CN114812622A - A High-Sensitivity Circuit Based on Third-Order Exception Points - Google Patents

Abstract

The invention belongs to the technical field of electronic circuits, and in particular relates to a high-sensitivity circuit based on third-order exception points. The circuit model of the present invention is a circuit with anti-PT symmetry, including a first unit, a second unit and a third unit, each unit is composed of an inductor, a resistor and a capacitor in parallel, the first unit and the second unit, the second unit The unit and the third unit are connected through a resistor, and then the circuit parameters are configured based on the third-order exception, so as to obtain a highly sensitive circuit. Through the different coupling modes between the three units, both wired and wireless sensors can be designed, showing ultra-sensitive response and ultra-high resolution, and can generate external disturbances according to the non-Hermitian degeneracy at abnormal points. nonlinear response to improve sensitivity.

Description

A High-Sensitivity Circuit Based on Third-Order Exception Points

technical field

The invention belongs to the technical field of electronic circuits, and in particular relates to a high-sensitivity circuit based on a third-order exception point.

Background technique

In recent decades, sensors with high sensitivity and high quality factor have received extensive attention in the fields of hygiene inspection and environmental monitoring due to their wide range of applications.

A sensor is a detection device that can sense the measured information, and can transform the sensed information into electrical signals or other required forms of information output according to certain rules, so as to meet the requirements of information transmission, processing, storage and display. , recording and control requirements. In 1967, C. Collins realized the first compact passive wireless sensor based on inductive (L) and capacitive (C) resonant circuits. With the development of technology, wireless sensors based on passive LC oscillator circuits are becoming more and more popular and can be used to measure pressure, temperature, chemical reaction, etc.

Parity-time (PT) symmetric systems are a class of non-Hermitian Hamiltonian systems that are symmetric under the joint action of the parity operator P and the time-reversal operator T. Due to interesting fundamental properties and promising applications, it is got a lot of attention. This system has been studied in many fields such as quantum mechanics, optics, electronic circuits and magnetic systems. The PT-symmetric non-Hermitian Hamiltonian can exhibit full real harmonics and spontaneous symmetry breaking, with simultaneous harmonic phase transitions at the exception point (EP) for both real and complex numbers. EPs are spectral singularities in parameter space where two or more eigenvalues and their corresponding eigenvectors are merged simultaneously. Near the EP, the eigenfrequency difference obeys the 1/N power exponential relationship of the external disturbance, where N is the order of the EP, that is, the number of merged eigenvalues is N. This theory is experimentally verified in optical and electronic circuits. . The relationship between the anti-PT symmetric system and the PT symmetric system is Hamiltonian H ^APT =±iH ^PT , so it is a feasible way to design a high-order EP to improve the sensitivity of the sensor.

SUMMARY OF THE INVENTION

The purpose of the present invention is to use the characteristics near EP3 (third-order exception) to design a sensor, that is, a high-sensitivity circuit, mainly as follows: first construct a circuit system model with anti-PT symmetry, and then use Kirchhoff's current-voltage law The equation of the circuit model is constructed, and based on the third-order exception point, the eigenvalue equation is solved by Shengjin's formula, and the parameters of the circuit are obtained.

The technical scheme of the present invention is:

A high-sensitivity circuit based on third-order exception points, as shown in FIG. 1, includes a first unit, a second unit and a third unit, wherein the first unit consists of a first inductor L ₁ , a first resistor R ₁ , The first capacitor C ₁ is formed, the second unit is formed by the parallel connection of the second inductor L ₂ , the second resistor R ₂ , the second capacitor C ₂ , and the third unit is formed by the parallel connection of the third inductor L ₃ , the third resistor R ₃ and The _third capacitor C3 is formed; one end of the first unit is connected to one end of the second unit through the _fourth resistor R4, one end of the second unit is connected to one end of the third unit through the _fifth resistor R5, the other end of the first unit, The other end of the second unit and the other end of the third unit are both grounded; the voltage at one end of the first unit is defined as V ₁ , the voltage at one end of the second unit is V ₂ , and the voltage at one end of the third unit is V ₃ . The design method of parameters is:

Build the circuit equation:

ω_n＝(L_nC)^-0.5，Vn＝Vn*exp(-iωt)，n＝1、2、3，可得：Definition R=R1 _{= R3} =R4= _R5 = _2R2 , C= _C1 = _{C2 = C3} ,

ω _n =(L _n C) ^-0.5 , Vn=Vn*exp(-iωt), n=1, 2, 3, we can get:

的行列式为零，得：because

The determinant of is zero, we get:

aω ⁶ +bω ⁴ +cω ² +d=0

where a=-1, b=ω ₁ ² +ω ₂ ² +ω ₃ ² -2α ² , c=α ² ω ₁ ² +α ² ω ₃ ² -ω ₁ ² ω ₂ ² -ω ₁ ² ω ₃ ² -ω ₂ ² ω ₃ ² , d=ω ₁ ² ω ₂ ² ω ₃ ² ; let ω=ω ² , and substitute the condition ω ₁ ² +ω ₃ ² =2ω ₂ ² into the equation to simplify:

aω ³ +bω ² +cω+d=0

where a=-1, b=3ω ₂ ² -2α ² , c=2α ² ω ₂ ² -2ω ₂ ⁴ -2ω ₂ ² ω ₃ ² +ω ₃ ⁴ , d=2ω ₂ ⁴ ω ₃ ² -ω ₂ ² ω ₃ ⁴ ;

Solve the equation based on the third-order exception point, specifically: the third-order exception point means that the cubic equation has triple roots, indicating that there are three eigenfrequencies coincident, first determine the parameters of R, C, L ₂ , and then solve the one-dimensional cubic by the Shengjin formula Equation, using the multiple root discriminant A=B=0, where A=b ² -3ac, B=bc-9ad, to obtain the L ₃ parameter, through the condition ω ₁ ² +ω ₃ ² =2ω ₂ ² , obtain the L in the circuit ₁ parameter, all the component parameters under the third-order exception point are obtained at this time.

The beneficial effect of the present invention is that, through different coupling modes between the three units, both wired and wireless sensors can be designed, showing ultra-sensitive response and ultra-high resolution, and can be based on non-Hermitian degeneracy at abnormal points. A nonlinear response to external perturbations is generated to improve sensitivity.

Description of drawings

FIG. 1 is a schematic diagram of the circuit structure of the present invention.

FIG. 2 is an evolution diagram of eigenvalues in the embodiment.

FIG. 3 is a schematic diagram of a fitted straight line in the embodiment.

FIG. 4 is an evolution diagram of the real part of the eigenfrequency in the embodiment.

FIG. 5 is a trend diagram of relative error versus resistance in the embodiment.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

Example

其中

计算ΔR从0增大到50欧姆，Δω＝(ω₁-ω₂)的变化，通过对理论数据进行拟合，得到了lgΔω＝0.3326lg-δ+2.862的拟合直线如图3，可以得到Δω∝δ^1/3，在扰动下的本征频率实部演化图4。可见在此电路下实现的三阶EP，使得在EP3处发生微扰时都能使得特征频率以1/3指数形式劈裂，优于传统的二阶EP电路。This embodiment is used to verify the solution in the content of the present invention, let R= _20000Ω , C ₌ 47nF, L2 ₌ ^1mH , through the condition _ω12 ⁺ _{ω32 =} ^2ω22 , change L1 and L3, it can be The evolution diagram of the obtained eigenvalues is shown in Figure 2, in which the solid line is the real part evolution of the eigenvalues, the dotted line is the imaginary part evolution of the eigenvalues, and the point is EP3. Take the parameters at EP3: L ₁ ≈ 989.913uH, L ₂ =1mH, L ₃ ≈ 1.01mH, R = 20000Ω, C = 47nF, under this set of parameters, the circuit system is at an abnormal point, and a micrometer is applied to the R ₂ resistance. disturb, order

in

Calculate the change of ΔR from 0 to 50 ohms, Δω=(ω ₁ -ω ₂ ). By fitting the theoretical data, the fitted straight line of lgΔω=0.3326lg-δ+2.862 is obtained as shown in Figure 3, it can be obtained Δω∝δ ^1/3 , the real part evolution of the eigenfrequency under perturbation Fig. 4. It can be seen that the third-order EP realized under this circuit can make the characteristic frequency split in the form of 1/3 exponential when the perturbation occurs at EP3, which is better than the traditional second-order EP circuit.

