1. Introduction Through years of research on soft starters, I have found that soft starters somewhat oversimplify motor overload protection. Although it is described as inverse-time protection, it actually uses a time-segmented method, which sometimes results in false trips and sometimes burns out the motor. For intermittent motor overload protection, the motor is already overheated, thus its overload capacity is reduced. A cold motor has a much greater overload capacity than a hot motor. To truly reflect the motor's overload capacity and provide effective overload protection, thermal integration and thermal memory functions are necessary. This ensures both system reliability and protection sensitivity. 1.1 Two typical mathematical models Soft starters have functions such as control, protection and monitoring for motors. The inverse time protection characteristic used for thermal overload protection of motors has a variety of mathematical models, among which there are two typical types: (1) Time current characteristic of equal I2t (2) Mathematical model recommended by IEC 60255-3[1] In the above formula: Ir — current setting value I — actual current value t — action time (s) K — constant characterizing the characteristic α — function exponent 1.2 Control method of trip unit The control method of trip unit can be: (1) Integral method Taking two typical mathematical models as examples, calculate the integral value respectively: Set the action value of K1 or K2 and control the action time t. (2) Look-up table method Set the I-t lookup table and control the action time t according to the current I. However, both methods have drawbacks in actual operation. If the integral method is used, the two mathematical models mentioned above may cause false action when the current is lower than the action value; if the look-up table method is used, it is difficult to reasonably control the delay time of overload trip under the condition that the current is constantly changing. In order to better solve the delay control of the long-delay trip in the intelligent controller of low-voltage circuit breaker, this paper attempts to analyze and discuss it according to the basic principle of thermal protection. 2 Basic requirements of thermal protection According to the thermal balance relationship, the heat generation of electrical equipment should be equal to the sum of heat dissipation and heat storage, that is (1) Where: P — heat generation power; Kr — heat dissipation coefficient; S — heat dissipation surface area; τ — temperature rise; c — specific heat; G — weight of heat generation body; t — time. The solution of the differential equation is: The overload protection element should operate at the set value less than the allowable temperature rise of the protected electrical equipment and disconnect the circuit. 3 Setting the long-delay trip of overload according to the thermal balance principle 4 Calculation of action value and thermal time constant 4.1 Action value According to the requirements of motor starter and circuit breaker, k[sub]2[/sub] should be less than 1.2 and 1.3 respectively. In order to meet these two requirements at the same time and leave a margin, k[sub]2[/sub] can be taken as 1.1～1.15. From equation (11), we can take K=k[sub]2[/sub][sup]2[/sup]T (12). Using K as the cutoff value of equation (6) or (7), the controller will operate when A≥K, realizing the long delay protection function. Equations (9) and (10) can be converted to: 4.2 Calculation of thermal time constant. Given the required tr value under any N value, T can be calculated. 4.3 Calculation of delay time. Calculate the delay time under different overload currents according to equation (13), and consider the influence of current measurement error. The calculation results are shown in Table 1 (T=642s is taken in the calculation). 5 Measurement and calculation of action value . In order to measure the A value when the intelligent trip unit is energized in the real state, the numerical integration method can be used to measure the current at equal intervals and calculate the A value and compare it with the K value. Let the measurement interval be Δt, and the initial temperature rise be 0. From equations (6) and (7), N can be a variable in the above equations. Calculate successively and compare with k successively until the controller operates when Ax≥k. Then... When there is an auxiliary power supply, the value of A gradually decreases until the soft starter restarts and the value of A starts to increase again; or the auxiliary power supply is disconnected and the value of A is cleared to zero. To prevent the soft starter from being reconnected and disconnected again in a short time after overload tripping, a certain recovery time can be set to ensure that the soft starter does not start within the recovery time. 6 Measurement error analysis Differentiate from equation (8): Corresponding to the calculated value tr in Table 1, the corresponding values ​​of p and f are listed in Table 2. [align=center]Table 2 p and f values ​​corresponding to the calculated value tr in Table 1[/align] The estimated value of the error transmission coefficient f in Table 2 is basically consistent with the calculation result in Table 1. It can be seen from Table 1 and Table 2 that the delay time error caused by the current measurement error is large at lower overload multiples. 