ENE 428
Microwave
Engineering
Lecture 9 Scattering parameters and
their properties.
1
Impedance and Admittance
Matrices
• Consider an arbitrary N-port network below,
Vn  Vn  Vn
I n  I n  I n
2
The impedance [Z] matrix relates
voltages and currents.
 V1 
Z
Z

11
12
 
 V2   Z 21
 
  
   Z N 1
 VN 
I
Z1N   1 
  I2 
 
 
 
Z NN   
 IN 
So we can write [V] =[Z][I]
V1 = Z11I1 + Z12I2
V2 = Z21I1 + Z22I2,
etc.
3
The admittance [Y] matrix relates
currents and voltages.
 I1 
   Y11 Y12
 I 2   Y21
 
  
   YN 1
 IN 
V
Y1N   1 
  V2 
 
 
 
YNN   
 VN 
So we can write [I] =[Y][V]
I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2,
etc.
4
Zij or Yij can be found by o/c or
s/c at all other ports
Vi
Zij 
Ij
I k 0 for k  j
and
Ii
Yij 
Vj
Vk 0 for k  j
• Zij can be found by driving port j with the current Ij, open
-circuiting all other ports and measuring the open-circuit
Voltage at port i.
• Yij can be found by driving port j with the voltage Vj,
short-circuiting all other ports and measuring the shortcircuit current at port i.
5
Reciprocal Network
• Many practical networks are reciprocal (not containing
any nonreciprocal media such as ferrites or plasmas, or
active devices)
• Impedance and admittance matrices are symmetric, that
is
Zij  Z ji
and
Yij  Yji .
6
Lossless Network
• If the network is lossless, then the net real power
delivered to the network must be zero. Thus, Re{Pav} = 0.
• Then for a reciprocal lossless N-port junction we can
show that the elements of the [Z] and [Y] matrices must be
pure imaginary
Re{Zmn }  0
where m, n = port index.
7
Single- and Two-port
Networks
• The analysis can be done easily through simple input-output
relations.
• Input and output port parameters can be determined without the
need to know inner structure of the system.
• At low frequencies, the z, y, h, or ABCD parameters are basic
network input-output parameter relations.
• 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.
8
Basic definitions
I1
I2
+
+
V1
Port 1
Two-port
network
V2
Port 2
• Assume the port-indexed current flows into the respective port
and the associated voltage is recorded as indicated.
9
Ex of h and ABCD parameters
for two-port network
• H parameters
 V1   H11 H12  I1 
 
 
I
H
H
 2   21
22  V2 
• ABCD parameters
 V1   A B   V2 
 



 I1   C D    I 2 
These two-port representations (Z, Y, H, and ABCD) are
very useful at low frequencies because the parameters
are
readily measured using short- and open- circuit tests at
the terminals of the two-port network.
10
Two-port connected in series
a
b  a
b

v
v

v
z

z
 1
1
1
11
11



   a b  a
b
v
v

v
z

z
 2   2 2  21 21
a
b 
z12
 z12
 i1 

a
b i 
z22  z22   2 
11
Two-port connected in shunt
a
b  a
b

i
i

i
y

y
 1
1
1
11
11



   a b  a
b
i
i

i
y

y
 2   2 2  21 21
a
b 
y12
 y12
 v1 

a
b v 
y22  y22   2 
12
Two-port connected in cascade
fashion
a
a


v
v
A
 1
1


   a    a
 i1   i1   C
B a   v2a   Aa
  a    a
a 
D   i2   C
B a  Ab

a  b
D  C
Bb   v2b 


b  b 
D   i2 
13
Disadvantages of using these
parameters at RF or microwave
frequency
• Difficult to directly measure V and I
• Difficult to achieve open circuit due to stray capacitance
• Active circuits become unstable when terminated in
short- and open- circuits.
14
Scattering Matrix (1)
• The scattering matrix relates the voltage waves incident
on the ports to those reflected from the ports
• Scattering parameters can be calculated using network
analysis techniques or measured directly with a network
analyzer.
 V1 
    S11
 V2   S
    21
  
  S
V    N1
 N
S12
 V1 
S1N   
  V2 
 
 
 
S NN    
 VN 
15
Scattering Matrix (2)
• A specific element of the [S] matrix can be determined
as

Vi
Sij  
Vj
V  0 for k  j
k
.
• Sii is the reflection coefficient seen looking into port i
when all other ports are terminated in matched
loads.
• Sij is the transmission coefficient from port j to port i
when all other ports are terminated in matched
loads.
16
Reciprocal networks and lossless
networks
• [S] matrix for a reciprocal network is symmetric,
[S]=[S]t.
• [S] matrix for a lossless network is unitary that means
[S ]  {[S ]t }1.
17
Ex1 A two-port network has this following
scattering matrix
 0.15 0
[S ]  
0.85 45
0.85 45 

