Double-circuit parametric amplifier. Semiconductor Parametric Amplifiers Parametric Gain in Detuning Mode

circuit diagram a dual-frequency or, as it is often called, a dual-circuit amplifier is shown in Fig. 10.16. The first, signal, circuit is tuned to the central frequency of the signal spectrum (resonant frequency), and the second, “idle”, circuit is tuned to the frequency of the sora, which is quite different from.

The pump frequency is chosen from the condition

(10.43)

When choosing a frequency, one proceeds from the condition that the signal frequency is outside the transparency band of the auxiliary contour. But the combination frequency must be outside the operating band of the signal circuit.

When these conditions are met, there will be only one frequency voltage on the signal circuit, and frequencies on the auxiliary circuit. Assuming the amplitudes of these voltages to be small compared to , we can replace the nonlinear capacitance , together with the pump generator, with a linear parametric capacitance , varying with frequency , as was done in § 10.5.

Rice. 10.16. Dual Frequency Parametric Amplifier

Then, under the influence of the signal voltage in the variable capacitance circuit, a current arises (in addition to other components that are not of interest in this case)

[cm. 10.36)]. Here .

On the resistance of the idle circuit, the current creates a voltage drop

We write the equivalent emf acting on the capacitance C, as in § 8.16 [see. (8.99)], in the form

The combination current due to this EMF, by analogy with expression (10.44) will be

Note that the pump phase and frequency (he in expression (10.45) are absent.

Taking into account the above relation for the last equality can be written in the form

As you can see, with respect to the signal circuit, the nonlinear capacitance, together with the pump generator and the idle circuit, can be replaced by a conductivity that takes into account the found current

The complex amplitude of this current

Complex amplitude of the voltage across the signal circuit Therefore, the conductance shunting the signal circuit will be

(10.46)

where is the complex conjugate function of the function

For resonance, when, therefore, the resistance of the auxiliary circuit will be and formula (10.46) takes the form

In the equivalent circuit shown in fig. 10.17, the elements located to the left of the dashed line correspond to the signal circuit of the amplifier, and to the right - to the nonlinear capacitance together with the auxiliary circuit. The resulting circuit essentially coincides with the circuit of a single-loop amplifier (see Fig. 10.15). The difference is only in the method of determining the equivalent negative conductivity.

The details related to the definition of combinational oscillations are given in order to draw attention to the following advantages of a two-loop amplifier:

a) the equivalent negative conductance, and hence the power gain, does not depend on the phase of the pump voltage.

b) it is not required to comply with a certain ratio between frequencies

Both of these properties of the two loop amplifier are explained by the fact that the total phase of the combination current in expression (10.45), which determines the nature of the equivalent conductivity, is essentially the phase difference of the pump voltages. The first of them has the form and the second (without taking into account ). When the difference is formed, it drops out, and the difference frequency in any case coincides with the signal frequency (since ).

The gain of a two-circuit amplifier at the resonant frequency can be determined from an expression similar to formula (10.40):

where is calculated by the formula (10.46), is the load conductivity of the signal circuit.

When the signal frequency deviates from the resonant frequency and, accordingly, the frequency from the resistance modulus decreases, which leads to a decrease in the modulus and, consequently, the power gain.

Based on expression (10.46), you can calculate the frequency response and bandwidth of a two-loop amplifier.

The amplifier stability condition in this case can be written in the form

Consider the energy balance in a two-frequency amplifier depending on the ratio of frequencies. Let the frequency and power of the signal at the input of the amplifier be given. Since with an increase in the auxiliary frequency, the modulus of a negative value increases [see. (10.46)], then it also grows [see (10.48)]. Amplifier Output Signal Power

To determine the required power of the pump generator Pson, as well as the power allocated in the auxiliary circuit, we use the Manley-Row theorem. Based on expression (7.104), the following relations can be written:

(The minus sign in the last expression is omitted, since it is obvious that this power is taken from the pump generator.) The power ratio is illustrated in Fig. 10.18. It can be seen from this figure that more power is released on the auxiliary circuit than on the signal circuit. Thus, although the power increases with increasing frequency, the distribution of the power taken from the pump generator changes in favor of the frequency. weak signal it is not the degree of power utilization that matters, but the ratio of power

To illustrate the quantitative relationships in a two-frequency parametric amplifier, we give the following example.

