Some Results of Theoretical and Experimental Study of Low-Frequency Inductive Parametrons in Frequency Repetition Mode
Abstract Generated abstract
This paper presents theoretical and experimental results on low-frequency inductive parametrons operating in frequency-repetition mode, a regime previously treated with limited attention to higher harmonics and phase variation. Using experimental phase measurements and harmonic-balance analysis, the authors derive relations for fundamental and third-harmonic voltages, examine instantaneous power and resonance properties, and obtain formulas for the excitation threshold. The study indicates that phase relationships in the device can be exploited in automation components such as phase shifters, two-phase converters, rectifiers, and photoelectric or thermoelectric relays. It also proposes an engineering method for analysis and synthesis of parametrons operating near no-load conditions.
Full Text
Some Results of a Theoretical and Experimental Study of Low-Frequency Inductive Parametrons in Frequency Repetition Mode
E. I. Goldshtein, V. N. Stan’evko
(Presented by the seminar of the Department of Instruments and Devices of Automation Systems)
Despite the great interest of engineers and physicists in questions concerning the study of inductive parametric systems [1, 2], low-frequency inductive parametrons (I.P.) in the frequency-repetition mode have still been insufficiently investigated. Thus, in [3], as well as in [4, 5], an analysis of I.P. is carried out without taking higher harmonics into account. In [6] the presence of a phase shift between the first harmonics of the magnetic inductions of the magnetic cores (Fig. 1) is indicated; in [7] it is asserted that the magnitude of this phase-shift angle at the excitation point is equal to 90°. However, neither in [6] nor in [7] are variations in the magnitude of the angle revealed when the input voltage changes, nor are possibilities indicated for using this interesting property. The methodology for engineering calculation of an I.P. at the excitation point also cannot be considered fully developed.
Fig. 1.
In light of the above, the present work gives the results of theoretical and experimental studies of I.P., which to a certain extent remedy the indicated shortcomings.
1. Study of Phase Relationships
The circuit of the inductive parametron under consideration is shown in Fig. 1. As follows from the circuit, active losses and leakage flux were not taken into account in the analysis.
Experimental studies of the phase relationships between the first harmonics of the voltages on windings \(W_1\) were carried out on a number of models; specially developed amplitude-phase harmonic analyzers were used for this purpose. As a result, the dependence of the phase between the first harmonics of the voltages for the windings \(W_1\) of chokes \(A\) and \(B\), represented by curve 1 (Fig. 2), was established, as was the same dependence for the voltages on windings \(W_2\) (curve 2, Fig. 2) on the magnitude of the input voltage.
The results of the study of phase relationships were used to develop a number of devices. Thus, for example, a continuously operating phase shifter, a discrete-action phase shifter for amplitude-phase harmonic analyzers, a converter of a single-phase system into a two-phase one, a single-phase–two-phase rectifier, etc., were obtained.
Fig. 2
Analysis of the known properties of the I.P., such as, for example, the property of generating oscillations provided that the loss energy in the parametric circuit is less than the energy introduced into the circuit, made it possible to develop such devices as photoelectric and thermoelectric relays of direct and reverse action.
The need to develop an engineering calculation method for designing the above-mentioned devices required certain theoretical investigations.
In the steady-state process, the following relationships hold for the I.P.:
\[ \begin{cases} U_{\mathrm{in}}(t)=U_A(t)+U_B(t),\\ U_c(t)=U'_A(t)+U'_B(t), \end{cases} \tag{1} \]
\(U_{\mathrm{in}}(t),\ U_A(t),\ U_B(t)\) are the instantaneous values of the input voltage and of the voltages on windings \(W_1\) of chokes A and B.
\(U'_A(t),\ U'_B(t)\) are the corresponding voltages on windings \(W_2\),
\(U_c(t)\) is the instantaneous value of the voltage on the capacitor.
Using the harmonic-balance method, the first equation of system (1) for the third harmonics takes the form:
\[ 0=U_{A3m}\sin(3\omega t+\varphi_{A3})+U_{B3m}\sin(3\omega t+\varphi_{B3}), \]
whence it follows:
\[ \begin{cases} U_{A3m}=U_{B3m}\\ \varphi_{A3}=\varphi_{B3}+180^\circ. \end{cases} \tag{2} \]
Analogously, from the second equation of system (1) we obtain:
\[ U_{c3m}\sin(3\omega t+\varphi_{c3})= U'_{A3m}\sin(3\omega t+\varphi'_{A3})+ U'_{B3m}\sin(3\omega t+\varphi'_{B3}), \tag{3} \]
where
\(U_{c3m},\ U'_{A3m},\ U'_{B3m}\) are the amplitude values of the third harmonics of the voltages on the capacitor and on windings \(W_2\),
\(\varphi_{\mathrm{c}3}\), \(\varphi'_{\mathrm{A}3}\), \(\varphi'_{\mathrm{B}3}\) are the phases of the third harmonics of the voltages, respectively.
