Networks - Rowan Universityusers.rowan.edu/~krchnavek/Rowan_University/RF... · 2018. 2. 1. ·...

Single- and Multiport Networks

RF Electronics Spring, 2018

Robert R. Krchnavek Rowan University

Objectives

• Generate an understanding of the common network representations of Z, Y, h, and ABCD.

• To be able to obtain matrix parameters by suitable tests on the networks.

• To be able to convert from one network representation to another.

• Understand the implications of interconnecting networks.

• Develop a working knowledge of S-parameters.

Single- and Multiport Networks

• Single- and multiport network representations consist of input/output relationships without knowing the internal network.

• The relationships can be determined experimentally.

• Provides a means of designing/analyzing networks with overall system performance in mind.

• Very important in RF/MW work because detailed modeling of individual devices requires complex EM work.

Single- and Multiport Networks Voltage and Current Definitions

One-port

NetworkTwo-port

Network

Multiport

Network

+

v1

–

+

v2

–

+

v1

–

+v1–

i1

Port 1

+v2–

i2

Port 2

+vN–

iN

Port N

+vN-1–

i1

i2

i1

iN-1

Port N-1

Port 2Port 1

Z-Matrix Representation

The voltage at each port is given by

This is easily represented as a matrix

3

Z(d) =V (d)I(d)

= Z0

1 + �0

e�2↵�d

1� �0

e�2↵�d

�(d) = �0

e�2↵�d

Z(d) = Z0

1 + �(d)1� �(d)

Z(d) = Z0

ZL + �Z0

tan�d

Z0

+ �ZL tan�d

SWR =|V

max

||V

min

| =|I

max

||I

min

|

SWR =1 + |�

0

|1� |�

0

|

Z(d) = Z0

1 + �(d)1� �(d)

z(d) =Z(d)Z

0

=1 + �(d)1� �(d)

y(d) =Y (d)Y

0

=Z

0

Z(d)=

1� �(d)1 + �(d)

�(d) = �0

e�2↵�d

1 + �0

e�2↵�d

1� �0

e�2↵�d⇤ 1 + �

0

e�↵(2�d+⇥)

1� �0

e�↵(2�d+⇥)

=1� �

0

e�2↵�d

1 + �0

e�2↵�d=

1� �(d)1 + �(d)

v1

= Z11

i1

+ Z12

i2

+ . . . + Z1N iN

v2

= Z21

i1

+ Z22

i2

+ . . . + Z2N iN

......

vN = ZN1

i1

+ ZN2

i2

+ . . . + ZNN iN⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Z11

Z12

· · · Z1N

Z21

Z22

· · · Z2N

......

. . ....

ZN1

ZN2

· · · ZNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧

Znm =vn

im

��ik=0 (for k ⇥=m)

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Y11

Y12

· · · Y1N

Y21

Y22

· · · Y2N

......

. . ....

YN1

YN2

· · · YNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧

Ynm =invm

��vk=0 (for k ⇥=m)

3

Z(d) =V (d)I(d)

= Z0

1 + �0

e�2↵�d

1� �0

e�2↵�d

�(d) = �0

e�2↵�d

Z(d) = Z0

1 + �(d)1� �(d)

Z(d) = Z0

ZL + �Z0

tan�d

Z0

+ �ZL tan�d

SWR =|V

max

||V

min

| =|I

max

||I

min

|

SWR =1 + |�

0

|1� |�

0

|

Z(d) = Z0

1 + �(d)1� �(d)

z(d) =Z(d)Z

0

=1 + �(d)1� �(d)

y(d) =Y (d)Y

0

=Z

0

Z(d)=

1� �(d)1 + �(d)

�(d) = �0

e�2↵�d

1 + �0

e�2↵�d

1� �0

e�2↵�d⇤ 1 + �

0

e�↵(2�d+⇥)

1� �0

e�↵(2�d+⇥)

=1� �

0

e�2↵�d

1 + �0

e�2↵�d=

1� �(d)1 + �(d)

v1

= Z11

i1

+ Z12

i2

+ . . . + Z1N iN

v2

= Z21

i1

+ Z22

i2

+ . . . + Z2N iN

......

vN = ZN1

i1

+ ZN2

i2

+ . . . + ZNN iN⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Z11

Z12

· · · Z1N

Z21

Z22

· · · Z2N

......