Let it be required to amplify the signal at a frequency with a spectrum width

Initial data of the first (signal) circuit: characteristic resistance Ohm; internal resistance signal source, shunt circuit, ; load resistance.

Initial data of the second (idle) circuit: resonant frequency ; characteristic resistance Ohm; load resistance.

Before calculating the required variation of the varicap capacitance, we find the limiting conductivity value that can be connected to the signal circuit for a given signal spectrum width

The maximum quality factor of the signal circuit (when shunted with negative conductivity), obviously, should not exceed

At , the resulting conductivity shunting the primary circuit must be at least

In conclusion, we note the main advantages and disadvantages of a parametric amplifier.

An important advantage of a parametric amplifier is the relatively low noise level compared to transistor or tube amplifiers. In § 7.3 it was noted that the main source of noise in transistor and tube amplifiers is the shot effect due to the chaotic transfer of discrete charges of electrons and holes (in a transistor). In a parametric amplifier, a similar effect occurs in a device that modulates a parameter. For example, a change in the capacitance of a varicap occurs due to the movement of electrons and holes. However, the intensity of the flow of electricity carriers in a varicap is many times less than in a transistor or a lamp. In the latter, the flow intensity directly determines the power of the useful signal emitted in the load circuit, and in the varicap, it is only the effect of parameter modulation. The weakening of the influence of the shot effect is so significant that the noise level in a parametric amplifier is determined mainly by thermal noise. In this regard, cooling of the parametric diode is often used up to 5 ... 10.

The disadvantage of a parametric amplifier is the complexity of decoupling the pump and signal circuits.

In the circuit shown in Figure 10.14, a, which is typical for meter-range parametric amplifiers, decoupling is carried out using isolation capacitors and blocking chokes. In the microwave range, where parametric amplifiers are especially widely used, one has to resort to very complex structures that combine in one node a two-frequency oscillatory circuit in the form of hollow resonators, a varicap, and special decoupling elements (circulator, directional coupler, absorber, surge filter). These issues are addressed in special courses.

Maybe not everyone has tried to think about what constitutes reinforcement.

We cannot amplify electrical vibrations without expending a certain amount of power. Amplified oscillations will have a large amplitude, their energy will increase. Excess energy cannot come from nothing. It must be entered from outside.

This is how it actually happens. The amplifier cannot operate without power, without input of energy into it, and the energy must be introduced into the system in such a way that the electrical oscillations present in it are amplified. The input of energy must take place in time with the oscillations, otherwise the existing oscillations can be dampened rather than increased.

New types of amplifiers include the so-called parametric amplifiers. Let's get to know their work.

The oscillatory circuit consists of an inductor and a capacitor. The values ​​of inductance and capacitance are one of the parameters of the circuit. Recall what the voltage across the capacitor is when a charge is applied to it. It is equal to:

where and - voltage on the capacitor; q is its charge and C is its capacitance.

The voltage is directly proportional to the amount of charge and inversely proportional to the capacitance of the capacitor. It follows from this expression that in order to increase the voltage on the capacitor, it is not necessary to increase its charge, that is, to give it an additional portion of electricity. This can also be achieved by reducing the capacitance of the capacitor.

If electrical oscillations occur in the circuit, then the charge and, consequently, the voltage across the capacitor change sinusoidally. Twice during the period, the charge on the capacitor plates will be the largest.

And what happens if we just at these moments reduce the capacitance of the capacitor? The charge of the capacitor will not change from this, but the voltage across the capacitor will increase by the same amount as the capacitance of the capacitor has decreased.