7. Slope Adjustment of Protection Characteristics 7.1 Establishment of Mathematical Model In order to meet different matching needs, some manufacturers now provide the function of adjusting the slope of long-delay protection characteristics[2] or provide protection characteristics with different mathematical models according to IEC 60255 standard. In order to achieve slope adjustment of protection characteristics, this paper recommends a scheme of using two mathematical models together. (1) Basic Mathematical Model Through comparative analysis, we can use equation (7) as the basic mathematical model of basic protection characteristics. (2) Mathematical Model for Slope Adjustment The mathematical model recommended by national standard GB 14598.7 (equivalent to IEC 60255-3) can be used for slope adjustment. According to GB 14598.7: (16) Where: N=I/Ir The exponent α can be selected and K is a constant. Now, take the protection characteristics with three slopes as examples: ● Type A inverse time tr=K/(N0.02-1) (17) ● Type B inverse time tr=K/(N-1) (18) ● Type C inverse time tr=K/(N4-1) (19) The value of K can be set according to the protection requirements, or refer to the time tr corresponding to the basic protection characteristic N1r (such as N=2 or N=6) mentioned above. 7.2 Measurement and control of action value Transform equations (17), (18), and (19) into A=t(N0.02-1) (20) A=t(N-1) (21) A=t(N4-1) (22) In actual operation, the value of A can be accumulated once after each equal interval Δt, and the value of A can be calculated successively and compared with the value of K successively until the set value of K is reached, and the delay time tr can be calculated. Taking equation (21) as an example, it is assumed that the corresponding equations (20) and (22) can be calculated and controlled in the same way. However, there are two problems that need to be solved when using this method: (1) The threshold of N is usually within the set value range of K. Under the condition of N=1.05, the calculated value tr is likely to be less than 1h, which cannot meet the requirements of the soft starter. In order to prevent false tripping at 1.05Ir and below, a threshold needs to be set, such as setting Nd=1.15. When N≤Nd, the basic mathematical model can still be used for control and calculation. (2) The conversion of the mathematical model above and below the threshold is as follows: When N>Nd, the calculation and control are performed according to the mathematical model of equation (20) to (22). Here is an example: ● The protection characteristic is taken as equation (21), and K is set to 13.5. The T value is calculated according to equation (12), and k2=1.15. T=13.5/1.152=10.2 When N≤Nd, the calculation and control are performed according to the method described in Section 4 above. When N>Nd, the calculation is performed according to the mathematical model of equation (21). If the current drops again before the action value is reached, making N≤Nd, and the current A value is Ay, then the A value needs to be calculated by accumulating according to the basic mathematical model: (24) ………… The initial value Ay is the A value retained under the original mathematical model. The calculation and control are performed according to the method described in Section 4 above. If the N>Nd condition is restored, the calculation and control should be performed again according to the mathematical model of equation (21). The K value and the current A value do not need to be changed when repeatedly switching mathematical models. ● The protection characteristic is taken from equation (22), and K is set to 1200. The T value is calculated according to equation (12), and k2 = 1.15. T = 1200/1.152 = 907.4 When N ≤ Nd, the calculation and control are performed according to the method described in Section 4 above. When N > Nd, the calculation is performed according to the mathematical model of equation (22). If the current drops to N ≤ Nd before the action value is reached, and the current A value is Ay, then the A value needs to be calculated according to equation (24). If the N > Nd condition is restored afterward, the calculation and control should be performed again according to the mathematical model of equation (22). The K value and the current A value do not need to be changed when repeatedly switching mathematical models. 7.3 Error analysis The calculation values ​​of the transmission coefficients between the time relative error and the current relative error of the three mathematical models (16), differential equations (19), (20) and (21) are shown in Table 3. [align=center]Table 3 Calculated values ​​of the transmission coefficient between the relative error of time and the relative error of current in three mathematical models[/align] As can be seen from Table 3, when α=0.02 and α=1, it is not difficult to meet the requirement that the delay time error does not exceed ±10% when Nr≥1.5; however, when α=4, due to the large slope of the characteristic curve, it is difficult to achieve the same index. Even if the current measurement error is ±2%, considering the control error of K and numerical rounding, the delay time error may be greater than ±10%. 8 Conclusion This paper proposes a real-time measurement and control method using numerical integration to solve the inverse time protection characteristics. This method can provide a variety of protection characteristics in a reasonable and convenient way, and can also solve the thermal memory problem under constantly changing load conditions. It also helps to improve the anti-interference capability of long delay control units. Since microprocessors cannot complete some complex mathematical operations of functions in a very short time in real-time control, some calculation formulas and parameters in this paper need to be transformed and processed in engineering calculations. They have been applied in the CMC series soft starters and have achieved ideal results through actual operation.