. . ....

ZN1

ZN2

· · · ZNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧

Znm =vn

im

��ik=0 (for k ⇥=m)

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Y11

Y12

· · · Y1N

Y21

Y22

· · · Y2N

......

. . ....

YN1

YN2

· · · YNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧

Ynm =invm

��vk=0 (for k ⇥=m)

Z-Matrix Representation

Each element in the Z-matrix is determined as follows

Note: This involves setting the current on all the other ports equal to 0. This is done using an open circuit.

Recall, true open circuits are difficult to create at RF/MW frequencies because of capacitance.

3

Z(d) =V (d)I(d)

= Z0

1 + �0

e�2↵�d

1� �0

e�2↵�d

�(d) = �0

e�2↵�d

Z(d) = Z0

1 + �(d)1� �(d)

Z(d) = Z0

ZL + �Z0

tan�d

Z0

+ �ZL tan�d

SWR =|V

max

||V

min

| =|I

max

||I

min

|

SWR =1 + |�

0

|1� |�

0

|

Z(d) = Z0

1 + �(d)1� �(d)

z(d) =Z(d)Z

0

=1 + �(d)1� �(d)

y(d) =Y (d)Y

0

=Z

0

Z(d)=

1� �(d)1 + �(d)

�(d) = �0

e�2↵�d

1 + �0

e�2↵�d

1� �0

e�2↵�d⇤ 1 + �

0

e�↵(2�d+⇥)

1� �0

e�↵(2�d+⇥)

=1� �

0

e�2↵�d

1 + �0

e�2↵�d=

1� �(d)1 + �(d)

v1

= Z11

i1

+ Z12

i2

+ . . . + Z1N iN

v2

= Z21

i1

+ Z22

i2

+ . . . + Z2N iN

......

vN = ZN1

i1

+ ZN2

i2

+ . . . + ZNN iN⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Z11

Z12

· · · Z1N

Z21

Z22

· · · Z2N

......

. . ....

ZN1

ZN2

· · · ZNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧

Znm =vn

im

��ik=0 (for k ⇥=m)

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Y11

Y12

· · · Y1N

Y21

Y22

· · · Y2N

......

. . ....

YN1

YN2

· · · YNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧

Ynm =invm

��vk=0 (for k ⇥=m)

Y-Matrix Representation

An alternative representation is the admittance or Y-matrix representation.

And, the individual elements are determined by

3

Z(d) =V (d)I(d)

= Z0

1 + �0

e�2↵�d

1� �0

e�2↵�d

�(d) = �0

e�2↵�d

Z(d) = Z0

1 + �(d)1� �(d)

Z(d) = Z0

ZL + �Z0

tan�d

Z0

+ �ZL tan�d

SWR =|V

max

||V

min

| =|I

max

||I

min

|

SWR =1 + |�

0

|1� |�

0

|

Z(d) = Z0

1 + �(d)1� �(d)

z(d) =Z(d)Z

0

=1 + �(d)1� �(d)

y(d) =Y (d)Y

0

=Z

0

Z(d)=

1� �(d)1 + �(d)

�(d) = �0

e�2↵�d

1 + �0

e�2↵�d

1� �0

e�2↵�d⇤ 1 + �

0

e�↵(2�d+⇥)

1� �0

e�↵(2�d+⇥)

=1� �

0

e�2↵�d

1 + �0

e�2↵�d=

1� �(d)1 + �(d)

v1

= Z11

i1

+ Z12

i2

+ . . . + Z1N iN

v2

= Z21

i1

+ Z22

i2

+ . . . + Z2N iN

......

vN = ZN1

i1

+ ZN2

i2

+ . . . + ZNN iN⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Z11

Z12

· · · Z1N

Z21

Z22

· · · Z2N

......

. . ....

ZN1

ZN2

· · · ZNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧

Znm =vn

im

��ik=0 (for k ⇥=m)

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Y11

Y12

· · · Y1N

Y21

Y22

· · · Y2N

......

. . ....

