conventional linear two port network parameters

ELE-6256 Active RF Circuits

A

Seminar Report on

Conventional Linear Two-port Network Parameters

Author: Bijaya Shrestha

Student Id: 217370

Date: 10.11.2010

Table of Contents

Page

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Chapter

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Network Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 One-port Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Multiport Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Linear Two-port Network Parameters . . . . . . . . . . . . . . . . . . . . 3

2 z-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Derivations of z-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Reciprocal Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 y-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Derivations of y-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 10


3.3 π-Equivalent Reciprocal Model . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 h-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Derivations of h-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 14


4.3 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

ii

4.4.1 Terminated Equivalent Two-port . . . . . . . . . . . . . . . . . . . 16

4.4.2 Parameters of Common Emitter BJT . . . . . . . . . . . . . . . . . 17

5 ABCD-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 Derivation of ABCD-parameters . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 Applications & Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3.1 Cascaded Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3.2 Finding Length of Microstrip Line . . . . . . . . . . . . . . . . . . . 22

6 Two-port Parameter Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.1 Expressing y-parameters in Terms of z-parameters . . . . . . . . . . . . . . 24

6.2 Expressing h-parameters in terms of z-parameters . . . . . . . . . . . . . . 25

6.3 Conversion Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

iii

Chapter 1

Introduction

1.1 Network Basics

An electrical circuit or device can be referred to as a network because it consists of different

electrical components or devices interconnected to each other. The devices are made to

make our lives easier. An information is provided to the device and gets processed to

produce the required result. There are different types of devices for fulfilling varieties of

applications. The most commonly used device is the amplifier; it is highly used in the

communications circuits to overcome the losses during signal propagation. One-port and

multiport network concepts are used to simplify the complicated circuits and determine

their performance in a convenient way. A device can be treated as a black box and its

properties can be obtained without knowing its internal structure by determining the input

and output port parameters. Port means a pair of terminals carrying equal currents in

opposite directions. In this paper, impedances are represented by resistors in all the figures.

1.2 One-port Network

If only the relationship between port voltage and current is of interest then a one-port

[1] network model is used. Such networks are therefore used for finding only the input-

output properties of a device. A resistor, capacitor, and inductor are the one-port devices

satisfying the current-voltage relationships vR = RiR, iC = CdvC/dt, and vL = LdiL/dt

respectively. A one-port device may contain any number of resistors, capacitors, inductors,

and other devices interconnected to each other. Thevenin and Norton equivalent circuits

1

[1] are used to determine one-port models.

I1

bb-V1

+-I1

One-port

Figure 1.1: One-port Model.

1.3 Multiport Network

A network having more than one pairs of terminals is called the multiport network. Two-

port networks are linear models and widely used to characterize different active and passive

devices; transformers and amplifiers are the typical examples. Power dividers and circula-

tors consist of more than two ports.

IN−1

bb-

VN−1

+

-IN−1

Port N − 1

I1

bb-V1

+

-

I1

Port 1

Multiport

-

IN

bb-VN

+

IN

Port N

-

I2

bb-V2

+

I2

Port 2

rrr rrr

Figure 1.2: Multiport Network with N Ports [2].

2

1.4 Linear Two-port Network Parameters

The following figure is the two-port linear model comprising of two ports. V1 and I1 are

respectively voltage and current of port 1 and V2 and I2 are respectively voltage and current

of port 2. The conventional directions and polarities of voltages and currents are as shown

in the figure below.

I1

bb-V1

+

-I1

Two-port

-

I2

bb-V2

+

I2

Figure 1.3: Two-port Network.

Modeling a two-port means definining a relationship among these variables. The net-

work is linear because this model gives any two of the variables i.e., dependent variables

as the linear combinations of the other two variables i.e., independent variables. Different

parameters are defined according to the choice of currents and voltages being dependent

or independent as tabulated below.

Table 1.1: Two-port Network Parameters.

Dependent Variables Independent Variables Description

V1, V2 I1, I2 z-parameters

I1, I2 V1, V2 y-parameters

V1, I2 I1, V2 h-parameters

I1, V2 V1, I2 g-parameters

V2, I2 I1, I1 ABCD-parameters

V1, I1 V2, I2 inverse t-parameters

There are six conventional linear two-port network parameters as listed in Table 1.1.

3

Inverse hybrid-parameters or g-parameters and inverse ABCD parameters or inverse t-

parameters are generally not used from applications point of view. z-parameters, y-

parameters, h-parameters, and ABCD-parameters are extensively used and are the major

topics to be discussed in this report.

