1

Chapter 18 Two-Port Circuits

18.1 The Terminal Equations18.2 The Two-Port Parameters18.3 Analysis of the Terminated Two-Port

Circuit 18.4 Interconnected Two-Port Circuits

2

Motivation

Thévenin and Norton equivalent circuits are used in representing the contribution of a circuit to one specific pair of terminals.

Usually, a signal is fed into one pair of terminals (input port), processed by the system, then extracted at a second pair of terminals (output port). It would be convenient to relate the v/i at one port to the v/i at the other port without knowing the element values and how they are connected inside the “black box”.

3

How to model the “black box”?

We will see that a two-port circuit can be modeled by a 22 matrix to relate the v/i variables, where the four matrix elements can be obtained by performing 2 experiments.

Source (e.g. CD

player)

Load (e.g.

speaker)

4

Restrictions of the model

No energy stored within the circuit.

No independent source.

Each port is not a current source or sink, i.e.

No inter-port connection, i.e. between ac, ad, bc, bd.

. , 2211 iiii

5

Key points

How to calculate the 6 possible 22 matrices of a two-port circuit?

How to find the 4 simultaneous equations in solving a terminated two-port circuit?

How to find the total 22 matrix of a circuit consisting of interconnected two-port circuits?

6

Section 18.1 The Terminal Equations

7

s-domain model

The most general description of a two-port circuit is carried out in the s-domain.

Any 2 out of the 4 variables {V1 , I1 , V2 , I2 } can be determined by the other 2 variables and 2 simultaneous equations.

8

Six possible sets of terminal equations (1)

matrix; admittance theis ;

matrix; impedance theis ;

1-

2

1

2221

1211

2

1

2

1

2221

1211

2

1

ZYVV

yyyy

II

ZII

zzzz

VV

matrix; siona transmis is ;

matrix; siona transmis is ;

1-

1

1

2221

1211

2

2

2

2

2221

1211

1

1

ABIV

bbbb

IV

AIV

aaaa

IV

9

Six possible sets of terminal equations (2)

matrix; hybrida is ;

matrix; hybrida is ;

1-

2

1

2221

1211

2

1

2

1

2221

1211

2

1

HGIV

gggg

VI

HVI

hhhh

IV

Which set is chosen depends on which variables are given. E.g. If the source voltage and current {V1 , I1 } are given, choosing transmission matrix [B] in the analysis.

10

Section 18.2 The Two-Port Parameters

1. Calculation of matrix [Z]2. Relations among 6 matrixes

11

Example 18.1: Finding [Z] (1)

Q: Find the impedance matrix [Z] for a given resistive circuit (not a “black box”):

By definition, z11 = (V1 /I1 ) when I2 =0, i.e. the input impedance when port 2 is open. z11 = (20 )//(20 )=10 .

2

1

2221

1211

2

1

II

zzzz

VV

12

Example 18.1: (2)

By definition, z21 = (V2 /I1 ) when I2 =0, i.e. the transfer impedance when port 2 is open.

When port 2 is open:

. 5.7) 10(

75.0

, 10

, 10

,75.0 15 5

15

1

1

1

221

1111

1

1

112

VV

IVz

VIzIV

VVV

13

. 5.7) 375.9(

8.0

, 375.9

, 375.9

,8.0 5 02

02

2

2

2

112

2222

2

2

221

VV

IVz

VIzIV

VVV

Example 18.1: (3)

By definition, z22 = (V2 /I2 ) when I1 =0, i.e. the output impedance when port 1 is open. z22 = (15 )//(25 ) =9.375 .

z12 = (V1 /I2 ) when I1 =0,

14

Comments

When the circuit is well known, calculation of [Z] by circuit analysis methods shows the physical meaning of each matrix element.

When the circuit is a “black box”, we can perform 2 test experiments to get [Z]: (1) Open port 2, apply a current I1 to port 1, measure the input voltage V1 and output voltage V2 . (2) Open port 1, apply a current I2 to port 2, measure the terminal voltages V1 and V2 .

15

Relations among the 6 matrixes

If we know one matrix, we can derive all the others analytically (Table 18.1).

[Y]=[Z]-1, [B]=[A]-1, [G]=[H]-1, elements between mutually inverse matrixes can be easily related.

