CN103969965B - The device of accurate control submersible photoetching apparatus soaking liquid temperature and Temp. control method thereof - Google Patents

Abstract

The invention discloses a device for accurately controlling the temperature of the immersion liquid of an immersion photolithography machine, which includes a booster pump; a primary heat exchanger, the inlet of the immersion liquid communicates with the outlet of the booster pump, and the immersion liquid performs heat exchange therein to realize immersion The primary control of the temperature of the immersion liquid; the secondary heat exchanger, whose immersion liquid inlet is connected with the immersion liquid outlet of the primary heat exchanger, in which the immersion liquid performs heat exchange again to realize the secondary control of the immersion liquid temperature; the flow servo valve, whose inlet It communicates with the outlet of the immersion liquid of the primary heat exchanger and forms a parallel connection with the secondary heat exchanger to adjust the flow rate of the immersion liquid entering the secondary heat exchanger. By coordinating the control of the refrigerant flow rate entering the primary heat exchanger and the secondary heat exchanger, and combining the flow servo valve to adjust the flow rate of the immersion liquid entering the secondary heat exchanger, the two-stage regulation of the immersion liquid temperature can be achieved, thereby obtaining Temperature stable and precise immersion liquids for immersion lithography processes.

Description

Device for precisely controlling the temperature of immersion liquid in immersion lithography machine and its temperature control method

technical field

The invention relates to an immersion photolithography machine, in particular to a device for precisely controlling the temperature of the immersion liquid of the immersion photolithography machine and a temperature control method thereof.

Background technique

In the traditional lithography technology, the medium between the lens and the photoresist is air. The immersion technology replaces the air medium with a liquid. After the light passes through the liquid medium, the wavelength of the light source is shortened. According to the following Rayleigh criterion:

$\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\frac{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; NA</span>}}$

Among them, R is the lithography resolution of the optical lithography system, k ₁ is the process factor, λ is the exposure wavelength, and NA is the numerical aperture. According to the Rayleigh criterion, when the exposure wavelength is shortened and the process factor and numerical aperture remain unchanged, R decreases, that is, the resolution of lithography is improved, that is, the resolution of lithography can be improved by using immersion technology.

The immersion liquid is filled between the last optical surface and the silicon wafer in the immersion lithography machine. On the one hand, the temperature change of the immersion liquid will cause the change of the refractive index and viscosity of the immersion liquid, which will lead to the shift of the exposure focal plane and cause the numerical value Changes in the NA value of the aperture will reduce the resolution of the lithography machine. On the other hand, changes in the temperature of the immersion liquid will lead to changes in the temperature of the silicon wafer and the projection objective lens, causing optical imaging aberrations, which will further reduce the resolution of the immersion lithography machine. . Therefore, controlling the temperature of the immersion liquid and maintaining its stability is a key technology that needs to be solved for the immersion lithography machine. The flow field of the immersion liquid in the actual immersion lithography machine requires the temperature control accuracy of the immersion liquid to reach +/-0.01°C.

The Chinese patent application number 201020596742.2 discloses an immersion liquid temperature control device for lithography machine, which uses thermoelectric refrigeration to stabilize the flow field temperature of the immersion liquid, and can measure the temperature of the immersion liquid in real time. In actual situations, the immersion lithography machine has extremely high requirements on the immersion liquid, and deionized and degassed ultrapure water needs to be used. Using thermoelectric refrigeration to control the temperature of the immersion liquid is not conducive to the pollution control of the immersion liquid, and it is harmful to the immersion liquid. The liquid adopts the method of primary temperature control, which lacks the ability of secondary temperature control, and its temperature control method cannot make the temperature of the immersion liquid accurate and stable.

Contents of the invention

Aiming at the above defects or improvement needs of the prior art, the present invention provides a device for precisely controlling the temperature of the immersion liquid of an immersion lithography machine and a temperature control method thereof, which uses the principle of heat exchange, adopts multiple heat exchangers and multi-stage servo flow Control, can accurately and stably control the temperature of the immersion liquid, and has no pollution to the immersion liquid.

In order to achieve the above purpose, the device and temperature control method for accurately controlling the temperature of the immersion lithography machine of the present invention, one aspect of the technical solution of the present invention,

A device for precisely controlling the temperature of the immersion liquid of an immersion photolithography machine, characterized in that it includes:

Booster pump, the immersion liquid to be treated is pumped through the booster pump and then output with a certain pressure;

The primary heat exchanger, the inlet of the immersion liquid communicates with the outlet of the booster pump, and the immersion liquid after being processed and output by the booster pump exchanges heat with the refrigerant in the primary heat exchanger to achieve the primary temperature of the immersion liquid. regulation;

The secondary heat exchanger, the immersion liquid inlet of which communicates with the immersion liquid outlet of the primary heat exchanger, is used for the refrigerant to perform heat exchange again on the immersion liquid after the secondary heat exchange treatment, so as to adjust the temperature of the immersion liquid again, Realize the secondary regulation of the temperature of the immersion liquid;

a flow servo valve, the inlet of which communicates with the immersion outlet of the primary heat exchanger, and the outlet of which communicates with the outlet of the secondary heat exchanger, so as to form a parallel connection with the secondary heat exchanger so that from the primary heat exchanger The immersion liquid treated by the device can respectively enter the secondary heat exchanger and the flow servo valve to adjust the flow rate of the immersion liquid entering the secondary heat exchanger for heat exchange with the refrigerant to realize the regulation of the temperature of the immersion liquid;

By coordinating the control of the refrigerant flow rate entering the primary heat exchanger and the secondary heat exchanger, combined with the adjustment of the flow rate of the immersion liquid entering the secondary heat exchanger by the flow servo valve, the temperature of the immersion liquid can be adjusted. Two-level regulation to obtain a temperature-stable and precise immersion solution for immersion lithography.