4

Chapter 2

z-parameters

For determining z-parameters of a two-port linear network V1 and V2 are written as the

linear combinations of I1 and I2. The coefficients of the resulting equations are called the

z-parameters or impedance parameters because they all have the units of impedance.

V1 = z11I1 + z12I2 (2.1)

V2 = z21I1 + z22I2 (2.2)

In matrix form, they can be written asV1

V2

=

z11 z12

z21 z22

I1I2

.2.1 Derivations of z-parameters

The parameters can be determined by open circuiting the ports one at a time. When port

2 is open circuited and port 1 is provided an excitation, I2 becomes zero. From equation

2.1

z11 =V1

I1

∣∣∣∣ I2 = 0 (2.3)

and from equation 2.2

z21 =V2

I1

∣∣∣∣ I2 = 0. (2.4)

Similarly, when port 1 is open circuited and port 2 is excited I1 becomes zero. From

equation 2.1

z12 =V1

I2

∣∣∣∣ I1 = 0 (2.5)

5


z22 =V2

I2

∣∣∣∣ I1 = 0. (2.6)

Since all the z-parameters are obtained either by open-circuiting port 1 or port 2 they are

also called open-circuit impedance parameters. Moreover, z11 is called the driving-point

input impedance, z22 the driving-point output impedance, and z12 and z21 the transfer

impedances.

2.2 Equivalent Circuit Model

Equations 2.1 and 2.2 can be realized by an equivalent circuit model [1] consisting of two

dependent current-controlled voltage sources as shown below.

A A A

z11

@@z12I2

+

−bb

+

-V1

-I1

@@z21I1

+

−

A A A

z22

b-V2

+bI2

Figure 2.1: Equivalent Circuit Modeled by z-parameters.

Considering input section of the above figure and applying Kirchhoff’s Voltage Law

(KVL), voltage V1 is the sum of voltage drop across z11 and current-controlled voltage

source z12I2 i.e., V1 = z11I1 + z12I2 as given by equation 2.1. This equation thus models the

input port of the network in terms of z-parameters. Similarly, the output port is modeled

by equation 2.2. This equivalent circuit helps to find the voltage gains, input and output

impedances of terminated two-port networks.

6

2.3 Reciprocal Networks

A network is said to be reciporcal if the voltage appearing at port 2 due to a current

applied at port 1 is the same as the voltage appearing at port 1 when the same current is

applied to port 2. Networks are reciprocal if they contain only linear passive elements (i.e.,

resistors, capacitors, and inductors) and the presence of dependent or independent sources

makes them non-reciporcal [3]. In terms of z-parameters, the networks can be treated as

reciporcal if z12=z21. And, the network can be represented by an equivalent T model.

Since z12=z21, equations 2.1 and 2.2 can be written as

V1 = z11I1 + z12I2 (2.7)

V2 = z12I1 + z22I2 (2.8)

Adding and subtracting the right hand side of equation 2.7 by z12I1 and equation 2.8

by z12I2,

V1 = (z11 − z12)I1 + z12(I1 + I2) (2.9)

V2 = z12(I1 + I2) + (z22 − z12)I2. (2.10)

And, the equivalent T network is shown in Figure 2.3.

b-V1

+b-I1 A A A

z11 − z12

HHH

z12

A A A

z22 − z12

b-V2

+bI2

Figure 2.2: T-Equivalent Circuit Modeled by z-parameters for a Reciprocal Two-port.

7

2.4 Examples

Let z-parameters of a two-port network be available. What are the expressions for input

impedance, output impedance, and gains of the following terminated network?

A A AZs

+

−Vs

A A A

z11

@@z12I2

+

−bb

+

-V1

-I1

@@z21I1

+

−

A A A

z22

b-V2

+bI2

HHH

ZL-

Zin

Zout

Figure 2.3: Terminated Two-port Network Modeled by z-parameters.