E.g.

.det where

,1

21122211

1121

12221

2221

1211

2221

1211

yyyyYy

yyyy

yyyyy

zzzz

16

Represent [Z] by elements of [A] (1)

[Z] and [A] are not mutually inverse, relation between their elements are less explicit.

By definitions of [Z] and [A],

the independent variables of [Z] and [A] are {I1 , I2 } and {V2 , I2 }, respectively.

Key of matrix transformation: Representing the distinct independent variable V2 by {I1 , I2 }.

, ,2

2

2221

1211

1

1

2

1

2221

1211

2

1

IV

aaaa

IV

II

zzzz

VV

17

Represent [Z] by elements of [A] (2)

By definitions of [A] and [Z],

)2()1(

2222211

2122111

IaVaIIaVaV

.det where,1

122

11

212221

1211 Aaa

aaazz

zz

),3(1)2( 222121221

221

212 IzIzI

aaI

aV

)4(

1)3(),1(

21211121221

22111

21

11

212221

221

21111

IzIzIaaaaI

aa

IaIaaI

aaV

18

Section 18.3 Analysis of the Terminated Two-Port Circuit

1. Analysis in terms of [Z]2. Analysis in terms of [T][Z]

19

Model of the terminated two-port circuit

A two-port circuit is typically driven at port 1 and loaded at port 2, which can be modeled as:

The goal is to solve {V1 , I1 , V2 , I2 } as functions of given parameters Vg , Zg , ZL , and matrix elements of the two-port circuit.

20

Analysis in terms of [Z]

Four equations are needed to solve the four unknowns {V1 , I1 , V2 , I2 }.

ons terminati todue equations constraint)4(

)3(

equationsport -two)2()1(

22

11

2221212

2121111

L

gg

ZIVZIVV

IzIzVIzIzV

,

0

00

010001

1001

2

1

2

1

2221

1211

g

L

g VIIVV

ZZ

zzzz

{V1 , I1 , V2 , I2 } are derived by inverse matrix method.

21

Thévenin equivalent circuit with respect to port 2

Once {V1 , I1 , V2 , I2 } are solved, {VTh , ZTh } can be determined by ZL and {V2 , I2 }:

)2(

)1(

22

2

Th

Th

ThLTh

L

ZVVI

VZZ

ZV

.1

;1 2

21

2

2

2

2

2

2

VZV

IVZ

ZV

VZV

ZV

IVZ LL

Th

ThL

Th

ThL

22

Terminal behavior (1)

The terminal behavior of the circuit can be described by manipulations of {V1 , I1 , V2 , I2 }:

Input impedance:

Output current:

Current gain:

Voltage gains:

;22

211211

1

1

Lin Zz

zzzIVZ

;))(( 21122211

212 zzZzZz

VzI

Lg

g

;22

21

1

2

LZzz

II

;))((

;

21122211

212

11

21

1

2

zzZzZzZz

VV

zZzZz

VV

Lg

L

g

L

L

23

Terminal behavior (2)

Thévenin voltage:

Thévenin impedance:

;11

21g

gTh V

ZzzV

;11

211222

gTh Zz

zzzZ

24

Analysis in term of a two-port matrix [T][Z]

If the two-port circuit is modeled by [T][Z], T={Y, A, B, H, G}, the terminal behavior can be determined by two methods:

Use the 2 two-port equations of [T] to get a new 44 matrix in solving {V1 , I1 , V2 , I2 } (Table 18.2);

Transform [T] into [Z] by Table 18.1, borrow the formulas derived by analysis in terms of [Z].

25

Example 18.4: Analysis in terms of [B] (1)

Q: Find (1) output voltage V2 , (2,3) average powers delivered to the load P2 and input port P1 , for a terminated two-port circuit with known [B].