0.2 0 
Determine if the network is reciprocal, and lossless
18
Introduction of generalized
scattering parameters (S
parameters)
1.Measure power and phase
2.Use matched loads
3.Devices are usually stable with matched loads.
S- parameters are power wave descriptors that permits us to define inputoutput relations of a network in terms of incident and reflected power
waves
19
Introduction of the normalized
notation (1)
Let’s define
we can write
v ( x )  a ( x )  b( x )
V ( x)
v( x) 
Z0
i ( x)  Z 0 I ( x)

V ( x)
a( x) 
 Z 0 I  ( x)
Z0
i ( x )  a ( x )  b( x )
and
b( x)  ( x)a( x).
V  ( x)
b( x ) 
 Z 0 I  ( x).
Z0
20
Introduction of the normalized
notation (2)
We can also show a(x) and b(x) in terms of V(x) and I(x) as
1
1
a( x)  [v( x)  i( x)] 
[V ( x)  Z0 I ( x)]
2
2 Z0
and
1
1
b( x)  [v( x)  i( x)] 
[V ( x)  Z0 I ( x)]
2
2 Z0
21
Normalized wave generalization
• For a two-port network, we can generalize the relationship
between b(x) and a(x) in terms of scattering parameters. Let
port 1 has the length of l1 and port 2 has the length of l2, we
can show that
b1 (l1 )  S11a1 (l1 )  S12 a2 (l2 )
b2 (l2 )  S21a1 (l1 )  S22 a2 (l2 )
or in a matrix form,
 b1 (l1 )   S11 S12  a1 (l1 ) 




 b2 (l2 )   S21 S22  a2 (l2 ) 
Observe that a1(l1), a2(l2), b1(l1), and b2(l2) are the values of incident and reflected waves at the specific locations denoted as
port 1 and port 2.
22
The measurement of S parameters
(1)
a1(x)
Input
port
Output
port
a1(l1)
a2(l2)
Two-port
network
Z01
b1(x)
a2(x)
Z02
b1(l1)
b2(l2)
Port 1
x1=l1
Port 2
x2=l2
b2(x)
• The S parameters are seen to represent reflection and
transmission coefficients, the S parameters measured at the
specific locations shown as port 1 and port 2 are defined in the
following page.
23
The measurement of S parameters
(2)
b1 (l1 )
S11 
|a2 (l2 ) 0
a1 (l1 )
(input reflection coefficient with output
properly terminated)
b2 (l2 )
S 21 
|a2 (l2 ) 0
a1 (l1 )
(forward transmission coefficient with
output properly terminated)
S 22
b2 (l2 )

|a1 (l1 )0
a2 (l2 )
b1 (l1 )
S12 
|a2 (l1 ) 0
a2 (l2 )
(output reflection coefficient with input
properly terminated)
(reverse transmission coefficient with
input properly terminated)
24
The advantages of using S
parameters
•
They are measured using a matched termination.
S11 
b1 (l1 )
a1 (l1 )
a2 ( l2 ) 0
a1(l1)
Z1=Z01
+
E1
-
•
ZOUT
a2(l2)=0
Two-port
network
Z01
Z02
b1(l1)
b2(l2)
Port 1
x1=l1
Port 2
x2=l2
Z2=Z02
Using matched resistive terminations to measure the S parameters
of a transistor results in no oscillation.
25
The chain scattering parameters or scattering
transfer parameters (T parameters) (1)
• The T parameters are useful in the analysis of cascade connections
of two-port networks.
 a1 (l1 )   T11 T12  b2 (l2 ) 




b
(
l
)
T
T
a
(
l
)
 1 1   21 22  2 2 
• The relationship between S and T parameters can be developed.
Namely,
 T11

 T21
 1
T12   S21
  
T22 
S11
S
 21


.
S11S22 
S12 
S21 

S22
S21
26
The chain scattering parameters or
scattering transfer parameters (T
parameters) (2)
and
 S11

 S21
T21T12 
 T21
T22 

T11
T11 
S12 
.
  

S22 
T12
1

T

T
 11
11

We can also write
x
x
y
y
 a1x   T11 T12  T11 T12   b2 y 
.
 y
 

    x
x
y


 b1x   T21 T22 
 T21 T22   a2 y 
27
Review (2)
• 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 ) 
28
Shifting reference planes
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.
29
Scattering matrix of the shifting
planes
• At the reference planes at port 1 and port 2, we write the scattering
matrix as
 b1 (l1 )   S11 S12   a1 (l1 ) 




 b2 (l2 )   S21 S22   a2 (l2 ) 
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 2 q2   a (0) 
S22e
 2 
30