YN1

YN2

· · · YNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧

Ynm =invm

��vk=0 (for k ⇥=m)

3

Z(d) =V (d)I(d)

= Z0

1 + �0

e�2↵�d

1� �0

e�2↵�d

�(d) = �0

e�2↵�d

Z(d) = Z0

1 + �(d)1� �(d)

Z(d) = Z0

ZL + �Z0

tan�d

Z0

+ �ZL tan�d

SWR =|V

max

||V

min

| =|I

max

||I

min

|

SWR =1 + |�

0

|1� |�

0

|

Z(d) = Z0

1 + �(d)1� �(d)

z(d) =Z(d)Z

0

=1 + �(d)1� �(d)

y(d) =Y (d)Y

0

=Z

0

Z(d)=

1� �(d)1 + �(d)

�(d) = �0

e�2↵�d

1 + �0

e�2↵�d

1� �0

e�2↵�d⇤ 1 + �

0

e�↵(2�d+⇥)

1� �0

e�↵(2�d+⇥)

=1� �

0

e�2↵�d

1 + �0

e�2↵�d=

1� �(d)1 + �(d)

v1

= Z11

i1

+ Z12

i2

+ . . . + Z1N iN

v2

= Z21

i1

+ Z22

i2

+ . . . + Z2N iN

......

vN = ZN1

i1

+ ZN2

i2

+ . . . + ZNN iN⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Z11

Z12

· · · Z1N

Z21

Z22

· · · Z2N

......

. . ....

ZN1

ZN2

· · · ZNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧

Znm =vn

im

��ik=0 (for k ⇥=m)

⇥

⌃⌃⌃⌃⌅

i1

i2

...iN

⇤

⌥⌥⌥⌥⇧=

⇥

⌃⌃⌃⌃⌅

Y11

Y12

· · · Y1N

Y21

Y22

· · · Y2N

......

. . ....

YN1

YN2

· · · YNN

⇤

⌥⌥⌥⌥⇧

⇥

⌃⌃⌃⌃⌅

v1

v2

...vN

⇤

⌥⌥⌥⌥⇧

Ynm =invm

��vk=0 (for k ⇥=m)

Z- and Y-Matrix

and

4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0

⇤ reflected power wave at port 1incident power wave at port 1

��NO incident power at port 2

S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



ABCD and h-matrix Representation

The hybrid, or h-matrix for a two-port network is given by

The definition for the individual h parameters is

The ABCD matrix for a two-port network is given by

The h-matrix is often used to characterize a transistor (at low frequencies) and the importance of the ABCD matrix will

be seen shortly.

4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



Interconnecting Networks

• Networks can be connected to other networks to create larger networks.

• The larger network representations can often be determined by the interconnection pattern of the sub-networks.

• Series, parallel and cascading interconnections are common.

• Cascading networks are the most common in RF/MW design.

Cascading Networks and the ABCD Representation

4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



Figure 4-9

Cascading two networks.

port 1 port 2

+

+

v1

v2

i1

i2

-

–

v1 v2

i1 i2

' "A

'

'

'

'

" "

" "+

–

'B'C 'D

A BC D

"""

+

v2

i2

-

v1

i1'

'

"

"v2

i2"

"

+

–

+v1

i1

–'

'

ABCD-Representation for Common Two-Port

Networks

Table 4-1

ABCD-

Parameters of Some Useful Two-Port Circuits.

Circuit

ABCD

-Parameters

Zi1 i2

v1 v2

A 1=

C 0=

B Z=

D 1=

Y

i1 i2

v1 v2

A 1=

C Y=

B 0=

D 1=

ZA ZB

ZC

i1 i2

v1 v2

A 1ZA

ZC------+=

C1

ZC------=

B ZA ZBZAZB

ZC-------------+ +=

D 1ZB

ZC------+=

YA

YB

YC

i1 i2

v1 v2

A 1YB

YC

------+=

C YA YB

Y AYB

YC

-------------+ +=

B1

YC

------=

D 1Y A

YC

------+=

l

Z ,0 β

i1 i2

v1 v2

d

A βlcos=

Cj βlsin

Z0

----------------=

B jZ0

βlsin=

D βlcos=

N:1i1 i2

v1 v2

A N=

C 0=

B 0=

D1

N----=

Conversions Between Matrix Representations

Table 4-2

Conversion between Different Network Representations

[Z] [Y] [h] [ABCD]

[Z]

[Y]