Considering the EP3 in systems with different resistance sizes, it is theoretically found that as the resistance increases, the relative error at the EP3 point is smaller, and the relative error changes with the resistance as shown in Figure 5. However, due to the condition of ω ₁ ² +ω ₃ ² =2ω ₂ ² , the eigenvalue evolution is symmetric with ω ₂ ² as the center, so as R increases, the imaginary part region of the eigenvalue evolution will become narrower. Therefore, when the relative error is small, choosing an appropriate resistance value can realize EP3 to enhance the sensitivity of the system.

Claims (1)

1. A high-sensitivity circuit based on three-order exception points is characterized by comprising a first unit, a second unit and a third unit, wherein the first unit is composed of a first inductor L connected in parallel ₁ A first resistor R ₁ A first capacitor C ₁ The second unit is composed of a second inductor L connected in parallel ₂ A second resistor R ₂ A second capacitor C ₂ The third unit is composed of a third inductor L connected in parallel ₃ A third resistor R ₃ And a third capacitance C ₃ Forming; one end of the first unit passes through a fourth resistor R ₄ One end of the second unit is connected with one end of the second unit through a fifth resistor R ₅ The other end of the first unit, the other end of the second unit and the other end of the third unit are all grounded; defining the voltage at one end of the first unit as V ₁ The voltage at one end of the second unit is V ₂ The voltage at one end of the third unit is V ₃ The design method of the element parameters in the circuit comprises the following steps:

establishing a circuit equation:

definition of R ═ R ₁ ＝R ₃ ＝R ₄ ＝R ₅ ＝2R ₂ ，C＝C ₁ ＝C2＝C ₃ ，

ω _n ＝(L _n C) ^-0.5 Where Vn is Vn x exp (-i ω t), n is 1, 2, 3, we can obtain:

due to the fact that

The determinant of (a) is zero, and the following result is obtained:

aω ⁶ +bω ⁴ +cω ² +d＝0

wherein a is-1, b is omega ₁ ² +ω ₂ ² +ω ₃ ² -2α ² ，c＝α ² ω ₁ ² +α ² ω ₃ ² -ω ₁ ² ω ₂ ² -ω ₁ ² ω ₃ ² -ω ₂ ² ω ₃ ² ，d＝ω ₁ ² ω ₂ ² ω ₃ ² (ii) a Let omega become omega ² And the condition ω is set ₁ ² +ω ₃ ² ＝2ω ₂ ² The substitution equation is simplified to obtain:

aω ³ +bω ² +cω+d＝0

wherein a is-1, b is 3 omega ₂ ² -2α ² ，c＝2α ² ω ₂ ² -2ω ₂ ⁴ -2ω ₂ ² ω ₃ ² +ω ₃ ⁴ ，d＝2ω ₂ ⁴ ω ₃ ² -ω ₂ ² ω ₃ ⁴ ；

Solving an equation based on the third-order exception point pairs, specifically: the third-order example exterior point means that the cubic equation has triple roots, which indicates that three characteristic frequencies are coincided, firstly R, C, L is determined ₂ The parameter(s) is obtained by solving a simple cubic equation through a prime equation and utilizing a multiple root discriminant formula of A ═ B ═ 0, wherein A ═ B ² -3ac, B ═ bc-9ad, giving L ₃ By the condition ω ₁ ² +ω ₃ ² ＝2ω ₂ ² Obtaining L in the circuit ₁ Is thereby obtained atAll component parameters at the third order exception point.

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