Two-Port Network

1

2

TWO PORT NETWORKSTwo port networks are a useful tool for describing idealized components. The basic device schematics are simple, but each set of parameters views the device differently.

The basic device is seen below, along with the various parameter sets,

I1 I2

s-domain circuitv1

v2

+

-

+

-

3

2221212

2121111

IzIzV

IzIzV

2221212

2121111

VyVyI

VyVyI

2222211

2122111

IaVaI

IaVaV

1221212

1121112

IbVbI

IbVbV

z-parameters

y-parameters

a-parameters

b-parameters

4

PARAMETER VALUES

Obviously some of the parameters are impedance, while others are admittance. They can be easily determined by setting other parameters to zero, and measuring relevant voltages/currents.

2221212

2121111

IgVgV

IgVgI

2221212

2121111

VhIhI

VhIhV

h-parameters

g-parameters

5

z-Parameters (impedance)

The values are as below,

6

The equivalent circuit for the z-parameters is shown below,

z11I2z21I1

z11z22

V1 V2

I1 I2

+

-

+

-

+

-

+

-

7

Find the z parameters for the circuit shown

EXAMPLE

SOLUTION

1040

2020

01

111

2II

Vz

8

When port 1 is open, I1 is zero and the voltage V1 is

5.710/

75.0

1

1

01

221

2V

V

I

Vz

I

375.940

2515

02

222

1II

Vz

22

1 8.020205

VV

V

375.92

2

VI

9

The following measurements pertain to a two-port circuit operating in the sinusoidal steady state. With port 2 open, a voltage equal to 150 cos 4000t V is applied to port 1. The current into port 1 is 25 cos (4000t - 45°) A, and the port 2 voltage is 100 cos (4000t + 15°) V. With port 2 short-circuited, a voltage equal to 30 cos 4000t V is applied to port 1. The current into port 1 is 1.5 cos (4000t + 30°) A, and the current into port 2 is 0.25 cos (4000t + 150°) A. Find the a parameters that can describe the sinusoidal steady-state behavior of the circuit.

5.7375.9/

8.0

2

2

02

112

1V

V

I

Vz

I

10

SOLUTION

.0 ;15100

;4525 ;0150

20

2

01

01

AIVV

AIVV

.15025.0 ;0

;305.1 ;0300

22

01

01

AIVV

AIVV

00

0

02

111 155.1

15100

0150

2

I

V

Va

SV

Ia

I

00

0

02

121 6025.0

15100

4525

2

11

The following measurements were made on a two-port resistive circuit. With 10 mV applied to port 2 and port 1 open, the current into port 2 is 025 µA, and the voltage across port 1 is 5 µV. With port 2 short-circuited and 50 mV applied to port 1, the current into port 1 is 50 µA, and the current into port 2 is 2 mA. Find the h parameters of the network.

Drill Exercise

00

0

02

112 30120

15025.0

030

2VI

Va

00

0

02

121 606

15025.0

305.1

2

V

I

Ia

12

RELATIONSHIPS AMONG THE TWO-PORT PARAMETERS

;

,

2221212

2121111

VyVyI

VyVyI

212

122

2221

1211

222

121

1 Iy

yI

y

y

yy

yy

yI

yI

V

211

121221

111

2 Iy

yI

y

y

y

Iy

Iy

V

13

To find the z parameters as functions of the a parameters,

;

,

2221212

2121111

IzIzV

IzIzV

221

221

212

1I

a

aI

aV

2222211

2122111

IaVaI

IaVaV

; ; 1212

2211 y

yz

y

yz

. ; 1122

2121 y

yz

y

yz

14

21221

22111

21

111 Ia

a

aaI

a

aV

21

2222

2121

2112

21

1111

;1

. ;

a

az

az

a

az

a

az

15

EXAMPLE

1.

2.

16

1.

kx

x

I

Vh

V

2.11020

10246

3

01

111

2

501020

106

3

01

221

2

xI

Ih

V

The parameters h12 and h22 cannot be obtained directly from the open circuit test.

;1025.040

1010

;1025.040

1010

66

02

121

33

02

111

2

2

Sxx

V

Ia

xx

V

Va

I

I

17

.102010

1020

;2410

1024

33

6

02

122

3

3

02

112

2

2

xx

I

Ia

x

I

Va

V

V

. ; 22

2122

2211 a

ah

a

ah

621122211 10 aaaaa

Sx

xh

xx

h

5.121020

1025.0

1051020

10

3

6

22

53

6

12

18

Analysis Of The Terminated Two Port Circuit

Two-port model of networkVg

Zg

I1

V1

+

-

I2

V2

+

-

ZL

Six characteristics of the terminated two-port circuit define its terminal behavior:

1. The input impedance Zin = V1/I1, or the admittance Yin =I1/V1;

2. The output current I2 ;

3. The Thevenin voltage and impedance (VTh , ZTh ) with respect to port 2 ;

4. The current gain I2 / I1 ;

5. The voltage gain V2 / V1 and 6. the voltage gain V2 / Vg.

19

.

