Network Fundamentals

A two-port network (a kind of four-terminal network ) is an electrical network (circuit) or

device with two pairs of terminals to connect to external circuits. Two terminals constitute
a port if the currents applied to them satisfy the essential requirement known as the port
condition: the electric current entering one terminal must equal the current emerging from the
other terminal on the same port.[1][2] The ports constitute interfaces where the network connects
to other networks, the points where signals are applied or outputs are taken. In a two-port
network, often port 1 is considered the input port and port 2 is considered the output port.

Active network contains 1 or more than one source of emf. It consists of active elements like
battery or a transistor. Active elements can inject power to the circuit, provide power gain,
control the current flow within the circuit.

Passive network it does not contain any source of emf. It consists of passive elements like R,
L, C. These elements are not capable of doing the functions done by the active elements.

1. T-Section

When a electrical network section looks like a “T”, it is known as T-Section. Figure 1 (a) and
(b) represent asymmetrical and symmetrical T-sections.
Symmetrical asymmetrical
 The figure indicates that in a symmetrical section, the series arm impedance is equal in both
the sides of the shunt arm impedance, it may be observed that in unsymmetrical section, the
impedances Z1 and Z3 are not equal to each other. The net series arm impedance of the
unsymmetrical section is (Z1 + Z3) ohm while that for the symmetrical section is (2Z1) ohm.
The shunt arm impedance in both the cases is given by Z2 ohm.

Balanced and Unbalanced configuration of a T-section has been shown in figure 2. The
balanced T-section has also been termed as H-section.

2. π-Section

In an identical manner, symmetrical and asymmetrical π section (i.e., the section whose
structure looks like a π) can be configured (Figure 3). In asymmetrical π section, the shunt arm
impedances are not identical while for the symmetrical π section, the shunt arm impedances
must be identical.

Here, the net series arm impedances being Z1 in each case, the shunt arm impedance is the
parallel combination of the individual shunt arms.

Figure 4 represents the unbalanced and balanced π section in exactly similar way to that in
T-section.
An asymmetrical π section can also be represented in the form of balanced and unbalanced
modes and the equivalence between modes exists till the mode of connection is same. A
balanced form of π network is known as a “O” section also.

T- π network Conversion

a T section is known (or vice versa) three equations will be written. For the T section, the
impedance Z12 measured between terminals

Figure 5. Circuits for deriving network transformation equations.

1-2 is Z12 = Z1 + Z3, and for the π section the impedance Z12 = ZA(ZB + ZC)/(ZA + ZB + ZC).
When these are equated,

The corresponding equations for impedances measured between terminals 3-4 are

The corresponding equations for impedances measured between terminals 1-3 are
These three equations can be solved simultaneously for Z1, Z2, and Z3 in terms of ZA, ZB, and
ZC. By adding equations 1 and 3, and subtracting equation 2, it is found that

Likewise, adding equations 2 and 3, and subtracting equation 1 gives

and adding equations 1 and 2, and subtracting equation 3 gives

These equations are for transformation from a π to a T network. To make transformations from
T to a π network, the equations prove to be

and

Image Impedance

It is the impedance which when connected to the input and the output of the transducer, it will
make both the impedances equal at the input and the output terminal. It is basically the concept
which is used in the field of the network analysis and design and also in the field of the filter
design. It applies to the seen impedance which is determined by looking through the ports of
the network.
The Two-port network can be properly used to describe the concept of the image impedance in
the better way.

Two port network

The impedance zi1 – when considered from the port 1
Zi2 –image impedance when considered from the port 2

The image impedance will not be equal until the network is the symmetrical network or anti-
symmetrical with respect to the ports.

Iterative Impedance

It is defined as the particular value of the load impedance which has the ability to produce an
input impedance with the value as same as the value of the load impedance. In the two port
system when it is connected at the one end then it produces equal impedance when looking at
each other.

