Toward the Development of an Engineering Methodology for the Calculation and Design of a Low-Frequency Inductive Parametron Without a DC Component in the Excitation Current
Abstract Generated abstract
This paper develops an approximate engineering calculation procedure for a low-frequency two-frequency inductive parametron excited from a 50 Hz source without a dc component in the excitation current. Based on experimental studies over varied core dimensions, winding data, capacitances, load resistance, excitation voltage, and output power, the authors derive empirical relations for stable operation in first-resonance and second-resonance modes and for switching between them by changing only the resonant capacitance. The proposed method gives formulas for selecting capacitor values, magnetic-core size, transformation ratio, and winding turns, followed by a worked design example for a telesignaling application. Comparison with tests of the fabricated sample shows generally satisfactory agreement, except for the excitation voltage, and the authors present the relations as a first-approximation design method requiring further refinement.
Full Text
TOWARD THE DEVELOPMENT OF AN ENGINEERING METHOD FOR THE CALCULATION AND DESIGN OF A LOW-FREQUENCY INDUCTIVE PARAMETRON WITHOUT A DC COMPONENT IN THE EXCITATION CURRENT
E. I. Goldshtein, A. M. Alin
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
At the Department of Automation and Telemechanics of TPI, possibilities are being considered for the application of an inductive parametron excited from an industrial-frequency source of 50 Hz as an element of automation and telemechanics systems.
In devices for remote-control and telemechanical control it is most expedient to use a two-frequency inductive parametron retuned by changing the capacitance of the capacitor of the resonant circuit. Figure 1 shows the circuit of the parametron under consideration, where
\(U_1\) — excitation voltage of frequency \(f\);
\(I_1\) — rms value of the current in the excitation circuit;
\(W_1\) and \(W_2\) — number of turns, respectively, of the excitation winding and the resonant winding;
\(U_2\) — output voltage;
\(I_2\) — rms value of the current in the resonant circuit;
\(C_1\) and \(C_2\) — capacitors of the corresponding capacitance;
\(K_1\) and \(K_2\) — control switches;
\(R_{\text{н}}\) — load resistance.
With an appropriate choice of the circuit parameters, it is possible to ensure a transition from the first-resonance mode \((f_{\text{out}} = f)\) to the second-resonance mode \((f_{\text{out}} = 2f)\) solely by changing the capacitance of the capacitor, with the excitation voltage unchanged. The authors carried out an experimental study of an inductive parametron with variation
Fig. 1
of the capacitance of the capacitor.
its main parameters (the cross sections of the core steel, the number of turns, the load resistance, the capacitance of the capacitors, the excitation voltage, etc.) over a wide range. All cores were shell-type; they were assembled from stamped plates of Э41 steel, 0.35 mm thick. In the experiments, currents, voltages, output power (power in the load), and the critical resistance \((R_{\mathrm{cr}})\) of the resonant circuit were recorded—that is, the resistance at which the occurrence of parametric oscillations is still possible for a given capacitor capacitance.
Processing of the experimental data made it possible to establish certain relationships among the basic parameters of the circuit under which a stable, long-duration operating mode of the inductive parametron into a known load is ensured.
For the first-resonance mode, the capacitive reactance of the resonant circuit is related to the active resistance of the circuit \(R_{\mathrm{k}}\) as follows:
\[ X_{C_1} \geq K_1 R_{\mathrm{k}}, \tag{1} \]
where the coupling coefficient \(K_1\), depending on the output power, varies within the limits from 8 (for \(P_{\mathrm{out}} = 50\) W) to 4 (for \(P_{\mathrm{out}} = 500\) W), i.e.,
\[ K_1 = (8 \div 4). \tag{2} \]
The resistance of the circuit consists of the load resistance and the resistance of the windings \(W_2\):
\[ R_{\mathrm{k}} = R_2 + R_{\mathrm{l}}. \tag{3} \]
As a first approximation it may be assumed that
\[ R_{\mathrm{k}} = (1.1 \div 1.2) R_{\mathrm{l}}. \tag{4} \]
For the second-resonance mode,
\[ X_{C_2} \geq K_2 R_{\mathrm{k}}; \qquad K_2 = 12 \div 10. \tag{5} \]
The transition from the first-resonance mode to the second-resonance mode at \(U_1 = \mathrm{const}\) is possible only for a definite relation between \(X_{C_1}\) and \(X_{C_2}\):
\[ X_{C_2} = (3.5 \div 5)X_{C_1}; \qquad C_2 = \frac{C_1}{7 \div 10}. \tag{6} \]
Analysis of the experimental data showed that, for a known calculated power of the resonant circuit \((P_2)\), the product of the core cross section and the window area is fixed, i.e.,
\[ S_C S_0 = K \frac{P_2}{2}, \tag{7} \]
where the coupling coefficient \(K\) depends on the absolute value of the power \(P_2\) and can be determined from the graph in Fig. 2.
