DATE: 27/11/06
ENE 490
Applied Communication Systems
Lecture 3 Stub matching, single- and two-port networks

 Loaded quality and node quality factor
Q
L
 f
0
BW
– At each node of the L-matching networks , there is an equivalent series input impedance, denoted by R
S
+jX
S
. Hence a circuit node Q defined at each node as n can be
Q n

X
S 
B
R
S
G
P
P
– The relationship between circuit node Q n loaded Q
L is
Q
L

Q n
2 and

 By adding another element, T and Pi matching networks, more flexibility in designing Q
L can be done.
 The loaded quality factor of the match network is usually estimated as simply the maximum circuit node quality factor Q n
.

a) Find the load reflection coefficient, the input impedance, and the VSWR in the transmission line shown below. The length of transmission line is

/8 and its characteristic impedance is 50

.

b) Evaluate V( and P(0).
 /8), I(  /8), V(0), I(0),

c) Find the length in cm of the  /8 T.L. at f = 1 GHz.

Design of a matching network with lumped and distributed components.
 popular design l
2 l
1
C
Z in
 have large tuning capabilities but very sensitive to the placement of the capacitor along the transmission line. Even a small deviations result in drastic changes in the input impedance

 Consider matching networks that consists of series transmission line connected to a parallel open-circuit or short-circuit stub.
a) short-circuit stub b) open-circuit stub

Zo=50
 y x y = ± jb s y in
1. Consider Y Smith chart since the stub is connected in shunt with the circuit.
2. The normalized admittance of shunt stub is written in the form of y sc
= jb s
( b s
> 0) for a capacitive susceptance and y sc
= -jb s susceptance, the length of
( b s l
1
> 0) for an inductive determines value of
 jb s
.
The admittance y x is given by y x
= y
L
+ y sc
= g
L
 jb s

3. Select b s such that admittance y in
, y x y x and is transformed to the y in are on a constant
 circle.
4. The length of series microstrip line change y x to y in
. l
2 is designed to
5. If an open-circuit stub is used, the design procedure is quite similar except that the length l
1 is read starting from an open-circuited termination. (i.e., starting from y = 0)

Ex2 For a load impedance of Z
L
= 60 - j45  , design a single stub matching networks that transform the load to a Z = 75 + j90  input impedance. Assume that both stub and transmission line have a characteristic impedance of Z
0
= 75  .





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.

I
1
+
V
1
-
Port 1
Two-port network
I
2
+
V
2
-
Port 2
 Assume the port-indexed current flows into the respective port and the associated voltage is recorded as indicated.

z Parameters v
1
= z
11 i
1
+ z
12 i
2 v
2
= z
21 i
1
+ z
22 i
2 or in the matrix form v
1 v z
11 z
12 i z
21 z
22 i

y, h,
ABCD
 y parameters i i
1
 h parameters y
11 y
12 v y
21 y
22 v i v
1 h
11 h
12 i h
21 h
22 v
 ABCD parameters i v
 



A B
C D



 v i
2
2


These two-port representations 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.

v v v v
1 a a
2

 v v
1 b b
2

 z z a
11 a
21

 z z b
11 b
21 z z a
12 a
22

 z z b
12 b
22

 i

i i
  i i
1 a  i i
1 a b
2

2 b

 y y a
11 a
21

 y y b
11 b
21 y y a
12 a
22

 y y b
12 b
22

 v

Two-port connected in cascade fashion i v
 

 
1 a i v
1



A a
B a
C a
D a




 v
 i a
2 a
2






A a
B a


A b
B b
C a
D a
C b
D b




 v
 i b
2 b
2




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.

Introduction of 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 input-output relations of a network in terms of incident and reflected power waves

Introduction of the normalized notation
(1)
Let’s define

Z
0
( )

( )

V

( )
Z
0


V

( )
Z
0

Z I

( )
Z I

( ).
we can write
( )

( )

( )
( )

( )

( ) and b x
  x a x

Introduction of the normalized notation
(2)
We can also show a(x) and b(x) in terms of V(x) and I(x) as a x

1
2 v x
 i x

2
1
Z
0
[ ( )

( )] and b x

1
2 v x
 i x

2
1
Z
0
[ ( )

( )]

 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 l
1 and port 2 has the length of l
2
, we can show that b l

S a l

12 2
( )
2
( )

S a l

22 2
( ) or in a matrix form, 
 b l S
S
11
21
S
S
12
22

 a l
2
( )


Observe that a
1
(l
1
), a
2
(l
2
), b
1
(l
1
), and b
2
(l
2
) are the values of incident and reflected waves at the specific locations denoted as port 1 and port 2.

The measurement of S parameters (1)
Input port a
1
(l
1
)
Output port a
2
(l
2
) a
1
(x)
Z
01 b
1
(x)
Two-port network b
2
(l
2
)
Port 2 x
2
=l
2 a
2
(x) b
2
(x)
Z
02 b
1
(l
1
)
Port 1 x
1
=l
1
 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.

The measurement of S parameters (2)
S
11
 b l a l
| a l

S
21
 a l
S
22

|
| a l
 a l

(input reflection coefficient with output properly terminated)
(forward transmission coefficient with output properly terminated)
(output reflection coefficient with input properly terminated)
S
12
 b l
| a l

(reverse transmission coefficient with input properly terminated)

The advantages of using S parameters
 They are measured using a matched termination.
S
11
 a l
 Z
OUT a
1
(l
1
) a
2
(l
2
)=0
Z
1
=Z
01
+
E
1
-
Z
01 b
1
(l
1
)
Two-port network b
2
(l
2
)
Z
02
Z
2
=Z
02
Port 1 x
1
=l
1
Port 2 x
2
=l
2
 Using matched resistive terminations to measure the
S parameters of a transistor results in no oscillation.

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.

 a l b l
T
T
11
21
T
T
12
22


2
2
( )
( )


 The relationship between S and T parameters can be developed. Namely,


T
11
T
12
T
21
T
22
 



S
21
S
1
11
S
21
S
12


S
S
22
21
S
21





The chain scattering parameters or scattering transfer parameters (T parameters) (2) and


S
11
S
12
S
21
S
22
 



T
21
T
11
T
1
11
We can also write
T
22

T
11

T
12
T
11




.

 a b
1 x
1 x


T x
11
T x
12


T y
11
T y
12
T x
21
T x
22
T y
21
T y
22


 b a
2
2 y y



.