Input Impedance: For the input section, V1 = z11I1 + z12I2. And, for the output

section, V2 = z21I1 + z22I2. Also, V2 = −I2ZL. So, z21I1 + z22I2 = −I2ZL. After simplifying

few steps for the last expression, I2 = −z21

z22+ZLI1. Substituting I2 in the first expression

results V1 = I1

(z11 − z12z21

z22+ZL

). Thus input impedance is found to be

Zin =V1

I1= z11 −

z12z21

z22 + ZL

(2.11)

Output Impedance: For determining output impedance, input voltage source is short

circuited so that KVL in input section gives 0 = (Zs +z11)I1 +z12I2, or I1 = −z12

Zs+z11I2. KVL

in output section results V2 = z21I1 + z22I2. By substituting I1 in this expression, output

impedance can be obtained as

Zout =V2

I2= z22 −

z12z21

z11 + Zs

. (2.12)

Gain: Voltage gain for the given network can be expressed as

Gv =V2

V1

=V2

V1

V1

Vs

(2.13)

By using voltage division rule,V1

Vs

=Zin

Zin + Zs

(2.14)

8

and,

V2 =ZL

ZL + z22

z21I1 (2.15)

=ZL

ZL + z22

z21V1

Zin

.

Therefore,V2

V1

=ZL

ZL + z22

z21

Zin

. (2.16)

Finally, using equations 2.13, 2.14, and 2.16, the voltage gain of the network is obtained as

Gv =Zin

Zin + Zs

ZL

ZL + z22

z21

Zin

(2.17)

=ZL

ZL + z22

z21

Zin + Zs

.

Now the above derived formulas can be used to find input and output impedance and

voltage gain of an amplifier or of any circuit if the z-parameters are known.

2.5 Limitations

The impedance parameters can not be defined for all kinds of two-port networks. For

examples, an ideal transformer and the following circuit don’t have z-parameters.

A A A

R

Figure 2.4: A Circuit Having No z-parameters .

It is obvious from this circuit that when any port is open-circuited, both the port

currents must be zero. When supply is provided at any port current will flow which violates

the port condition for determining z-parameters. For an ideal transformer, voltages V1 and

V2 can not be expressed as functions of I1 and I2 [1]. Consequently z-parameters can not

be defined.

9

Chapter 3

y-parameters

The y-parameters are determined by short circuiting the input and output ports one at

a time. Therefore, they are also called short-circuit parameters. Currents I1 and I2 are

expressed as

I1 = y11V1 + y12V2 (3.1)

I2 = y21V1 + y22V2 (3.2)

In matrix form, I1I2

=

y11 y12

y21 y22

V1

V2

where, the coefficients y11, y12, y21, and y22 are called the y-parameters or the short-circuit

admittance parameters.

3.1 Derivations of y-parameters

When port 2 is short circuited and port 1 is provided an excitation, V2 becomes zero. From

equation 3.1

y11 =I1V1

∣∣∣∣V2 = 0 (3.3)


y21 =I2V1

∣∣∣∣V2 = 0. (3.4)

Similarly, when port 1 is short circuited and port 2 is excited V1 becomes zero. From

equation 3.1

y12 =I1V2

∣∣∣∣V1 = 0 (3.5)

10


y22 =I2V2

∣∣∣∣V1 = 0. (3.6)


The equations 3.1 and 3.2 can be modeled by an equivalent circuit [1] with two dependent

voltage controlled current sources and two admittances as shown in figure below.

b-V1

+b-I1

HHH

y11 ?@@y12V2 ?@

@y21V1

HHH

y22

b-V2

+bI2

Figure 3.1: Equivalent Circuit Modeled by y-parameters.

3.3 π-Equivalent Reciprocal Model

If the network is reciprocal, then y12 = y21. Equations 3.1 and 3.2 can be rewritten as

I1 = y11V1 + y12V2 (3.7)

I2 = y12V1 + y22V2 (3.8)

Adding and subtracting the right hand side of equation 3.7 by y12V1 and equation 3.8 by

y12V2,

I1 = y11+y12V1 − y12(V1 − V2) (3.9)

I2 = y12(V2 − V1) + (y22 + y12)V2. (3.10)

These equations lead to the equivalent π-network as shown in Figure 3.3.

11

b-V1

+b-I1

HHH

y11 + y12

A A A

−y12

HHH

y22 + y12

b-V2

+bI2

Figure 3.2: π-Equivalent Circuit Modeled by y-parameters for a Reciprocal Two-port.

3.4 Examples

If y-parameters of a two-port network are considered then the formulas of input and output

admittance and voltage gain for the terminated case can be derived. Let us consider voltage

Vs with source admittance Ys be applied at port 1 and port 2 be terminated by load

admittance YL as shown in Figure 3.3. This is the configuration used in the real practice.

A A A

Ys

+

−Vs b-V1

+b-I1

HHH

y11 ?@@y12V2 ?@

@y21V1

HHH

y22

b-V2

+bI2

HHH

YL

Figure 3.3: Terminated Equivalent Circuit Modeled by y-parameters.

