UNIT: 3

NETWORKS: Definition of Z, Y, T and Hybrid parameters for 2-port networks,

Network Elements, Classification of Networks, parameters of two port networks,
Topologies of 2 port network- T, Pi, Ladder network, Bridged T, Lattice, Parallel T
networks, Symmetrical networks, Reciprocal networks, Symmetrical network
parameters.

Two Port Networks:

A two port network is one of the most fundamental concepts in network theory and circuit
analysis. It refers to a network that comprises of two connection points known as ports through
which input and output signals can pass. Understanding two port networks is essential for
analyzing more complex signal processing and electronic systems. In this article, we will
discuss in detail two port networks, their properties, important parameters, and representations.
We will also explore various connection configurations and applications of two port networks.

A two port network can be simply defined as a "pair of two-terminal electrical networks in
which, the current enters through one terminal and leaves through another terminal of each
port." In other words, it is a network that has two ports for input/output with each port
consisting of two terminals.

The figure below shows a basic representation of a two port network:

Basic Representation of Two Port Network

Fig- Basic Representation of Two Port Network

As seen, a two port network contains four terminals divided into two ports. Port 1 comprises
terminals 1 and 1' while port 2 contains terminals 2 and 2'. Here, current I1 flows into
terminal 1 and exits from terminal 1'. Similarly, for port 2, current I2 enters through terminal
2 and leaves through terminal 2'.
The two port network allows modeling relationships between input and output quantities like
voltages (V1, V2) and currents (I1, I2).

Properties of Two Port Network

Some important properties exhibited by two port networks are:

 Linear and Lumped: A two port network demonstrates linear behavior meaning
superposition holds. It also follows lumped element approximation.
 Reciprocal: If the network is linear and passive, it satisfies the condition of reciprocity
where its mixed partial parameters are equal (Z12=Z21, Y12=Y21, and so on).
 Causal: The output cannot precede the input. It simply describes input-output
relationships.
 Time-invariant: Network behavior does not change over time for a fixed set of inputs.
 Deterministic: Specific inputs will always produce the same outputs.

Two Port Network Parameters

The input-output relationships in a two port network can be described through various
parameter sets. Based on which variables are considered dependent/independent, different
parameter representations are possible.

Some common two port network parameters include:

Z parameters of two port network:

The Z parameters, also known as impedance parameters, represent the relationship between the
voltage and current at each port of the network. They are defined by the voltage and current
ratios at the input and output ports.

With voltages dependent and currents independent, this gives impedance parameters Z11, Z12,
Z21, and Z22
  Z parameters are called as impedance parameters because these are simply the ratios of
voltages and currents. The units of Z parameters are Ohm (Ω).
 We can calculate two Z parameters, Z11 and Z21, by doing open circuit of port2.
Similarly, we can calculate the other two Z parameters, Z12 and Z22 by doing open
circuit of port1. Hence, the Z parameters are also called open-circuit impedance
parameters.

Y parameters

The Y parameters, or admittance parameters, describe the conductance and susceptance of the
network at each port. They are the reciprocal of Z parameters and are useful for analysing
networks in terms of current rather than voltage.

Taking currents dependent and voltages independent yields admittance parameters Y11, Y12,
Y21, and Y22.When considering the variables I1 and I2 as dependent and V1 and V2. as
independent, we can derive a set of two equations representing the behaviour of the two port
networks.

These equations are typically written in matrix form as:

The Y parameters are defined as:

Y11 is the input admittance seen when port 2 is short circuited.

Y12 is the transfer admittance with port 1 short circuited.

Y21 is the reverse transfer admittance with port 2 short circuited.

Y22 is the output admittance with port 1 short circuited.

In simpler terms -

Y11 = Current I1/Voltage V1 when V2 is zero

Y12 = Current I1 /Voltage V2 when V1 is zero

And so on for Y21 and Y22

Since they represent ratios of currents to voltages, the units of Y parameters are Siemens (S).

We can measure the Y parameters by doing a short circuit (connecting the terminals) at one
port while measuring the other. So Y parameters describe the two port network under short
circuit conditions.