For an IP in the frequency-repetition mode, the following relations are obvious:
\[ \left\{ \begin{aligned} U'_{\mathrm{A}3}&=K_{\mathrm{tr}}U_{\mathrm{A}3},\\ U'_{\mathrm{B}3}&=K_{\mathrm{tr}}U_{\mathrm{B}3}, \end{aligned} \right. \tag{4} \]
where the transformation ratio is
\[ K_{\mathrm{tr}}=\frac{W_2}{W_1}. \]
If one takes into account that the windings \(W_1\) are connected in series and aiding, while \(W_2\) are connected in series and opposing, then the following relations will be obvious:
\[ \left\{ \begin{aligned} \varphi'_{\mathrm{A}3}&=\varphi_{\mathrm{A}3}\\ \varphi'_{\mathrm{B}3}&=\varphi_{\mathrm{B}3}+180^\circ . \end{aligned} \right. \tag{5} \]
Starting from (3), taking (4) and (5) into account, one may write:
\[ \begin{aligned} U_{\mathrm{c}3m}&=2K_{\mathrm{tr}}U_{\mathrm{A}3m}=2K_{\mathrm{tr}}U_{\mathrm{B}3m}\\ \varphi_{\mathrm{c}3}&=2\varphi_{\mathrm{A}3} \end{aligned} \tag{6} \]
The obtained relations (2) and (6) prove useful in calculating the IP with allowance for higher (third) harmonics.
Analysis of Energy Relations
In accordance with the propositions advanced in [8], the relations between the instantaneous powers in the IP were investigated. The results of the investigation made it possible to obtain a number of relations, in particular:
\[ \begin{aligned} I_{11m}&=K_{\mathrm{tr}}I_{21m},\\ B_{\mathrm{A}1m}&=B_{\mathrm{B}1m},\\ \varphi_{\mathrm{c}1}&=90^\circ,\\ \varphi_{\mathrm{A}1}&=\varphi_{\mathrm{B}1}=45^\circ, \end{aligned} \tag{7} \]
where
\(I_{11m}\) and \(I_{21m}\) are the amplitudes of the fundamental harmonics of the currents \(i_1(t)\) and \(i_2(t)\), respectively,
\(B_{\mathrm{A}1m}\) and \(B_{\mathrm{B}1m}\) are the amplitudes of the fundamental harmonics of the inductions, respectively, in chokes A and B.
\(\varphi_{\mathrm{c}1}\) is the phase of the fundamental harmonic of the voltage on the capacitor relative to the input voltage.
\(\varphi_{\mathrm{A}1}\), \(\varphi_{\mathrm{B}1}\) are the phases of the fundamental harmonics of the voltages, respectively,
\(U_{\mathrm{A}}(t)\) and \(U_{\mathrm{B}}(t)\).
Let us consider the derivation of several relations for the proposed calculation procedure.
From analysis of the vector diagram [7], the following expressions can be obtained:
\[ U_{\mathrm{in}\,m}=S_{\mathrm{st}}\cdot W_1\cdot \omega B_{1m}\cdot \sqrt{2}, \tag{8} \]
\[ U_{c1m}=K_{\mathrm{tr}}U_{\mathrm{in}\,m}, \tag{9} \]
where
\[
S_{\mathrm{st}} \text{ is the active cross section of the steel,}
\]
\[
\omega \text{ is the angular frequency.}
\]
Taking into account that
\[ \left\{ \begin{aligned} H_A(t)\cdot l_c &= W_1\cdot i_1 + W_2\cdot i_2,\\ H_B(t)\cdot l_c &= W_1\cdot i_1 - W_2\cdot i_2, \end{aligned} \right. \tag{10} \]
and applying the harmonic-balance method to (10), one may write the following expression for the first harmonics:
\[ \left\{ \begin{aligned} H_{11A}(t)\,l_c &= i_{11}W_1+i_{21}W_2,\\ H_{11B}(t)\,l_c &= i_{11}W_1-i_{21}W_2, \end{aligned} \right. \tag{11} \]
where
\[ l_c \text{ is the length of the mean magnetic line.} \]
The vector diagram corresponding to (11) is shown in Fig. 3.