By proceeding the same way as done in section 2.3 the following properties [1] can be

obtained.

Input Admittance:

Yin = y11 −y12y21

y22 + YL

. (3.11)

Output Admittance:

Yout = y22 −y12y21

y11 + Ys

. (3.12)

12

Voltage Gain:

Gv =

(Ys

Ys + Yin

)(−y21

y22 + YL

). (3.13)

These formulas can be applied for any two-port network defined by y-parameters.

3.5 Limitations

In an ideal transformer currents I1 and I2 can’t be expressed as the linear combinations of

voltages V1 and V2. Therefore, ideal transformer doesn’t have y-parameters. The following

circuit also doesn’t have admittance parameters. When one of the ports is short-circuited

HHH

R

Figure 3.4: A Circuit Having No y-parameters .

both the port voltages V1 and V2 are zero. Connecting a source in any port means non zero

terminal voltages. Therefore y-parameters can’t be defined here as well.

13

Chapter 4

h-parameters

Hybrid parameters or h-parameters are determined by expressing voltage V1 and current I2

as the linear combinations of current I1 and voltage V2 as given by the following equations.

V1 = h11I1 + h12V2 (4.1)

I2 = h21I1 + h22V2. (4.2)

In matrix form, they can be written asV1

I2

=

h11 h12

h21 h22

I1V2

where, the coefficients are called the h-parameters.

4.1 Derivations of h-parameters

Short-circuiting port 2, V2 is zero. Then from equations 4.1 and 4.2, h11 abd h21 can be

obtained.

h11 =V1

I1

∣∣∣∣V2 = 0 (4.3)

and,

h21 =I2I1

∣∣∣∣V2 = 0. (4.4)

Similarly, open-circuiting port 1, I1 = 0. Then,

h12 =V1

V2

∣∣∣∣ I1 = 0 (4.5)

14

and,

h22 =I2V2

∣∣∣∣ I1 = 0. (4.6)

Here, h11 is the ratio between input voltage and input current and is determined when

port 2 is short-circuited. Therefore, it is known as the short-circuit input impedance. h21

is the ratio between output current and input current and is determined by short-circuiting

port 2. So, h21 is termed as the short-circuit forward current gain. h12 is given by the

ratio between input voltage and output voltage when port 1 is open-circuited. It is hence

termed as the reverse open-circuit voltage gain. The last parameter h22 it the ratio between

output current and output voltage when port 1 is open-circuited. So, h21 is referred to as

the open-circuit output admittance. All the parameters are not of same kind. They include

different properties: impedance, admittance, current gain, and voltage gain. Also, they are

obtained only when both open-circuit and short-circuit conditions are applied. That’s the

reason why they are called hybrid parameters.


The mathematical expressions given by equations 4.1 and 4.2 can be realized by an equiv-

alent circuit [1] as shown below.

b-V1

+b-I1 A A A

h11

@@h12V2

+

−?@

@h21I1

HHH

h22 b-V2

+bI2

Figure 4.1: Equivalent Circuit Modeled by h-parameters.

Actually, this is the simplified model of a common emitter configuration of a bipolar

junction transistor (BJT). The h-parameters are therefore extensively used for character-

15

izing the transistors at low frequencies. At high or microwave frequencies, scattering or

s-parameters are used which is out of scope for this topic.

4.3 Reciprocity

In Chapter 2, the equivalent T-model for a reciprocal two-port was discussed in terms

of z-parameters whereas the equivalent π-model was discussed in terms of y-parameters in

Chapter 3. This section only presents the condition for reciprocity in terms of h-parameters.

If h12 = −h21, then the two-port can be said reciprocal.

4.4 Examples

4.4.1 Terminated Equivalent Two-port

In real life, a device is terminated in both the ports. One port is connected to a voltage

source (Vs) or a current source (Is) having internal impedance of Zs and the other port is

terminated with a load as shown in figure below.

16

A A A

+

−Vs

Zs

-

Zin

b-V1

+b-I1 A A A

h11

@@h12V2

+

−?@

@h21I1

HHH

h22 b-V2

+bI2

HHH

YL

Yout

Figure 4.2: Terminated Equivalent Circuit Modeled by h-parameters.

From this circuit one can easily derive expressions [1] for input impedance, output

admittance, and voltage gain which are directly written here.

Input Impedance:

Zin = h11 −h12h21

h22 + YL

. (4.7)

Output Admittance:

Yout = h22 −h12h21

h11 + Zs

. (4.8)

Voltage Gain:

Gv = −(

1

Zin + Zs

)(h21

h22 + YL

). (4.9)

These formulas can be used to characterize a two-port network if its h-parameters are given.