ABCD or T parameters

Using voltages and currents as dependent-independent defines transmission parameters A, B,

C, and D.

We will get the following set of two equations by considering the variables V1 & I1 as
dependent and V2 & I2 as independent.

The coefficients of V2 and -I2 are called T parameters.

The T parameters are:

The parameters commonly known as T parameters in a two-port network are also referred to
as transmission parameters or ABCD parameters. Among these parameters, A and D are
dimensionless, lacking any units. However, the units of parameters B and C are ohms and
mhos, respectively.

To determine the values of parameters A and C, an open circuit is implemented at port 2,

allowing for the calculation of the resulting parameters. Conversely, parameters B and D can
be obtained by performing a short circuit at port 2, facilitating the determination of their values

g parameters

Considering inverse hybrid representation leads to inverse hybrid parameters g11, g12, g21,
g22.

When considering the variables I1 and V2 as dependent and V1 and I2

as independent, we can derive a set of two equations representing the behavior of the two-port
network. These equations are typically expressed in matrix form as:

The g-parameters are:

g-parameters are called inverse hybrid parameters. The parameters, g12, and g21 are
dimensionless, since they are unit less. The units of parameters, g11, and g22 are mho and ohm
respectively.

By doing an open circuit of port2, we can calculate two parameters, g11, and g21. Similarly,
we can calculate the other two parameters, g12 and g22 by doing a short circuit of port1.

h parameters (hybrid parameters)

Considering the variables V1 and I2 as dependent and I1 and V2 as independent, we can

formulate a set of two equations describing the behavior of the two-port network. These
equations can be represented in matrix form as:

The h-parameters are:

h-parameters are called hybrid parameters. The parameters, h12 and h21, do not have any units,
since those are dimension-less. The units of parameters, h11, and h22, are Ohm and Mho
respectively.

We can calculate two parameters, h11 and h21 by doing a short circuit of port2. Similarly, we
can calculate the other two parameters, h12, and h22 by doing an open circuit of port1.

The h-parameters or hybrid parameters are useful in transistor modeling circuits (networks).
Application of two port network

Two port networks serve as basic but useful models with wide-ranging applications across
different domains:

 Circuit analysis: Used to represent circuits involving resistors, capacitors, coils, and
basic electronic components.
 Transmission lines: Characterize voltage and current propagation along transmission
lines.
 Antennas: Employed to model radiation properties and matching networks of antennas.
 Amplifiers: Describes small signal behavior of transistors in amplifiers.
 Networks: Model communication systems, control systems, mechanical and acoustic
systems.
 Acoustics: Analogous to electric circuits in modeling acoustical/mechanical systems.
 Electrical machines: Represent transformers, motors, and generators through suitable
parameter models.

Hence, the simple yet powerful two port network concept allows for addressing broader
interdisciplinary problems in engineering. Its parameters provide designers valuable insights
into system operation.

Network Elements:

A network element is the basic building block of an electrical network. The network element
is sometimes also called a circuit element or circuit component.

A network element can be defined as a mathematical model of an electrical device and is

characterized by its voltage and current relationship. Also, a network element or circuit element
cannot be further divided into a device. Thus, the network element is the most fundamental
component of an electrical system.

The behavior of the entire network depends on the behavior and characteristics of its elements.
Based on such characteristics electrical network can be classified as below.

Active Elements
Passive Elements
Bilateral Elements
Unilateral Elements
Linear Elements
Non-Linear Elements
Lumped Elements
Distributed Elements

1.Linear Network Elements:

A circuit or network whose parameters i.e. elements like resistances, inductances and
capacitances are always constant irrespective of the change in time, voltage, temperature etc.
is known as linear network. The Ohm’s law can be applied to such network. The mathematical
equations of such network can be obtained by using the law of superposition.

An electric circuit element that follows linearity property (i.e. homogeneity and superposition
properties) for the relationship between excitation (input) and response (output), is called a
linear element.