[h]

[ABCD]

Z11Z12

Z21Z22

Z22

ZΔ-------

Z12

ZΔ-------–

Z21

ZΔ-------–

Z11

ZΔ-------

ZΔ

Z22

-------Z12

Z22

-------

Z21

Z22

-------–1

Z22

-------

Z11

Z21

-------ZΔ

Z21

-------

1

Z21

-------Z22

Z21

-------

Y22

YΔ--------

Y12

YΔ--------–

Y21

YΔ--------–

Y11

YΔ--------

Y11Y12

Y21Y22

1

Y11

--------Y12

Y11

--------–

Y21

Y11

--------YΔ

Y11

--------

Y22

Y21

--------–1

Y21

--------–

YΔ

Y21

--------–Y11

Y21

--------–

hΔ

h22

-------h12

h22

-------

h21

h22

-------–1

h22

-------

1

h11

-------h12

h11

-------–

h21

h11

-------hΔ

h11

-------

h11h12

h21h22

hΔ

h21

-------–h11

h21

-------–

h22

h21

-------–1

h21

-------–

A

C----

ABCDΔ

C--------------------

1

C----

D

C----

D

B----

ABCDΔ

B--------------------–

1

B---–

A

B---

B

D----

ABCDΔ

D--------------------

1

D----–

C

D----

A B

C D

Conclusion #1

• For Z, Y, h, and ABCD matrix representations, the individual matrix elements are easily determined by selectively shorting or opening the various ports of the network.

• PROBLEM: In RF/MW design, short-circuits and open-circuits are difficult to achieve so it is correspondingly difficult to obtain the matrix representation.

• SOLUTION: The scattering or S-parameters provide a method of solving this problem.

Scattering or S-parameter Representation

The Scattering or S-parameter representation solves the problem of not being able to achieve the perfect open- or short-circuits required to determine the matrix elements for the previously defined representations (Z, Y, h, ABCD).

It does this by considering how a wave acts when it is incident on the Device Under Test (DUT). Since we are now dealing with wave phenomena, terminating transmission lines (attached to the DUT) with their characteristic impedance can be used to determine the S-parameters.

We begin by defining the terminal parameters. Previously, this was either a voltage or a current. Now, it is a normalized power.

S-parameters – Definitions

Consider the network above. We define an as follows and we want to

determine the significance of this definition.

Note: The subscript n can take on any port number and in the two-port network above, we would have a1 and a2.

an =1

2√

Z0

(Vn + Z0In)

Figure 4-14 Convention used to define S-parameters for a two-port network.

b1 b2

[ ]S

a1 a2


For port 1, we would have

4

[v] = [Z] [i] and [i] = [Y ] [v][v] = [Z] [Y ] [v]

[Z] [Y ] =

⌃

⌦⌦⌦⌦�

1 0 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

⌥

↵↵↵↵ = [1]

⌅v1

i2

⇧

=⌅

h11 h12

h21 h22

⇧ ⌅i1v2

⇧

h11 =v1

i1

��v2=0

⌅v1

i1

⇧

=⌅

A BC D

⇧ ⌅v2

�i2

⇧

⌅v1

i1

⇧

=⌅

v⇥1i⇥1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥2�i⇥2

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅v⇥⇥1i⇥⇥1

⇧

⌅v1

i1

⇧

=⌅

A⇥ B⇥

C ⇥ D⇥

⇧ ⌅A⇥⇥ B⇥⇥

C ⇥⇥ D⇥⇥

⇧ ⌅v⇥⇥2�i⇥⇥2

⇧

an =1

2⌅

Z0(Vn + Z0In)

a1 =1

2⌅

Z0(V1 + Z0I1)

V1 = V +1 + V �

1

I1 = I+1 + I�1 =

V +1

Z0� V �

1

Z0

a1 =1

2⌅

Z0

⇥

V +1 + V �

1 + Z0

⇥V +

1

Z0� V �

1

Z0

⇤⇤

a1 =V +

1⌅Z0

bn =1

2⌅

Z0(Vn � Z0In)

⌅b1

b2

⇧

=⌅

S11 S12

S21 S22

⇧ ⌅a1

a2

⇧

S11 =b1

a1

��a2=0



S12 =b1

a2

��a1=0



S21 =b2

a1

��a2=0



S22 =b2

a2

��a1=0



We see that a1 is the incident voltage (V+) normalized to the square root of the characteristic impedance. Since power is

V2/R, we call this an incident normalized power wave.