Knowing \(S_C S_0\), it is easy to select a suitable standard size of core from reference data. To determine \(P_2\), we shall use the obvious relation
\[ P_2 = I_2^2 \sqrt{(R_{\mathrm{l}} + R_2)^2 + (X_{C_1})^2}, \tag{8} \]
In an approximate determination of the numbers of turns \(W_1\) and \(W_2\), we shall take as a basis the calculation expressions obtained using the method of calculation by rms values and vector diagrams [1, 2]. Without
Without giving the entire derivation because of its bulkiness, we shall confine ourselves to the final design expression for the transformation coefficient
\[ K_{\mathrm{tr}}=\frac{W_1}{W_2}, \tag{9} \]
\[ K_{\mathrm{tr}}= \frac{U_1}{ \sqrt{ \left| \left(\frac{S_0 K_{\mathrm{m}}\delta}{2 I_2}\right)^2 \cdot 2 S_c^2 \omega^2 B_p^2 K_c^2 - U_2^2 \right| } }, \tag{10} \]
where \(K_{\mathrm{m}}\) is the window filling factor;
\(\omega\) is the angular frequency;
\(\delta\) is the current density;
\(B_p\) is the design induction;
\(K_c\) is the core filling factor.
For the parametrons we investigated, with an output power of \(50 \div 500\ \mathrm{VA}\),
\[ B_p=(1 \div 1.1)\ \mathrm{T}. \tag{11} \]
It should be noted that in deriving expression (10), losses in the steel, leakage, and the resistance of the excitation circuit were not taken into account. Comparison of the calculation results with experimental data showed that the value of the transformation coefficient should lie in the range
\[ K_{\mathrm{tr}}\leq(0.3 \div 0.5). \tag{12} \]
If condition (12) is not satisfied, we recommend installing a matching transformer between the parametron and the power source, or changing to a larger standard size.
The number of turns of the excitation winding is determined from the expression
\[ W_1= \sqrt{ \frac{U_1^2+U_2^2 K_{\mathrm{tr}}^2} {2 S_c^2 \omega^2 B_p^2 K_c^2} }. \tag{13} \]
Fig. 2
When using expressions (10) and (13), it is necessary to know the voltage \(U_2\),
\[ U_2=I_2\sqrt{X_{C_1}^{\,2}+(R_n+R_2)^2}. \tag{14} \]
The design expressions (10) and (13) considered above make it possible to carry out a first-approximation calculation and to choose indicative parameters of the circuit, which are refined in the course of a second-approximation calculation taking into account the resistance of the excitation circuit. The authors are not aware of a method for exact calculation; however, calculation using effective values with the aid of vector diagrams [1, 2] gives sufficiently satisfactory agreement.
Example: Let us show the application of the method considered to the approximate calculation of an inductive parametron of an object telesignaling system without a power source [3]. In calculating the system, it was assumed
The initial data for calculating the parametron were obtained: \(U_1=12\ \mathrm{V}\); \(R_{\mathrm{l}}=72\ \Omega\); \(I_{\mathrm{l}}=0.052\ \mathrm{A}\); \(X_{c1}=1260\ \Omega\); \(C_1=2.53\ \mu\mathrm{F}\).
- We check the possibility of exciting oscillations at the fundamental frequency according to condition (1), taking (4) into account,
\[ 1260 > (8 \div 4)\cdot(1.1 \div 1.2)\cdot 72. \]
The condition is satisfied.