For a linear element, the characteristics curve is always a straight line that passes through the
origin. The most common example of a linear element is an ohmic resistor.

2.Nonlinear Network Elements:

A circuit whose parameters change their values with change in time, temperature, voltage etc.
is known as nonlinear network. The Ohm’s law may not be applied to such network. Such
network does not follow the law.

The circuit elements that do not follow the linearity property (i.e. homogeneity and
superposition) for the relationship between input and output are called non-linear elements.

In simple words, those elements which are not linear are called non-linear elements.

For the non-linear elements, the characteristics curve may not be a straight line or it may not
pass through the origin.
Examples of non-linear elements include diodes, transistors, vacuum tubes, etc.

3.Bilateral Network Elements:

A circuit whose characteristics, behavior is same irrespective of the direction of current through
various elements of it, is called bilateral network. Network consisting only resistances is good
example of bilateral network.

The elements for which the relationship between voltage and current remains the same for
current flowing in either direction are known as bilateral elements.

Therefore, for the bilateral elements, the characteristics curve is similar in the opposite
quadrants. Examples of bilateral elements are resistors, inductors, and capacitors

4.Unilateral Network Elements:

A circuit whose operation, behavior is dependent on the direction of the current through various
elements is called unilateral network. Circuit consisting diodes, which allows flow of current
only in one direction is good example of unilateral circuit.

The circuit elements that exhibit the different relationship between current and voltage for two
directions (forward and reverse) of the current are called unilateral elements.

Hence, for the unilateral elements, the characteristics curve is different in opposite quadrants
A PN junction diode is the most common example of a unilateral circuit element.

5.Active Network Elements:

A circuit which contains a source of energy is called an energy source may be a voltage or
current source.

When a network element or circuit element has the ability to deliver electrical energy or to
produce power gain in the circuit, then the element is called an active element.

In other words, a circuit element for which the slope of its characteristics curve at any point is
negative then the element is called an active element.

Common examples of active elements are generators, batteries, other independent sources,
transistors, Op-Amps, etc.

The active elements are able to provide power or power gain to the electric circuit for an infinite
duration of time.

Note – The transistor (BJT) can provide power amplification or power gain in the circuit so it
is an active element, while the transformer has the same power at input terminals and output
terminals, hence the transformer is not an active element.

6.Passive Network Elements:

A circuit which contains no energy source is called passive. There are two forms of circuits in
which two types of voltages are used. One alternating i.e. ac. while second is direct i.e. d.c.
The alternating current (a.c.) circuits contains voltages which are periodically varying and
hence the currents also vary periodically. The direct current circuits (d.c) contains fixed voltage
sources having polarities +ve and -ve.

A circuit element that can only absorb electric power is called a passive element. The passive
elements are not able to deliver the electric power or power gain to the circuit.

In other words, if the slope of the characteristics curve of an electric circuit element is positive
at all the points, then the element is a passive element.

Examples of passive elements are resistors, inductors, capacitors, transformers, etc.

Note – The charged inductor and capacitor provide power to the circuit but for a very small
time, i.e. they cannot provide power or power gain for an infinite duration of time which is
why they are passive circuit elements.

7.Lumped Network Elements:

A network in which all the network elements are physically separable is known as lumped
network. Most of the electric networks are lumped in nature.

A circuit element is considered a lumped element is its physical size is very small with respect
to the wavelength of the signal. Therefore, the separate elements that are very small in size are
called lumped elements.

Examples of lumped elements are resistors, capacitors, and inductors

8.Distributed Network Elements:

A network in which the circuit elements like resistance, inductance etc. cannot be physically
separable for analysis purposes, is called distributed network.
The best example of such a network is a transmission line where resistance, inductance and
capacitance of a transmission line are distributed all along its length and cannot be shown as a
separate element, anywhere in the circuit.

A distributed element is one that is distributed over the entire length of the circuit and is not
electrically separable. One practical example of a distributed element is transmission lines.

In the case of transmission lines, the resistance, capacitance, and inductance are distributed
over the entire length of the line and it is not possible to consider it at a single point.