[

b1

b2

]

=

[

S11 S12

S21 S22

] [

a1

a2

]

bn =1

2√

Z0

(Vn − Z0In)

In a similar fashion, we can define a reflected normalized power wave as follows

The S-parameters for a two-port network are then defined as

where . . . . .

S11 =b1

a1

∣

∣

∣

∣

a2=0

≡

reflected power wave at port 1

incident power wave at port 1

∣

∣

∣

∣

NO incident power at port 2

S12 =b1

a2

∣

∣

∣

∣

a1=0

≡



∣

∣

∣

∣


S21 =b2

a1

∣

∣

∣

∣

a2=0

≡



∣

∣

∣

∣


S22 =b2

a2

∣

∣

∣

∣

a1=0

≡



∣

∣

∣

∣


The requirement that no incident power appears at either port 1 or port 2 is achieved by terminating the transmission line in its characteristic impedance.

S11 and S21

No incident power is seen at port 2 because the transmission line load, ZL, is equal to Z0.

S11 is the reflection coefficient and S21 is the forward voltage gain.

S11 =b1

a1

∣

∣

∣

∣

a2=0

=V

−

1

V+1

= Γin =Zin − Z0

Zin + Z0

S21 =b2

a1

∣

∣

∣

∣

a2=0

=

V−

2√

Z0

1

2√

Z0

(V1 + Z0I1)=

2V−

2

VG1

=2V2

VG1

Figure 4-15 Measurement of S11 and S21by matching the line impedance Z0 at port 2 through a corresponding load impedance ZL = Z0.

b1 b2

[S]

a1a2 = 0

Z 0 ZLZ 0

Z 0

VG1

S12 and S22

No incident power is seen at port 1 because ZG is set to Z0.

S22 =b2

a2

∣

∣

∣

∣

a1=0

=V

−

2

V+2

= Γout =Zout − Z0

Zout + Z0

S12 =b1

a2

∣

∣

∣

∣

a1=0

=

V−

1√

Z0

1

2√

Z0

(V2 + Z0I2)=

2V−

1

VG2

=2V1

VG2

Figure 4-16 Measurement of S22 and S12 by matching the line impedance Z0 at port 1 through a corresponding input impedance ZG = Z0.

b1 b2

[ ]S

a2a1 = 0

Z0ZG Z0

Z0

VG2

S-parameters and Cascading Networks

• S-parameters do not directly apply to cascaded networks.

• Cascaded networks are common in RF systems.

• The chain scattering matrix works with cascading networks. The T-parameters are obtained from S-parameters.

S-parameters and Cascading Networks

[

b1

b2

]

=

[

S11 S12

S21 S22

] [

a1

a2

]

[

a1

b1

]

=

[

T11 T12

T21 T22

] [

b2

a2

]

Figure 4-14 Convention used to define S-parameters for a two-port network.

b1 b2

[ ]S

a1 a2

Chain Scattering Matrix

[

a1

b1

]

=

[

T11 T12

T21 T22

] [

b2

a2

]

This matrix is similar to the ABCD matrix and is useful for cascading networks.

Figure 4-18 Cascading of two networks A and B.

port 1 port 2

b

b

[ ]T

a 1 a b

b

[ ]T

a

aA1A

b 1A

2A

1B

2B

2A

1B

2B

A B

Chain Scattering Matrix

[

a1

b1

]

=

[

T11 T12

T21 T22

] [

b2

a2

]

T11 =a1b2

��a2=0

=1

S21

Summary• Traditional network representations (Z, Y, h,

ABCD) do not work at RF/MW frequencies because we cannot experimentally determine the matrix elements. This is because we cannot create perfect opens or shorts which is required to determine the different elements.

• S-parameters solve this problem by using transmission line segments terminated with a matched load to prevent reflections.

• For cascaded networks, T-parameters should be used. T-parameters are easily derived from S-parameters and vice-versa.

Networks - Rowan Universityusers.rowan.edu/~krchnavek/Rowan_University/RF... · 2018. 2. 1. ·...

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