Chapter 14 Two-port networkscc.ee.ntu.edu.tw/~thc/course_ckt/note/chap14.pdf · Chapter 14 Two-port networks 14.1 Two-ports and impedance parameters two-port concept, impedance parameters,

電路學講義第14章14-1

Chapter 14 Two-port networks14.1 Two-ports and impedance parameters

two-port concept, impedance parameters, reciprocal networks 14.2 Admittance, hybrid, and transmission parameters

admittance parameters, hybrid parameters, transmission parameters, parameter conversion

14.3 Circuit analysis with two-portsterminated two-ports, two-ports in cascade, two-ports in series, two-ports in parallel


14.1 Two-ports and impedance parameters Basics1. Two-port network

• a four-terminal network with input port and output port• the network characteristics is completely described by• a useful method to analyze filter, amplifier,….• can be extended to multi-port networks

4231 , iiii ==

2211 ,,, iviv

No independent sources are in the two-port network and load.concernednot are and 4321 vv


2. O.C. impedance parameters

[ ]

[ ]

o.c. 2port with impedance transfer forward:

o.c. 1port with impedance transfer reverse:

o.c. 1port with 2port at impedanceinput :

o.c. 2port with 1port at impedanceinput :

matrixparameter impedance o.c.:

:domain sin

,:responses,,:sources

01

221

02

112

02

222

01

111

2221

1211

2

1

2

1

2221212

2121111

2121

2

1

1

2

=

=

=

=

=

=

=

=

⎥⎦

⎤⎢⎣

⎡≡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎩⎨⎧

+=+=

−

I

I

I

I

IVz

IVz

IVz

IVz

zzzz

z

II

zVV

IzIzVIzIzV

vvii


Discussion1. Most two-port networks are three-terminal networks.

2. Equivalent circuit expressed in z-parameters

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎩⎨⎧

+=+=

2

1

2221

1211

2

1

2221212

2121111

II

zzzz

VV

IzIzVIzIzV


3. Use definition to determine the z-parameter of a two-port network.4. Ex. 14.1 find z-parameters of a “symmetrical” network

21

22

2212

222020102

112

22

2

11

212111

11,

211

11

111

1)1//(

11

1

zsRCCsR

sRCCsRR

sRCsRCz

sRCsRCz

zIsRC

sRCV

sCR

RVIVz

zsRC

CsRR

sRCsC

RsC

RR

RsC

Rz

III

=+

=++

+=

+=→

+=

+==

=++

=

++

=

++

=+=

===

2112

2211

zzzz

==

02

222

01

221

02

112

01

111

2

1

2221

1211

2

1

21

12

12

,

,

,:sources

==

==

==

==

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

II

II

IVz

IVz

IVz

IVz

II

zzzz

VV

II


5. Ex. 14.2 find z-parameters of an “active” network

100

10023,50

60

)5010(0o.c., 2port)1(

1

221

1

21

1

111

11

2

−==→

−=−=+−==

==→

+==−

ivz

ivvvviv

ivz

ivi

x

xxx

122

112

21

112

222

22

1

50

50

100

50,20o.c., 1port)2(

zivz

ivv

zivz

ivvvi

x

xx

≠==→

==

≠−==→

=−==−

02

222

01

221

02

112

01

111

2

1

2221

1211

2

1

21

12

12

,

,

,:sources

==

==

==

==

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

ii

ii

ivz

ivz

ivz

ivz

ii

zzzz

vv

ii


6. Reciprocal circuit

21121

2

2

1:network reciprocal zzI

VI

V ococ =→=

21121

2

2

1:network reciprocal yyVI

VI scsc =→=


7. Any linear network containing no controlled sources is a reciprocal network.∵node equation and mesh equation have symmetrical forms8. Ex. 14.3 T-network

2

112121211

2

2221

121

221121211

1

1112

,,0)2(

,,0)1(

IVzzzz

IVzI

zIVzzzz

IVzI

=+−===

==+−===

1222

121211 ,zzZ

zZzzZ

b

ca

−==−=

02

222

01

221

02

112

01

111

2

1

2221

1211

2

121

1212

,,,

,:sources

====

====

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

IIII IVz

IVz

IVz

IVz

II

zzzz

VV

II


14.2 Admittance, hybrid, and transmission parametersBasics1. Not all two-ports posses meaningful or measurable z-parameters. →other parameters2. Admittance parameter