But an increase in the voltage on the capacitor means an increase in the amplitude of oscillations, their amplification. Thus, to amplify the oscillations in the circuit, it is possible to reduce its capacitance at the moments of the greatest charge of the capacitor, so that at the moments of the full discharge of the capacitor, the capacitance of the capacitor can be returned to its initial value. Twice during the period of fluctuations, the capacitance will have to be increased and 2 times returned to its original value. This should be done in time with the oscillations - exactly at the moments of the greatest charge and full discharge - and in phase with them - decrease at the moments of full charge and increase at the moments of full discharge.

Using this method, it is possible to amplify the oscillations in the circuit. Since the amplification is carried out by changing one of the parameters of the circuit, this method is calledparametric gain.

Naturally, amplification does not occur here without the expenditure of energy. In a capacitor between the plates there is an electric field, and in order to push the plates apart, it is necessary to expend a known energy (equal to.

This energy increases the field of the capacitor, as a result of which the voltage across it increases. At the moments of the full discharge of the capacitor, an increase in its capacitance to the initial value will not be accompanied by the communication of any additional energy to it, since the convergence of the plates does not meet the opposition of the field, which is absent (for simplicity, we do not take into account other types of energy losses to restore the initial capacitance of the capacitor).

The practical implementation of a parametric amplifier is not particularly difficult. For this purpose, you can use, for example, a semiconductor diode. The diode has a barrier layer in which there are no free charge carriers. This layer is located between layers of different conductivity. So a diode is essentially a capacitor. The distance between the "plates" of this capacitor, i.e., the thickness of the barrier layer, depends on the sign and magnitude of the voltage in both layers. When voltage is applied in the "forward" direction, the layer thickness decreases, when voltage is applied in the opposite direction, it increases. By changing the voltage on the layers of the diode, it is possible to change the capacitance of the “capacitor”, which is the diode, as needed. The diode is a "variable capacitor" in which the change in capacitance can be controlled by the same oscillations to be amplified, and it receives its power from an oscillator, which is often called a pump generator.

An increase in the amplitude of oscillations, their amplification cannot be infinite. Upon reaching a certain limit, the device will begin to generate oscillations - it will turn into a parametric generator.

Modern diodes allow parametric amplifiers to operate at very high frequencies - up to several tens of thousands of megahertz.Parametric amplifierscharacterized by very low self-noise. If some negative bias is applied to the barrier layer, then there will be practically no free charge carriers in this layer and the noise will be reduced to an insignificant value.

As the reader has probably noticed, parametric amplifiers have a lot in common with regenerative amplifiers. This similarity extends even further. Perhaps the device is a kind of "super-regenerative" parametric amplifiers. The operating principles of superparametric and superregenerative amplifiers are essentially the same. The parametric amplifier is brought to generation a certain number of times per second, which is immediately extinguished (the super-regenerator works in the same way). The parametric super-regenerator makes it possible to amplify the signal power in some cases by tens of millions of times.

The ability of controlled reactive two-terminals under certain conditions to play a role active elements The circuit served as the basis for the creation of a special type of radio engineering devices called parametric amplifiers. These amplifiers have found application mainly in the microwave range as input stages of highly sensitive radio receivers. The main advantage of parametric amplifiers is the low level of intrinsic noise, which is associated with the absence of shot current fluctuations in them.

Implementation of parametrically controlled reactive elements.

The possibility of parametric amplification of signals was theoretically predicted at the beginning of the century.

However, the plastic implementation of this idea became possible only in the 50s after the first successful designs of parametric semiconductor bottoms were created. The operation of these diodes, also called varactors, is based on the following effect. If a voltage of reverse polarity is applied to the junction of the diode, then the divided charge q in the blocking layer is a non-linear function of the applied voltage u. The dependence is called the volt-coulomb characteristic of such a nonlinear capacitor. When the voltage changes in the locked junction of the bottom, a bias current arises

Here, is the differential capacitance of the varactor, which is approximately described by the formula

where k - dimensional coefficient; - contact potential difference.

The more the junction is blocked, the less its differential capacitance.

Modern varactors have very advanced characteristics and are capable of operating up to frequencies of several tens of gigahertz, which corresponds to the millimeter wavelength range.