Classification of Networks:

Symmetrical and asymmetrical network

Symmetrical is one whose electrical characteristics do not change when its input and output
terminals are interchanged.

Symmetrical network offers same impedance at the input and output terminals

Asymmetrical is one whose electrical characteristics changes when its input and output
terminals are interchanged.

Asymmetrical network doesn't offer same impedance at the input and output terminals.
Balanced and unbalanced network

Balanced is one, in which all the series impedances are identical/same and also these are
symmetrical with respect to ground. Thus, the corresponding series arm impedances must be
equal.

All impedances in the network are equal, and symmetrical currents or voltages exist in all parts
of the circuit.

Unbalanced network refers to an electrical circuit or network where the impedances are not the
same in all branches or legs of the system. This is a common occurrence in many practical
electrical and electronic systems and can have significant implications for system performance.

Impedances are unequal, leading to asymmetry in currents or voltages.

Ladder network

Ladder is one which consists of a large number of similar networks connected one after other.
A ladder network may be balanced network or unbalanced network.
When all the impedances are connected in series, admittances are connected in parallel in the
given network, then it is called as a ladder network.
Topologies of 2 port network

Many topology names relate to their appearance when drawn diagrammatically. Most circuits
can be drawn in a variety of ways and consequently have a variety of names. For instance, the
three circuits shown in Figure 1.1 all look different but have identical topologies.

Figure 1.1. T, Y and Star topologies are all identical.

This example also demonstrates a common convention of naming topologies after a letter of
the alphabet to which they have a resemblance. Greek alphabet letters can also be used in this
way, for example Π (pi) topology and Δ (delta) topology.

Series and parallel topologies

A network with two components or branches has only two possible topologies: series and
parallel.

Figure 1.2. Series and parallel topologies with two branches

Even for these simplest of topologies, the circuit can be presented in varying ways.

Figure 1.3. All these topologies are identical. Series topology is a general name. Voltage
divider or potential divider is used for circuits of that purpose. L-section is a common name for
the topology in filter design.

Figure 1.4. Series and parallel topologies with three branches

The topologies shown in figure 1.5 are commonly used for filter and attenuator designs. The
L-section is identical topology to the potential divider topology. The T-section is identical
topology to the Y topology. The Π-section is identical topology to the Δ topology. These kinds
of circuits are commonly analyzed and characterized in terms of a two-port network.

Figure 1.5. Common balanced and unbalanced filter network topologies

T network topology structure Resembles the letter "T." It consists of three elements: one
series element and two shunt elements.

Pi network topology structure Resembles the Greek letter π (pi). It consists of three elements:
two shunt elements and one series element.
Bridge topology

Bridge topology is an important topology with many uses in both linear and non-linear
applications, including, amongst many others, the bridge rectifier, the Wheatstone bridge and
the lattice phase equalizer. Bridge topology is rendered in circuit diagrams in several ways.
The first rendering in figure 1.6 is the traditional depiction of a bridge circuit. The second
rendering clearly shows the equivalence between the bridge topology and a topology derived
by series and parallel combinations. The third rendering is more commonly known as lattice
topology.

Lattice topology Consists of two diagonally crossed branches between input and output ports,
forming a diamond or lattice shape.

Figure 1.6

Bridged T topology

Bridged T topology is derived from bridge topology

Figure 1.7. Bridged T topology

Infinite topologies (ladder topologies)

Ladder topology can be extended without limit and is much used in filter designs. There are
many variations on ladder topology. Ladder topology structure Alternates between series and
shunt elements, resembling the rungs of a ladder.

Reciprocal networks

A network is said to be reciprocal 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. Exchanging voltage and current results in an equivalent definition of reciprocity.

A two-port network is said to be symmetrical if the input and output ports can be interchanged
without altering the port voltages and currents. A network is said to be reciprocal if the ratio of
the response to the excitation is invariant to an interchange of the positions of the excitation
and response of the network.

A reciprocal network is a multi-port network in which the power losses are the same between
any pair of ports regardless of direction of propagation.