RU2568374C1 - Method of generating high-frequency signals and apparatus therefor - Google Patents

Abstract

FIELD: radio engineering, communication.SUBSTANCE: method includes converting the energy of a dc voltage source into the energy of a high-frequency signal by discontinuous variation of the amplitude of the dc voltage source when said source is turned on, amplifying and limiting the amplitude of the high-frequency signal by setting up internal feedback in a nonlinear element in the form of an active two-terminal nonlinear element with a negative differential resistance. Further, the method includes satisfying excitation conditions in the form of an amplitude balance and a phase balance, which respectively determine the amplitude and frequency of the generated high-frequency signal, and conditions of matching the nonlinear element with a load through a four-terminal element. Excitation conditions in the form of an amplitude balance and a phase balance and matching conditions are simultaneously satisfied on a given number of frequencies through the interaction of high-frequency signals with a radio circuit in the form of a two-terminal nonlinear element with a negative differential resistance, connected between an additional composite two-terminal element and the input of the composite four-terminal element in a longitudinal circuit.EFFECT: wider range of generated oscillations when using composite four-terminal elements with lumped parameters.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, which allows you to generate complex signals and create effective radio communications with a given number of radio channels.

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 an external positive feedback between the load and the control electrode of a three-pole nonlinear element, fulfilling the excitation conditions in the form of a balance of amplitudes and phase balance, which respectively determine the amplitude and frequency of the generated high-frequency signal, and the conditions for matching a nonlinear element with a load (see Gonorovsky I.S. Radio Engineering kie chains and signals - M: “Bustard”, - 2006, p. 383-401).

A device for generating a high-frequency signal, consisting of a constant voltage source that sets the operating point in the middle of the quasilinear section of the current-voltage characteristic of the transistor, reactive four-terminal, load in the form of a parallel oscillatory circuit, RC-circuit of the external positive feedback between the load and the control electrode of the transistor, the parameters of the circuit, transistor and varicap are selected from the condition of ensuring the given amplitude and frequency of the generated high frequency signal (see IS Gonorovsky. Radio engineering circuits and signals - M: “Bustard”, - 2006, p. 383-401).

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 a positive feedback circuit, the oscillation with a frequency equal to the resonant frequency of the oscillatory circuit is supplied to the control electrode of the transistor, which, by matching with the reactive four-terminal device, starts to operate in the amplification mode until the amplitude of this oscillation increases to the level at which saturation mode (amplitude limits). There is a stationary mode.

The closest in technical essence and the achieved result (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 internal feedback in a non-linear element by using a bipolar non-linear element with negative differential resistance, performing excitation conditions in the form of a balance of amplitudes and a balance of phases, which determine respectively the amplitude and frequency from the generated high-frequency signal, and the conditions for matching the non-linear element with the load (see Gonorovsky IS Radio engineering circuits and signals - M: “Bustard”, - 2006, pp. 414-417).

The closest in technical essence and the achieved result (prototype) is a device for generating a high-frequency signal, consisting of a constant voltage source that sets an operating point in the middle of the falling section of the current-voltage characteristic of a bipolar nonlinear element with negative differential resistance, a reactive four-terminal, load in the form of a parallel oscillatory circuit, the parameters of the circuit, the bipolar nonlinear element and the varicap are selected from the conditions to ensure the specified amplitude and frequency of the generated high-frequency signal, (see Gonorovsky IS Radio-technical 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 disadvantages 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. Moreover, with the help of reactive four-terminal it is not always possible to provide the conditions for the generation, since they have certain areas of physical feasibility (areas of variation of the real and imaginary components of the load resistance), within which the matching conditions are realized (A. Golovkov. Complex electronic devices. M .: Radio and communications, 1996. - 128 p.).

The technical result of the invention is to increase the range of generated oscillations when using complex four-terminal with concentrated parameters, the generation of high-frequency signals at a given number of frequencies and the expansion of the field of physical feasibility, 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.