Further, a throttling valve is included, the inlet of which communicates with the outlet of the immersion liquid of the secondary heat exchanger, and the outlet of which communicates with the outlet of the flow servo valve, so as to control the flow rate of the immersion liquid flowing out of the secondary heat exchanger. Take control.

Further, it also includes a primary flow servo valve, the inlet of which is connected to the refrigerant output end, and the outlet of which is connected to the refrigerant inlet of the primary heat exchanger, and the primary flow servo valve is used to control the flow of refrigerant flowing into the primary heat exchanger itself, thereby The primary regulation of temperature is achieved by controlling the heat exchange capacity.

Further, it also includes a secondary flow servo valve, the inlet of which is connected to the outlet of the refrigerant of the primary heat exchanger, and the outlet of which is connected to the inlet of the refrigerant of the secondary heat exchanger, so as to control the flow of refrigerant from the primary heat exchanger through itself into the secondary heat exchanger. The refrigerant flow rate of the device can be controlled to control the heat exchange capacity and realize the secondary regulation of temperature.

Further, it also includes a first overflow valve, which communicates with the inlet of the primary flow servo valve and the refrigerant recovery port, so as to ensure the inlet pressure value of the primary flow servo valve; it also includes a second overflow valve, which communicates with the inlet of the secondary flow servo valve and Refrigerant recovery end to ensure the inlet pressure value of the secondary flow servo valve.

Further, the immersion liquid flowing out from the outlet of the flow servo valve and the immersion liquid flowing out from the outlet of the throttle valve converge at the outlet of the throttle valve and enter two sub-pipelines respectively, and part of the immersion liquid flows from one sub-pipeline to In the immersion photolithography process, the remaining immersion liquid flows from another sub-pipeline to the inlet of the booster pump, and flows into two heat exchangers again to form an immersion liquid circuit, thereby ensuring the stability of the temperature in the immersion liquid circuit.

Further, it also includes a one-way valve, the outlet of which is connected to the inlet of the booster pump, and the waste immersion liquid used in the immersion lithography process flows to the inlet of the one-way valve after the degassing and impurity removal process, and passes through the one-way valve. And the booster pump enters the immersion circuit again to realize the recycling of the immersion liquid.

Further, it also includes a secondary temperature sensor installed at the entrance of the immersion photolithography process for measuring the temperature of the immersion liquid that passes through the immersion photolithography process after the two-stage temperature regulation; it also includes a primary temperature sensor installed At the outlet of the immersion liquid of the primary heat exchanger, it is used to measure the temperature of the immersion liquid after the primary temperature regulation. The primary temperature sensor and the secondary temperature sensor are preferably precision temperature sensors; a flow sensor is also included, installed in the submerged The photolithography process provides a sub-pipeline for the immersion liquid to ensure the flow of the immersion liquid into the immersion photolithography process.

In the second aspect of the technical solution of the present invention, a method for controlling the temperature of an immersion lithography machine using a device for precisely controlling the temperature is characterized in that,

Set the target values of the temperature two-level regulation respectively, and use the manual reset factor to characterize the relationship between the two-level regulation target values. The temperature two-level regulation target value is expressed by the following formula:

SV2=SV1+MR

In the formula, SV2 is the target value of the temperature secondary regulation, that is, the final target temperature of the regulation, SV1 is the target value of the primary temperature regulation, and MR is the manual reset factor. The fuzzy logic self-adaptive method is used to adjust the size of the manual reset factor MR. It is used to coordinate the target temperature of the two-level control, so as to ensure that the relationship between the target temperatures of the two-level control is reasonable and feasible, and then ensure the efficiency of the primary temperature control and the accuracy of the secondary temperature control.

Further, the membership function in the fuzzy logic adaptive method is used to calculate the size of ΔMR, and within a specified time period, the kth value of MR is calculated using the following formula:

MR(k)=MR(k-1)+ΔMR

In the formula: MR(k) is the kth preset value of the manual reset factor, ΔMR is the increment of the kth preset value of the manual reset factor relative to the k-1th preset value of the manual reset factor, k= 0,1,2...R, R is infinite, the process of calculating MR(k) is completed by the main controller cycle calculation;

The target value SV2 of temperature secondary regulation is also the final target temperature of regulation, which is calculated by the following formula:

SV2=SV1(k)+MR(k)

In the formula, SV1(k) is the kth preset value of the primary temperature regulation. By continuously adjusting MR(k) and SV1(k), the relationship between the target value SV2 of the temperature secondary regulation and the target value SV1 of the temperature primary regulation is realized. Reasonable.

