**General information about quadripoles**

**A pole** is an output (contact, terminal, clamp) of an electrical circuit to which something is connected (signal source, load, another circuit).

Electrical circuits are classified according to the number of poles into:

two-terminal networks (resistance, inductance, capacitance, ideal and real signal sources, semiconductor diode);

three-terminal networks (for example, a three-phase voltage source without a neutral, a three-phase load when connected with a “triangle” in Fig. ** 1.1** , a bipolar transistor);

quadripoles (signal amplifiers, filters).

Circuits with a large number of poles occur much less frequently.

Rice. 1.1

**A quadripole** is an electrical circuit with four poles, divided into a pair of input and a pair of output poles, as shown in ** Fig** . 1.2.

Rice. 1.2

The input poles are usually shown on the left with index *1* , and the output poles are on the right with index *2* . The input and output currents are most often referred to as ** flowing** into the quadripole.

There are linear (containing only linear elements) and non-linear (which include at least one non-linear element) quadripoles (Fig. ** 1.3** ).

Rice. 1.3

The properties of linear quadripoles are considered under ** harmonic** effects, the calculation of which is conveniently carried out by the method of complex amplitudes.

In a linear four-terminal network with harmonic action, all currents and voltages are also harmonic with the same frequency, and their amplitudes and initial phases depend on the frequency of the action.

**Input and output resistance of a quadripole**

The frequency characteristics are considered to be the input and weekend resistance as a function of signal frequency. By definition, for a given load resistance of a quadripole connected to its output,

. (2.1)

The output impedance is determined by the known internal resistance of the input signal source ,

. (2.2)

Knowledge of these characteristics is necessary when analyzing the possibilities of connecting a real signal source and load to a four-terminal network.

Consider a quadripole with a real voltage source connected to it with internal resistance and load , as shown in ** Figure 2.1** .

Fig.2.1

The equivalent circuit of the input circuit of the quadripole is shown in ** Fig** . 2.2.

Rice. 2.2

Here is the input impedance of the quadripole.

**If it is necessary to ensure the maximum amplitude of the input voltage , then according to Ohm’s law we get**

. (2.3)

Representing and , can be written

. (2.4)

For an ideal voltage source and the input voltage is equal to the EMF of the source.

**If we provide the maximum condition** *on*

, (2.5)

then from (2.4) it follows

. (2.6)

and subject to input voltage quadripole will be greater than the EMF of the signal source.

**This is due to resonant phenomena in the input circuit** , which will be considered later.

**For maximum power , consumed by the quadripole from the signal source,** from the general expression

, (2.7)

where is ** the complex conjugate** amplitude of the input current, we get

, (2.8)

where – complex-conjugate EMF of the source;

– operator for calculating the real part of a number.

The product of complex conjugate numbers is equal to the square of their modulus

, (2.9)

as a result we get

. (2.10)

From the expression obtained, it is easy to obtain the condition for the maximum power consumed by the four-terminal network (Fig. ** 3.6** ) (

*matching condition)*(2.11)

A similar analysis can be carried out for the output circuit of the quadripole.

In accordance with the theorem on the equivalent voltage source, the equivalent circuit of the output circuit has the form shown in ** Fig. 2.3, a** , where and – equivalent EMF and internal resistance of the active two-terminal network shown in

**, a .**

*Fig. 2.3, b*Rice. 2.3

The power in the load, similarly to (2.10), is equal to

, (2.12)

where

and .

As a result, it is easy to obtain the condition for transmitting the maximum power from the signal source through the quadripole to the load ( ** the condition for matching the quadripole with the load** ),

(2.13)

As an example, consider the circuit in Fig. ** 2.4** , which includes an input signal source (a real source of harmonic voltage with a complex amplitude , internal resistance and frequency ), RC – quadripole and load .

Rice. 2.4

The circuit diagram for determining the input resistance of a loaded quadripole is shown in Fig. ** 2.5** .

Rice. 2.5

Value is defined by the expression

.

With active load , multiplying the numerator and denominator of the fraction by the complex conjugate factor, we get

,

the input resistance modulus is equal to

,

but active and reactive components can be written as

,

.

Figure ** 2.6** shows the dependencies on the frequency of the module and active component input impedance of a quadripole at and .

Rice. 2.6

Figure ** 2.7** shows the frequency dependence of the reactive component of the input impedance of the quadripole.

Rice. 2.7

As you can see, the input impedance of the quadripole changes significantly in the selected frequency range and has a capacitive character. The module and the active component of the resistance decrease with increasing frequency from the value at before at an infinite frequency (at high frequencies, the capacitance shunts the load).

Consider what the internal resistance of the signal source should be in order to ensure the transfer of maximum power to the four-terminal network at a frequency rad/s.

In this case, the input impedance of the quadripole is Ohm. Then, in accordance with (2.11), the signal source must have an internal resistance equal to Ohm.

Substituting into (2.10) the expressions for the active and reactive components for the circuit Fig. ** 2.4** , we get the dependence of the power consumed by the four-terminal on the frequency of the signal. It is shown in

**with and .**

*Figure 2.8*Rice. 1.12

As can be seen, at the frequency rad/s when Ohm and the matching condition (2.11) is satisfied, there is a maximum power consumed by the four-terminal network.

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