1. The specified result is achieved by the fact that in the known method of generating high-frequency signals, which consists in the fact that the energy of a constant voltage source is converted into energy of a high-frequency signal by abruptly changing the amplitude of the constant voltage source at the time of its inclusion, amplify and limit the amplitude of the high-frequency signal using organization of internal feedback in a nonlinear element by using an active bipolar nonlinear element with negative differential differential resistance, the excitation conditions are satisfied in the form of a balance of amplitudes and phase balance, which respectively determine the amplitude and frequency of the generated high-frequency signal, and the conditions for matching a nonlinear element with a load using a four-terminal device, additionally, the excitation conditions in the form of a balance of amplitudes and phase balance and matching conditions are simultaneously satisfied at a given number of frequencies due to the fact that the interaction of high-frequency signals with the radio circuit in the form of two of a non-linear non-linear element with negative differential resistance, connected between the introduced additional complex two-terminal and the input of the four-terminal into the longitudinal circuit, the four-terminal is made of a complex of reactive and resistive elements, the load is connected to the output of the four-terminal, the element z _{11n of} the resistance matrix of the complex four-terminal is selected from the condition for providing a stationary mode generation in the form of equal to zero denominator of the transmission coefficient simultaneously at all the given frequencies of the generated high-frequency signals with a constant amplitude of the constant voltage source in accordance with the following mathematical expression:

where z _22n , z _21n are the given values of the corresponding elements of the resistance matrix of the complex quadrupole at a given number of frequencies; z _{0n are the} set values of the complex resistances of the additional two-terminal network at a given number of frequencies; z _nn - set values of the complex load resistances at a given number of frequencies; z _n - set values of the complex resistances of the bipolar active nonlinear element at a given number of frequencies at a given amplitude of the constant voltage; n = 1, 2 ... - frequency numbers.

2. This result is achieved by the fact that in the device for generating high-frequency signals, consisting of a constant voltage source, an active bipolar nonlinear element with negative differential resistance, a four-terminal and a load, an additional four-terminal is made complex in the form of a T-shaped connection of three two-terminal with complex resistances z _1n , z _2n , z _3n , the first two-terminal T-shaped connection is formed from the series-connected first resistive two-terminal with resistance of R ₁ , the first coil with inductance Z, and a second resistive bipolar with resistance R ₂ and a second coil with inductance L ₂ connected in parallel, a bipolar nonlinear element is connected between the input of an additional bipolar and the input of the complex four-terminal into the longitudinal circuit, to the output of the complex four-terminal the load is connected, the values of the parameters of the first two-terminal T-shaped connection are determined in accordance with the following mathematical expressions:

- оптимальные значения сопротивления первого комплексного двухполюсника комплексного четырехполюсника на двух частотах; z_2n, z_3n - заданные значения сопротивления второго и третьего комплексных двухполюсников комплексного четырехполюсника на двух частотах; z_0n -заданные значения комплексных сопротивлений дополнительного двухполюсника на двух частотах; z_нn - заданные значения комплексных сопротивлений нагрузки на двух частотах; z_n - заданные значения комплексных сопротивлений двухполюсного активного нелинейного элемента на двух частотах при заданной амплитуде постоянного напряжения; ω_1,2=2πf_1,2; n=1, 2 - номера заданных двух частот f_1,2.where r ₁ , r ₂ , x ₁ , x ₂ are the optimal values of the real and imaginary components of the resistance of the first complex two-terminal complex four-terminal at two frequencies;

- the optimal resistance values of the first complex two-terminal complex four-terminal at two frequencies; z _2n , z _3n - set resistance values of the second and third complex two-terminal complex four-terminal on two frequencies; z _{0n are the} set values of the complex resistances of the additional two-terminal network at two frequencies; z _нn - set values of the complex load resistances at two frequencies; z _n - set values of the complex resistances of the bipolar active nonlinear element at two frequencies at a given amplitude of the constant voltage; ω _1,2 = 2πf _1,2 ; n = 1, 2 - numbers of the given two frequencies f _1,2 .