Further, the difference between the target value SV1 of the primary temperature regulation and the actual temperature obtained by the primary regulation measured by the primary temperature sensor is used as feedback, and no overshoot control is adopted in which the actual temperature of the primary regulation is not greater than the target value of the primary regulation. The output volume of the primary flow servo valve is adjusted to control the flow of refrigerant entering the primary heat exchanger, so as to realize the primary regulation of the primary heat exchanger quickly approaching the target value SV1 of the primary regulation.

Further, the difference between the target value of the temperature secondary regulation and the actual temperature obtained by the secondary temperature control measured by the secondary temperature sensor is used as feedback, and a stable PI is used to control the output volume of the secondary flow servo valve port, By adjusting the output of the valve port of the secondary flow servo valve, the refrigerant flow rate entering the secondary heat exchanger is adjusted, so that the secondary regulation occurring in the secondary heat exchanger can accurately approach the target value SV2 of the secondary regulation.

Further, the proportional gain limiting factor α and the integral time correction factor β are used to correct the refrigerant output volume μ(t) passing through the primary flow servo valve (20) and the secondary flow servo valve (21) at time t, and the flow rate at time t is The output volume μ(t) in the servo valve is calculated by the following formula:

$\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&alpha;K</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; \&beta;T</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; edt</span>}<span\; class="notranslate">\; ]</span>$

Among them, r is the target temperature of temperature control, y is the system output, e is the temperature deviation, K _p is the proportional gain, T _i is the integral time, α is the proportional gain limiting factor, and β is the integral time correction factor;

The proportional gain K _p and the integral time T _i are obtained by the ZN critical proportionality method, the formula is,

Kp _{= 0.45Ku}

T _i =0.83T _d

In the formula, K _u is the critical gain, and T _d is the critical oscillation period;

The calculation of the proportional gain limiting factor α and the integral time correction factor β is completed by the following formula:

$\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}{\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 0.12</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\end{array}\right.$

$\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}{\mathrm{<span\; class="notranslate">\; 9</span>}}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\end{array}\right.$

In the formula, the standard pure lag τ _b =τ/T, τ is the lag time, T is the system time constant, K _u is the critical gain.

The technical solution of the present invention has the following advantages:

1. The heat exchanger is used for heat exchange, which prevents too much contact between the immersion liquid and the thermoelectric cooler device in the thermoelectric refrigeration, thereby avoiding pollution;

2. The use of secondary temperature control devices and temperature control methods ensures the stability and accuracy of temperature control;

3. The temperature control method without overshoot is adopted, so that the temperature control adjustment time is short, and the required process temperature can be obtained quickly.

Description of drawings

FIG. 1 is a schematic diagram of a device for accurately controlling the temperature of an immersion lithography machine in an embodiment of the present invention;

Fig. 2 is a schematic diagram of the circulation process of the immersion liquid in the device for precisely controlling the temperature of the immersion liquid of the immersion lithography machine in the embodiment of the present invention;

Fig. 3 is the average output volume of the primary flow servo valve valve port in the manual reset factor MR adaptive adjustment in the embodiment of the present invention The membership function used;

Fig. 4 is the average output volume of the valve port of the secondary flow servo valve in the manual reset factor MR adaptive adjustment in the embodiment of the present invention The membership function used;

Fig. 5 is the flow chart of temperature control in the embodiment of the present invention;

Fig. 6 is an example diagram of a control effect applying the temperature control method in the embodiment of the present invention;

Fig. 7 is a graph showing an example of a control error using the temperature control method in the embodiment of the present invention.

In all the drawings, the same reference numerals represent the same technical features, wherein 10-immersion liquid circulation inlet, 11-immersion photolithography machine inlet, 12-refrigerant output port, 13-refrigerant recovery end, 20-primary flow Servo valve, 21-secondary flow servo valve, 22-flow servo valve, 23-primary heat exchanger, 24-secondary heat exchanger, 31-first relief valve, 32-second relief valve, 33- Check valve, 34-booster pump, 35-throttle valve, 36-throttle valve, 40-temperature sensor, 41-pressure sensor, 42-secondary precision temperature sensor, 43-flow sensor, 44-pressure sensor, 45-temperature sensor, 46-temperature sensor, 47-primary precision temperature sensor, 48-flow sensor, 49-temperature sensor, 50-temperature sensor, 60-degassing and impurity removal process, 61-workpiece table, 62-silicon wafer, 63-immersion liquid, 64-projection objective lens.

Detailed ways

In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the embodiments of the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described here are only used to explain the embodiments of the present invention, and are not intended to limit the embodiments of the present invention. In addition, the technical features involved in the various implementations of the embodiments of the present invention described below may be combined as long as they do not conflict with each other.

