RU2487444C2 - Method of generating high-frequency signals and apparatus for realising said method - Google Patents

Abstract

FIELD: radio engineering, communication.

SUBSTANCE: method involves interaction of high-frequency signals with a three-terminal nonlinear element, an arbitrary four-terminal element, an external feedback circuit, a load in form of a first two-terminal element with complex impedance and a reactive four-terminal element, the input of which is connected in a transverse circuit to a second two-terminal element with complex impedance which imitates resistance of a generator signal source in amplification mode, excitation conditions in form of amplitude balance and phase balance. Apparatus has a dc voltage source (2), a three-terminal nonlinear element (1), a reactive four-terminal element, a load in form of a first two-terminal element, a second two-terminal element with complex impedance which imitates resistance of the input high-frequency signal source of the generator in amplification mode and an arbitrary four-terminal element. The reactive four-terminal element is in form of stage-connected two reverse L-shaped connections of four reactive two-terminal elements, where the two-terminal elements of each L-shaped connection contain an oscillatory circuit.

EFFECT: generating compound signals and designing an efficient generator for radio communication equipment with a given number of radio channels for any given frequency characteristics of the load.

2 cl, 4 dwg

Description

The invention relates to the field of radio communications and can be used to create devices for generating high-frequency signals at a given number of frequencies for arbitrary frequency characteristics of the load in any frequency range.

A known method of generating a high-frequency signal, based on the conversion of the energy of a constant voltage source into the energy of a high-frequency signal, organizing internal feedback in a non-linear element by using a bipolar non-linear element with negative differential resistance, fulfilling the excitation conditions in the form of amplitude balance and phase balance, which determine respectively, the amplitude and frequency of the generated high-frequency signal, and non-linear matching conditions element with a load (see IS Gonorovsky. Radio engineering circuits and signals. - M: “Bustard”, - 2006, p. 414-417).

A device for generating a high-frequency signal, consisting of a constant voltage source that sets the operating point in the middle of the falling section of the current-voltage characteristics of a bipolar nonlinear element with negative differential resistance, a four-terminal reactive load in the form of a parallel oscillatory circuit, while the parameters of the bipolar nonlinear element and varicap are selected from conditions for ensuring the given amplitude and frequency of the generated high-frequency signal ala (see. Gonorovsky, I. S. Radio engineering circuits and signals. - M: "Bustard", - 2006, p. 414-417). The principle of operation of this device is as follows. When a constant voltage (current) source is turned on, due to an abrupt change in the amplitude, oscillations arise in the entire circuit, the spectrum of which occupies the entire frequency radio range. The amplitudes of these oscillations decay quickly. However, due to the presence of internal feedback in a bipolar nonlinear element, a negative differential resistance arises in the section with a falling current-voltage characteristic, which, by matching with a reactive four-terminal, compensates for losses in the circuit. Due to this, the oscillation with a frequency equal to the resonant frequency of the oscillatory circuit is amplified until the amplitude of this oscillation increases to a level at which the amplitude goes beyond the falling section of the current-voltage characteristic. There is a stationary mode.

The closest in technical essence to the claimed method (prototype) is a method of generating a high-frequency signal based on the conversion of the energy of a constant voltage source into the energy of a high-frequency signal, organizing external positive feedback between the load and the control electrode of a three-pole nonlinear element, and fulfilling the excitation conditions in the form of an amplitude balance and phase balance, respectively determining the amplitude and frequency of the generated high-frequency signal, and conditions according to popping the nonlinear element to the load (. cm Gonorovsky IS Radio circuits and signals - M:. "Bustard" - 2006, s.383-401).

The closest in technical essence to the claimed device (prototype) is a device for generating a high-frequency signal (see Gonorovsky IS Radio engineering circuits and signals. - M: "Bustard", - 2006, p. 383-401), the structural diagram of which is given in figure 1.

и $${\stackrel{\text{'}}{Z}}_H$$

в поперечные цепи. Все это вместе образует цепь внешней обратной связи. Первый двухполюсник 8 подключен к нагрузке 7. Второй двухполюсник 10 подключен к управляющему электроду VT 1.The prototype device contains a direct transmission circuit in the form of a three-pole nonlinear element (VT) 1 connected to a constant voltage source 2, the first matching filtering device (SFU) 3, which uses a reactive four-terminal or matching four-terminal, and load 7 in the form of an oscillatory contour on the elements L-4, R-5, C-6. The first SFU 3 is connected between the output electrode VT 1 and the load 7. Between the load 7 and the control electrode VT 1, the second SFU 9 (which is used as a reactive four-terminal or matching four-terminal) is connected with the first two-terminal 8 connected to its input and the second two-pole connected to the output 10 with corresponding complex resistances $${\stackrel{\text{'}\text{'}}{Z}}_0$$

and $${\stackrel{\text{'}\text{'}}{Z}}_H$$

in transverse chains. All this together forms an external feedback circuit. The first two-terminal 8 is connected to the load 7. The second two-terminal 10 is connected to the control electrode VT 1.

The principle of operation of the prototype device is as follows.

, второго СФУ 9 и второго двухполюсника 10 с комплексным сопротивлением $${\stackrel{\text{'}}{Z}}_H$$

) нагрузки 7 и управляющего электрода VT 1 компенсируются потери в контуре L, R, C. Благодаря этому, обратная связь становится положительной, и реализуются условия баланса фаз и амплитуд - условия возбуждения электромагнитных колебаний. В результате колебание с частотой, равной резонансной частоте колебательного контура, подается на управляющий электрод VT 1, который на начальном этапе работает в режиме усиления. Амплитуда этого колебания усиливается до момента ее увеличения до уровня, при котором наступает режим ограничения трехполюсного нелинейного элемента VT 1. Наступает стационарный режим генерации.When you turn on the constant voltage source 2 due to the abrupt change in the amplitude in the entire circuit oscillations occur, the spectrum of which occupies the entire frequency radio frequency range. The amplitudes of these oscillations decay quickly. However, due to the presence of external feedback, coordination with the first SFU 3 of the output electrode VT 1 and load 7 (direct transmission circuit), coordination with the feedback circuit (first two-terminal 8 with complex resistance $${\stackrel{\text{'}\text{'}}{Z}}_0$$

, the second SFU 9 and the second two-terminal 10 with complex resistance $${\stackrel{\text{'}\text{'}}{Z}}_H$$

) load 7 and the control electrode VT 1 compensates for losses in the circuit L, R, C. Due to this, the feedback becomes positive, and the conditions of the balance of phases and amplitudes are realized - the conditions for the excitation of electromagnetic waves. As a result, the oscillation with a frequency equal to the resonant frequency of the oscillatory circuit is supplied to the control electrode VT 1, which at the initial stage operates in amplification mode. The amplitude of this oscillation is amplified until it increases to a level at which the limiting mode of the three-pole nonlinear element VT 1 sets in. The stationary generation mode sets in.