An element with a parametrically controlled inductance can also be created. It is an inductive coil having a core made of a ferromagnetic material with a pronounced dependence of the induction B on the magnetizing current I. Such elements are not widely used at radio frequencies due to the large inertia of the processes of magnetization reversal of the material.

Single-loop parametric amplifier.

Consider a signal generator formed by a parallel connection of an element with active conductivity and an ideal source of harmonic current with amplitude and frequency . A resistive load having conductivity is connected to the generator. At the terminals of the generator there is a voltage with an amplitude in the load active power is released

As is known from the theory of circuits, in the mode of matching the load with the generator, when the value reaches its maximum value:

(12.37)

Obviously, the power in the load can be increased by reducing the generator conductivity in some way. This can be achieved, for example, by connecting a parametric capacitor (varactor) in parallel with the generator.

Rice. 12.4. Schemes of a single-circuit parametric amplifier: a - basic; b - equivalent

The capacitance of the varactor should change with frequency. The initial phase of the pump generator should be chosen so that the resistance [see. formula (12.34)] was negative.

On fig. 12.4, a, b shows the circuits of the simplest single-circuit parametric amplifier that implements this principle.

Inductive element L together with a capacitor [see. formula (12.27)] form a parallel oscillatory circuit tuned to the signal frequency. The input resistance of this circuit is so high that it practically does not shunt the negative active conductivity

introduced by the varactor.

Referring to fig. 12.4, b, we note that the power released in the load will also be maximum in the matching mode, i.e. when

The ratio of this value to that which is determined by formula (12.37) in the absence of a parametric element is commonly called the nominal gain

For example, let . Then or in logarithmic units.

Stability of a parametric amplifier.

If the negative conductance of the varactor fully compensates for the sum of the generator and load conductances, then the parametric amplifier becomes unstable and self-excited.

From the equivalent circuit shown in fig. 12.4, b, it follows that the critical value of the introduced negative conductivity

Assuming that the phase relations of signal and pump oscillations are optimal in the sense that from formulas (12.34), (12.41) we find the critical depth of capacitance modulation:

Example 12.3. A single-circuit parametric amplifier operates at a frequency ), the signal generator and the load have the same conductivity, varactor capacitance Determine the limiting limits of capacitance change, upon reaching which the amplifier is self-excited.

According to the formula (12.42) we determine

Thus, a parametric amplifier is self-excited if the capacitance of the varactor, changing in time according to the harmonic law, varies from to

Parametric gain in detuning mode.

In real conditions, it is difficult, and sometimes impossible, to fulfill the synchronism condition exactly. If the signal frequency is somewhat detuned relative to the required value, that is, then they say that the parametric amplifier operates in an asynchronous mode. In this case, the value Ф, which determines, according to (12.34), the active resistance introduced, depends on time: The resistance introduced, changing according to the law

periodically acquires different signs. As a consequence, there are deep changes in the level of the output signal, similar in nature to beats. This shortcoming of single-loop amplifiers largely hinders their practical use.

Double-circuit parametric amplifier.

Work aimed at improving the performance of parametric amplifiers has led to the creation of fundamentally different devices, free from the above disadvantage. The so-called two-loop amplifier is capable of operating at an arbitrary ratio of signal and pump frequencies, regardless of the initial phases of these oscillations. This effect is achieved through the use of auxiliary oscillations that occur at one of the combination frequencies.

The diagram of a two-circuit parametric amplifier is shown in fig. 12.5.

The amplifier consists of two oscillatory circuits, one of which, called the signal circuit, is tuned to the frequency and the other, the so-called idle circuit, to the idle frequency. The connection between the circuits is carried out using the parametric capacitance of the varactor, which changes in time according to the harmonic law with the pump frequency:

Rice. 12.5. Diagram of a two-loop parametric amplifier

Usually, the quality factors of the signal and idle circuits are high. Therefore, in the stationary mode, the voltages on these circuits are quite accurately described by the harmonic functions of time:

with some amplitudes and initial phases.