FIG. 1 is a schematic diagram of a device for precisely controlling the temperature of an immersion liquid in an immersion photolithography machine in an embodiment of the present invention. The outlet of the booster pump 34 is connected to the immersion liquid inlet of the primary heat exchanger 23, the immersion liquid inlet of the secondary heat exchanger 24 is connected to the immersion liquid outlet of the primary heat exchanger 23, and the flow servo valve 22 is connected to the immersion liquid outlet of the primary heat exchanger 23 and The outlet of the secondary heat exchanger 24 is connected in parallel with the secondary heat exchanger 24. After the immersion liquid is pumped by the booster pump 34, it has a certain pressure to be output to the primary heat exchanger. The refrigerant enters the heat exchanger, and the temperature is regulated, that is, the primary regulation of the temperature of the immersion liquid is realized, and then enters the secondary heat exchanger 24 and the flow servo valve 22 respectively, and the immersion liquid entering the secondary heat exchanger 24 is heated again with the refrigerant. exchange, to further regulate the temperature, that is, the secondary regulation of the temperature, the flow servo valve 22 can regulate the flow of the immersion liquid entering the secondary heat exchanger 24, thereby controlling the flow of the immersion liquid for heat exchange to realize temperature regulation.

By coordinating the control of the flow rate of the refrigerant entering the primary heat exchanger 23 and the secondary heat exchanger 24, combined with the adjustment of the flow rate of the immersion liquid entering the secondary heat exchanger 24 by the flow servo valve 22, two-stage control of the immersion liquid temperature can be realized. control to obtain a temperature-stable and precise immersion solution for immersion lithography processes.

The inlet of the throttle valve 36 communicates with the outlet of the immersion liquid of the secondary heat exchanger 24 , and its outlet communicates with the outlet of the flow servo valve 22 , and the throttle valve 36 controls the flow of the immersion liquid flowing out of the secondary heat exchanger 24 . The immersion liquid flowing out from the outlet of the flow servo valve 22 and the immersion liquid flowing out from the outlet of the throttle valve 36 converge at the outlet of the throttle valve 36 and enter two sub-pipes respectively, and part of the immersion liquid flows from one sub-pipe to the immersion lithography process , the remaining immersion liquid flows from another sub-pipeline to the inlet of the booster pump 34, and flows into the two heat exchangers again to form an immersion liquid circuit, thereby ensuring the stability of the temperature in the immersion liquid circuit.

The inlet of the primary flow servo valve 20 is connected to the refrigerant output port, and its outlet is connected to the refrigerant inlet of the primary heat exchanger 23. The primary flow servo valve 20 controls the flow of refrigerant flowing into the primary heat exchanger 23 by itself, thereby controlling the heat exchange capacity to achieve temperature primary regulation.

The inlet of the secondary flow servo valve 21 is connected to the refrigerant outlet of the primary heat exchanger, and the outlet is connected to the refrigerant inlet of the secondary heat exchanger 24 to control the flow of refrigerant flowing from the primary heat exchanger 23 into the secondary heat exchanger 24 through itself, thereby controlling the thermal The exchange capacity realizes the secondary regulation of temperature.

The first overflow valve 31 is connected to the inlet of the primary flow servo valve 20 and the refrigerant recovery end to ensure the inlet pressure value of the primary flow servo valve; the second overflow valve 32 is connected to the inlet of the secondary flow servo valve 21 and the refrigerant recovery end to ensure the secondary The inlet pressure value of the stage flow servo valve.

The outlet of the one-way valve 33 is connected to the inlet of the booster pump 34. After the degassing and impurity removal process 60, the waste immersion liquid used in the immersion lithography process flows to the inlet of the one-way valve 33, and enters the immersion pump again through the one-way valve and the booster pump. Liquid circuit to realize the recycling of immersion liquid.

The secondary temperature sensor 42 is installed at the front end of the entrance 11 of the immersion lithography process to measure the temperature of the immersion liquid which is passed into the immersion lithography process after two-stage temperature regulation, and the primary temperature sensor 47 is installed at the primary heat exchanger 23 The outlet of the immersion liquid is used to measure the temperature of the immersion liquid after the primary temperature regulation, and the primary temperature sensor and the secondary temperature sensor are precision temperature sensors. The temperature sensor 40 is installed on the pipeline between the one-way valve 33 and the booster pump 34, and is used to measure the temperature of the immersion liquid entering the immersion liquid circuit after the degassing and impurity removal process. A temperature sensor 45 is installed at the immersion liquid inlet of the primary heat exchanger 23 for measuring the temperature of the immersion liquid entering the primary heat exchanger. The temperature sensor 50 is installed at the outlet of the immersion liquid of the secondary heat exchanger 24 for measuring the temperature of the immersion liquid after the secondary temperature regulation. The temperature sensor 46 is installed at the refrigerant inlet of the primary heat exchanger for measuring the initial temperature of the refrigerant for heat exchange. The temperature sensor 49 is installed at the refrigerant inlet of the secondary heat exchanger for measuring the temperature of the refrigerant entering the secondary heat exchanger.

The pressure sensor 41 is installed on the pipeline between the temperature sensor 40 and the booster pump 34 to ensure the pressure of the immersion liquid entering the booster pump 34 . The pressure sensor 44 is installed on the pipeline between the booster pump and the temperature sensor 45 for monitoring the pressure of the immersion liquid flowing into the primary heat exchanger 23 .

The flow sensor 48 is installed on the pipeline between the temperature sensor 47 and the immersion liquid inlet of the secondary heat exchanger 24 to ensure the flow of immersion liquid flowing into the secondary heat exchanger 24 . The flow sensor 43 is installed on the branch pipeline supplying the immersion liquid to the immersion lithography process, so as to ensure the flow rate of the immersion liquid flowing into the immersion lithography process.