- We select the capacitor according to condition (6) to ensure the second resonance:
\[ X_{C_2}=3.5\cdot1260=4410\ \Omega. \qquad C_2=0.36\ \mu\mathrm{F}. \]
- We determine the calculated power from expression (8), taking (4) into account,
\[ P_2=0.052^2\sqrt{(1.2\cdot72)^2+(1260)^2}=3.4\ \mathrm{VA}. \]
-
From the graph in Fig. 2 we determine \(K=0.4\cdot10^{-8}\).
-
From expression (7),
\[ S_C S_0=0.4\cdot10^{-8}\frac{3.4}{2}=0.68\cdot10^{-8}\ \mathrm{m}^4. \]
-
According to reference data [4], we select a ShU \(10\times10\) magnetic core; \(S_C=1.0\ \mathrm{cm}^2\); \(S_0=1.15\ \mathrm{cm}^2\); \(S_0S_C=1.15\ \mathrm{cm}^4\).
-
From Tables 4-3, 4-2, 4-4 [5] we find
\[ K_{\mathrm{m}}=0.24;\qquad \delta=5\cdot10^6\ \mathrm{A/m^2};\qquad K_C=0.91. \]
- From expression (14),
\[ U_2=0.052\sqrt{(1260)^2+(1.2\cdot72)^2}=65.5\ \mathrm{V}. \]
- From expression (10), taking (11) into account,
\[ K_{\mathrm{tr}}= \frac{12}{ \sqrt{ \left[ \left( \frac{1.15\cdot10^{-4}\cdot0.24\cdot5\cdot10^6}{2\cdot0.052} \right)^2 \cdot2\cdot1.0^2\cdot314^2\cdot1.05^2\cdot10^{-8}\cdot0.91^2 -65^2 \right] } } = \]
\[ =0.355. \qquad \text{We select } K_{\mathrm{tr}}=0.4. \]
- From expression (13), taking (11) into account,
\[ W_1= \sqrt{ \frac{12^2+65.5^2\cdot0.4^2} {0.91^2\cdot2\cdot1.0\cdot10^{-8}\cdot314^2\cdot1.05^2} } =680\ \text{turns}. \]
- From expression (9),
\[ W_2=\frac{680}{0.4}=1700. \]
After determining the main parameters of the circuit, \(U_1\) is refined taking into account the voltage drop across the resistance of the excitation windings, and, if necessary, the number of turns \(W_1\) is refined. For the problem under consideration, no refinement was carried out, since the circuit is supplied through a matching transformer, which facilitates the selection of
Table
| \(U'_1\) | \(I_1\) | \(U_2\) | \(I_2\) | \(f_{\mathrm{out}}\) | \(C\) | \(R_{\mathrm{cr}}\) |
|---|---|---|---|---|---|---|
| V | A | V | A | Hz | \(\mu\mathrm{F}\) | \(\Omega\) |
| 24.5 | 0.15 | 64 | 0.052 | \(f\) | 2.5 | 290 |
| 24.6 | 0.067 | 57 | 0.017 | \(2f\) | 0.4 | 390 |
of the required value of \(U_1\). The table gives the experimental data obtained in testing the fabricated sample.
Conclusions
-
Comparison of the experimental data with the calculated data shows satisfactory agreement for all indices except \(U_1\).
-
The relations presented in the article may be used in an approximate calculation of a two-frequency inductive parametron without a dc component in the excitation current.
-
It appears advisable to continue further studies of inductive low-frequency parametrons with the aim of developing a simple, yet sufficiently accurate, method for their calculation and design.
References
-
R. A. Vainshtein, E. I. Goldshtein, A. I. Kutyavin, A. V. Shmoilov. Theoretical and experimental study of a low-frequency inductive parametron without biasing. Abstracts of papers of the XI All-Union Conference on Magnetic Elements, Tallinn, 1966.
-
E. I. Goldshtein, A. M. Alin. On the calculation of a parametron with a repeated frequency. Abstracts of papers of the II Conference of the Faculty of Automation and Computer Engineering, TPI, 1966.
-
E. I. Goldshtein, A. N. Gavrilova, V. P. Kaz’min. Use of parametrons without dc biasing in automation and telemechanics devices. Abstracts of papers of the II Conference of the Faculty of Automation and Computer Engineering, TPI, 1966.
-
R. Kh. Bul’yan. Low-power transformers. Sudpromgiz, 1961.
-
I. I. Belopol’skii, L. G. Pikalova. Calculation of low-power transformers and chokes. Gosenergoizdat, 1963.