Fig. 3
Since
\[ H_{1Am}\cdot l_c=H_{1Bm}\cdot l_c, \]
the following relation is valid:
\[ H_{1Am}=H_{1Bm}=\frac{1}{l_c}\sqrt{(I_{11m}\cdot W_1)^2+(I_{21}\cdot W_2)^2}. \tag{12} \]
From (8) and (12), taking into account the Robinson approximation
\[ B=\frac{H}{\alpha+\beta H}, \tag{13} \]
where
\[ \alpha \text{ and } \beta \text{ are approximation coefficients,} \]
we shall have:
\[ U_{\mathrm{in}\,m}= \frac{2S_{\mathrm{st}}(W_1\cdot\omega\cdot K_{\mathrm{tr}})^2\cdot C-\alpha l_c} {\sqrt{2}\beta W_1\omega K_{\mathrm{tr}}^2\cdot C}. \tag{14} \]
Study of the Resonance Properties of a Parametric Circuit
As is known, the nonlinear inductance varies according to the law:
\[ L(t)=L_0+\sum_{k=1}^{\infty}\dot{L}_{km}\sin(K\omega t+\gamma_k). \tag{15} \]
From (15) the following equality follows:
\[ \mu_d(t)=\mu_{d0}+\sum_{k=1}^{\infty}\mu_{dk}\sin(K\omega t+\gamma_k), \tag{16} \]
where
\[
\mu_d(t)
\]
is the instantaneous value of the magnetic dynamic permeability,
\[
\mu_{d0}
\]
is the constant component of the magnetic dynamic permeability.
For a series resonant circuit, the resonant frequency is related to the circuit parameters as follows:
\[ \omega_{\mathrm{res}}^2=\frac{1}{L_k C}, \tag{17} \]
where
\[ L_k=\frac{S_{\mathrm{ct}}\cdot W^2}{l_c}\mu. \tag{18} \]
It is known that one of the conditions for the occurrence of parametric oscillations is the proximity of the resonant frequency of the circuit to the frequency of the oscillations. It is also known that, with unchanged parameters of the IP circuit, oscillations with frequencies that are multiples of the excitation-voltage frequency can arise in it.
It has been assumed that the quantity \(\mu\) in (18), not being constant, is in the first approximation determined by the quantity \(\mu_{d0}\), which, in turn, in the absence of oscillations in the parametric circuit, is determined by the magnitude of the excitation current. In this case
\[ \omega_{\mathrm{res}}^2= \frac{l_c}{\mu_{d0}S_{\mathrm{ct}}W^2C}. \tag{19} \]
To determine \(\mu_{d0}\), it is convenient to use the power approximation
\[ b=\alpha_1 h^{\beta_1}, \]
where
\(b\) and \(h\) are the instantaneous values of magnetic induction and field strength,
\(\alpha_1,\beta_1\) are approximation coefficients.
In this case we obtain:
\[ \mu_{d0}=\frac{\omega}{\pi}\int_0^{\pi/\omega} \frac{\alpha_1}{2}h^{-\frac{1}{2}}\,dt = \frac{\alpha}{2}\sqrt{\frac{l_c}{I_{1m}\cdot W_1}}, \tag{20} \]
where
\[
I_{1m}
\]
is the value of the first harmonic of the excitation current in the absence of oscillations in the parametric circuit.
Repeated experimental studies have shown that $\mu$ from (18) and $\mu_{d0}$ from (16) are sufficiently close to one another. From (19) and (20) the expression
\[ \omega_{\text{res}}^{2}=-\frac{4}{a}\frac{\sqrt{l_c I_{1m} W_1}}{S_{\text{st}} W_2^{2}\cdot C}. \tag{21} \]
was obtained.
Using the obtained relations (9), (14), and (21), it proves possible to carry out both the analysis and the synthesis of an IP.
The solution of the analysis problem for an IP can be carried out either with the aid of expressions (9) and (14), or with the aid of expressions (9) and (21). The solution of the synthesis problem for an IP can be carried out with the aid of expressions (9), (14), and (21). To obtain a unique solution in the synthesis problem, one may use the relation between $l_c$ and the dimensions of the core
\[ l_c=K_l\sqrt{S_{\text{ok}}}, \]
where
$K_l$ is a well-defined quantity for a normalized core.
As the experiments conducted have shown, excitation of parametric oscillations is observed at an induction
\[ B=(0.8\div 1)\ \text{T}, \]
which makes it possible to use expression (8) as an additional equation in IP synthesis problems.
Conclusions
-
The study of the properties of a low-frequency IP operating in the frequency-repetition mode made it possible to develop a number of automation elements and devices.
-
The study of the energy relations and resonance properties of the IP made it possible to obtain comparatively simple and clear relations for its excitation threshold.
-
An engineering method has been developed for the analysis and synthesis of an IP operating in a mode close to no-load operation.
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