2.0mS 2k 320

B-b12

-b22

RLVg

Rg

26

Example 18.4 (2)

Use the voltage gain formula of Table 18.2:

.V 016.26305001910

,1910

...)k 5)(2.0()k 5.0)(20()k 3()k 5)(2(

,264)mS 2)(k 3()2.0)(20(

;

2

2

21122211

21221112

2

V

VV

bbbbbZZbZbZbb

bZVV

g

LgLg

L

g

27

Example 18.4 (3)

The average power of the load is formulated by

.W 93.6k 5

V 016.26321

21

222

2

LRV

P

The average power delivered to port 1 is formulated by .Re

21 2

11 inZIP

W. 55.41)33.133()789.0(21

A, 0789.0) 33.133() 500(

V 0500

; 33.13320)k 5)(mS 2(

)k 3()k 5)(2.0(

21

1

1121

1222

1

1

P

ZZV

I

bZbbZb

IVZ

ing

g

L

Lin

28

Section 18.4 Interconnected Two-Port Circuits

29

Why interconnected?

Design of a large system is simplified by first designing subsections (usually modeled by two-port circuits), then interconnecting these units to complete the system.

30

Five types of interconnections of two-port circuits

a. Cascade: Better use [A].

b. Series: [Z]

c. Parallel: [Y]

d. Series-parallel: [H].

e. Parallel-series: [G].

31

Analysis of cascade connection (1)

Goal: Derive the overall matrix [A] of two cascaded two-port circuits with known transmission matrixes [A'] and [A"].

A A

Overall two-port circuit [A]=?

)2(

,

2

2

2

2

2221

1211

1

1

2

2

2221

1211

2

2

1

1

I

VA

I

V

aa

aa

I

V

I

V

aa

aa

I

VA

I

V

32

Analysis of cascade connection (2)

)1(1

1

2

2

1

1

I

VA

I

VA

I

V

.)(

)( ,

, ),2(),1(By

2222122121221121

2212121121121111

2221

1211

2

2

2

2

1

1

aaaaaaaa

aaaaaaaa

aa

aaAAA

I

VA

I

VAA

I

V

33

Key points

How to calculate the 6 possible 22 matrices of a two-port circuit?

How to find the 4 simultaneous equations in solving a terminated two-port circuit?

How to find the total 22 matrix of a circuit consisting of interconnected two-port circuits?

34

Practical Perspective Audio Amplifier

35

Application of two-port circuits

Q: Whether it would be safe to use a given audio amplifier to connect a music player modeled by {Vg =2 V (rms), Zg =100 } to a speaker modeled by a load resistor ZL =32

with a power rating of

100 W?

36

Find the [H] by 2 test experiments (1)

;2

1

2221

1211

2

1

V

I

hh

hh

I

V

Definition of hybrid matrix [H]:

Test 1:I1 =2.5 mA (rms) V2 =0 (short)

. 500mA 5.2

V 25.1 ,01

1111111

2

VI

VhIhV

.1500mA 5.2

A 75.3 ,01

2211212

2

VI

IhIhI

V1 = 1.25 V (rms) I2 =3.75 A (rms)

Input impedance

Current gain

37

Find the [H] by 2 test experiments (2)

;2

1

2221

1211

2

1

V

I

hh

hh

I

V

Definition of hybrid matrix [H]:

Test 2:I1 = 0 (open) V2 =50 V (rms)

.10V 05

mV 50 , 3

02

1122121

1

IV

VhVhV

.) 20(V 05A 5.2 , 1-

02

2222222

1

IV

IhVhI

V1 =50 mV (rms)

I2 =2.5 A (rms)

Voltage gain

Output admittance

38

Find the power dissipation on the load

For a terminated two-port circuit:

,Re)(ReRe 22

*22

*22 LLL ZIIZIIVP

the power dissipated on ZL is

where I2 is the rms output current phasor.

39

Method 1: Use terminated 2-port eqs for [H]

By looking at Table 18.2:

(rms), A 98.1))(1( 21121122

212

LgL

g

ZhhZhZhVh

I

. 32 , 100 ,(rms) V 2

;) 20(1500

10 500 where 1-

3

2221

1211

Lgg ZZV

hhhh

.W 100W 126)32()98.1(Re 222 LL ZIP

Not safe!

40

Method 2: Use system of terminated eqs of [Z]

Transform [H] to [Z] (Table 18.1):

.W 100W 126)32()98.1(Re 222 LL ZIP

. 20000,3002.0470

11

21

12

222221

1211

h

hhhzz

zz

.

A 98.1mA .43

V 63.5V 66.1

0

00

010001

1001 1

2221

1211

2

1

2

1

g

L

g VZ

Zzzzz

IIVV

By system of terminated equations: