
Two-port network
- At low frequencies, the z, y, h, or ABCD parameters are
basic network input-output
parameter relations. The
parameters are readily measured using short- and opencircuit tests at the terminals.
- At RF or microwave frequency, these parameter are
difficult to measure
- At high frequencies (in microwave range), scattering
parameters (S parameters) are defined in terms of
traveling waves and completely characterize the
behavior of two-port networks.
- S parameters can be easily measured using matched
loads which ensure the stability of the network.
 Normalized notation of the incident a(x) and reflected
waves b(x) are defined as
V  ( x)
a( x) 
 Z0 I  ( x )
Z0
V  ( x)
b( x ) 
 Z0 I  ( x )
Z0
 The relationship between the incident and reflected waves
and the scattering matrix of the two-port network,
 b1 (l1 )   S11


 b2 (l2 )   S 21
S12   a1 (l1 ) 


S 22   a2 (l2 ) 
Reference
planes
q1bl1
a1(0)
a1(l1)
q2bl2
a2(l2)
a2(0)
b2(l2)
b2(0)
Two-port
network
b1(0)
Port 1'
x1=0

b1(l1)
Port 1
x1=l1
Port 2
x2=l2
Port 2'
x2=0
S parameters are measured using traveling
waves, the positions where the measurements
are made are needed to be specified. The
positions are called reference planes.

At the reference planes at port 1 and port 2, we
write the scattering matrix as
 b1 (l1 )   S11 S12   a1 (l1 ) 




S
S
b
(
l
)
a
(
l
)
 2 2   21 22   2 2 
and at port 1’ and port 2’ as
'
 b1 (0)   S 11

   '
 b2 (0)   S 21

S '12   a1 (0) 

'   a (0) 
S 22   2 
We can show that
 j 2q
 b1 (0)   S11e 1

  
 j ( q q )
 b2 (0)   S21e 1 2
S12e j (q1 q2 )   a1 (0) 

 j 2q2   a (0) 
S22e
 2 

The average power associated with incident
wave on the primed i th port (i.e., at x1=0 and x2
= 0) can be expressed as
1




Pi (0)  Re[Vi,rms (0)( Ii,rms (0)) ]  Re[Vi  (0)( Ii (0)) ]
2
1
2
 ai (0) .
2
Similarly, the average reflected power is
1
2
Pi (0)  bi (0)
2

ZT1
I1(0)
Z1=Z01
+
E1
-
+
V1(0)
Port 1'
x1=0

a1(x)
I1(l1)
I2(l2)
+
+
Z01
V1(l1)
b1(x)
Port 1
x1=l1
Two-port
network
I2(0)
a2(x)=0
V2(l2)
Z02
-
b2(x)
Port 2
x2=l2
+
V2(0)
Port 2'
x2=0
Z2=Z02
-
When port 1’ is excited by the sinusoidal voltage
source with source impedance Z1 = Z01 and port
2’ is matched (Z2 = Z02), we can show that
2
E
1
2

P1 (0)  a1 (0)  1  PAVS
2
8Z01
Note: PAVS represents the power available from
the source
If Z1 is not equal to Z01, the net power delivered to port
1’ or to port 1 (the line is lossless, the delivered power
is then equal) is then

1
1
2
2
a1 (0)  b1 (0)
2
2
1
2
P1 (0)  P1 (l1 )  PAVS  b1 (0)
2
P1 (0) 
or

The power delivered to the load when Z2 = Z02 can
then be derived similarly and can expressed as
P2 (0) 
1
2
b2 (0)
2
Note: PAVS represents the power available from the
source
2

S11 represents the ratio of the power reflected from
port 1 to the power available at port 1.
From
b (l )
S11  1 1
a1 (l1 )
a2 (l2 ) 0
V1 (l1 )
 
V1 (l1 )
V2 (l2 ) 0
then we can write
Zin  Z 0
S11  in 
Zin  Z 0
and
 S21 represents a forward voltage transmission
coefficient from port 1 to port 2.
2
P1 (l1 )  P1 (0)  PAVS (1  S11 )
b2 (l2 )
S21 
a1 (l1 )
a2 ( l2 ) 0