Characteristic impedance
The characteristics impedance also known as the surge impedance is usually considered in the
case of the transmission line and is represented as Z0. The characteristics impedance is defined
as the ratio of the amplitude of the voltage and the current taking the consideration of the single
wave through the line. The surge impedance is usually allocated through the transmission line
with its geometry and the material. It is to be noted that this impedance is independent of the
line length.SI unit – ohm
Characteristic impedance derivations
The symmetrical T and π network are the most frequently used networks. The condition in the
symmetrical T network is that the total series arm impedance and shunt arm impedance must
be Z1 and Z2 respectively. To have a total series arm impedance of Z1, the two series arm
impedances must be selected as Z1/2 each as shown in the Fig. 8.7.
Let us derive the expressions for the characteristic impedance (Z0) and propagation constant
(γ) in terms of the network elements.
Characteristic Impedance (Z0) of T section:
(A) In terms of series and shunt arm impedances
Consider a symmetrical T network terminated at its output terminal with its characteristic
impedance as shown in the Fig. 8.8.

By the property of the symmetrical network, the input impedance of such network terminated
in Z0 at other port is equal to Z0.
The input impedance of a T network is given by,

Characteristic Impedance (Z0) of π network:

(A) In terms of series and shunt arm impedances
Consider a symmetrical π network terminated at its output terminals with its characteristic
impedance Z0 as shown in the Fig. 8.16.

By the property of the symmetrical network, the input impedance of such network terminated
with Z0 at other port is equal to Z0.
The input impedance of a symmetrical π network is given by

Multiplying numerator and denominator by the factor Z1/4,

Here the characteristic impedance of it section is indicated by Z0π and that of T section by Z0T.
Taking square root on both the sides,

Characteristic Impedance (Z0) of T section:

(B) In terms of open and short circuit impedances
For symmetrical networks, impedances measured at any pair of terminals with other pair of
terminals either open circuit or short circuit are of same value. Consider symmetrical T network
with terminals 2-2′ either open circuit or short circuit as shown in the Fig. 8.9.

Consider Fig. 8.9 (a),

Consider Fig. 8.9 (b),

Multiplying equations (3) and (4), we can write,

Characteristic Impedance (Z0) of 𝝅 network:

(B) In terms of open and short circuit impedances
Consider Symmetrical pi Network in Network Analysis shown in the Fig. 8.17 (a) and Fig.
8.17 (b).

Consider Fig. 8.17 (a),

Consider Fig. 8.17 (b),

Multiplying equations (3) and (4), we can write,

Thus the characteristic impedance of symmetrical π network is given by

Propagation constant

This constant is usually considered for the wave and is defined as change in the phase angle
with respect to the per unit change in the distance travelled by the wave. In other words we can
say as the rate of the change in the phase of wave with distance. The constant is represented
as the term K.

Electromagnetic waves propagate in a sinusoidal fashion. The measure of the change in its
amplitude and phase per unit distance is called the propagation constant. Denoted by the
Greek letter 𝜸. The terminologies like Transmission function, Transmission constant,
Transmission parameter, the Propagation coefficient, Propagation parameter are synonymous
with this quantity. Sometimes 𝜶 and 𝜷 are referred collectively as Propagation parameters or
Transmission parameters.
For any given system, the Propagation constant can be mathematically expressed as-

Propagation Constant (γ)= ratio of the Complex amplitude at the source of the wave(A0) to
the Complex amplitude at distance x (Ax) such that,
𝐴𝑜
= 𝑒 𝛄𝐱
𝐴𝑥

The complex entity can be written as γ=α+iβ

Where,

𝜶, is the real part called the attenuation constant.

𝜷, is the imaginary part called phase constant.

The phase can be calculated using Euler’s formula as –

eiθ= cosθ +i sinθ

The equation is sinusoidal in nature. Here phase varies according to 𝜽 whereas, the amplitude
remains invariant as ∣eiθ∣ = [cos2θ+sin2θ]1/2 = 1
The angles are measured in radian.