[ ]

s.c. 2port with admittance transfer forward:

s.c. 1port with admittance transfer reverse:

s.c. 1port with 2port at admittanceinput :

s.c. 2port with 1port at admittanceinput :

matrixparameter admittance s.c.:

01

221

02

112

02

222

01

111

2221

1211

2

1

1

2

=

=

=

=

=

=

=

=

⎥⎦

⎤⎢⎣

⎡≡

V

V

V

V

VIy

VIy

VIy

VIy

yyyy

y

[ ] ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎩⎨⎧

+=+=

2

1

2

1

2221212

2121111

2121 ,:responses,,:sources

VV

yII

VyVyIVyVyI

IIVV


3. Hybrid parameter

[ ]

s.c. 2port with ratiocurrent forward:

o.c. 1port with ratio voltagereverse:

o.c. 1port with 2port at impedanceinput :1

s.c. 2port with 1port at admittanceinput :1

matrixparameter hybrid:

01

221

02

112

2202

222

1101

111

2221

1211

2

1

1

2

=

=

=

=

=

=

==

==

⎥⎦

⎤⎢⎣

⎡≡

V

I

I

V

IIh

VVh

zVIh

yIVh

hhhh

h

[ ] ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎩⎨⎧

+=+=

2

1

2

1

2221212

2121111


VI

hIV

VhIhIVhIhV

IVVI


4. Transmission parameter

[ ]

s.c. 2port with ratiocurrent forward:

o.c. 2port with admittance transfer reverse:

s.c. 2port with impedance transfer reverse:

o.c. 2port with ratio voltagereverse:

matrixparameter on transmissi:

02

1

02

1

02

1

02

1

2

2

2

2

=

=

=

=

−=

=

−=

=

⎥⎦

⎤⎢⎣

⎡≡

V

I

V

I

IID

VIC

IVB

VVA

DCBA

T

[ ] ⎥⎦

⎤⎢⎣

⎡−

=

⎥⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎩⎨⎧

−=−=

2

2

2

2

1

1

221

221

1122

,:responses,,:sources

IV

T

IV

DCBA

IV

IDVCIIBVAV

IVIV


Discussion1. Equivalent circuit expressed in y-parameters

12122

112

1212222

222

1

12121

221

1212111

111

2

11,0)2(

11,0)1(

yyV

Iy

yyyVIy

V

yyV

Iy

yyyVIy

V

=−

−==

−+===

=−

−==

−+===

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎩⎨⎧

+=+=

2

1

2221

1211

2

1

2221212

2121111

21,:sources

VV

yyyy

II

VyVyIVyVyI

VV


2. Ex. 14.5 find y-parameters of an “active” network

20

40223

4040

0s.c., 2port)1(

1

221

11112

1

11111

2

sVIy

VsIIII

sVIysVI

V

==→

==−=

==→=

=−

202

202

1020

102

103

4040

0s.c., 1port)2(

2

222

222

21

2112

212

11221

1

sVIy

VsVVs

VIVIII

ysVIysVI

V

−==→

−=+−=

+=+−=

≠−==→−=

=−

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

2

1

2221

1211

2

1

21,:sources

VV

yyyy

II

VV


4. Equivalent circuit expressed in h-parameters

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎩⎨⎧

+=+=

ce

b

oefe

reie

c

be

vi

hhhh

iv

CE

vi

hhhh

iv

vhihivhihv

ivvi

2

1

2221

1211

2

1

2221212

2121111


3. H-parameters are applied to transistor because they are measured physical quantities.

bevcev

bici


5. Ex. 14.6 find h-parameters of an “active” network

2

23

4040

0s.c., 2port)1(

1

221

1112

1

111

11

2

==→

=−=

==→

=

=−

IIh

IIIIsI

Vh

sVI

V

1.0

10

1

0o.c., 1port)2(

2

222

22

2

11221

1

==→

=

==→=

=−

VIh

IVVVhVV

I

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎩⎨⎧

+=+=

2

1

2221

1211

2

1

2221212

2121111

21

21

,:responses,:sources

VI

hhhh

IV

VhIhIVhIhV

IVVI


6. Ex. 14.7 find ABCD-parameters of an “active” network

sVVA

Vs

Is

VV

VIC

VIVII

I

21

)21(40201

20,

103

0o.c., 2port)1(

2

1

2121

2

1

21

211

2

−==→

−=+=

−==→

−=+=

=−

sIVB

Is

Is

V

IID

IIIIIV

20

204021

2,30s.c., 2port)2(

2

1

211

2

1

21211

2

−=−

=→

==

−=−

=→

=−==−

⎥⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→

⎩⎨⎧

−=−=

2

2

1

1

221

221

11

22

,:responses,:sources

IV

DCBA

IV

IDVCIIBVAVIV

IV


7. All the 2-port parameters are related as given in Table 14.2.8. Conversion between z-parameters and y-parameters

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ] 211222111121

1222

1

211222111121

1222

1

2

1

2

11

2

1

2

1

2

1

2221

1211

2

1

,

,

parameter

yyyyyyy

yy

yz

zzzzzzz

zz

zy

VV

yVV

zII

II

zII

zzzz

VV

z

y

yy

yy

z

zz

zz

−==∆

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆−

∆−

∆==

−==∆

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∆∆−

∆−

∆==⇒

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡→⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

−

−

−


9. Derivation of h-parameters from z-parameters

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

∆

=

−=∆+∆

=+−+=→

+−=→⎩⎨⎧

+=+=

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

2222

21

22

12

22

21122211222

121

222

221

22

21121111

)3(

222

122

212

2221212

2121111

2

1

2221

1211

2

1

2

1

2221

1211

2

1

1

,)1()1(

)3...(1)2( )2...()1...(

parameter ,parameter

zzz

zz

zh

zzzzVzzI

zV

zI

zzzIzV

Vz

IzzI

IzIzVIzIzV

VI

hhhh

IV

hII

zzzz

VV

z

z

zz


10. Derivation of z-parameters from T-parameters

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ ∆

=

−=∆∆

+=−+=→

+=→⎩⎨⎧

−=−=

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

CD

C

CCA

z

BCADIC

ICAIBI

CDI

CAV

ICDI

CV

IDVCIIBVAV

II

zzzz

VV

zI

VDCBA

IV

T

T

TT

1

,)1()1(

)3...(1)2( )2...()1...(

parameter ,parameter

212211

)3(

212221

221

2

1

2221

1211

2

1

2

2

1

1

11. For a reciprocal 2-port network, 1,,, 211221122112 =−=∆−=== BCADhhyyzz T


14.3 Circuit analysis with two-portsBasics1. Terminated two-ports using z-parameters

Lz

L

L

L

LL

v

LL

LL

Li

LL

L

Ss

ZzZz

zzzzZzZz

Zz

Zzzzzz

VVH

VZzV

ZzzV

zzV

VZz

zVz

IVZzIzV

zZz

IIH

IzZIzIzIzIZ

IZVVIZV

IzIzVIzIzV

11

21

2211211211

21

22

11

2112

11

21

1

2

222

211

121

11212

(5)

211

121

1112

121111

(4)22

21

1

2

2221212221212

)4(

22

11

2221212

2121111

1function transfer voltage

)1()2(

)5...(1,(1)

function ansfer current tr

)(,)2(

)4........()3...(

,)2...()1...(

+∆=

+−=

+−=≡⇒

−+=→

+=−=→

+−

=≡⇒

+−=+=−→

⎩⎨⎧

−=+=

⎩⎨⎧

+=+=


11

22

11

2112221122

02

2

222211

12212

)8(

211

1212121111

)7(

11

22

11

22

2112221111

1

1

122

21121111

(6)22

21

1

2

22

11

2221212

2121111

impedanceoutput equivalent

)()2(

)8...()1(),7...()3(,0

impedanceinput equivalent

(1)

)6...(

)4........()3...(

,)2...()1...(

zZZz

zZzzzzZz

IVZ

IzIzZ

zzV

IzZ

zIIzIzIZIZVV

zZZz

zZzzzzZz

IVZ

IzZ

zzIzV

zZz

IIH

IZVVIZV

IzIzVIzIzV

S

Sz

S

S

Vo

S

SSSs

L

Lz

L

Li

L

Li

L

Ss

s+

+∆=

+−+

=≡⇒

++

−=→

+−=→+=−→−=→=

++∆

=+

−+=≡⇒

+−

−=→

+−

=≡

⎩⎨⎧

−=+=

⎩⎨⎧

+=+=

=


2. Cascade connection using T-parameters

[ ] [ ]

