S
b aS S
    
=    
    
l l
l l
( )0iP+
( )0iP−
( )a x
( )b x
9/25 Department of Electronic Engineering, NTUT

10. ( ) ( )
21
0 0
2
i iP a+
=
( ) ( ) ( ){ } ( )
( ) ( )
2
22
,
01 1 1
0 Re 0 0 0 0
2 2 2
i
i i i i i rms
oi
V
P V I b b
Z
−
∗− − −
 = ⋅ = = = 
• ith port (x1=0, x2=0)
•
( ) ( )0i i iP P+ +
= l
( ) ( )0i i iP P− −
= l
( ) ( )
2 21 1
0
2 2
i i ia a x=
( ) ( )
2 21 1
0
2 2
i i ib b x=
( ) ( )
21
0 0
2
i iP b−
=
• ith port (x1=0, x2=0)
( ) ( ) ( ){ } ( )
( ) ( )
2
22
,
01 1 1
0 Re 0 0 0 0
2 2 2
i
i i i i i rms
oi
V
P V I a a
Z
+
∗+ + +
 = ⋅ = = = 
( )
( )
0
V x
a x
Z
+
=
( )
( )
0
V x
b x
Z
−
=
10/25 Department of Electronic Engineering, NTUT

11. ( ) ( )2 2 20 0oV Z I= −
( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 2
2 2
1 1
0 0 0 0 0 0
2 2
o o o
o o
a V Z I Z I Z I
Z Z
= + = − + =      
Two-port
Network
+
1oZ
Port 1
1 1x = l
Port 2
2 2x = l
Port 1'
1 0x =
( )1 0I
Port 2'
2 0x =
−
( )1 0V
( )2 0I
+
−
( )2 2V l
2oZ
( )1 1I l ( )2 2I l
( )1 1V l
+
−
( )2 2a x
( )2 2b x
( )1 1a x
( )1 1b x
+
−
( )2 0V
+
−
1E
1 1oZ Z=
2 2oZ Z=
• x2=0
( ) ( )1 1 1 10 0oV E Z I= −
( ) ( ) ( ) 1
1 1 1 1
1 1
1
0 0 0
2 2
o
o o
E
a V Z I
Z Z
= + =   ( )
2
2 1
1
1
0
4 o
E
a
Z
=• x1=0
Vpp
matched
• E1 (available power) x1=0 (
Z1=Zo1 )
( ) ( )
2
2 1
1 1
1
1
0 0
2 8
AVS
o
E
P P a
Z
+
= = = ( ) ( )
2 2
1 1 1
1 1
0
2 2
a a= l
matched
11/25 Department of Electronic Engineering, NTUT

12. ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2 2 21 1 1 1 1 1 2
1 1 1 1 1 1 1 1 1 1
1 1
0 0 0 01 1
0 0 0 0 0 0 0
2 8 8
o o
o o o
o o
V Z I V Z I
a V Z I V Z V I Z I
Z Z
∗
∗ ∗
+ +        = = + + +
 
( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2 22
1 1 1 1 1 1 1 1 1 1
1
1 1
0 0 0 0 0 0 0
2 8
o o o
o
b V Z I V Z I V Z I
Z
∗ ∗ = − − +
 
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }
2 2
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1
0 0 0 0 0 0 0 0 0 Re 0 0
2 2 4 2
P P P a b I V I V I V+ − ∗ ∗ ∗
 = − = − = + = 
• Z1 Zo1
• port 1' ( port 1 )
( ) ( )
2
1 1
1
0 0
2
AVSb P P= − ( ) ( )
2
1 1 1 1
1
2
AVSb P P= −l l• port 1 ( port 1')
( ) ( ) ( )
2
1 1 1 1
1
0 0
2
AVSP P P b= = −l
• ZT1=Zo1 0 ZT1≠Zo1
port 1
( )
( ) ( )
( )
2 2
2
2 1 1 1 1
11 2
1 1 0
AVS
AVS
a
b P P
S
Pa
=
−
= =
l
l l
l
( ) ( ) ( )2
1 1 1 110 1AVSP P P S= = −lor
12/25 Department of Electronic Engineering, NTUT

13. (Power Waves) (I)
•
• (Z0)
Rs
s sV E Z I= −
LZ
+
−
sE
sZ
V
I
+
−
source
impedance
load
impedance
Department of Electronic Engineering, NTUT
L s
L s
Z Z
Z Z
∗
−
Γ =
+
• ( )*
L sZ Z= Γ = 0
slide 46
13/25

14. (Power Waves) (II)
( )
1
2
p s
s
a V Z I
R
= + ( )1
2
p s
s
b V Z I
R
∗
= −
LZ
+
−
sE
sZ pa
pb
V
I
+
−
s
p s L s
p s L s
s
V
Zb V Z I Z ZI
Va V Z I Z ZZ
I
∗
∗ ∗−− −
Γ = = = =
+ ++
p pa bΓ =
• (Normalized power waves)
0pb =L sZ Z∗
=
• (Available power)
( )
1
2
p s
s
a V Z I
R
= + s sV E Z I= −
( )
1
2 2
s
p s s s
s s
E
a E Z I Z I
R R
= − + =
2
2
4
s
p
s
E
a
R
=
2
2 2
,
1
2 8
s
AVS p p rms
s
E
P a a
R
= = =
• ( )
{ } { }
2
21 1
Re Re
2 2
s
L L L
s L
E
P I Z Z
Z Z
= =
+
2
,max
1
8
s
L AVS
s
E
P P
R
= =
14/25 Department of Electronic Engineering, NTUT

15. ( )( ) ( )( ) { }
2 2 *1 1 1 1 1
Re
2 2 8 8 2
L p p s s s s
s s
P a b V Z I V Z I V Z I V Z I V I
R R
∗∗ ∗ ∗
= − = + + − − − =
2 2 21 1 1
2 2 2
L p p AVS pP a b P b= − = −
21
2
p AVS Lb P P= −
= ( – )
•
= ( – )
15/25 Department of Electronic Engineering, NTUT

16. − (Traveling Waves)
0
0
s
s
s
Z Z
Z Z
−
Γ =
+
0
0
L
L
L
Z Z
Z Z
−
Γ =
+
1 11 1 12 2b S a S a= +
2 21 1 22 2b S a S a= +
Transistor
[S]
2a
2b
1a
1b
Port 1 Port 2
+
−
sE
sZ
outΓ
LZ
inΓ
sΓ LΓ
•
[S] Z0
sΓ LΓ
?
+
−
sE
sZ
sΓ
LZ
LΓ
Transistor
[S]
1b
1a 2a
2b
16/25 Department of Electronic Engineering, NTUT

17. 1
1
in
b
a
Γ =
2 2La b= Γ
2 21 1 22 2Lb S a S b= + Γ 21 1
2
221 L
S a
b
S
=
− Γ
• inΓ
[ ]SLΓ
1 12 21
11
1 221
L
in
L
b S S
S
a S
Γ
Γ = = +
− Γ
12 21
1 11 1 12 2 11 1 1
221
L
L
L
S S
b S a S b S a a
S
Γ
= + Γ = +
− Γ
a1 b1
1 11 1 12 2b S a S a= +
a1 b1 = a2
a2 = b2
Transistor
[S]
2a
2b
1a
1b
+
−
sE
sZ
outΓ
LZ
inΓ
sΓ LΓ
1 11 1 12 2b S a S a= +
2 21 1 22 2b S a S a= +
inΓ
17/25 Department of Electronic Engineering, NTUT

18. 2
2 0s
out
E
b
a =
Γ =
1 1sa b= Γ
1 11 1 12 2sb S b S a= Γ + 12 2
1
111 s
S a
b
S
=
− Γ
12 21
2 21 1 22 2 2 22 2
111
s
s
s
S S
b S b S a a S a
S
Γ
= Γ + = +
− Γ
12 212
22
2 110
1
s
s
out
sE
S Sb
S
a S=
Γ
Γ = = +
− Γ
• outΓ
[ ]SsΓ
Transistor
[S]
2a
2b
1a
1b
+
−
sE
sZ
outΓ
LZ
inΓ
sΓ LΓ
1 11 1 12 2b S a S a= +
2 21 1 22 2b S a S a= +
outΓ
outΓ inΓ
2 21 1 22 2b S a S a= +and
18/25 Department of Electronic Engineering, NTUT