Z 02 I 2 (l2 )
I 2 ( l2 ) 0

Z 01 I1 (l1 )
ZT1
I1(l1)
Z01
+
E1,TH
-
+
I2(l2)
a1(l1)
Two-port
network
V1(l1)
-
a2(l2)=0
b1(l1)
+
V2(l2)
b2(l2)
Port 2
x2=l2
Port 1
x1=l1
2 Z 01 V2 (l2 )
S 21 
Z 02 E1,TH
Z02

2
S 21 represents the ratio of power deliver to the load
Z02 to the power available from the source,
E1, TH. The ratio is known as transducer power gain GT.
2
2
S21 
1 V2 (l2 )
Z02
2
E1,TH
2
8Z01

S22  out
Z out  Z 0

Z out  Z 0
represents the ratio of power
reflected from port 2 to the power available at port 2,
PAVN or power available from the network.

S12 represents a reverse voltage transmission
coefficient from port 2 to port 1.
2 Z 02 V1 (l1 )
S12 
Z 01 E2,TH
2
 S12
represents a reverse transducer power
gain.
2
1 V1 (l1 )
Z 01
2
2
S12 
2
E2,TH
8Z 02

These S parameters are measured in Z0 system. If Z1
and Z2 are arbitrary then the gain GT is no longer
2
S
equal to 21 .


The analysis of lumped circuits (from one-port to nport lumped circuits) in terms of a new set of waves,
called power waves.
One-port network:
+
ZS
V
+
ES
-
ap
ZS
ZL
-
1
[V  Z S I ]
2 RS
1
bp 
[V  Z S I ]
2 RS
ZL
+
ES
-
bp
ap 
where RS = Re[ZS].
2
1
These definitions are such that the quantity a p is
2
equal to the power available from the source, and
the reflected power wave bp is zero when the load
impedance is conjugately matched to the source
impedance (i.e. when ZL = ZS* )
2
ES
2
1
PAVS  a p 
2
8RS
 Power delivered to the load
2
1 2
1 ES
PL  I Re  Z L  
Re  Z L .
2
2 ZS  ZL
It can be shown as

2
1
1 2
PL  a p  bp .
2
2


The reflected power
2
1
bp  PAVS  PL
2
A power-wave reflection coefficient P
Z L  Z S
P 

a p Z L  Z S
bp

Voltage and current can be expressed as a function of
incident and reflected power waves as
V
1
[Z S a p  Z S bp ]
RS
I
1
[a p  bp ].
RS
and

We can also define incident and reflected voltage and
currents and relate them to the power waves. That is, let
V  V p  V p
where
V p

Z S
RS
ap
and
V p

ZS
RS
bp .
I  I p  I p
and
where
I p

ap
RS
and
I p

bp
RS
.

A voltage reflection coefficient V can be defined
as
V p  Z S Z L  Z S
V    
.

Vp
ZS ZL  ZS

A current reflection coefficient I can be defined as
I 

I p
Ip


bp
ap
 P.
When the normalizing impedance ZS is real and
positive and equal to Z0 or ZS = ZS* = Z0,
Z L  Z0
0   P  V   I 
.
Z L  Z0
Z T1
I1
+
a p1
Z1
+
E1
-
I2
ap2
Two-port
Network
V1
+
V2
(Sp parameters)
-
bp1
bp2
-
Two-port network representation in terms of
generalized S parameters.
Z2
+
E2
-
Z T1
I1
+
a p1
Z1
+
E1
-
I2
ap2=0
Two-port
Network
V1
+
V2
(Sp parameters)
-
bp1
bp2
-
Z2

Generalized scattering parameters (Sp parameters)
denoted by Sp11, Sp12, Sp21, and Sp22 can be shown in
terms of power waves as follows:
bp1  S p11a p1  S p12a p 2
bp2  S p21a p1  S p22a p2
The input power to the two-port network can be
expressed as
2
2
2
1
1
PIN  a p1  bp1  PAVS (1  S p11 )
2
2
ZT 1  Z1
where S p11 
is measured when ap2 = 0 or E2 = 0.
ZT 1  Z1


The power delivered to the load Z2 is
2
2
2
1
1
PL  bp 2  S p 21 a p1 .
2
2

Therefore the transducer power gain GT is given by
2
PL
GT 
 S p21 .
PAVS

If we let Z1 = Z2 = Z0, it follows that
2
2
GT  S p21  S21 .