The task of ACS analyzing in ECAD in the large-scale signal mode was considered. New models based on continuous time domain functions allow us to solve the problem of transient analysis in non-stationary nonlinear ACS. This approach does not require a transition to the frequency domain by calculating the transfer functions for each block, while the model retains its nonlinearity. it was found that such modeling in ACS in ESAV is quasi-causal indeed According to the chosen approach, dynamic nonlinear macromodels for subsystems of ACS for determine magnetic susceptibility were developed: for the actuator/stepper motor, PWM-controller, Choke with movable core, etc.
For measure of magnetic susceptibility, which has a high informative value when
conducting research on the structural state of austenitic materials, various installations
are used, for example, magnetometric scales. The block diagram of the automated
system for magnetic susceptibility measuring is given in [
Traditionally, for the analysis of such ACS is used approach, in which for the each
block it was necessary to obtain transfer functions by converting the equations of the
time domain to the equations of the frequency domain [
However, it is quite often necessary to calculate the time of transients, response delays, it is also necessary to take into account the nonlinearity of characteristics to predict the behavior of systems in real time and to write and debug programs for generating control signals by microcontroller. 2
In ACS the microcontroller performs measurements, calculations and control, so, the main requirements for the designed model are data representation in the time domain and the ability to determine the nonlinearity of parameters in full scale variation of variables. This approach does not require calculation the transfer functions for each block. The model must retain its nonlinearity.
Consider an example of building a dynamic non-linear model of an ACS adapted
to the capabilities of the Electronic Computer Aided Design (ECAD) mathematical
processor Micro Cap v.11 (MС 11) [
Simulation of ACS as multi-domain dynamical systems, for example, mechatronic, is
one of the most powerful tools of system research [
Simulation of ACS is used in the design of microcontroller systems at the stage of
Control Law Design [
There are different approaches to the study of ACS and various levels of model
abstraction, and, accordingly, different software (CAD/CAE). There are many tools
from CAE and CAD market for automated mechatronic systems simulation [
The capabilities of many packages are largely overlapping and approaches to solving the same tasks are roughly the same.
The computer simulation model is built visually through block diagrams. Usually
for the study of ACS, it is necessary to obtain transfer functions for its subsystems by
converting the functions of time and transition to the frequency domain. Conversion
(e.g., Laplace transform) allows to obtain simple model as algebraic equations with
complex coefficients. However, the area of the adequacy of such models is small,
because for an adequate transition to the frequency domain, it is necessary to linearize
the models and make an assumption that the processes are stationary. Transients in
nonlinear systems with this approach cannot be modeled. As a rule, in CAE programs,
blocks are built on the basis of transfer functions, however, SimApp [
Analysis of ECAD programs [
MС 11 contains many features that help you grow the model sophistication to the
level needed for realistic results [
The optimality of the designed system was checked by mathematical modeling, which was carried out in two stages: the construction of the model according to the selected criteria (modeling) and the implementation of the model experiment (simulation). The purpose of the simulation is to determine the quality of the system (stability, dynamic characteristics, sensitivity) and choose the directions of its parametric (structural) optimization.
Information-measuring system is an AСS (magnetometric scales with the registration of the zero position on the basis of determining the frequency changes of LC generator). ACS is designed to compensate for the displacement of the rod caused by the magnet's impact on the magnetic material. The resulting scheme of ACS is shown in fig. 1.
1
2 7
2 Current PWM y c n e u q e r F 4 6 1 Position2 n o iit s o P
PWM 5
The total model of ACS is built visually through block diagrams and consists of subsystems: 1 – Experimental subsystem / Power Magnet; 2 – tested sample; 3 – the movable rod; 4 – Measurement subsystem on Colpitts variable frequency oscillator; 5 – Compensating Subsystem (implemented on an inductive choke); 6 – Microprocessor Control System; 7 – SMPS (Switch Mode Power Supply) with DAC (Digital – to Analog Converter).
Microcontroller allowed increasing the accuracy and stability of automated measurement of magnetic susceptibility. ACS measuring and fixing the current of a power magnet (CURRENT), resulting displacement of the rod (POSITION1 – displacement of the rod (ejection/retraction of the tested sample from the Power Magnet); POSITION2 – result displacement (ejection/retraction from Compensating Subsystem)), frequency difference (FREQUENCY), duty circle of pulse width modulation signal (PWM-signal) for the compensating choke (PWM) and for SPMS, which generated signal for Power Magnet.
The rod, to which the ampule with the test materials attached, is pushed out or retracted by the magnetic field in the Experimental Subsystem, accordingly, its position changes relative to the coil in Measuring Subsystem. The subsystem for measuring the displacement of the rod is realized on the composition of the alternator generator and the electronic frequency meter, which transmits the data about frequency of oscillations changing relative to the base frequency f0 to the microprocessor system, which, based on these data, determines the Duty Cycle (D.C.) of PWM output voltage for the Compensation System to return the rod in start position.
Operations of the measuring and compensating subsystems are based on the variation of the chokes` inductance, for which a specific choke model was developed. During the development of the system, the simulation based on a-causal block modeling principle was applied. 5
The biasing sensor is actually implemented on the Colpitts oscillator with an
inductance, which changes its value by changing the core`s depth of penetration into the
coil [
All parameters have been optimized by the Powell algorithm in MC11 [
The inductance of the generator`s coil depends on the current position of the core ( l ) and the magnetic field parameters ( VAR ):
LVAR L0 0 VAR k 2 l S ,
N k l , where 0 – absolute magnetic permeability; VAR – variating magnetic permeability; L0 – initial value, calculated by (1) on 300 kHz; N – number of turns; l, S – length and area of the solenoid cross, respectively; l – variation of the rod`s penetration depth.
To take into account the change in the magnetic permeability µVAR from the
intensity of the magnetic field H [
The scheme for modeling the Stoletov's curve and nonlinear inductance is shown in fig. 2. Basic parameters of approximation predefined. The field intensity changes (node with the name H) is emulated by the generator V1. The subsystem of LVAR (determining the inductance of the measuring oscillator by the formula (2)) is consist of: generator X1, which defines a step change in the depth of penetration of the rod into the coil (node with the name DELTA_L); resistors R2, R3; inductance LVAR , through which the pulsating current flows from the generator V2 and resistor Rmu, which variate its resistivity by the formula (3). The results of LVAR simulation in the full range of field intensity variation are shown in fig. 3.
.define MU0 1.26E-6 .define MUmax 500 .define L0 6 .define k 6E5 .define S 12.6E-6 .define DELTA_L 1E-3
H V1
R2 1 9.60 8.80 8.00 7.20 6.40 5.600.00K
L(L1)(H) .define Lvar L0+MU0*R(Rmu)*(k**2)*V(DELTA_L)*S
DELTA_L X1
R3 1
V2
Lvar L1 E1
Rmu
MU0+MUmax*SQRT(V(H))*exp(-0.01*V(H))
The dependence of the resonant frequency FRES on the value of the inductance is approximated by the results of a multivariate analysis in the following way: FRES 465 L(LCOIL)0.2 , (4) where L(LCOIL) – variable choke inductance.
The characteristics obtained by simulation are shown in fig. 5.
The upper graph shows the output voltage oscillogram, the frequency of which varied by the variation of the inductance: from the base frequency at 300 kHz initially increased by 48kHz, then decreased by 28kHz, as illustrated by the lower graph DELTA_FREQ vs T (Time). The initial frequency is set after the full charge of resonance circuit capacitance.
Inductance`s changing leads to the frequency change in the oscillation circuit. Current value of measured frequency is compared with the reference frequency 300 kHz. The difference between the measured and the base frequency DELTA_FREQ is transmitted to the voltage source and then sent to the subsystem, which generates certain pulse duration of the PWM-signal.
According to the simulation results, the program for the microcontroller has been adjusted. 6
Model of the Compensating Subsystem (nonlinear solenoid with moving core) is
based on the relay model [
1
LCOIL where LCOIL – initial choke inductance;
The FLUX is determined by the magnetizing FORCE, by the formula (given in functional source E3):
FORCE KFORCE KFLUX
V (FLUX )2
, AREA where KFORCE, KFLUX – coefficients are given by operators .define; V(FLUX) – flux emulation, received as the voltage of the “FLUX” node; AREA – the cross-sectional area of the rod is given by the operator .define.
By Faraday's law of induction, any change in flux through a circuit induces an
electromotive force (EMF) or voltage in the circuit, proportional to the rate of change
of flux. Any alteration to a circuit which increases the flux (total magnetic field)
through the circuit produced by a given current increases the inductance. The force
produced by a magnetic field [
This result is proved in textbooks such as Hammond, and also Smith [
There is another way to determine this FORCE. Force affects to the rod by pushing it, or pulling it at the speed VELOCITY, which is determine through the second law of Newton. The current position of the rod (variable POSITION2) is determined by the integration of VELOCITY. 7
The simplified model of ACS is shown on fig.6.
Total model consist of macros: 1. Measurement subsystem on Colpitts oscillator; 2. Finding the frequency difference − DELTA_FREQ; displacement of the rod − POSITION1; 3. Calculation of Current/voltage for compensating system − V_COIL; 4. Emulation of PWM signal − PWM_RES; 5. Finding the result displacement in compensating system − POSITION2; 6. Stoletov`s curve fitting.
PWM controller model [26] in which compensating signal produced as a constant continuous signal with variation of its amplitude (staircase signal). This assumption is acceptable provided if the output signal is smoothing effectively by filter (Choke’s inductance). Since the PWM signal regulates the average value of the voltage by changing the duty cycle, it is possible to directly use the variation of the mean/average value of PWM signal. The output PWM signal is control the current of the choke with inductance 10 mH. The compensation signal stops changing when the values of POSITION1 and POSITION2 become equal.
Verification during the simulation was carried out by comparing the founded values of the rod movement (POSITION1), which should be the same in the measuring and compensating subsystems The displacement of the rod in the measuring system (POSITION1 − lower graph) and its response in the compensation system (POSITION2 − upper graph) is shown on fig.7. Fig. 7. Displacement of the rod in the measuring system (POSITION1) and response in the compensation system (POSITION2)
The variation of the inductance in the range 8040 ± 40 nH has changes the frequency (± 800 Hz), the displacement of the rod (± 3.2 μm), and the voltage on the compensation circuit (± 3.2 mV).
A-causal models work together with causal models very well due to the
development of the simulation SPICE algorithm [
This system is implemented on the ATmega8 microcontroller family. It has 23 inputs, six of which can receive an analog signal through the built-in ten-bit analog-to-digital converter (ADC). This controller is capable of generating a PWM signal by hardware, as it has three built-in PWM controllers. One PWM signal is used for stepped pumping of a Power Magnet, and the other is for regulating the current in a Compensating Subsystem.
The PWM controller is tuned by setting the control bits in the corresponding registers. The PWM controller operates in Fast PWM mode, in this mode the counter counts from zero to 255, after an overflow is reached, it is reset to zero and the counting starts again. The timer has a special OCR comparison register, the reset also occurs when the value in the counting register is compared with the value written to the comparison register. The D.C. factor of the signal is determined with an accuracy of 2-8 by writing a number from 0 to 255 in the OCR register. With a clock frequency of 8 MHz, PWM frequency will be equal to 31.25kHz. If frequency dividers from 8 to 1024 is used, the frequency will decrease proportionally, but the accuracy of the interval setting will not improve.
The current measurement of the Power Magnet is made using a Hall sensor. At the output, it produces a voltage proportional to the current flowing. The signal enters the controller via an ADC. In the Atmega8 controller, the ADC has the following characteristics: resolution of 10 bits; the time of conversion of one indication from 13 to 250 µs depending on the measurement bit depth, as well as on the clock frequency of the controller generator; start support by interrupts, which allows you to automatically process data and save them to the correct register, for processing them in the main cycle.
The change in the state of the system and the processing of information occur after system initialization and after calculating the frequency of the Colpitts generator. The signal from the generator after converting it to a rectangular view is fed to the input of the controller. To determine the frequency, two interrupt vectors are used: the timer overflow interrupt and the external pulse interrupt. An interrupt from an external pulse increases the register value by one. The higher the timers interrupt frequency, the faster the controller response, but the frequency measurement accuracy is lower. However, there is a limit value of the frequency at which the controller does not have time to make calculations. The minimum frequency of the timer 244.14 Hz, which is quite suitable for this ACS, has been determined.
The speed of the measurements is determined mainly by the transition process in the compensating coil, which is about ten times longer than the period of the microcontroller timer (46 ms vs 4.1 ms). Measurement delay is also associated with transitions to a stationary state in a power coil. The system is stable, oscillations of transient processes are not revealed either during modeling, or in «field» experiments.
A new way of modeling ACS is based on dynamic models for its subcircuits. This will allow you to remain inside the time domain for a full study of the dynamics even for non-stationary systems. Different levels of abstraction Macromodels for all subsystems of the Automated Magnetic Susceptibility Measurement System have been developed. Analysis of static and dynamic characteristics in the environment of the ECAD program Micro-Cap 11 has been conducted.
The resulting model is dynamic, nonlinear, quasi-causal [
The relative error of the model is not more than 20%, so it can be recommended for the study of such systems.
After the model experiment, in which the parameters of the system has been
optimized and refined on the basis of physical features of the real hardware, a prototype
of compensation system was designed [
New approach, chosen for measurement system modeling lies within the multidomain modeling paradigm, which allows us to investigate a wider spectrum of characteristics of nonlinear dynamic systems, in contrast to the standard approach adopted in the theory of automatic regulation, which focuses on the analysis of stationary processes in linearized systems. The developed macromodels are distinguished by minimalism and simplicity and therefore, may be recommended for the study of ACS on the different levels of abstraction in ECAD and in other similar mathematical processors.
References.
http://info.ee.surrey.ac.uk/Workshop/advice/coils/force.html 26. Vasylenko, O.V., Snizhnoi, G.V.: PWM controller's models for investigation ACS in SPICE-family ECAD programs. Radio Electronics, Computer Science, Control. 1, 64-71 (2018). 27. Kendall, R.: Modular macromodeling techniques for Spice simulators (2002). https://www.edn.com/design/analog/4348119/Modular-macromodeling-techniques-forSpice-simulators