The throttle valve 35 is installed between the secondary temperature sensor 42 and the inlet 11 of the immersion lithography process, and is used to accurately ensure the flow rate of the immersion liquid entering the immersion lithography process.

2 is a schematic diagram of the immersion liquid circulation process in the device for accurately controlling the immersion liquid temperature of the immersion lithography machine in the embodiment of the present invention. The immersion liquid enters the immersion lithography process, and the immersion liquid 63 immerses the projection objective lens 64 and the silicon wafer 62, and the immersion liquid Enter the immersion liquid cooling loop from the entrance 10 of the immersion liquid cooling circuit to be cooled, and enter the photolithography machine using the immersion liquid after the temperature reaches a specified value, immerse the projection objective lens 64 and the silicon wafer 62, and perform the photolithography process. The exhausted waste gas immersion liquid passes through the degassing and impurity removal process 60, and enters the immersion liquid cooling circuit through the check valve 33 again.

In the embodiment of the present invention, in the temperature control method using the device for accurately controlling the temperature of the immersion lithography machine, the target values of the two-level regulation of the temperature are respectively set, and the manual reset factor is used to represent the relationship between the target values of the two-level regulation. The level control target value is expressed by the following formula:

SV2=SV1+MR

In the formula, SV2 is the target value of temperature secondary regulation, that is, the final target temperature of regulation, SV1 is the target value of temperature primary regulation, and MR is the manual reset factor. An appropriate manual reset factor MR can coordinate the target temperature of the two-level regulation, so as to ensure that the relationship between the target temperatures of the two-level regulation is reasonable and feasible, thereby ensuring the efficiency of the primary temperature regulation and the accuracy of the secondary temperature regulation.

After the target value SV1 of the primary temperature control and the target value SV2 of the secondary temperature control are set, the output volume of the primary flow servo valve 20 and the output volume of the secondary flow servo valve 21 are respectively adjusted to adjust the output volume of the primary flow servo valve respectively. The amount of refrigerant in the heat exchanger and the amount of refrigerant in the secondary heat exchanger, so that the actual temperature achieved by the primary temperature control and the temperature achieved by the actual secondary temperature control are respectively close to the target value SV1 of the primary temperature control and the target of the secondary temperature control Value SV2.

The manual reset factor MR is introduced to ensure that the primary flow servo valve and the secondary flow servo valve have a good output, that is, have a good valve port size. If the MR setting is too large, the adjustable range of the primary flow servo valve 20 will decrease, and the primary temperature control efficiency will deteriorate. If the MR setting is too small, the adjustable range of the flow servo valve 21 will become small, and the secondary temperature control accuracy Affected. In order to make the two-stage flow servo valves always in the appropriate adjustment range, so as to better realize the primary temperature control and secondary temperature control, fuzzy logic is used to adaptively adjust the size of the manual reset factor MR. In order to characterize the size of MR, first use The following formula characterizes the average output volume of the flow servo valve

${\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}\underset{\mathrm{<span\; class="notranslate">\; k</span>}}{\overset{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>$

In the formula, u _i (k) is the kth output of the i-level flow servo valve, k=1, 2, 3...R, R is infinite, i=1 or 2, 1 represents the primary, 2 Represents the secondary, u ₁ is the output of the primary flow servo valve, u ₂ is the output of the secondary flow servo valve.

In order to ensure the stability of temperature control, u ₁ is limited, as u _1max = 80%, u _1min = 8%, if Then MR is too large, if Then MR is too small, similarly, if Then MR is too small, if Then the MR is too large. for ${\stackrel{\&OverBar;}{u}}_1<30\%$ and ${\stackrel{\&OverBar;}{u}}_2>80\%$ and ${\stackrel{\&OverBar;}{u}}_1>70\%$ and ${\stackrel{\&OverBar;}{u}}_2<20\%,$ MR has a definite size state, that is, the former is too large and the latter is small. However, for intermediate states outside these two categories, such as when and etc., the status of MR is uncertain.

The membership function in the fuzzy logic adaptive method is used to first calculate the size of ΔMR, and within a specified time period, the kth value of MR is calculated by the following formula:

MR(k)=MR(k-1)+ΔMR

In the formula: MR(k) is the kth preset value of the manual reset factor, ΔMR is the increment of the kth preset value of the manual reset factor relative to the k-1th preset value of the manual reset factor, k= 0, 1, 2...R, R is infinite, the process of calculating MR(k) is completed by the main controller cycle calculation.

Figure 3 and Figure 4 are the average value of the output volume of the primary flow servo valve for MR adaptive adjustment and secondary flow servo output average The membership function used. The method of using fuzzy logic to adaptively adjust the size of MR is as follows: Divide the current state of MR into five states: large, too large, suitable, too small, and small, and the corresponding ΔMRs are NB, NM, Z0, PM, PB, and For the corresponding state, the fuzzy rules shown in Table 1 are adopted, where ΔV is the change in the opening of the valve port of the flow servo valve 22, and ΔV=+v indicates that the opening of the valve port increases by v, and vice versa.