In FIG. 1 shows a diagram of a device for generating high-frequency signals (prototype) that implements the prototype method.

In FIG. 2 shows a structural diagram of the proposed device according to p. 2, which implements the proposed method according to p. 1.

In FIG. 3 shows a diagram of a complex quadrupole included in the proposed device, a diagram of which is shown in FIG. 2.

In FIG. 4 is a diagram of a second complex two-terminal network included in a four-terminal network, the circuit of which is shown in FIG. 3.

The prototype device (Fig. 1), which implements the prototype method, contains a non-linear element - 1 with negative differential resistance, connected to a voltage source - 2 with low internal resistance, matching filtering device - 3 (reactive four-terminal or matching four-terminal), oscillatory the circuit on the elements L - 4, R - 5, C - 6, which is the load - 7. The principle of operation of the device for generating high-frequency signals (prototype) that implements the prototype method is as follows.

When a constant voltage source-2 is turned on, due to an abrupt change in the amplitude, oscillations occur 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, for example, tunnel diode-1, a negative differential resistance appears in the section with a falling current-voltage characteristic, which, by matching with a reactive four-terminal-3, compensates for losses in the circuit L - 4, R - 5, C - 6. Due to this, the oscillation with a frequency equal to the resonant frequency of the oscillating circuit is amplified until the amplitude of this oscillation increases to a level at which the amplitude goes beyond for the falling section of the current-voltage characteristic. There is a stationary mode.

The disadvantages of the prototype method and device for its implementation are described above. The proposed device according to claim 2 (Fig. 2), which implements the proposed method according to claim 1, contains a non-linear element-1 with a known negative differential resistance z _n = r _n + jx _n at two given frequencies of the generated signals, connected to a constant voltage source - 2 with low internal resistance and connected in high frequency in the longitudinal circuit between the additional two-terminal device - 8 with a given resistance z _0n = r _0n + jx _0n at two given frequencies, which simulates the resistance of a high-frequency oscillation source in amplification mode of arising when the constant voltage source - 2 is turned on at the moment of a jump in the amplitude of its voltage, and the input of a complex four-terminal or matching four-terminal (matching filtering device (SFU)) - 3. A load-9 is connected to the output of the four-terminal, with given arbitrary resistances z _нn = r _nn + jx _nn at given two frequencies. The four-terminal - 3 is made in the form of a T-connection of three complex two-terminal (Fig. 3) with resistances z _1n - 10, z _2n - l1, z _3n - 12. In the generation mode, a jumper is switched on instead of a source of high-frequency oscillations. Synthesis of a complex four-terminal network (selection of resistance values-10 of the first two-terminal T-shaped connection (Fig. 3) at two preset frequencies (n = 1, 2 is the frequency number), a circuit for the formation of this two-terminal network from a series of connected first resistive two-terminal devices with resistance R ₁ - 13, the first coil with inductance L ₁ - 14 and a parallel-connected between a second two-terminal resistive with a resistance R ₂ - 15 and the second coil with inductance L ₂ - 16 (. Figure 4) and the said parameters values) both implemented according to the criterion baking the amplitude balance and the phase balance equal to zero by implementing the denominator of the transfer coefficient generating device simultaneously at two predetermined frequencies of the generated signals at a constant amplitude DC voltage.

The proposed device operates as follows.

When you turn on the DC voltage source - 9 due to the abrupt change in amplitude in the entire circuit, oscillations arise, 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, for example, a tunneling diode - 1, a negative differential resistance arises in the section with a falling current-voltage characteristic, which, due to the synthesis of a four-terminal-3, compensates for losses in the entire circuit simultaneously at two given frequencies . The oscillation amplitudes with given frequencies are amplified to certain levels and then limited. The synthesis of a four-terminal - 3 was carried out according to the criterion for the coincidence of the real frequency dependences of the resistance of the first - 10 two-terminal at two frequencies with optimal characteristics that ensure the denominator of the transmission coefficient (the condition of the stationary generation mode) to zero simultaneously at two given frequencies.