4.4.2 Parameters of Common Emitter BJT

If h-parameters of a BJT in the following configuration are h11 = 1.6 kΩ, h12 = 2e−4,

h21 = 110, and h22 = 20 µS, find the input impedance, output impedance, and voltage gain

of the given circuit? [4]

Solution: The hybrid equivalent circuit of the given circuit is given in Figure 4.4. The

Thevenin equivalent in the input section gives ZTh = Zs||470 k ≈ Zs and VTh ≈ Vs.

17

+

−Vs

A A A

1 kΩ

R@@

HHH

470 kΩ

HHH

4.7 kΩ

bVCC

HHH

3.3 kΩ

Figure 4.3: Common Emitter Amplifier [4].

A A A

+

−Vs

Zs

-

Zin

HHH

470 kΩ

b-V1

+b-I1 A A A

h11

@@h12V2

+

−?@

@h21I1

HHH

h22 b-V2

+bI2

HHH

4.7 k||3.3 k

Yout

Figure 4.4: Hybrid Equivalent Circuit.

Now, by using equation 4.7, input impedance is s

Zin = 1.6e3 − 2e−4.110

20e−6 + 5.15e−4= 1.6 kΩ

Using equation 4.8, output admittance is

Yout = 20e−6 − 2e−4.110

1.6e3 + 1e3= 1.2e−5 S.

And hence,

Zout =1

Yout

= 86.7 kΩ.

18

Using equation 4.9, voltage gain is

Gv = −(

1

1.6e3 + 1e3

)(110

20e−6 + 5.15e−4

)= −79.

19

Chapter 5

ABCD-parameters

ABCD-parameters are also called transmission parameters or t-parameters because they are

normally used in transmission line analysis. These parameters are related by the following

equations.

V1 = AV2 −BI2 (5.1)

I1 = CV2 −DI2. (5.2)

They are represented in matrix form byV1

I1

=

A B

C D

V2

−I2

The negative sign associated with I2 is for indicating that current in the second port is also

directed along right side. It can be seen that all the ABCD-parameters are some kinds of

transfer functions. They relate directly between input and output. These parameters are

very helpful for cascaded networks.

5.1 Derivation of ABCD-parameters

Open-circuiting port 2,

A =V1

V2

∣∣∣∣ I2 = 0 (5.3)

C =I1V2

∣∣∣∣ I2 = 0. (5.4)

Short-circuiting port 2,

B = −V1

I2

∣∣∣∣V2 = 0 (5.5)

20

D = −I1I2

∣∣∣∣V2 = 0. (5.6)

5.2 Reciprocity

For ABCD-parameters of a two-port network, the reciprocity can be checked with the

value of determinant (AB −CD) (i.e.,|AB −CD|) [5]. If it is 1, the network is reciprocal,

otherwise non-reciprocal.

5.3 Applications & Examples

5.3.1 Cascaded Networks

Let us consider two two-port networks in cascade as shown in Figure 5.1. First network

has ABCD-parameters of A1, B1, C1, and D1 and second network has corresponding

parameters A2, B2, C2, and D2.

bb-V1

+

-I1

N/W 1 bb

-V2

+

-−I2

-V3

+

-I3

N/W 2 bb

-V4

+

-−I4

Figure 5.1: Cascaded Two-port Networks.

For first network,

V1

I1

=

A1 B1

C1 D1

V2

−I2

(5.7)

For second network,

V3

I3

=

A2 B2

C2 D2

V4

−I4

(5.8)

21

Since, V2 = V3 and −I2 = I3, expression 5.7 can be written as

V1

I1

=

A1 B1

C1 D1

A2 B2

C2 D2

V4

−I4.

(5.9)

From the last expression it is seen that the ABCD-parameters of the overall system is

the product of matrices of individual’s ABCD-parameters. Therefore analysis is easy for

cascaded networks with ABCD-parameters.

5.3.2 Finding Length of Microstrip Line

When a small series inductance is needed along the transmission line, its small portion can

be made narrower so that it behaves like an inductor. ABCD- or transmission parameters

can be used to find the length of this portion for the required value of inductance.

Z0 Z0

Z1, l, βL

Figure 5.2: Microstrip and Corresponding Inductance Model.