As shown in the dotted line in Figure 3 , the corresponding ΔMR is: 33.3% possibility is NB, 33.3% possibility is NM, if at this time The quantization is 75%, as shown in Figure 4, and its corresponding ΔMR is: 50% possibility is NM, 50% possibility is Z0, according to the linear interpolation method and the fuzzy rules in Table 1, the final calculation method of ΔMR is :

ΔMR=33.3%×50%NB+33.3%×50%NM+33.3%×50%NM+33.3%×50%NM

Through the above ΔMR, the calculation formula of MR(k) in the current two-stage temperature control method is:

MR(k)=MR(k-1)+ΔMR

Then the primary temperature control target value SV1(k) can be adjusted as:

SV1(k)=SV2+MR(k);

The secondary temperature control target value SV2 is the target temperature of temperature control.

Table 1

The difference between the target value SV1 of the primary temperature regulation and the actual temperature obtained by the primary regulation measured by the primary temperature sensor is used as feedback, and no overshoot control is adopted in which the actual temperature of the primary regulation is not greater than the target value of the primary regulation. By adjusting the primary flow The output volume of the valve port of the servo valve controls the flow of refrigerant entering the primary heat exchanger, and realizes the rapid primary regulation of the primary heat exchanger.

The difference between the target value of the temperature secondary regulation and the actual temperature obtained by the secondary temperature control measured by the secondary temperature sensor is used as feedback, and a stable PI is used to control the output of the secondary flow servo valve port. The output volume of the valve port of the primary flow servo valve is used to adjust the refrigerant flow rate entering the secondary heat exchanger, so as to realize the accuracy of the secondary regulation occurring in the secondary heat exchanger.

The proportional gain limiting factor α and the integral time correction factor β are used to correct the refrigerant output volume μ(t) passing through the primary flow servo valve (20) and the secondary flow servo valve (21) at time t, and the flow servo valve at time t The output μ(t) is calculated by the following formula:

$\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&alpha;K</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; \&beta;\; T</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; edt</span>}<span\; class="notranslate">\; ]</span>$

Among them, r is the target temperature of temperature control, y is the system output, e is the temperature deviation, K _p is the proportional gain, T _i is the integral time, α is the proportional gain limiting factor, and β is the integral time correction factor;

The proportional gain K _p and the integral time T _i are obtained by the ZN critical proportionality method, the formula is,

Kp _{= 0.45Ku}

T _i =0.83T _d

In the formula, K _u is the critical gain, and T _d is the critical oscillation period;

The calculation of the proportional gain limiting factor α and the integral time correction factor β is completed by the following formula:

$\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}{\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 0.12</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\end{array}\right.$

$\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}{\mathrm{<span\; class="notranslate">\; 9</span>}}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\end{array}\right.$

In the formula, the standard pure lag τ _b =τ/T, τ is the lag time, T is the system time constant, K _u is the critical gain.

In the embodiment of the present invention, the output volume of the valve port of the primary flow servo valve and the output volume of the valve port of the secondary flow servo valve are adjusted by adjusting the size of the valve port.

The flow chart of the temperature control method based on the device in the embodiment of the present invention is shown in Figure 5:

Step S00: turn on the device;

Step S01: setting the temperature control target value SV of the device, the target value SV2 of the secondary temperature control = SV;

Step S02: The main controller calculates the optimal PI parameters for two-level temperature regulation, reads the default manual reset factor MR, and the default MR is the optimal MR saved last time;

Step S03: Calculate primary temperature control target value SV1=SV+MR;

Step S04: Calculate the proportional gain limiting factor α and the integral time correction factor β in the embodiment of the present invention by the Z-N critical proportionality method;

Step S05: Determine whether the absolute value of the deviation between the current temperature T1 at the temperature sensor 47 and the primary temperature control preset value SV1 is greater than em, if yes, go to step S06, otherwise, go to step S07. em is the threshold for entering primary temperature control, em=2°C in this embodiment;

Step S06: adjust the valve ports of the primary flow servo valve 20 and the secondary flow servo valve 21 to the maximum value respectively, the maximum valve port value of the flow servo valve 20 is 80%, and the maximum valve port value of the flow servo valve 21 is 100%;

Step S07: Determine whether the absolute value of the deviation between the current temperature T1 at the temperature sensor 47 and the preset target value SV1 of the primary temperature regulation is between es and em, if yes, go to step S08, otherwise go to step S09, the selection of es directly affects The temperature control efficiency and temperature control accuracy of this embodiment, in this device, select em=0.1;

Step S08: Set the valve port size of the primary flow servo valve according to the aforementioned primary temperature control target value SV1, primary temperature control PI parameters, proportional gain limiting factor α, and integral time correction factor β;

Step S09: Determine whether the absolute value of the deviation between the current temperature T1 at the temperature sensor 47 and the primary temperature control target value SV1 is less than or equal to es, if yes, enter step S10, otherwise, repeatedly enter steps S05, S06, S07, and S08;

Step S10: Set the valve port size of the secondary flow servo valve according to the secondary PI parameter in step S02;

Step S11: According to the average value of the output volume of the servo valve according to the characteristic flow rate Calculate the average output volume of the primary flow servo valve 20 and the secondary flow servo valve 21 using the formula and

Step S12: Judging the average value of the output volume and Whether it meets the requirements, for the primary flow servo valve 20, the valve port size is In order to meet the requirements, similarly, for the secondary flow servo valve 21, the size of the valve port is time to meet the requirements. If the requirements are met, proceed to step S09 and continue the two-stage temperature control adjustment, if the requirements are not satisfied, proceed to step S13;

Step S13: According to the output average value obtained in step S11 and Adjust the size of MR, the adjustment method adopts the above-mentioned fuzzy adaptive method, after obtaining a new MR in step S13, re-enter step S03, and calculate the two-level temperature control target values SV1 and SV2.