The disadvantage of these methods and devices is the generation of a high-frequency signal at only one frequency. In addition, it does not indicate how it is necessary to choose the values of the parameters of the reactive four-port network at which the excitation mode and the stationary mode occur. This question arises especially sharply when designing generation devices in the HF and UHF bands, on which the reactive components of the parameters of nonlinear elements must be taken into account. Currently, the classical theory of radio circuits does not take this into account. Another disadvantage should be considered the lack of the possibility of generation at arbitrary complex load resistances.

The objective of the invention is to achieve a technical result, which consists in increasing the range of generated oscillations, generating high-frequency signals at a given number of frequencies for arbitrary complex load resistances, which allows you to generate complex signals and create efficient generation devices for radio communications with a given number of radio channels for any given frequency load characteristics, such as antennas. The possibility of using various options for including a three-pole non-linear element with respect to a matching four-terminal and various types of feedback expands the possibilities of the physical feasibility of this result.

The specified technical result is achieved by the fact that in the method of generating high-frequency signals, including converting the energy of a constant voltage source into the energy of a high-frequency signal, the interaction of a high-frequency signal with a direct transmission circuit made of a three-pole nonlinear element and a four-terminal, load and external feedback circuit, the fulfillment of the excitation conditions in the form of a balance of amplitudes and phase balance, respectively determining the amplitude and frequency of the generated high-frequency signal c, the conditions for matching the direct transmission circuit with the load and the conditions for matching the load with the control electrode of a three-pole nonlinear element, according to the invention, the four-terminal device is reactive, another, arbitrary four-terminal device connected to the three-pole nonlinear element in series-parallel circuit is used as an external feedback circuit, a three-pole nonlinear element and an external feedback circuit as a single unit cascade between the output of the reactive four-terminal and the load short, the load is performed in the form of a first two-terminal with complex resistance, a second two-terminal with complex resistance imitating the resistance of the signal source of the generator in the amplification mode is connected to the input of the reactive four-terminal in the transverse circuit; the parameters of the reactive four-terminal network are selected from the conditions for ensuring the stationary generation mode in the form of equal to zero the denominator of the transmission coefficient in the amplification mode simultaneously at all given frequencies of the generated high-frequency signals with a constant amplitude of the constant voltage source in accordance with the following mathematical expressions:

α = (- E + x _0m ) γ-D;

β = Fγ-Ex _0m ,

, $$\beta =\frac{b}{d}$$

- оптимальные значения отношений соответствующих элементов классической матрицы передачи на заданных частотах;Where $$\alpha =\frac{a}{d}$$

, $$\beta =\frac{b}{d}$$

- the optimal values of the relations of the corresponding elements of the classical transmission matrix at given frequencies;

- заданные отношения соответствующих элементов классической матрицы передачи на заданных частотах; $$\gamma =\frac{c}{d}$$

- given relations of the corresponding elements of the classical transmission matrix at given frequencies;

a, b, c, d - elements of the classical transmission matrix;

$$D=\frac{r_{0m}(r_{22\mathrm{<figure-callout\; id="2"\; label="n\; m"\; filenames="00000070.png"\; state="\{\{state\}\}">n\; </figure-callout>}m}^2+x_{22nm}^2)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}};$$

; $$E=-\frac{r_{0m}(x_{\mathrm{eleven}nm}{r}_{22nm}-r_{\mathrm{eleven}nm}{x}_{22nm})}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}}$$

;

; $$F=\frac{-r_{0m}(r_{\mathrm{eleven}nm}^2+x_{\mathrm{eleven}nm}^2)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}}$$

;

r _11nm = r _11m -A ₁ r _nm + B ₁ x _nm ;

x _11nm = x _11m -B ₁ r _nm + A ₁ x _nm ;

r _22nm = 1-r _{22m rnm} + x _22m x _nm ;

x _22nm = -x _nm r _22m -r _nm x _22m ;

A ₁ = r _11m r _22m -x _11m x _22m -r _12m r _21m + x _12m x _21m ;

B ₁ = r _11m x _22m -x _11m r _22m -r _12m x _21m + x _12m r _21m ;

r _0m , x _0m - set values of the real and imaginary components of the resistance of the source of the input high-frequency signal of the generator in the amplification mode at a given number of frequencies;

r _nm , x _nm - set values of the real and imaginary component of the load resistance at a given number of frequencies;

r _11m , x _11m , r _12m , x _12m , r _21m , x _21m , r _22m , x _22m are the given total values of the real and imaginary components of the mixed matrix H of the three-pole nonlinear element for a given amplitude of the constant voltage and the corresponding real and imaginary components mixed matrix H of the external feedback circuit h _11m = r _11m + jx _11m , h _12m = r _12m + jx _12m , h _21m = r _21m + jx _21m , h _22m = r _22m + jx _22m at given frequencies;

m = 1, 2, ... N - frequency numbers;

Using the obtained values of the parameters of the reactive four-port network, at a given number of frequencies, the excitation conditions in the form of a balance of amplitudes and phase balance are simultaneously satisfied, as well as the conditions for matching the direct transmission circuit with the load and the load with the control electrode of a three-pole nonlinear element.

The indicated result is also achieved by the fact that in the device for generating high-frequency signals containing a constant voltage source, a direct transmission circuit of a three-pole nonlinear element and a four-terminal, the load and the external feedback circuit, according to the invention, the four-terminal is made reactive, the external feedback circuit is made in the form of an arbitrary a four-terminal connected to a three-pole non-linear element in a series-parallel circuit, a three-pole non-linear element and an external circuit In this connection, as a single node, they are cascaded between the output of the reactive four-terminal and the load, the load is made in the form of the first two-terminal with complex resistance, a second two-terminal with complex resistance is connected to the input of the reactive four-terminal in the transverse circuit, which simulates the resistance of the source of the input high-frequency signal of the generator in amplification mode, reactive four-terminal is made in the form of cascade-connected two inverse L-shaped connections of four reactive two-terminal in with resistances X _1m , X _2m , X _3m , X _4m , moreover, the two-terminal with resistance X _1m , X _{2m are} formed in the form of a parallel oscillatory circuit of elements with parameters L _1k , C _1k , connected in series with an arbitrary reactive two-terminal with resistance x _km , and the values of the parameters are determined from the matching condition according to the criterion of providing a stationary mode of generation at two frequencies using the following mathematical expressions:

; $$L_{\mathrm{one}k}=\frac{{\omega }_{\mathrm{one}}{X}_{k\mathrm{one}}{x}_{k\mathrm{one}}\left(X_{k2}-x_{k2}\right)+{\omega }_2{X}_{k2}{x}_{k2}\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)}{\left(X_{k2}-x_{k2}\right)\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)\left({\omega }_{\mathrm{one}}^2-{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000070.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2^2\right)}$$

;

; $$C_{\mathrm{one}k}=\frac{\left({\omega }_{\mathrm{one}}^2-{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000070.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2^2\right)\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)\left(X_{k2}-x_{k2}\right)}{{\omega }_{\mathrm{one}}{\omega }_2\left[{\omega }_{\mathrm{one}}{X}_{k2}{x}_{k2}\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)+{\omega }_2{X}_{k\mathrm{one}}{x}_{k\mathrm{one}}\left(X_{k2}-x_{k2}\right)\right]}$$

;

; $$X_{\mathrm{one}m}=\frac{X_{\mathrm{four}m}^{3}D+\left(Q-x_{0m}-\mathrm{four}E\right)X_{\mathrm{four}\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="00000070.png"\; state="\{\{state\}\}">m</figure-callout>}}^2+\left[r_{0m}^2-\mathrm{four}F-x_{0m}E+Q\left(x_{0m}+E\right)\right]X_{\mathrm{four}m}+F\left(Q-x_{0m}\right)}{X_{\mathrm{four}\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="00000070.png"\; state="\{\{state\}\}">m</figure-callout>}}^2+\left(E-Q\right)X_{\mathrm{four}m}+F}$$

;

; $$X_{2m}=\frac{X_{\mathrm{four}m}\left[2F-X_{\mathrm{four}m}^{2}D+X_{\mathrm{four}m}\left(3E-Q\right)\right]}{\left(\mathrm{one}+D\right)X_{m\mathrm{four}}^2-2E{X}_{\mathrm{four}m}-F}$$

;

X _3m = X _4m ;

, $$Q=\pm \sqrt{-r_{0m}^2-X_{\mathrm{four}m}^{2}D+\mathrm{four}\left(F+E{X}_{\mathrm{four}m}\right)}$$

,

;Where $$D=\frac{r_{0m}\left(r_{22\mathrm{<figure-callout\; id="2"\; label="n\; m"\; filenames="00000070.png"\; state="\{\{state\}\}">n\; </figure-callout>}m}^2+x_{22nm}^2\right)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}}$$

;

; $$E=-\frac{r_{0m}\left(x_{\mathrm{eleven}nm}{r}_{22nm}-r_{\mathrm{eleven}nm}{x}_{22nm}\right)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}}$$

;

; $$F=\frac{-r_{0m}\left(r_{\mathrm{eleven}nm}^2-x_{\mathrm{eleven}nm}^2\right)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}}$$

;

r _11nm = r _11m -A ₁ r _nm + B ₁ x _nm ;

x _11nm = x _11m -B ₁ r _nm -A ₁ x _nm ;

r _22nm = 1-r _{22m rnm} + x _22m x _nm ;

x _22nm = -x _nm r _22m -r _nm x _22m ;

A ₁ = r _11m r _22m -x _11m x _22m -r _12m r _21m + x _12m x _21m ;

B ₁ = r _11m x _22m + x _11m r _22m -r _12m r _21m -x _12m r _21m ;

r _0m , x _0m - set values of the real and imaginary components of the resistance of the source of the input high-frequency signal of the generator in the amplification mode at two frequencies ω _m = 2πf _m ;

ω _m = 2πf _m ;

m = 1, 2 - frequency number;

r _nm , x _nm - the specified values of the real component of the load resistance at two frequencies;

r _11m , x _11m , r _12m , x _12m , r _21m , x _21m , r _22m , x _22m are the set total values of the real and imaginary components of the mixed matrix H of the three-pole nonlinear element for a given amplitude of the constant voltage and the corresponding real and imaginary components mixed matrix H of the external feedback circuit h _11m = r _11m + jx _11m , h _12m = r _12m + jx _12m , h _21m = r _21m + jx _21m , h _22m = r _22m + jx _22m at given frequencies;

k = 1, 2 is an index characterizing the corresponding numbers of reactive two-terminal networks with resistances X _1m , X _2m ;

X _3m , X _4m - given equal values of the resistance of the third and fourth two-terminal circuits matching four-terminal in the form of cascade-connected two inverse L-shaped connections of four reactive two-terminal networks at given frequencies.

The invention is illustrated using the following drawings:

Figure 2 shows the structural diagram of the inventive device.

Figure 3 shows the structural diagram of the matching reactive quadrupole included in the inventive device.

In figure 4. a diagram of a reactive dvukhpolosnykh, realizing the second and third dvukhpolosnykh matching dvukhpolosnykh in the form of cascade-connected two reverse L-shaped connections of four reactive dvukhpolosnykh (shown in figure 3).

, $$h_{12m}^{VT}=r_{12m}^{VT}+j{x}_{12m}^{VT}$$

, $$h_{21m}^{VT}=r_{21m}^{VT}+j{x}_{21m}^{VT}$$

, $$h_{22m}^{VT}=r_{22m}^{VT}+j{x}_{22m}^{VT}$$

) на заданных частотах генерируемых сигналов m (m=1, 2 - номер частоты), подключенный к источнику постоянного напряжения (Е₀) 2 и соединенный по высокой частоте с цепью внешней обратной связи (ОС) по последовательно-параллельной схеме (входы соединены последовательно, а выходы - параллельно), выполненной в виде произвольного четырехполюсника 14, сформированного в общем случае на двухполюсниках с комплексными сопротивлениями.The inventive device (figure 2) that implements the inventive method contains a three-pole nonlinear element (VT) 1 with known elements of a mixed matrix H ^VT (where the matrix elements $$h_{\mathrm{eleven}m}^{VT}=r_{\mathrm{eleven}m}^{VT}+j{x}_{\mathrm{eleven}m}^{VT}$$

, $$h_{12m}^{VT}=r_{12m}^{VT}+j{x}_{12m}^{VT}$$

, $$h_{21m}^{VT}=r_{21m}^{VT}+j{x}_{21m}^{VT}$$

, $$h_{22m}^{VT}=r_{22m}^{VT}+j{x}_{22m}^{VT}$$

) at given frequencies of the generated signals m (m = 1, 2 is the frequency number) connected to a constant voltage source (E ₀ ) 2 and connected at high frequency to an external feedback (OS) circuit in a parallel-parallel circuit (inputs are connected in series , and the outputs are parallel), made in the form of an arbitrary four-terminal 14, formed in the general case on two-terminal with complex resistances.

In this case, VT 1 and the four-terminal 14 as a single unit are cascaded at a high frequency between the output of the reactive four-terminal (RF) 12 and the load, which is made in the form of the first two-terminal 13 with complex resistance z _n (where z _nm = r _nm + jx _nm ).

A second two-terminal 11 with a complex resistance z ₀ (where z _0m = r _0m + jx _0m ) is connected to the input of the RF 12 into the transverse circuit at given frequencies, simulating the resistance of the source of high-frequency oscillations that occur when the DC voltage source 2 is turned on at the moment of its amplitude voltage in generation mode.

Moreover, the RF 12 is made in the form of cascade-connected two reverse L-shaped connections of four reactive two-terminal networks with resistances X _1m 15, X _2m 16, X _3m 17, X _4m 18 at given frequencies m (Fig. 3). The synthesis of the generator (the choice of resistance values X _1m , X _2m ) and the scheme for the formation of these two-terminal circuits in the form of series-connected parallel circuit of elements with parameters L _1k , C _1k and an arbitrary reactive two-terminal with resistance x _km (Fig. 4) was carried out according to the criterion of ensuring amplitude balance and phase balance by implementing the equalization of the denominator of the transmission coefficient of the generation device in the amplification mode simultaneously at the given frequencies of the generated signals at a constant amplitude of constant voltage . To simplify the formulas for the resistances X _1m , X _{2m, the} resistances X _3m , X _4m were taken equal to each other.

An arbitrary quadripole 14 is also characterized by the known values of the elements of the mixed matrix H ^OC at given frequencies m:

, $$h_{12m}=r_{12m}^{OC}+j{x}_{12m}^{OC}$$

, $$h_{21m}^{OC}=r_{21m}^{OC}+j{x}_{21m}^{OC}$$

; $$h_{22m}^{OC}=r_{22m}^{OC}+j{x}_{22m}^{OC}$$

. $$h_{\mathrm{eleven}m}^{OC}=r_{\mathrm{eleven}m}^{OC}+j{x}_{\mathrm{eleven}m}^{OC}$$

, $$h_{12m}=r_{12m}^{OC}+j{x}_{12m}^{OC}$$

, $$h_{21m}^{OC}=r_{21m}^{OC}+j{x}_{21m}^{OC}$$

; $$h_{22m}^{OC}=r_{22m}^{OC}+j{x}_{22m}^{OC}$$

.

The selection of the resistance of the four-terminal 14 can be carried out arbitrarily or on the basis of any other physical considerations. In this invention, the resistance values of the complex two-terminal networks of the four-terminal network 14 are selected from the conditions of physical feasibility. In the generation mode, the input source of the high-frequency signal is turned off, and a short-circuit jumper is installed instead.

The inventive device operates as follows.

When you turn on the constant voltage source 2 due to the abrupt change in the amplitude in the entire circuit oscillations occur, the spectrum of which occupies the entire frequency radio frequency range. The amplitudes of these oscillations decay quickly. However, due to the presence of external feedback and due to the indicated choice of the resistance values X _1m , X _{2m of the} second and third two-terminal circuits of the matching RF 12 and the circuits for the formation of these two-terminal circuits, the feedback becomes positive, which is equivalent to the appearance of negative resistance in the circuit (r ₂₁ or r ₁₂ ), which compensates for losses in the entire circuit simultaneously at two given frequencies. Therefore, the oscillation amplitudes with given frequencies are amplified to certain levels and then limited. Due to this, the oscillations with the given two frequencies are amplified until the amplitudes of these oscillations increase to a level at which the amplitude goes beyond the quasilinear portion of the passage of the current-voltage characteristic. There is a stationary mode.

Finally, as a result of the interaction of signals at two frequencies with a nonlinear element in the generation mode, products of nonlinear interaction with combination frequencies arise:

ω _n = Iω ₁ ± Kω ₂ ,

where I, K = 0, 1, 2 ...

Let us prove the feasibility of implementing these properties.

The initial ones are also the dependences of the elements of the mixed matrix H ^{VT of the} three-pole nonlinear element and the matrix H ^{OC of} the feedback circuit on the frequency, which can be determined by the known (for example, measured or calculated) elements of the corresponding resistance, conductivity or transmission matrices:

, $$h_{12}^{VT}=r_{12}^{VT}+j{x}_{12}^{VT}$$

, $$h_{21}^{VT}=r_{21}^{VT}+j{x}_{21}^{VT}$$

, $$h_{22}^{VT}=r_{22}^{VT}+j{x}_{22}^{VT}$$

. $$h_{\mathrm{eleven}}^{VT}=r_{\mathrm{eleven}}^{VT}+j{x}_{\mathrm{eleven}}^{VT}$$

, $$h_{12}^{VT}=r_{12}^{VT}+j{x}_{12}^{VT}$$

, $$h_{21}^{VT}=r_{21}^{VT}+j{x}_{21}^{VT}$$

, $$h_{22}^{VT}=r_{22}^{VT}+j{x}_{22}^{VT}$$

.

, $$h_{12}^{OC}=r_{12}^{OC}+j{x}_{12}^{OC}$$

, $$h_{21}^{OC}=r_{21}^{OC}+j{x}_{21}^{OC}$$

, $$h_{22}^{OC}=r_{22}^{OC}+j{x}_{22}^{OC}$$

. $$h_{\mathrm{eleven}}^{OC}=r_{\mathrm{eleven}}^{OC}+j{x}_{\mathrm{eleven}}^{OC}$$

, $$h_{12}^{OC}=r_{12}^{OC}+j{x}_{12}^{OC}$$

, $$h_{21}^{OC}=r_{21}^{OC}+j{x}_{21}^{OC}$$

, $$h_{22}^{OC}=r_{22}^{OC}+j{x}_{22}^{OC}$$

.

With a series-parallel connection of the four-terminal network, the elements of their matrices add up. The total dependences of the elements of the matrices H of the direct transmission circuit in the form of a nonlinear element and feedback circuit on frequency:

h ₁₁ = r ₁₁ + jx ₁₁ , h ₁₂ = r ₁₂ + jx ₁₂ , h ₂₁ = r ₂₁ + jx ₂₁ , h ₂₂ = r ₂₂ + jx ₂₂ ,

where the dimensions of the elements of the matrix H are as follows:

h ₁₁ is the resistance;

h ₁₂ is dimensionless;

h ₂₁ is dimensionless;

h ₂₂ - conductivity.

The general mixed matrix H of the nonlinear element (VT) 1 and the four-terminal circuit of the OS circuit and the corresponding classical transmission matrix:

$$H=\left|\begin{array}{cc}{h}_{\mathrm{eleven}}& h_{12}\\ h_{21}& h_{22}\end{array}\right|;A=\left|\begin{array}{cc}\frac{-\left|h\right|}{h_{21}}& \frac{h_{\mathrm{eleven}}}{h_{21}}\\ \frac{-h_{22}}{h_{21}}& \frac{\mathrm{one}}{h_{21}}\end{array}\right|,\begin{array}{cccc}& & & \end{array}\left(\mathrm{one}\right)$$

where | h | = h ₁₁ h ₂₂ -h ₁₂ h ₂₁ .

For a variant of a cascade connection of a transistor between a reactive four-terminal and a load (Fig. 2), the four-terminal is described by a transfer matrix:

$$A_{\mathrm{one}}=d\left|\begin{array}{cc}\alpha & j\beta \\ j\gamma & \mathrm{one}\end{array}\right|,\begin{array}{cccc}& & & \end{array}\left(2\right)$$

; $$\beta =\frac{b}{d}$$

; $$\gamma =\frac{c}{d}$$

;Where $$\alpha =\frac{a}{d}$$

; $$\beta =\frac{b}{d}$$

; $$\gamma =\frac{c}{d}$$

;

a, b, c, d - elements of the classical transmission matrix.

The general normalized classical generator transfer matrix is obtained by multiplying the transfer matrices (2) and (1) taking into account normalization conditions:

$$A_{\mathrm{about}bu}=\left|\begin{array}{cc}\frac{-\left(\alpha \left|h\right|+j\beta h_{22}\right)}{h_{21}}\sqrt{\frac{z_n}{z_0}}& \frac{\left(\alpha h_{\mathrm{eleven}}+j\beta \right)}{h_{21}}\frac{\mathrm{one}}{\sqrt{z_0{z}_n}}\\ \frac{-\left(h_{22}+j\gamma \left|h\right|\right)}{h_{21}}\sqrt{z_0{z}_n}& \frac{\left(\mathrm{one}+j\gamma h_{\mathrm{eleven}}\right)}{h_{21}}\sqrt{\frac{z_0}{z_n}}\end{array}\right|.\begin{array}{cccc}& & & \end{array}\left(3\right)$$

Using the well-known connection between the elements of the scattering matrix and the elements of the classical transmission matrix (Feldstein A.L., Yavich L.R. Synthesis of four-terminal and eight-terminal devices on a microwave. M: Communication, 1971. P. 34-36) and a transmission matrix (3), taking into account the normalization conditions, we obtain the expression for the gain of the generator in amplification mode:

$$S_{21}=\frac{2h_{21}\sqrt{z_0{z}_n}}{\left[\left(h_{\mathrm{eleven}}-\left|h\right|z_n\right)\left(\alpha +j\gamma z_0\right)+\left(z_0+j\beta \right)\left(\mathrm{one}-z_n{h}_{22}\right)\right]d}.\begin{array}{ccc}& & \end{array}\left(\mathrm{four}\right)$$

We transform the denominator of the transmission coefficient and write it in the form corresponding to the immitance stability criterion (Kulikovsky A.A. Stability of active linearized circuits with amplifiers of a new type. M - L .: GEI, 1962. 192 p.):

, $$z_0+\frac{\alpha \frac{h_{\mathrm{eleven}}-\left|h\right|z_n}{\mathrm{one}-z_n{h}_{22}}+j\beta }{\mathrm{one}+j\gamma \frac{h_{\mathrm{eleven}}-\left|h\right|z_n}{\mathrm{one}-z_n{h}_{22}}}=0$$

,

where the first term is the resistance z _{0 of the} passive part of the generator;

трехполюсного нелинейного элемента со смешанной матрицей (1), нагруженного на сопротивление нагрузки z_n.the second term, taking into account the transfer matrices (1) and (2), is the input impedance of the active part of the generator in the form of a four-port reactive load impedance $$\frac{h_{\mathrm{eleven}}-\left|h\right|z_n}{\mathrm{one}-z_n{h}_{22}}$$

a three-pole nonlinear element with a mixed matrix (1), loaded on the load resistance z _n .

If this condition for the emergence of a stationary generation regime is written in the form of another equality:

$$\mathrm{one}-\frac{-(\alpha \frac{h_{\mathrm{eleven}}-\left|h\right|z_n}{\mathrm{one}-z_n{h}_{22}}+j\beta )}{(\mathrm{one}+j\gamma \frac{h_{\mathrm{eleven}}-\left|h\right|z_n}{\mathrm{one}-z_n{h}_{22}})z_0}=0$$

then it can be interpreted as a condition for the balance of amplitudes and phase balance 1-KB = 0 (Gonorovsky IS Radio engineering circuits and signals. - M: "Bustard", - 2006, p. 383-401) for an equivalent circuit with an external positive feedback communication. In this case, the four-terminal circuits of the feedback circuit and the equivalent circuit of the three-pole nonlinear element are connected in series-parallel, and in the transmission coefficient (4), instead of the elements of the matrix H ^{VT of the} nonlinear element, it is necessary to use the sum of the elements of this matrix and the elements of the matrix H ^{OC of} the feedback circuit.

For this type of generator and frequency modulator, the feedback gear ratio B and the forward link gain K are determined by the formulas:

; $$B=\frac{\mathrm{one}}{(\mathrm{one}+j\gamma \frac{h_{\mathrm{eleven}}-\left|h\right|z_n}{\mathrm{one}-z_n{h}_{22}})d}$$

;

. $$K=\frac{-(\alpha \frac{h_{\mathrm{eleven}}-\left|h\right|z_n}{\mathrm{one}-z_n{h}_{22}}+j\beta )}{z_0}d$$

.

Equate the denominator of the transmission coefficient to zero:

$${\text{[(h}}_{\text{eleven}}-\left|\text{h}\right|{\text{z}}_{\text{n}}\text{) (α}+\text{jγ}{z}_{\text{0}}\text{)}+{\text{(z}}_{\text{0}}+\text{jβ})-{\text{z}}_{\text{n}}{\text{h}}_{\text{22}}\text{)}=\text{0}\text{(5)}$$

We divide in (5) the real and imaginary parts and obtain a system of two algebraic equations:

$$\begin{array}{c}(\alpha -\gamma x_0)r_{\mathrm{eleven}n}-(x_0+\beta )x_{22n}-\gamma r_0{x}_{\mathrm{eleven}n}+r_0{r}_{22n}=0;\\ (x_0+\beta )r_{22n}+(\alpha -\gamma x_0)x_{\mathrm{eleven}n}+\gamma r_0{r}_{\mathrm{eleven}n}+r_0{x}_{22n}=0\end{array}(6)$$

where r _11n = r ₁₁ -A ₁ r _n + B ₁ x _n ;

x _11n = x ₁₁ -B ₁ r _n -A ₁ x _n ;

r _22n = 1-r ₂₂ r _n + x ₂₂ x _n ;

x _22n = -x _n r ₂₂ -r _n x ₂₂ ;

A ₁ = r ₁₁ r ₂₂ -x ₁₁ x ₂₂ -r ₁₂ r ₂₁ + x ₁₂ x ₂₁ ;

B ₁ = r ₁₁ x ₂₂ + x ₁₁ r ₂₂ -r ₁₂ x ₂₁ + x ₁₂ r ₂₁ .

The solution to system (6) has the form of optimal frequency dependences of the relationships between elements of the classical RF transmission matrix:

$$\alpha =(x_0-E)\gamma -D;\beta =F\gamma -E-x_0,(7)$$

Where $$D=\frac{r_0(r_{22\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="00000070.png"\; state="\{\{state\}\}">n</figure-callout>}}^2+x_{22n}^2)}{r_{\mathrm{eleven}n}{r}_{22n}+x_{\mathrm{eleven}n}{x}_{22n}};$$

; $$E=-\frac{r_0(x_{\mathrm{eleven}n}{r}_{22n}-r_{\mathrm{eleven}n}{x}_{22n})}{r_{\mathrm{eleven}n}{r}_{22n}+x_{\mathrm{eleven}n}{x}_{22n}}$$

;

. $$F=\frac{-r_0(r_{\mathrm{eleven}\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="00000070.png"\; state="\{\{state\}\}">n</figure-callout>}}^2+x_{\mathrm{eleven}n}^2)}{r_{\mathrm{eleven}n}{r}_{22n}+x_{\mathrm{eleven}n}{x}_{22n}}$$

.

To find the optimal dependences of the reactance of the two-terminal circuits that make up the matching four-terminal network, it is necessary to select a typical four-terminal scheme, find its transmission matrix, present its elements in the form (2), substitute the coefficients α, β, γ thus defined in (7) and solve the resulting system of equations with respect to some two parameters of the reactive matching quadripole.

Here is a solution to the synthesis problem for a typical circuit in the form of cascade-connected inverse two L-shaped links (figure 3):

$$\begin{array}{l}{X}_{\mathrm{one}}=\frac{X_{\mathrm{four}}^{3}D+(Q-x_0-\mathrm{four}E)X_{\mathrm{four}}^2+[r_0^2-\mathrm{four}F-x_{0}E+Q(x_0+E)]X_{\mathrm{four}}+F(Q-x_0)}{X_{\mathrm{four}}^2+(E-Q)X_{\mathrm{four}}+F};\\ {\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="00000070.png"\; state="\{\{state\}\}">X</figure-callout>}}_2=\frac{X_{\mathrm{four}}[2F-X_{\mathrm{four}}^{2}D+X_{\mathrm{four}}(3E-Q)]}{(\mathrm{one}+D)X_m^2-2E{X}_{\mathrm{four}}-F};(8)\\ X_3=X_{\mathrm{four}},\\ gdeQ=\pm \sqrt{-r_0^2-X_{\mathrm{four}}^{2}D+\mathrm{four}(F+E{X}_{\mathrm{four}})}.\end{array}$$

The implementation of the optimal approximating functions of the frequency dependences of the two-terminal resistances (8) can be carried out in various ways, for example, using the interpolation method by finding the values of the parameters of the selected reactive two-pole, at which their resistances at the given frequencies coincide with the optimal. The values of the parameters of the remaining two-terminal devices can be chosen arbitrarily or from providing any other conditions. Here is an example of the construction of these two-terminal networks for two interpolation frequencies that were used to synthesize the considered alternatives of the generators.

A parallel oscillatory circuit of elements with parameters L _1k , C _1k , connected in series with an arbitrary reactive two-terminal with resistance x _km (figure 4):

$$\begin{array}{l}\frac{{\omega }_m{L}_{\mathrm{one}k}}{\mathrm{one}-{\omega }_m^2{L}_{\mathrm{one}k}{C}_{\mathrm{one}k}}+x_{km}=X_{km},m=\mathrm{one,}2;\\ L_{\mathrm{one}k}=\frac{\left({\omega }_{\mathrm{one}}^2-{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000070.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2^2\right)\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)\left(X_{k2}-x_{k2}\right)}{{\omega }_{\mathrm{one}}{\omega }_2\left[{\omega }_{\mathrm{one}}\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)+{\omega }_2\left(X_{k2}-x_{k2}\right)\right]};(9)\\ C_{\mathrm{one}k}=\frac{{\omega }_2\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)+{\omega }_{\mathrm{one}}\left(X_{k2}-x_{k2}\right)}{\left(X_{k2}-x_{k2}\right)\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)\left({\omega }_{\mathrm{one}}^2-{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="00000070.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_2^2\right)}.\end{array}$$

For k = 1, we have parameter values for the first two-terminal network, and for k = 2 for the second two-terminal circuit in the form of two cascade-connected inverse L-shaped links. The index m must be introduced into the notation and other explicitly frequency-dependent quantities.

The implementation of optimal approximations of the frequency characteristics of the parameters of the matching quadrupole (6) and (7) with the help of (8) provides the implementation of the matching condition, amplitude balance and phase balance simultaneously at two given frequencies.

As a result of the interaction of signals at two frequencies with a nonlinear element, additional products of nonlinear interaction with combination frequencies arise:

ω _n = Iω ₁ ± Kω ₂ ,

where I, K = 0, 1, 2 ....

The proposed technical solutions have an inventive step, since it does not explicitly follow from the published scientific data and the known technical solutions that the claimed sequence of operations (execution of an external feedback circuit in the form of an arbitrary four-terminal connected to a three-pole nonlinear element in a series-parallel circuit, including a three-pole nonlinear feedback element and circuit as a single unit between the output of the reactive four-terminal network and the load; ki in the form of the first two-terminal with complex resistance, connecting to the input of the reactive four-terminal in the transverse circuit of the second two-terminal with complex resistance, which simulates the resistance of the source of the input high-frequency signal of the generator in amplification mode (figure 2), the implementation of the reactive four-terminal in the form of two cascade-connected inverse L-shaped links from four reactive two-terminal circuits (Fig. 3), the choice of frequency characteristics of the first and second two-terminal circuits of this circuit, the formation of their circuit in the above form (4), the choice of values of parameters of the conditions for ensuring a stationary lasing at two frequencies at a constant state of the nonlinear three-pole element provides both the formation (generation) of high frequency signals at predetermined frequencies.

The proposed technical solutions are practically applicable, since for their implementation three-pole non-linear elements (transistors or lamps) commercially available by the industry, reactive elements formed in the claimed schemes of reactive two-pole can be used (FIG. 4). The values of the parameters of the inductances and capacitances of these circuits can be uniquely determined using mathematical expressions given in the claims.

The technical and economic efficiency of the claimed device consists in simultaneously ensuring the generation of a high-frequency signal at two predetermined frequencies due to the choice of circuits and parameter values of two reactive two-terminal devices of the matching four-terminal network according to the criterion of ensuring the conditions of phase balance and amplitudes at these frequencies with the non-linear state of a three-pole element unchanged, which, taking into account nonlinear interaction allows you to generate complex signals and create radio communications that operate on a given the number of radio channels for a given frequency response of all other two-terminal and four-terminal.

Claims (2)

1. The method of generating high-frequency signals, including converting the energy of a constant voltage source into the energy of a high-frequency signal, the interaction of a high-frequency signal with a direct transmission circuit made of a three-pole nonlinear element and a four-terminal, load and external feedback circuit, the fulfillment of the excitation conditions in the form of a balance of amplitudes and balance phases determining respectively the amplitude and frequency of the generated high-frequency signals, the conditions for matching the direct transmission circuit with the load Oh and the conditions for matching the load with the control electrode of a three-pole non-linear element, characterized in that the four-terminal is reactive, another, arbitrary four-terminal connected to a three-pole non-linear element in a series-parallel circuit, a three-pole non-linear element and an external feedback circuit are used as an external feedback circuit communications as a single node cascade between the output of the reactive four-terminal and the load, the load is performed in the form of the first two-terminal with complex resistance, to the input of the reactive four-terminal in the transverse circuit connect the second two-terminal with complex resistance, simulating the resistance of the signal source of the generator in amplification mode; the parameters of the reactive four-terminal network are selected from the conditions for ensuring the stationary generation mode in the form of equal to zero the denominator of the transmission coefficient in the amplification mode simultaneously at all given frequencies of the generated high-frequency signals with a constant amplitude of the constant voltage source in accordance with the following mathematical expressions:
α = (- E + x _0m ) γ-D;
β = Fγ-Ex _0m ,
Where $$\alpha =\frac{a}{d},$$

$$\beta =\frac{b}{d}\text{-}$$

optimal values of the ratio of the corresponding elements of the classical transmission matrix at given frequencies;
$$\gamma =\frac{c}{d}\text{-}$$

predetermined relations of the corresponding elements of the classical transmission matrix at predetermined frequencies;
a, b, c, d - elements of the classical transmission matrix;
$$D=\frac{r_{0m}(r_{nm}^2+x_{22nm}^2)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}};$$

$$E=-\frac{r_{0m}(x_{\mathrm{eleven}nm}{r}_{22nm}-r_{\mathrm{eleven}nm}{x}_{22nm})}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}};$$

$$F=\frac{-r_{0m}(r_{\mathrm{eleven}nm}^2+x_{\mathrm{eleven}nm}^2)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}};$$

r _11nm = r _11m -A ₁ r _nm + B ₁ x _nm ;
x _11nm = x _11m -B ₁ r _nm + A ₁ x _nm ;
r _22nm = 1-r _{22m rnm} + x _22m x _nm ;
x _22nm = -x _nm r _22m -r _nm x _22m ;
A ₁ = r _11m r _22m -x _11m x _22m -r _12m r _21m + x _12m x _21m ;
B ₁ = r _11m x _22m -x _11m r _22m -r _12m x _21m + x _12m r _21m ;
r _0m , x _0m - set values of the real and imaginary components of the resistance of the source of the input high-frequency signal of the generator in the amplification mode at a given number of frequencies;
r _nm , x _nm - the specified values of the real and imaginary component of the load resistance at a given number of frequencies;
r _11m , x _11m , r _12m , x _12m , r _21m , x _21m , r _22m , x _22m are the given total values of the real and imaginary components of the mixed matrix H of the three-pole nonlinear element for a given amplitude of the constant voltage and the corresponding real and imaginary components the mixed matrix H of the external feedback circuit h _11m = r _11m + jx _11m , h _12m = r _12m + jx _12m , h _21m = r _21m + jx _21m , h _22m = r _22m + jx _22m at given frequencies;
m = 1, 2, ... N - frequency numbers;
Using the obtained values of the parameters of the reactive four-port network, at a given number of frequencies, the excitation conditions in the form of a balance of amplitudes and phase balance are simultaneously satisfied, as well as the conditions for matching the direct transmission circuit with the load and the load with the control electrode of a three-pole nonlinear element. 2. A device for generating high-frequency signals, containing a constant voltage source, a direct transmission circuit of a three-pole nonlinear element and a four-terminal network, a load and an external feedback circuit, characterized in that the four-terminal network is made reactive, the external feedback circuit is made in the form of an arbitrary four-terminal connected to a three-pole a nonlinear element in a series-parallel circuit, a three-pole nonlinear element and an external feedback circuit as a single node are cascaded before the output of the reactive four-terminal and the load, the load is made in the form of the first two-terminal with complex resistance, to the input of the reactive four-terminal in the transverse circuit is connected a second two-terminal with complex resistance, which simulates the resistance of the source of the input high-frequency signal of the generator in amplification mode, the reactive four-terminal is made in the form of a cascade connected by two reverse L-shaped connections of four two-terminal reactive with resistances X _1m, X _2m, X _3m, X _4m, wherein two olyusniki with resistances X _1m, X _2m are formed as a parallel oscillatory circuit element with the parameters L _1k, C _1k, connected in series with any reactive two-pole resistance x _km, and the parameter values are determined from the condition score matching ensure steady state lasing at two frequencies using the following mathematical expressions:
$$L_{\mathrm{one}k}=\frac{{\omega }_{\mathrm{one}}{X}_{k\mathrm{one}}{x}_{k\mathrm{one}}\left(X_{k2}-x_{k2}\right)+{\omega }_2{X}_{k2}{x}_{k2}\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)}{\left(X_{k2}-x_{k2}\right)\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)};$$

$$C_{\mathrm{one}k}=\frac{\left({\omega }_{\mathrm{one}}^2-{\omega }_2^2\right)\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)\left(X_{k2}-x_{k2}\right)}{{\omega }_{\mathrm{one}}{\omega }_2\left[{\omega }_{\mathrm{one}}{X}_{k2}{x}_{k2}\left(x_{k\mathrm{one}}-X_{k\mathrm{one}}\right)+{\omega }_2{X}_{k\mathrm{one}}{x}_{k\mathrm{one}}\left(X_{k2}-x_{k2}\right)\right]};$$

$$X_{\mathrm{one}m}=\frac{X_{\mathrm{four}m}^{3}D+\left(Q-x_{0m}-\mathrm{four}E\right)X_{\mathrm{four}m}^2+\left[r_0^2-\mathrm{four}F-x_{0}E+Q\left(x_0+E\right)\right]X_{\mathrm{four}m}+F\left(Q-x_{0m}\right)}{X_{\mathrm{four}m}^2+\left(E-Q\right)X_{\mathrm{four}m}+F};$$

$$X_{2m}=\frac{X_{\mathrm{four}m}\left[2F-X_{\mathrm{four}m}^{2}D+X_{\mathrm{four}m}\left(3E-Q\right)\right]}{\left(\mathrm{one}+D\right)X_{m\mathrm{four}}^2-2E{X}_{\mathrm{four}m}-F};$$

X _3m = X _4m ;
$$Q=\pm \sqrt{-r_{0m}^2-X_{\mathrm{four}m}^{2}D+\mathrm{four}\left(F+E{X}_{\mathrm{four}m}\right)},$$

Where $$D=\frac{r_{0m}\left(r_{nm}^2+x_{22nm}^2\right)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}};$$

$$E=-\frac{r_{0nm}\left(x_{\mathrm{eleven}nm}{r}_{22nm}-r_{\mathrm{eleven}nm}{x}_{22nm}\right)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}};$$

$$F=\frac{-r_{0m}\left(r_{\mathrm{eleven}nm}^2-x_{\mathrm{eleven}nm}^2\right)}{r_{\mathrm{eleven}nm}{r}_{22nm}+x_{\mathrm{eleven}nm}{x}_{22nm}};$$

r _11nm = r _11m -A ₁ r _nm + B ₁ x _nm ;
x _11nm = x _11m -B ₁ r _nm -A ₁ x _nm ;
r _22nm = 1-r _{22m rnm} + x _22m x _nm ;
x _22nm = -x _nm r _22m -r _nm x _22m ;
A ₁ = r _11m r _22m -x _11m x _22m -r _12m r _21m + x _12m x _21m ;
B ₁ = r _11m x _22m + x _11m r _22m -r _12m r _21m -x _12m r _21m ;
r _0m , x _0m - set values of the real and imaginary components of the resistance of the source of the input high-frequency signal of the generator in the amplification mode at two frequencies ω _m = 2πf _m ;
ω _m = 2πf _m ;
m = 1, 2 - frequency number;
r _nm , x _nm - the specified values of the real and imaginary component of the load resistance at two frequencies;
r _11m , x _11m , r _12m , x _12m , r _21m , x _21m , r _22m , x _22m are the given total values of the real and imaginary components of the mixed matrix H of the three-pole nonlinear element for a given amplitude of the constant voltage and the corresponding real and imaginary components the mixed matrix H of the external feedback circuit h _11m = r _11m + jx _11m , h _12m = r _12m + jx _12m , h _21m = r _21m + jx _21m , h _22m = r _22m + jx _22m at given frequencies;
k = 1, 2 is an index characterizing the corresponding numbers of reactive two-terminal networks with resistances X _1m , X _2m ;
X _3m , X _4m - given equal values of the resistance of the third and fourth two-terminal circuits matching four-terminal in the form of cascade-connected two inverse L-shaped connections of four reactive two-terminal networks at given frequencies.

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