19. +
−
sE
sZ
sΓ
1a
1b
•
inΓ
+
−
1V
1I
• 1 1s sa a b= + Γ
inΓ outΓ
Pin
1 1inb a= Γ
1 1 1s s s s ina a b a a= + Γ = + Γ Γ 1
1
s
s in
a
a =
− Γ Γ
( )
2
2 2 2 2 2
1 1 1 2
11 1 1 1
1
2 2 2 2 1
in
in in s
s in
P a b a a
− Γ
= − = − Γ =
− Γ Γ
•
2 2
2 2 2
2 2 22 2
1 11 1 1 1
2 2 2 11 1
in s
s s
AVS in s s s
ss s
P P a a a∗
∗
Γ =Γ
− Γ − Γ
= = = =
− Γ− Γ − Γ
( )( )2 22
2
2 2
1 111
2 1 1
s inin
in s AVS AVS s
s in s in
P a P P M
− Γ − Γ− Γ
= = =
− Γ Γ − Γ Γ
• Ms (source mismatch factor) (mismatch loss)
inΓ s o
s
o so
E Z
a
Z ZZ
=
+
19/25 Department of Electronic Engineering, NTUT

20. LZ
LΓ
outΓ
+
−
thE
outZ 2a
2b
+
−
LV
LI
LZ
LΓ
outΓ
( )2 2 2 2
2 2 2
1 1 1
1
2 2 2
L LP b a b= − = − Γ
• ZL
2
2
2
11
2 1
L
L th
out L
P b
− Γ
=
− Γ Γ
• ( source)
2
2
1 1
2 1L out
AVN L th
out
P P b∗
Γ =Γ
= =
− Γ
( )( )2 2
2
1 1
1
L out
L AVN AVN L
out L
P P P M
− Γ − Γ
= =
− Γ Γ
• ML (load mismatch factor) (mismatch loss)
( source)
outΓ
2 2La b= Γ
20/25 Department of Electronic Engineering, NTUT

21. Transistor
[S]+
−
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
• (power gain) L
p
in
P
G
P
=
• (transducer power gain) L
T p s
AVS
P
G G M
P
= =
• (available power gain) AVN T
A
AVS L
P G
G
P M
= =
p TG G>
A TG G>
• p T AG G G= =
21/25 Department of Electronic Engineering, NTUT

22. Gp (Operating Power Gain)
( )
( )
2 2
2
2 2
1
1
1
2
1
1
2
L
L
p
in
in
b
P
G
P a
− Γ
= =
− Γ
21 1
2
221 L
S a
b
S
=
− Γ
2
2
212 2
22
11
1 1
L
p
in L
G S
S
− Γ
=
− Γ − Γ
• The Operating Power Gain Gp
where
Transistor
[S]+
−
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
slide 17
22/25 Department of Electronic Engineering, NTUT

23. GT (Transducer Power Gain)
• The Transducer Power Gain GT
in inL L
T p p s
AVS in AVS AVS
P PP P
G G G M
P P P P
= = = =
2 2 2 2
2 2
21 212 2 2 2
22 11
1 1 1 1
1 1 1 1
s L s L
T
s in L s out L
G S S
S S
− Γ − Γ − Γ − Γ
= =
− Γ Γ − Γ − Γ − Γ Γ
( )( )2 2
2
1 1
1
s in
s
s in
M
− Γ − Γ
=
− Γ Γ
where
Transistor
[S]+
−
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
slide 19
23/25 Department of Electronic Engineering, NTUT

24. GA (Available Power Gain)
• The Available Power Gain GA
AVN AVN AVNL T
A T
AVS AVS L L L
P P PP G
G G
P P P P M
= = = =
2
2
212 2
11
1 1
1 1
s
A
s out
G S
S
− Γ
=
− Γ − Γ
Transistor
[S]+
−
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
( )( )2 2
2
1 1
1
L out
L
out L
M
− Γ − Γ
=
− Γ Γ
where
slide 20
24/25 Department of Electronic Engineering, NTUT

25. •
(1) ( )
(2) (power waves, [Sp])
(3) (traveling waves, [S])
{ }Re 2L L LP V I∗
=
• ( )
2
2 2
,
1
2 8
s
AVS p p rms
s
E
P a a
R
= = =
2 2 21 1 1
2 2 2
L p p AVS pP a b P b= − = −
•
L p inP G P= L T AVSP G P=
• (defined with traveling waves, circuitries are
separately measured in a Zo system) :
25/25 Department of Electronic Engineering, NTUT