Figure 6 and Figure 7 are the temperature control effect diagrams obtained by using the device of the embodiment of the present invention and the temperature control method respectively. In Figure 7, through the method of two-stage temperature control, the temperature error of the final output of the immersion liquid after the system is stable is within ±0.01 Within ℃, the average error is -0.00142℃, and the average absolute error MAE=0.0056℃; as shown in Figure 6, UPW is the immersion liquid. When the output flow rate of the immersion liquid changes from 0.5 to 5 L/min, the temperature of the immersion liquid is successfully controlled and stabilized by using the device and temperature control method of the embodiment of the present invention. The temperature control method of the embodiment of the present invention makes the immersion liquid The temperature reaches the target value within 300s.

It is easy for those skilled in the art to understand that the above descriptions are only preferred embodiments of the embodiments of the present invention, and are not intended to limit the embodiments of the present invention. Any modifications made within the spirit and principles of the embodiments of the present invention, Equivalent replacements and improvements should all be included within the protection scope of the embodiments of the present invention.

Claims (13)

1. A device for accurately controlling the temperature of an immersion lithography machine, characterized in that it comprises: A booster pump (34), the immersion liquid to be treated is pumped and processed by the booster pump to output after having a certain pressure; The primary heat exchanger (23), the inlet of the immersion liquid communicates with the outlet of the booster pump, and the immersion liquid after being processed and output by the booster pump exchanges heat with the refrigerant in the primary heat exchanger to realize the immersion liquid Primary regulation of temperature; The secondary heat exchanger (24), the immersion liquid inlet of which communicates with the immersion liquid outlet of the primary heat exchanger (23), is used for the refrigerant to perform heat exchange again on the immersion liquid after the secondary heat exchange treatment, so as to The temperature of the immersion liquid is adjusted again to realize the secondary control of the temperature of the immersion liquid; A flow servo valve (22), the inlet of which communicates with the outlet of the immersion liquid of the primary heat exchanger (23), and the outlet of which communicates with the outlet of the secondary heat exchanger (24) to form a connection with the secondary heat exchanger (24) in parallel, so that the immersion liquid treated from the primary heat exchanger (23) can enter the secondary heat exchanger (24) and the flow servo valve (22) respectively to regulate the flow into the secondary heat exchanger (24) The immersion liquid flow rate for heat exchange with the refrigerant realizes the regulation and control of the immersion liquid temperature; Through coordinated control of the flow of refrigerant entering the primary heat exchanger (23) and secondary heat exchanger (24), combined with the flow servo valve (22) to control the immersion flow into the secondary heat exchanger (24) The adjustment of the liquid flow can realize two-stage regulation of the temperature of the immersion liquid, so as to obtain a temperature-stable and accurate immersion liquid for the immersion photolithography process. 2. A device for precisely controlling the temperature of the immersion liquid of an immersion lithography machine according to claim 1, further comprising a throttle valve (36), the inlet of which is connected to the secondary heat exchanger (24) The outlet of the immersion liquid is communicated with the outlet of the flow servo valve (22) to control the flow of the immersion liquid flowing out of the secondary heat exchanger (24). 3. A device for accurately controlling the temperature of the immersion liquid of an immersion lithography machine according to claim 2, further comprising: The primary flow servo valve (20), its inlet is connected to the refrigerant output port, and its outlet is connected to the refrigerant inlet of the primary heat exchanger (23), and the primary flow servo valve (20) flows into the primary heat exchanger (23) through itself. The refrigerant flow rate is controlled, so as to control the heat exchange capacity and realize the primary regulation of temperature. 4. A device for precisely controlling the temperature of the immersion liquid of an immersion lithography machine according to claim 3, further comprising: The secondary flow servo valve (21), whose inlet is connected to the refrigerant outlet of the primary heat exchanger (23), and whose outlet is connected to the refrigerant inlet of the secondary heat exchanger (24), to control flow from the primary heat exchanger (23) ) flows into the secondary heat exchanger (24) through itself, so as to control the heat exchange capacity and realize the secondary regulation of temperature. 5. A device for precisely controlling the temperature of the immersion liquid of an immersion lithography machine according to claim 4, further comprising: The first overflow valve (31), which is connected to the inlet of the primary flow servo valve (20) and the refrigerant recovery end, so as to ensure the inlet pressure value of the primary flow servo valve; The second overflow valve (32) communicates with the inlet of the secondary flow servo valve (21) and the refrigerant recovery end to ensure the inlet pressure value of the secondary flow servo valve. 6. A device for precisely controlling the temperature of the immersion liquid of an immersion lithography machine according to claim 5, characterized in that the immersion liquid flowing out through the outlet of the flow servo valve (22) is connected with the flow rate from the throttle valve (36) The immersion liquid flowing out of the outlet is converged at the outlet of the throttle valve (36) and then enters two sub-pipelines respectively, part of the immersion liquid flows from one sub-pipeline to the immersion lithography machine, and the remaining immersion liquid flows from the other sub-pipeline to the immersion photolithography machine. The booster pump (34) enters and flows again into two heat exchangers to form an immersion liquid circuit. 