Taking into account the stress signs shown in Fig. 12.5, we find that the voltage on the varactor, from where the current through the varactor

(12.44)

Let us analyze the spectral composition of this current. Using the already encountered formula, we make sure that the current contains components at the signal frequency, at idle frequency and also at combination frequencies

In order to find the conductivity introduced into the signal circuit by the series connection of the varactor and the idle circuit, it is first of all necessary to isolate the current component at the signal frequency in formula (12.44):

(12.45)

Here, the first term is in time quadrature with the voltage and, therefore, is not associated with the introduction of active conduction into the circuit. The second term is proportional to the voltage amplitude on the idle circuit. To find this quantity, we single out in (12.44) the useful component of the current at idle frequency, which is proportional to the amplitude

If - the resonant resistance of the idle circuit, then the voltage on it, caused by oscillations at the signal frequency,

whence it follows that

(12.47)

Substituting the values ​​into the second term of formula (12.45), we obtain the expression for the useful current component at the signal frequency, which is due to the influence of the varactor and the idle circuit:

Thus, the conductivity introduced into the signal circuit by the series connection of the varactor and the idle circuit turns out to be active and negative:

The nominal gain is calculated using formula (12.40). Stability analysis is carried out in the same way as in the case of a single-loop amplifier.

Comparing formulas (12.38) and (12.49), it can be noted that in a two-circuit amplifier, the introduced negative conductivity is not related to the initial phases of the signal and pumping. In addition, a two-loop parametric amplifier is not critical to the choice of frequencies coc and the introduced conductivity will always be negative if

Power balance in multiloop parametric systems.

The insensitivity of parametric amplifiers using Raman oscillations to the ratio of the phases of the useful signal and the pump makes it possible to study such systems on the basis of simple energy relations. Let's turn to general scheme shown in fig. 12.6.

Here, three circuits are connected in parallel with the non-linear capacitor. Two of them contain signal and pump sources, the third is passive and serves as an idle circuit tuned to the combination frequency ( - integers). Each circuit is equipped with a narrow-band filter that passes only oscillations with frequencies close to, respectively. For simplicity, it is assumed that the signal and pump circuits have no ohmic losses.

Let one of the sources (signal or pump) be absent. Then, in the current flowing through the nonlinear capacitor, there will be no components with combination frequencies. The idle loop current is zero and the system as a whole behaves like a reactive circuit, drawing no power from the source on average.

If both sources are available, then a current component appears at the combination frequency; this current can only be closed through the idle circuit.

Rice. 12.6. To the derivation of energy relations in a two-loop parametric system

The load available here, on average, consumes power, and positive or negative resistances are introduced into the signal and pump circuits, the value and sign of which determine the redistribution of power between sources.

The system under consideration is closed (autonomous), and, based on the law of conservation of energy, the average powers of the signal, pump, and Raman oscillations are related by the relation

The power averaged over the oscillation period T can be expressed in terms of the energy E released in this time interval:

( - frequency in hertz). In this way,

or considering that

As is customary, we will consider the positive power released in the load, and the negative power given by the generator. From relations (12.54) it can be seen that since then So, if the idle circuit of the amplifier is tuned to a frequency, then both sources (signal and pump) give power to the idle circuit, where it is consumed in the load. Because the power gain

The advantage of this method of parametric amplification lies in the stability of the system, which is unable to self-excite at any signal and pump powers. The disadvantage is that the frequency of the output signal is higher than the frequency of the signal at the input. In the microwave range, this causes certain difficulties in the further processing of oscillations.

Regenerative parametric amplification.

Let i.e., the idle circuit tuning frequency. The Manly-Rowe equations take the form

As follows from the first equation, this mode both powers are positive. Thus, some part of the power taken from the pump generator enters the signal circuit, i.e., regeneration at the signal frequency is observed in the system. output power can be extracted both from the signal and from the idle circuit.

Equations (12.56) do not make it possible to determine the system gain, since the power contains both a part consumed from devices connected to the amplifier input and a part arising due to the regeneration effect. One can note the ability of such amplifiers to self-excitation, since under certain conditions a non-zero power will develop in the signal circuit even in the absence of a useful signal at the input.

