ENE 490Applied Communication

Systems

ENE 490Applied Communication

Systems

Lecture 3 Stub matching, single- and two-port networks

DATE: 27/11/06

Review (1)Review (1) Loaded quality and node quality factor

– At each node of the L-matching networks, there is an equivalent series input impedance, denoted by RS +jXS. Hence a circuit node Qn can be defined at each node as

– The relationship between circuit node Qn and loaded QL

is

0L

fQ

BW

S Pn

S P

X BQ

R G

2n

LQ

Q

Review (2)Review (2)

By adding another element, T and Pi matching networks, more flexibility in designing QL can be done.

The loaded quality factor of the match network is usually estimated as simply the maximum circuit node quality factor Qn.

Ex1 an extra T.L.s problemEx1 an extra T.L.s problema) 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(/8), I(/8), V(0), I(0), and P(0).b) Evaluate V(/8), I(/8), V(0), I(0), and P(0).

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

Design of a matching network with lumped and distributed components.

popular design

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

l

Z L

Z in

l

C

2 1

Single-stub matching networksSingle-stub matching networks

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

Procedures (1)Procedures (1)

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 ysc= jbs (bs > 0) for a capacitive susceptance

and ysc= -jbs (bs > 0) for an inductive susceptance, the

length of l1 determines value of jbs. The admittance yx

is given by

yx = yL + ysc = gL jbs

y

Zo=50

Z

Zo=

50

L

x

yin

y = ± jbsc s

Procedures (2)Procedures (2)

3. Select bs such that yx is transformed to the admittance yin,

yx and yin are on a constant circle.

4. The length of series microstrip line l2 is designed to

change yx to yin.

5. If an open-circuit stub is used, the design procedure is quite similar except that the length l1 is read starting from

an open-circuited termination. (i.e., starting from y = 0)

Ex2 For a load impedance of ZL = 60 - j45 , design a single stub matching networks that transform the load to a Zin = 75 + j90 input impedance. Assume that both stub and transmission line have a characteristic impedance of Z0 = 75 .

Ex2 For a load impedance of ZL = 60 - j45 , design a single stub matching networks that transform the load to a Zin = 75 + j90 input impedance. Assume that both stub and transmission line have a characteristic impedance of Z0 = 75 .

Single- and Two-port NetworksSingle- 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.

Basic definitions Basic definitions

Assume the port-indexed current flows into the respective port and the associated voltage is recorded as indicated.

Two-portnetwork

Port 1 Port 2

V1

+

-

V2

+

-

I1 I2

z Parametersz Parameters

v1 = z11i1 + z12i2v2 = z21i1 + z22i2

or in the matrix form

1 11 12 1

2 21 22 2

v z z i

v z z i

y, h, and ABCD parametersy, h, and ABCD parameters y parameters

h parameters

ABCD parameters

1 11 12 1

2 21 22 2

i y y v

i y y v

1 11 12 1

2 21 22 2

v h h i

i h h v

1 2

1 2

v vA B

i iC D

These two-portrepresentations are very useful at lowfrequencies becausethe parameters arereadily measured usingshort- and open- circuittests at the terminals ofthe two-port network.

Two-port connected in seriesTwo-port connected in series

1 1 1 11 11 12 12 1

2 22 2 21 21 22 22

a b a b a b

a b a b a b

v v v z z z z i

v iv v z z z z

Two-port connected in shuntTwo-port connected in shunt

1 1 1 11 11 12 12 1

2 22 2 21 21 22 22

a b a b a b

a b a b a b

i i i y y y y v

i vi i y y y y

Two-port connected in cascade fashionTwo-port connected in cascade fashion

1 1 2 2

1 1 2 2

a a ba a a a b b

a a ba a a a b b

v v v vA B A B A B

i i i iC D C D C D

Disadvantages of using these parameters at RF or microwave frequency

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)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)Introduction of the normalized notation (1)

0

0

00

00

( )( )

( ) ( )

( )( ) ( )

( )( ) ( ).

V xv x

Z

i x Z I x

V xa x Z I x

Z

V xb x Z I x

Z

we can write Let’s define

( ) ( ) ( )

( ) ( ) ( )

v x a x b x

i x a x b x

and

( ) ( ) ( ).b x x a x

Introduction of the normalized notation (2)Introduction of the normalized notation (2)

We can also show a(x) and b(x) in terms of V(x) and I(x) as

00

1 1( ) [ ( ) ( )] [ ( ) ( )]

2 2 a x v x i x V x Z I x

Z

and

00

1 1( ) [ ( ) ( )] [ ( ) ( )]

2 2 b x v x i x V x Z I x

Z

Normalized wave generalizationNormalized 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

1 1 11 1 1 12 2 2

2 2 21 1 1 22 2 2

( ) ( ) ( )

( ) ( ) ( )

b l S a l S a l

b l S a l S a l

or in a matrix form, 1 1 11 12 1 1

2 2 21 22 2 2

( ) ( )

( ) ( )

b l S S a l

b l S S a l

Observe that a1(l1), a2(l2), b1(l1), and b2(l2) are the values of in-cident and reflected waves at the specific locations denoted as port 1 and port 2.

The measurement of S parameters (1)The measurement of S parameters (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.

Two-portnetwork

Input port

Output port

Z01

Port 1x1=l1

a1(x)

b1(x)

a1(l1)

b1(l1)

Port 2x2=l2

Z02

a2(x)

b2(x)

a2(l2)

b2(l2)

The measurement of S parameters (2)The measurement of S parameters (2)

2 2

2 2

1 1

2 1

1 111 ( ) 0

1 1

2 221 ( ) 0

1 1

2 222 ( ) 0

2 2

1 112 ( ) 0

2 2

( )|

( )

( )|

( )

( )|

( )

( )|

( )

a l

a l

a l

a l

b lS

a l

b lS

a l

b lS

a l

b lS

a l

(input reflection coefficient with output properly terminated)

(forward transmission coefficient with output properly terminated)

(output reflection coefficient with input properly terminated)

(reverse transmission coefficient with input properly terminated)

The advantages of using S parametersThe advantages of using S parameters

They are measured using a matched termination.

Using matched resistive terminations to measure the S parameters of a transistor results in no oscillation.

Two-portnetwork

Port 1x1=l1

a1(l1)

b1(l1)

Port 2x2=l2

a2(l2)=0

b2(l2)E1

+

-

Z2=Z02

ZOUT2 2

1 111 ( ) 0

1 1

( )( ) a l

b lS

a l

Z1=Z01

Z01 Z02

The chain scattering parameters or scattering transfer parameters (T parameters) (1)

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.

The relationship between S and T parameters can be developed. Namely,

1 1 11 12 2 2

1 1 21 22 2 2

( ) ( )

( ) ( )

a l T T b l

b l T T a l

22

21 2111 12

21 22 11 11 2212

21 21

1

S

S ST T

T T S S SS

S S

The chain scattering parameters or scattering transfer parameters (T parameters) (2)

The chain scattering parameters or scattering transfer parameters (T parameters) (2)

21 21 1222

11 1111 12

21 22 12

11 11

.1

T T TT

T TS S

S S T

T T

and

We can also write

21 11 12 11 12

1 221 22 21 22

.

x x y yyx

x x y yx y

ba T T T T

b aT T T T