Where, Z1, l, and β are respectively the characteristic impedance, length, and phase con-

stant of the transmission line portion shown in the Figure 5.2. And, L is the the corre-

sponding inductance as shown in the right side of the figure. For a section of transmission

line, ABCD-parameters are given as

A = cos(βl), B = jZ1sin(βl), C =jsin(βl)

Z1

, D = cos(βl)[2].

And, for an inductor of inductance L Henry,

A = 1, B = jωL, C = 0, D = 1.

Both of them are equivalent; the parameter B can be equated. So,

jωL = jZ1sin(βl)

22

Assuming that βl is very very small, sin(βl) ≡ βl. Therefore,

jωL = jZ1βl.

Hence, the required length of the microstrip for given L is

l =ωL

βZ1

.

23

Chapter 6

Two-port Parameter Conversions

One set of parameters can be converted to another set because of linear relationships. This

chapter includes some of the conversions.

6.1 Expressing y-parameters in Terms of z-parameters

Since z-parameters are defined byV1

V2

=

z11 z12

z21 z22

I1I2

(6.1)

and, y-parameters are defined byI1I2

=

y11 y12

y21 y22

V1

V2

, (6.2)

Equation 6.1 can be rewritten asI1I2

=

z11 z12

z21 z22

−1 V1

V2

. (6.3)

By comparing equations 6.2 and 6.3,

y11 y12

y21 y22

=

z11 z12

z21 z22

−1

. Similarly, z-parameters

can be expressed in terms of y-parameters,

z11 z12

z21 z22

=

y11 y12

y21 y22

−1

.

Both sets of parameters exist if determinants z11z22 − z12z21 6= 0 and y11y22 − y12y21 6= 0.

24

6.2 Expressing h-parameters in terms of z-parameters

Recalling z-parameter equations,

V1 = z11I1 + z12I2 (6.4)

V2 = z21I1 + z22I2 (6.5)

Rearranging the equations by making left hand side with V1 and I2 terms and right hand

side with V2 and I1 terms,

V1 − z12I2 = z11I1 (6.6)

z22I2 = −z21I1 + V2 (6.7)

In the matrix form they can be written as,1 −z12

0 z22

V1

I2

=

z11 0

0 −z21

I1V2

.Taking the first matrix as the inverse to the right hand side and doing some matrix

manupulations [1],

V1

I2

=1

z22

z11z22 − z12z21 z12

−z21 1

I1V2

.And, z22 should not be zero. This is the matrix equation of h-parameters and hence the

h-parameters in terms of z-parameters. Same process can be applied to express again

h-parameters in terms of y-parameters.

25

6.3 Conversion Table

It is not possible to present here every conversion process. So a table is presented which

has interraltions among two-port parameters.

Table 6.1: Two-port Parameter Conversions [6].

26

Chapter 7

Conclusions

The two-port network parameters are helpful in analysing the complicated circuits. By

measuring terminal voltages and currents the network parameters can be determined which

then are used to find the characterisics of the circuit. The main concerns of a device

are about input impedance, output impedance and gains. These are determined without

dealing with internal components by using the network parameters. A device can therefore

be treated as a black box if some sets of two-port network parameters can be defined for

that device. The impedance, admittance, hybrid, and transmission parameters are the

conventional linear two-port network parameters. All the networks don’t have all sets of

parameters. Impedance and Admittance parameters don’t exist for an ideal transformer.

The hybrid parameters are normally used in the analysis of transistors. Transmission

parameters are extensively used in the transmission line analysis. Moreover, they are

helpful in solving many two-ports in cascade because the overall parameters is the product

of the parameters of the individuals. One set of parameters can be converted to another

set of parameters because all the expressions are linear.

27

References

[1] Raymond A. DeCarlo, “Linear Circuit Analysis,” Time Domain, Pha-

sor, and Laplace Transform Approaches, Oxford University Press, New York,

2001, pp. 800–836.

[2] Reinhold Ludwig and Gene Bogdanov, “RF Circuit Design,” Theory and Applications,

Pearson Prentice Hall, Second Edition, pp. 145–163.

[3] http://en.wikipedia.org/wiki/Two-port network

[4] Robert L. Boylestad and Louis Nashelsky ”Electronic Devices and Circuit Theory.”

Pearson Education Low Price Edition, Eight Edition, pp. 429.

[5] Willian H. Hayt, Jarck E. Kemmerly, “Engineering Circuit Analysis,” MCGraw-Hill,

INC., Fifth Edition, pp. 459–486.

[6] http://www.ece.ucsb.edu/Faculty/rodwell/Classes/ece2c/resources/two port.pdf

28

conventional linear two port network parameters

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