From the previous paragraph it follows that by introducing a variable capacitance or inductance into the oscillatory circuit, it is possible, with the appropriate law of parameter change, to amplify the oscillations. The simplest circuit single-circuit parametric amplifier with variable capacitance is shown in fig. 10.14 a. The non-linear capacitance is under the influence of two voltages: a signal voltage with a frequency and a pump voltage with a frequency .

Coupling capacitors protect the pump generator and signal source from the DC voltage used to establish the operating point on the varicap capacitance-voltage characteristic. The blocking choke blocks the path to the source circuit for high frequency currents.

Let us first consider the operating mode of the amplifier under exact observance of the condition . In this so-called synchronous mode, the combination frequency sleep - coincides with the frequency so that there is a current in the circuit only at a frequency. The equivalent circuit for the synchronous mode is shown in fig. 10.14, b for the case of the corresponding negative real conductivity

Rice. 10.14. Single-circuit parametric amplifier (a) and equivalent circuit (b)

The symbol indicates the sum of the capacitance of the circuit capacitor and the average capacitance of the varicap (corresponding to a constant voltage).

To simplify the analysis, the source of the EMF signal , included in the circuit in series, is replaced in fig. 10.15 by a current generator connected in parallel to the circuit and shunted with internal conductivity G. The load conductivity also includes conductivity, taking into account power losses in the circuit elements. Shunting the load conductance with negative conductance reduces the total conductance and thus increases the quality factor of the circuit. It turns out the effect of amplification.

Let us formulate an expression for the gain as the ratio of the signal power at the amplifier output to the maximum power that can be obtained in the absence of parametric modulation. As is known, the maximum power dissipated in the load conductance (in the absence of amplification) is achieved when the signal power

(I - generator current amplitude).

When connecting additional conductivity, the output voltage will be , and the power released in the load conductivity,

Hence the power gain

Recall that is a negative value.

This expression directly implies the stability condition of a parametric amplifier (in synchronous mode)

whence the critical value of the parametric modulation coefficient

where is the quality factor of the circuit, taking into account .

Note that at , i.e., when parametric modulation compensates for losses only in power gain, it is only four.

Rice. 10.15. Single-circuit parametric amplifier (to the circuit in Fig. 10.14, a)

In practice, when amplifying a real signal, the phase of which is not known, and the frequency can change in a certain band, it is impossible to comply with the conditions of the synchronous mode. Let the signal frequency be not exactly a, where is a small deviation that does not go out of the transparency band of the oscillatory circuit. Then the combination frequency will be dream -) In this case, two oscillations appear in the passband of the circuit: one with a frequency (useful signal) and the other with a frequency (combination frequency).

The ratio between the amplitudes of these two oscillations depends on the depth of capacitance modulation and on the value of . A detailed analysis, which is not given here, shows that for values ​​close to the critical value [see (10.42), and relatively small detuning Q, the amplitudes of both oscillations are approximately the same. There are beats and the consequences associated with this (pulsation of the amplitude and changes in the phase of the resulting oscillation). True, it can be shown that even with a frequency divergence, the average oscillation power over the period of beats is greater than in the absence of parametric action, i.e., that in this so-called biharmonic regime, signal amplification takes place. However, this mode of operation of the amplifier is not always acceptable.

From the shortcomings inherent in a single-circuit parametric amplifier, the circuit considered in the next paragraph is free.

A parametric amplifier (PA) is a device containing an oscillatory circuit in which, under the influence of an external source (pump generator), an energy-intensive parameter (capacitance or inductance) changes. And due to the appropriate organization of the oscillatory system, the signal is amplified.

Let us consider a system consisting of two charged plates representing a certain capacitance.

The value of the charge of this capacity:

A forced change in capacitance can be represented as a change (for example, increase) in the distance between the plates. Due to the fact that the capacitance is not closed, the amount of charge will be constant, and the voltage will increase. In this case, the capacitance charge energy will increase, equal to , and the energy (which is a kind of power source) spent on changing the distance between the capacitor plates is transformed into charge energy. Consequently, there will be an increase in the power released by such a capacitor when discharged through a certain load, that is, amplification.

A parametric amplifier functions in a similar way. The source of power (or energy to change the capacitance) for it is a certain high-frequency pump generator that modulates the capacitance or inductance of any element of the oscillatory circuit. With such a change in the energy-intensive parameter in oscillatory circuit there is a negative electrical resistance, so parametric amplifiers are a kind of regenerative amplifiers. A regenerative amplifier is an amplifier with a positive feedback, which is accompanied by the introduction of negative conductivity into the signal circuit. From an energy point of view, the introduction of negative conductivity into the signal circuit corresponds to the pumping of energy into it from the power supply of the amplifier, which makes it possible to provide power amplification.

Distinguish semiconductor, ferrite and electron beam PU. Semiconductor PU (PPU), built on the basis of parametric diodes (varicaps), are most widely used due to such parameters as the low power of the pump generator and the possibility of microminiaturization.

The main element of the PPU is a parametric diode (PD), which is a reverse biased p-n junction, included in an appropriate way in the oscillatory system, to which constant pressure displacement U SM and voltage from the pump generator, which modulates the capacitance of the PD.

If a pump voltage is applied to the reverse-biased p-n junction of the PD, then the change in the capacitance of the diode can be described by the expression

where M 1 \u003d C 1 / C 0, M 2 \u003d C 2 / C 0 is the modulation depth of the PD capacitance with respect to the corresponding harmonics of the pump frequency.

The capacitance modulation depth depends on the pump voltage and can be determined from the capacitance-voltage characteristic of the FP. Moreover, the greater the modulation depth, the more negative resistance is introduced into the circuit.

Due to the non-linear dependence of the PD capacitance on the applied voltage, currents of various combination frequencies f m,n = mf n + nf c, where m, n are integers, can occur in it.

If the capacitance has no losses, then the power distribution over the combination frequencies is determined by the Manly-Row relation:

	}

where P m,n is the power at frequency f m,n .

An analysis of this equality allows us to draw a number of conclusions about the properties of parametric amplifiers. For example, in the case when a nonlinear capacitance connects oscillatory circuits tuned to frequencies f c, f n and f 1,1 = f c + f n = f + , then, taking into account the Manly-Row relations, we obtain

And if power enters the nonlinear capacitance at frequencies f c and f n, then it is released at frequency f + , and at P c = 0 and P + = 0, i.e. the system is non-regenerative. In this case, the maximum gain

Parametric amplifiers of this type are called stable boost converters. Their use is limited by the fact that when amplifying microwave signals, it is difficult to achieve large gains, because f + and f n are very high.

Consider an example when a nonlinear capacitance connects oscillatory circuits tuned to frequencies f s, f n and f 1,-1 = f s - f n = f - , then, taking into account the Manly-Row relations, we obtain

,

Since the circuits of frequencies f c and f - are energetically equivalent from the point of view of parametric action, the power of the pump generator is pumped into both these circuits, or, in other words, negative resistance is introduced both at frequency f c and at frequency f -. Therefore, this type of amplifier is regenerative and can provide arbitrarily high gain.

Depending on the ratio of frequencies f c and f - = f c - f n, resonances can be in different oscillatory systems, or, if f c » f - , - in one oscillatory system. In the first case, the amplifier is called dual-circuit, in the second - single-circuit.

In the theory of regenerative amplifiers, it was shown that amplifiers of this type can be implemented according to two schemes - “for passing” and “for reflection”. The latter, ceteris paribus, make it possible to obtain a larger gain-bandwidth product at a lower noise figure, which determines the expediency of their practical use.

At present, double-loop reflective-type PPAs are most widely used, since, unlike single-loop ones, they do not require rigid phasing of the signal and pump frequencies and allow low noise temperatures in combination with good broadband to be realized.

It is possible to build a PPU, which will not only amplify the signal, but also transfer its frequency, while the pump generator also acts as a local oscillator. In this case, it is possible to convert the frequency both to the top, i.e. with inversion spectrum, and down, without inversion .