Principles of Analog Electronics

161

Two-Port Networks

5.1 DEFINITIONS

Let us consider the scheme shown in Figure 5.1.

It is known as a two-port network (or four-terminal network or quadripole or,

simply, 2-port), a sort of expansion with respect to the two-terminal component

previously detailed in Chapter 4. A 2-port is simply a network with four terminals

arranged into pairs called ports, which serve respectively as input and output for

voltages (v

i

and v

0

) and currents (i

i

and i

0

). Similarly to the aforementioned two-

terminal component, the two-port network can be made of a single device or by a

set of electrical devices, but also by part of a circuit or even by a complete circuit.

It is sometimes useful to consider a complete circuit, or a part of it, as made of

an ensemble of two or more two-port networks, so that the problem of solving the

complete circuit can be broken into a set of manageable sub-problems, to solve

them separately and then to link the sub-problem solutions together.

Examples of basic two-port networks are transformers (Section 5.2), while other

elementary quadripoles include matching networks (Section 6.8 in Chapter6),

lters (Section 7.3 in Chapter 7), and even transmission lines and coaxial cable

between cities.

The two-port network can be considered a functional block, a kind of “black

box” to which an input voltage and current, v

i

and i

i

, is supplied and from which an

output voltage and current, v

o

and i

o

, is obtained. Thus, it is called functional block

because it links output and input values functionally:

,,

vi fv i

oo

ii

))

((

=

5

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Principles of Analog Electronics

162

The knowledge of these two-port parameters (voltages and currents) enables

us to treat thetwo-port as a “black box” that can be embedded within a large

complete circuit.

A 2-port is called active if the output power is bigger than the input power (due

to the presence of sources or generators), and passive if the output power is

smaller than the input power (it could be that the voltage is bigger in the output

or that the current is higher in the output, but the two cannot happen at the

same time because the resulting power must be smaller). But, we will consider as

general conditions for the 2-port that

➜ No energy is stored within it.

➜ No independent sources or generators are inside it (eventually, only depen-

dent ones, according to the denition in Section 5.3).

➜ The same current must enter and leave the same port (as schematized in

Figure 5.1).

Actually, the two-port network is mostly referred to as a complete circuit used to

process a signal. The signal is furnished at the input port by another network or by

a signal source, then the two-port processes the signal, and the processed signal

is fed into a load connected at the output port (Figure 5.2).

R

L

i

o

i

i

i

o

i

i

+

+

–

v

i

v

s

R

s

+

–

v

o

∼

FIGURE 5.2 A voltage source furnishes a signal to a

2-port,which elaborates it and feeds a resistance load.

+

+

–

–

i

o

i

i

i

o

i

i

v

i

v

o

FIGURE 5.1 Schematic

representation of a generic

electric two-port network.

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Two-Port Networks

163

Curiosities

The analysis of two-port networks was pioneered in the 1920s by Breisig (Franz

Breisig, German mathematician, 1868–1934).

In engineering, a black box is any device or system that can be viewed solely

in terms of its input, output, and transfer characteristics. It means that you can

even ignore its real internal workings, and the inner components can be even-

tually unavailable for inspection.

In a two-port network, if one terminal is in common with the input and the out-

put port, then the network is said to be a three-terminal network (Figure 5.3).

In any case, it is commonly preferred to consider a three-terminal network as a

simple instance of a two-port network.

5.2 TRANSFORMERS

A transformer is an example of a two-port network formed by a single device.

It is based on Faraday’s law (Section 4.7.3 in Chapter 4) and on the ferromag-

netic properties (Section 2.13 in Chapter 2) of an iron core. The transformer in its

simplest form is made by a loop of steel laminations, a good conductor for mag-

netic elds, with two inductively coupled coils as in Figure 5.4.

i

o

i

i

i

o

i

i

+

–

v

i

+

–

v

o

FIGURE 5.3 Schematization of

a three-terminal network.

(a) (b)

1

3

2

4

lines of Ф

B

linking

primary & secondary coil

s

i

1

i

2

+

–

v

i

+

–

v

o

(b)

(c)

n

1

:n

2

(c)

FIGURE 5.4 (a) The transformer, (b) its scheme, and (c) its circuit symbol.

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