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 falling section 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 arise with Raman frequencies ω _n = Ιω ₁ ± Κω ₂ , I, K = 0, 1, 2 ...

Let us prove the feasibility of implementing these properties.

Let the dependences of the resistance z ₀ = r ₀ + jx _{0 of} an imaginary signal source occurring when the DC voltage source, the load z _n = r _n + jx _n and the nonlinear element z = r + jx (the real part of the resistance r are negative) on the frequency given amplitude of direct voltage (current). For simplicity of writing, the arguments ω = 2πf (circular frequency) and U, I (amplitudes of constant voltage or current) are omitted.

The nonlinear element and complex quadrupole are characterized by the following transfer matrices:

; z₁₁, z₂₁, z₂₂ - определитель и элементы матрицы сопротивлений СФУ с учетом условия взаимности z₁₂=-z₂₁.Where

; z ₁₁ , z ₂₁ , z ₂₂ - determinant and elements of the matrix of resistances of SFU taking into account the reciprocity condition z ₁₂ = -z ₂₁ .

Multiply the transfer matrix of the nonlinear element by the transfer matrix of the complex four-terminal network. Taking into account z0, zн (Feldstein A.L., Yavich L.R. Synthesis of four-terminal and eight-terminal on microwave. M .: Communication, 1971. S. 34-36) we obtain the expression for the normalized classical transfer matrix of the generation device:

Using the known relations between the elements of the classical transfer matrix and the elements of the scattering matrix (ibid.) Taking into account (2), we obtain the expression for the transfer coefficient in the amplification mode:

.A physically realized transfer function is related to the transfer coefficient by a simple relation:

.

We show that the condition for ensuring the stationary generation mode (the condition of the balance of amplitudes and phase balance) corresponds to the vanishing of the denominator of the transmission coefficient (3).

, где первое слагаемое - это сопротивление z₀ пассивной части генератора; второе слагаемое - сопротивление нелинейного элемента, третье слагаемое с учетом матриц передачи (1) - это входное сопротивление комплексного четырехполюсника, нагруженного на сопротивление z_н нагрузки. Второе и третье слагаемые - это входное сопротивление активной части генератора. Если это условие записать в виде другого равенства

, то ее можно трактовать как условие баланса амплитуд и баланса фаз 1-КВ=0 (Гоноровский И.С. Радиотехнические цепи и сигналы - М: «Дрофа», - 2006, с. 383-401) для эквивалентной цепи с внешней положительной обратной связью.The denominator of the gain in gain mode is represented in the form corresponding to the condition for the emergence of a stationary generation mode (A. Kulikovsky. Stability of active linearized circuits with amplifiers of a new type. M - L .: GEI, 1962. 192 s):

where the first term is the resistance z _{0 of the} passive part of the generator; the second term is the resistance of the nonlinear element, the third term, taking into account the transfer matrices (1), is the input impedance of a complex four-terminal network loaded with resistance z _{n of the} load. The second and third terms are the input resistance of the active part of the generator. If this condition is written as another equality

, 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.

The solution of the complex equation formed from the equal to zero denominator (3):

The obtained relationship of the elements of the SFU resistance matrix (4) taking into account the given frequency dependences z ₂₂ , z ₂₁ , z ₀ , z, z _n is the optimal approximating function of the frequency dependence of the corresponding element (z ₁₁ ) of the SFU resistance matrix. If this approximating function is realized within any frequency band or at separate frequencies, then in this frequency band or at these frequencies the conditions of phase and amplitude balance will be provided. For this, it is necessary to take any typical SFU circuit, find the resistance matrix of this circuit, and find the elements of this matrix found in this way, expressed in terms of the circuit parameters, substitute in (4) and solve the generated complex equation with respect to the resistance of the selected one two-terminal network. The frequency characteristics of the remaining parameters r ₀ , x ₀ , r _n , x _n , r, x and the remaining two-terminal SFUs can be selected arbitrarily or based on any other physical considerations.

In accordance with the above algorithm, expressions are obtained for finding the optimal approximation of the frequency dependence of the complex resistance of the first two-terminal SFU in the form of a T-shaped connection of three complex two-terminal:

where n = 1, 2 ... are the numbers of the interpolation frequencies. Resistances z _2n , z _3n can be chosen arbitrarily or on the basis of any other physical considerations. The index n must also be introduced in other notation of physical quantities that explicitly depend on the frequency. The physical meaning of solution (5) is that the frequency dependence of the complex resistance of the first two-terminal SibFU must be opposite in sign and equal in magnitude to the frequency dependence of the complex resistance of the series-connected resistances of the signal source in the amplification mode, the nonlinear element, and the total resistance of the parallel connection of the second two-terminal SFU with serially connected third SFU bipolar and load. In this case, generation would be ensured over the entire frequency spectrum. However, the implementation of (5) in a continuous, even very narrow frequency band, is not possible.

To implement the optimal approximation (5) on a finite number of frequencies by interpolation, it is necessary to form a two-terminal network with a resistance z _1n of at least 2Ν (Ν is the number of interpolation frequencies) of elements of type R, L, C, find expressions for their resistances, and equate their optimal values the two-terminal resistances at the given frequencies determined by formulas (5), and solve the system of 2N equations formed in this way with respect to the 2N selected parameters R, L, C. The values of the parameters of the remaining elements can be chosen arbitrarily or on the basis of any other physical considerations, such as the condition of physical realizability. Let the first SFU bipolar with resistance z _{1n be} formed from the first resistive bipolar with resistance R ₁ connected in series, the first coil with inductance L ₁ and the second resistive bipolar with resistance R ₂ and the second coil with inductance L ₂ connected in parallel (Fig. 4) . The complex resistance of the first bipolar SFU:

In (6), we divide the real and imaginary parts between us and for N = 2 we compose a system of four equations:

Decision:

r ₁ , r ₂ , x ₁ , x ₂ are the optimal values of the real and imaginary components of the resistance of the first complex two-terminal complex four-terminal at two frequencies.

The implementation of optimal approximations of the frequency characteristics of the four-terminal network (4) using the T-shaped link (5) and the first two-terminal network of this link using (6), (8) provides the implementation of the condition for the balance of amplitudes and phase balance simultaneously at two given frequencies. As a result of the interaction of signals at two frequencies with a nonlinear element in the generation mode, products of nonlinear interaction arise with Raman frequencies ω _n = Ιω ₁ ± Κω ₂ , 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 (the implementation of the generating device in the form shown in Fig. 2, the execution of the four-terminal system in the form of three connected in the form described above two-terminal (FIG. 3), forming a first two-terminal of the first resistor connected in series with a two-terminal resistance R _1, the first coil and inductance L ₁ and connected in parallel between a second resistive two-terminal resistance R ₂ and a second coil with inductance L ₂ (FIG. 4), the selection element z ₁₁ matrix resistances integrated quadripole, selecting parameters of the first two-pole SibFU values of conditions ensuring stationary lasing at two frequencies with the unchanged state of a nonlinear bipolar element with negative differential resistance), simultaneously provides the formation of high-frequency signals at given frequencies with a constant amplitude of a constant voltage source.

The proposed technical solutions are practically applicable, since active semiconductor diodes (Gunn diodes, tunnel diodes, avalanche-span diodes, etc.), inductors, resistors and capacitances formed in the claimed complex four-terminal circuit can be used for their implementation . The values of the parameters of the inductances and resistive elements included in the circuit of the first two-terminal SFU can be uniquely determined using mathematical expressions given in the claims.

The technical and economic efficiency of the proposed method and device consists in simultaneously ensuring the generation of a high-frequency signal at a given number of frequencies by selecting the circuit and parameter values of the elements of the complex, quadrupole, according to the criterion of providing conditions for the balance of phases and amplitudes at these frequencies with a non-linear bipolar state element with negative differential resistance, which allows you to generate complex signals and create radio communications, functioning on a given number of radio channels with arbitrary load characteristics.

Claims (2)

1. The method of generating high-frequency signals, which consists in the fact that the energy of a constant voltage source is converted into energy of a high-frequency signal by abruptly changing the amplitude of the constant voltage source at the time of its inclusion, amplify and limit the amplitude of the high-frequency signal by organizing internal feedback in a nonlinear element by the use of an active bipolar nonlinear element with negative differential resistance as it, the conditions excitation in the form of a balance of amplitudes and phase balance, respectively determining the amplitude and frequency of the generated high-frequency signal, and the conditions for matching a nonlinear element with a load using a four-terminal device, characterized in that the excitation conditions in the form of a balance of amplitudes and phase balance and matching conditions are simultaneously satisfied for a given number frequencies due to the fact that the interaction of high-frequency signals with a radio circuit in the form of a bipolar nonlinear element with a negative d the differential resistance included between the introduced additional complex two-terminal and the input of the four-terminal into the longitudinal circuit, the four-terminal is made complex of reactive and resistive elements, the load is connected to the output of the four-terminal, the values of the element z11n of the resistance matrix of the complex four-terminal from the condition of ensuring the stationary generation mode in the form of a zero value transmission coefficient simultaneously at all given frequencies generated by high-frequency x signals with a constant amplitude of the DC voltage source in accordance with the following mathematical expression:

where z _22n , z _21n are the given values of the corresponding elements of the resistance matrix of the complex quadrupole at a given number of frequencies; z _0n - set values of the complex resistances of the additional two-terminal network at a given number of frequencies; z _nn - set values of the complex load resistances at a given number of frequencies; z _n - set values of the complex resistances of the bipolar active nonlinear element at a given number of frequencies at a given amplitude of the constant voltage; n = 1, 2 ... - frequency numbers. 2. A device for generating high-frequency signals, consisting of a constant voltage source, an active bipolar nonlinear element with negative differential resistance, a four-terminal and a load, characterized in that the four-terminal is made complex in the form of a T-shaped connection of three two-terminal with complex resistances z _1n , z _2n , z _3n , the first T-pin bipolar is formed from the series-connected first resistive bipolar with resistance R ₁ , the first coil with ind by the inductance L ₁ and a second resistive two-terminal with resistance R ₂ and a second coil with an inductance L ₂ connected in parallel, a two-pole non-linear element is connected between the input of an additional two-terminal and the input of the complex four-terminal into the longitudinal circuit, the load is connected to the output of the complex four-terminal, the values of the parameters of the first two-terminal T-joints are defined according to the following mathematical expressions:

where r ₁ , r ₂ , x ₁ , x ₂ are the optimal values of the real and imaginary components of the resistance of the first complex two-terminal complex four-terminal at two frequencies;

- the optimal resistance values of the first complex two-terminal complex four-terminal at two frequencies; z _2n , z _3n - set resistance values of the second and third complex two-terminal complex four-terminal on two frequencies; z _0n - set values of the complex resistances of the additional two-terminal network at two frequencies; z _нn - set values of the complex load resistances at two frequencies; z _n - set values of the complex resistances of the bipolar active nonlinear element at two frequencies at a given amplitude of the constant voltage; ω _{1, 2} = 2πf ₁ , 2; n = 1, 2 - numbers of the given two frequencies f _{1, 2} .

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