;

;

;

22

11

2221212

2121111

L

gg

zIV

ZIVV

IzIzV

IzIzV

22

21121111

22

1212

/).1zZ

zzzZIVZ

zZ

IzI

Linin

L

g

g

Zz

IzVI

11

2121

20

21122211

212).2

zzZzZz

VzI

Lg

g

The Thevenin voltage with respect to port 2

11

12112102

2).3

z

VzIzVVa ThI

11111 zZ

VIZIVV

g

ggg

gg

ThIV

zZ

zVV

11

2102

2

Thevenin or output impedance is ratio V2 / I2 when Vg is replace by short cicuit.

21

gTh

V

g

Zz

zzzZ

I

Vb

Zz

IzI

IzIzV

g

11

211222

02

2

11

2121

2221212

).3

;

;

The current gain

22

21

1

2).4zZ

z

I

I

L

The voltage gain V2 / V1

LZ

VzIzV 2

221212

22

LZ

VzVIz 2

121111LZz

Vz

z

VI

11

212

11

11

zZz

Zz

zzzzZz

Zz

V

V

L

L

L

L

11

21

2112221111

21

1

2).5

The voltage ratio V2 / Vg

222

11

21

11

212212

1111

2121

VZ

z

Zz

Zz

ZzZ

VzzV

Zz

V

ZzZ

VzI

Lg

g

gL

g

g

gL

23

21122211

212).6zzZzZz

Zz

V

V

Lg

L

g

24

y parameters

LgLg

L

g

L

L

L

g

gTh

g

gTh

LggL

g

L

Lin

ZyZyZZyy

Zy

V

V

Zy

Zy

V

V

yZy

y

I

I

yZy

ZyZ

yZy

VyV

ZyZZyZy

VyI

Zy

ZyyyY

22112112

212

22

21

1

2

11

21

1

2

22

11

22

21

1122

212

22

211211

11

1

1

1

1

25 gLg

L

g

L

L

L

g

gTh

g

gTh

gLgL

g

L

Lin

ZaaZZaa

Z

V

V

aZa

Z

V

V

aZaI

I

Zaa

ZaaZ

Zaa

VV

ZaZZaaZa

VI

aZa

aZaZ

22122111

2

12111

2

22211

2

2111

2212

2111

222112112

2221

1211

1

a parameters

26

LgLg

L

g

L

L

L

g

gTh

g

gTh

LLgg

g

L

Lin

ZZbZbZbb

bZ

V

V

Zbb

bZ

V

V

Zbb

b

I

I

bZb

bZbZ

Zbb

bVV

bZbZZbZb

bVI

bZb

bZbZ

21221112

2

22121

2

21111

2

2221

1211

2122

122221112

1121

1222

b parameters

27

[b]

500Ω

5kΩ

+

-

V1

+

-

V2

00500

I1

I2

The two port circuit shown above is described in term of its b parameters, the values of which are

b11= -20, b12= -3000Ω, b21= -2 mS, and b22 = -0.2.

a. Find the phasor voltage V2.

b. Find the average power delivered to the 5 kΩ load.

c. Find the average power delivered to the input port.

d. Find the load impedance for maximum average power transfer.

Zg

ZL

Example

28

e. To find he maximum average power delivered to the load in (d)

Solution

a. 26410230002.020 3 xb

19

10

500050010250002.0500203000

500023

2

xV

V

g

VV 02 016.263500

19

10

b. The average power delivered to the 5000 Ω load is

WP 93.650002

16.263 2

2

c. To find the maximum average power delivered to the input port

33.1333

400

205000102

300050002.03x

Z in

29

mAI 47.78933.133500

5001

WP 55.4133.1332

78947.0 2

1

d. The load impedance for maximum power transfer

33.833,102.1

000,13

2.0500102

3000500203x

ZTh

33.833,10*ThL ZZ

e. The maximum average power delivered to ZL

WmaxP

VVV

V

g

01.833.833,10

67.416

2

1

67.4165008333.08333.0

2

2

22

30

The b parameters of the two-port network are :

b11 =2000/3, b12 =2/3 MΩ, b21 =1/15 S, and b22 =-100/3. The network is driven by a sinusoidal current source having a maximum amplitude of 100 μA and an internal impedance of 1000+j0 Ω. It is terminated in a resistive load of 10 kΩ.

a. Calculate the average power delivered to the load resistor.

b. Calculated the load resistance for maximum average power.

c. Calculate the maximum average power delivered to the resistor in (b).

Two port model of a networkVg

Zg

ZL

I1I2

V1 V2

+ +

- -