[ ] [ ]

[ ] [ ] [ ]bacas

ba

b

ba

a

aa

TTTI

VTT

IV

TI

VT

IV

=⇒

⎥⎦

⎤⎢⎣

⎡−

=

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡

2

2

1

1

2

2

1

1

3. Series connection using z-parameters

[ ] [ ]

[ ] [ ]

[ ] [ ] [ ]baser

ba

ba

b

b

a

a

zzzII

zz

II

zII

z

VV

VV

VV

+=⇒

⎥⎦

⎤⎢⎣

⎡+=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

2

1

2

1

2

1

2

1

2

1

2

1

)(


4. Parallel connection using y-parameters

[ ] [ ]

[ ] [ ]

[ ] [ ] [ ]bapar

ba

ba

b

b

a

a

yyyVV

yy

VV

yVV

y

II

II

II

+=⇒

⎥⎦

⎤⎢⎣

⎡+=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

2

1

2

1

2

1

2

1

2

1

2

1

)(


Discussion1. Relations of terminated two-ports in terms of z- y- h- and T-

parameters are given in Table 14.3. They are useful in network analysis.

2. Ex. 14.9 given load be a 2.5H inductor, find I2/V1 from T-parameters

)2)(4(4.0

824.0

205.2)21(

11

1)(

21

201

2021 ex.14.7 From

2

2.14

1

2

1

2

+−−

=

−−−

=−−

−=

+−

=

++

+−

====

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−=⎥

⎦

⎤⎢⎣

⎡

sss

sss

ss

sBAZ

BAZDCZ

DCZZH

IZI

VIsH

ssDCBA

L

L

L

L

Table

i

i

i


3. Ex. 14.10 find RL to give Ai=Iout/Is=-25 from transistor h-parameters

[ ]

L

Li

L

Lhi

h

YhYh

IIsH

YhYh

IVsZ

h

+==

++∆

==

=∆

⎥⎦

⎤⎢⎣

⎡

×=

−

−

22

21

1

2

22

11

1

1

3

3

)(

)(

05.0101.050

101000

Ω=→−=+−

=

+=

++

=

+−=

−==

kRR

A

RsH

RRsZ

ZRRH

II

II

IIA

LL

i

Li

L

Li

is

si

ss

outi

42525.03

10011.0

50)(,11.0105.0)(

11

2


4. Ex. 14.11 find Ai of two amplifiers of ex.14.10 in cascade

[ ] [ ] [ ]

[ ] [ ] [ ]

1100

16451)(,961)(

1044010042.042.01041

02.01022010

1,101.050101000

1

1

2

66

6

6

3

2121

22

21

11

213

3

=+

−=−

==

−=+

−==

++

=

⎥⎦

⎤⎢⎣

⎡

×××

==→

⎥⎦

⎤⎢⎣

⎡

−×−−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−∆−

==⎥⎦

⎤⎢⎣

⎡

×=

−−

−

−

−

−

−

is

si

ss

outi

Li

L

Li

bacas

h

ba

ZRRH

II

II

IIA

DCRsH

DCRBARsZ

TTT

hhh

hh

hTTh


5. Bridged-T connection

[ ]

[ ] ⎥⎦

⎤⎢⎣

⎡++−+−+

=⇒

−====

====⎥⎦

⎤⎢⎣

⎡=

====

bFbF

bFbF

FF

VVVV

yYyYyYyY

y

YyyYyy

VIy

VIy

VIy

VIy

yyyy

y

2221

1211

21122211

01

221

02

112

02

222

01

111

2221

1211

,

,, ,,2112


6. Ex.14.12 find Hv=V2/V1 of a high frequency transistor

[ ] [ ] [ ]

LoLo

m

Lo

m

Lv

om

i

om

ibaF

RRRRGCGsCgs

RRsCgsC

YyyH

RsCsCg

sCR

sCy

Rg

RysCsCsCsC

ysC

Z

///1/1/1//

/1/1

1

1

1

01

,,1

22

21

=+=+−

=++

−=

+−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

−+=⇒

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡−

−==

Documents

Chapter 14 Two-port networkscc.ee.ntu.edu.tw/~thc/course_ckt/note/chap14.pdf · Chapter 14 Two-port networks 14.1 Two-ports and impedance parameters two-port concept, impedance parameters,