Linear Interpolator for Automatic Discrete Recording of Continuous Distribution Curves of Technological Parameters Measured by EMF Sensors
Abstract Generated abstract
This paper addresses automatic recording of continuous spatial distribution curves for technological parameters, such as temperature or concentration, measured at discrete points by EMF sensors. It formulates sequential sensor polling as a time based linear interpolation problem and describes an electromechanical interpolator using a stepping commutator, rheochord, motor, and self-recording potentiometer to generate piecewise linear signals between adjacent sensor readings. The device records the reconstructed distribution curve on chart paper during a polling cycle, with technical characteristics including operation with standard thermocouples, 9 sensors expandable to 26, and a basic instrumental error of 1.5%.
Full Text
UDC 681.1
LINEAR INTERPOLATOR FOR AUTOMATIC DISCRETE RECORDING OF CONTINUOUS DISTRIBUTION CURVES OF TECHNOLOGICAL PARAMETERS MEASURED BY EMF SENSORS
V. F. Lyadik, V. I. Ses
(Submitted by the scientific seminar of the Faculty of Physics and Technology)
Problems are considered concerning the construction of analog linear interpolators based on sequential input of readings from sensors into a computing device. A description and technical characteristics are given for an electromechanical linear interpolator for recording on chart paper continuous distribution curves of technological parameters from discrete readings of EMF sensors. Illustrations 1; bibliography 2.
When recording spatially distributed technological parameters (temperature, concentration, etc.) by means of sensors installed along the length of the monitored object, the task arises of constructing a continuous curve of the distribution of the parameters \(P(l)\) from discrete readings.
The construction of such curves can be performed manually by plotting the sensor readings on a graph and connecting these points, for example, by straight-line segments (linear interpolation).
However, this method of constructing distribution curves in the discrete recording of distributed parameters is inconvenient and not very effective, especially in the operational monitoring and control of technological processes.
Below a device is described that is intended for the automatic representation and subsequent recording on chart paper of continuous distribution curves of the parameters \(P(l)\) from discrete readings of sensors arranged with spacing \(h\) along the length of an object \(L\), on the basis of the linear interpolation formula [1].
In sequential polling of the sensors in time, the formula of linear interpolation can be represented as a function of time:
\[ P(t)=P_1+(P_2-P_1)\frac{t-t_1}{t_2-t_1}, \tag{1} \]
where
\(P_1, P_2\) — values of the monitored distribution at the interpolation nodes (readings of two adjacent sensors),
\(t_1\) — the moment at which two adjacent sensors are connected to the device.
We rewrite expression (1) in a form more convenient for technical implementation, taking \(t_1=0\):
\[ P(t)=P_1+\Delta P_{21}\frac{t}{\tau}, \tag{2} \]
where
\[ \tau = t_2 - t_1 \]
is the time interval during which two sensors are connected to the device; in this case the value of \(\tau\) is expressed from the condition \(\tau = \frac{T}{L}h\), where \(T\) is the polling cycle of the entire group of sensors arranged along the length of the object \(L\) with spacing \(h\).
As is evident from formula (2), the linear interpolator can be implemented by means of switching, subtracting, forming a linear function \(\Delta P \frac{t}{\tau}\), and summing elements.
The schematic diagram of an electromechanical device that implements piecewise-linear interpolation according to (2), for the case of uniform placement of sensors, is shown in Fig. 1,
where \(d_1, d_2, \ldots, d_9\) are EMF sensors;
1, 2 are rows of contact plates (lamellae) of the commutator;
3 is the control winding of the commutator;
4 is the interpolating rheochord;
5 is the motor;
EPP is an electronic automatic self-recording potentiometer.
Fig. 1. Schematic diagram of an electromechanical linear interpolator.
The commutator in the device described is a stepping selector with palladium-plated contacts, ShI 25/8.
Generation of the linear signal \(\Delta P \frac{t}{\tau}\) over the time \(\tau\) is carried out by rheochord 4, whose movable contact is displaced by motor 5 (SD-54). At the moment when the movable contact of rheochord 4 returns from the end position K to the initial position H, the stepping selector makes the next step and connects the next pair of sensors. From the movable contact of rheochord 4 the signal is fed to the input of the electronic automatic self-recording potentiometer EPP, where the linear signal is summed with the reading of the previous sensor from the two connected sensors.
Thus, piecewise-linear interpolation of the distribution is carried out successively from one pair of sensors to the next. As a result, during the polling cycle of all sensors, a piecewise-linear distribution curve is plotted on the chart tape of the potentiometer.
The resistance value of the interpolating rheochord is determined on the basis of the required instrumental error and the resistance value of the sensors by the formula:
\[ R \geq \frac{2\left[(E_2 - E_1)_{\max} - \delta\right]}{\delta} R_d, \tag{3} \]
where \(\delta\) is the absolute instrumental error.
The main technical characteristics of the instrument are: sensors—standard thermocouples; number of sensors—9 (up to 26); basic instrumental error 1.5%; recording is performed on the chart tape of an EPPV-60 self-recording potentiometer.
The method for determining the required number of sensors when recording with this device, and for calculating the unavoidable interpolation error caused by the error of the readings and by the inaccuracy of sensor installation, is given in [2].
Conclusions
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The problems of constructing analog linear interpolators on the basis of sequential input of sensor readings into a computing device have been considered.
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A description and technical characteristics are given for an electromechanical linear interpolator for recording continuous curves on chart tape from discrete readings taken from EMF sensors.
References
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B. P. Demidovich; I. A. Maron. Fundamentals of Computational Mathematics. Moscow, Fizmatgiz, 1963.
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V. F. Dyadik; I. E. Naats. Application of linear interpolation in problems of recording distributed parameters. Tomsk, TSU Publishing House, Proceedings of TPI, vol. 168, 1969.