7. A device for precisely controlling the temperature of the immersion liquid of an immersion lithography machine according to claim 6, further comprising: A one-way valve (33), the outlet of which is connected to the inlet of the booster pump (34), and the waste immersion liquid used in the immersion photolithography process flows to the inlet of the one-way valve (33) after the degassing and impurity removal process, Flow through the one-way valve (33) to the inlet of the booster pump (34) to enter the immersion circuit again. 8. A device for precisely controlling the temperature of the immersion liquid of an immersion lithography machine according to claim 7, further comprising: A secondary temperature sensor (42), installed at the entrance of the immersion lithography machine, used to measure the temperature of the immersion liquid passed into the immersion lithography machine after the two-stage temperature regulation; A primary temperature sensor (47), installed at the outlet of the immersion liquid of the primary heat exchanger (23), used to measure the temperature of the immersion liquid after primary temperature regulation; The primary temperature sensor (47) and the secondary temperature sensor (42) are preferably precision temperature sensors. 9. A method for temperature control using the device according to any one of claims 1-8, characterized in that it comprises Set the target values of the temperature two-level regulation respectively, and use the manual reset factor to characterize the relationship between the two-level regulation target values. The temperature two-level regulation target value is expressed by the following formula: SV2=SV1+MR In the formula, SV2 is the target value of the temperature secondary regulation, that is, the final target temperature of the regulation, SV1 is the target value of the primary temperature regulation, and MR is the manual reset factor; Among them, the fuzzy logic adaptive method is used to adjust the size of the manual reset factor MR to coordinate the difference between the target temperatures of the two-level temperature control, thereby ensuring the efficiency of the primary temperature control and the accuracy of the secondary temperature control. 10. A temperature control method as claimed in claim 9, characterized in that, the membership function in the fuzzy logic self-adaptive method is used to calculate the size of ΔMR, and within a specified period of time, the kth value of MR adopts Calculated with the following formula: MR(k)=MR(k-1)+ΔMR In the formula: MR(k) is the kth preset value of the manual reset factor, ΔMR is the increment of the kth preset value of the manual reset factor relative to the k-1th preset value of the manual reset factor, k= 0,1,2...R, R is infinite, the process of calculating MR(k) is completed by the main controller cycle calculation; The target value SV2 of temperature secondary regulation is also the final target temperature value of regulation, which is calculated by the following formula: SV2=SV1(k)+MR(k) In the formula, SV1(k) is the kth preset value of the primary temperature regulation. By continuously adjusting MR(k) and SV1(k), the target value SV2 of the control temperature secondary regulation and the target value SV1 of the temperature primary regulation are realized. The size of the difference. 11. A temperature control method as claimed in claim 10, characterized in that, the primary regulation of temperature is realized by adopting no overshoot control mode, specifically, the target value SV1 of the primary regulation of temperature and the primary temperature sensor (47) The difference between the actual temperatures obtained by the primary regulation measured in real time is feedback, and the primary regulation actual temperature is not greater than the primary regulation target temperature. The refrigerant flow rate of the heat exchanger realizes the primary regulation of the temperature. 12. A kind of temperature control method as claimed in claim 11, is characterized in that, adopts stable PI control mode to realize the secondary regulation and control of temperature, with the target value SV2 of temperature secondary regulation and described secondary temperature sensor (42 ) The difference between the actual temperatures obtained by the secondary temperature control measured in real time is feedback, using a stable PI to control the output of the secondary flow servo valve (21) valve port, by adjusting the output of the secondary flow servo valve (21) valve port The volume thus adjusts the flow rate of the refrigerant entering the secondary heat exchanger to realize the secondary regulation of the temperature. 13. A temperature control method according to claim 12, characterized in that the following formula is used to calculate the refrigerant output volume μ( t): $\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&alpha;K</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; \&lsqb;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; \&beta;\; T</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; \&rsqb;</span>$ Among them, r is the target temperature of temperature control, y is the system output, e is the temperature deviation, K _p is the proportional gain, T _i is the integral time, α is the proportional gain limiting factor, and β is the integral time correction factor; The proportional gain K _p and the integral time T _i are obtained by the ZN critical proportionality method, and the formula is: K _p =0.45K _u T _i =0.83T _d In the formula, K _U is the critical gain, and T _d is the critical oscillation period; The calculation of the proportional gain limiting factor α and the integral time correction factor β is completed by the following formula: $\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}{\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 0.12</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\end{array}\right.$ $\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}}{\mathrm{<span\; class="notranslate">\; 9</span>}}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 0.58</span>}\end{array}\right.$ In the formula, the standard pure lag τ _b = τT, τ is the lag time, T is the system time constant, K _u is the critical gain.