Analysis of a Frequency Doubler as an Element of an Active Compensation Smoothing Filter
Abstract Generated abstract
This paper analyzes a ferromagnetic frequency doubler used as the basic element of an active compensation smoothing filter intended to improve ripple suppression by compensating pulsations from an auxiliary source. Using a hyperbolic sine approximation of the magnetization curve and neglecting hysteresis, leakage flux, and winding resistance, the authors derive expressions for operating modes with sinusoidal variation of induction and of current or voltage, and calculate harmonic components through trigonometric interpolation of Fourier coefficients. The resulting formulas give primary excitation harmonics and secondary output voltage harmonics in explicit form and are presented as a practical tabulatable calculation method. Experimental measurements on a sample doubler are reported to agree satisfactorily with the analytical values.
Full Text
ANALYSIS OF A FREQUENCY DOUBLER AS AN ELEMENT OF AN ACTIVE COMPENSATION SMOOTHING FILTER
V. V. Sorokovikov, E. I. Goldstein
(Presented by the scientific seminar of the Department of Instruments and Devices of Automation Systems)
To increase the effectiveness of smoothing filters, the use of the principle of compensating pulsations from an auxiliary source appears promising. To implement such a principle, circuits of compensation filters based on static frequency multipliers developed at the Tomsk Polytechnic Institute have been proposed [1].
Taking into account that ACSFs are substantially nonlinear devices and that their calculation—and still more their design—is impossible without carrying out detailed theoretical and experimental investigations of the basic element of an ACSF, the frequency doubler, an analysis of the basic modes is given below; on its basis calculation expressions are derived for an ACSF doubler (Fig. 1).
The proposed analytical calculation method is based on an approximate determination of the coefficients of a Fourier series, which can be represented by an analytic function expressing the relation between the magnetic induction and the magnetic-field strength. For the calculation, formulas of approximate harmonic analysis are used, based on replacing a definite integral by a finite sum by the method of trigonometric interpolation [2].
Fig. 1. Schematic diagram of an ACSF
The investigations are carried out without taking into account hysteresis, leakage fluxes, and the active resistance of the magnetizing circuits. For approx-
in describing the magnetization curves, the hyperbolic-sine equation in relative units is used
\[ h=\operatorname{sh}\Theta, \tag{1} \]
where
\[ h=\frac{IW}{al} \]
and
\[ \Theta=\beta B, \tag{2} \]
\(I\) is the current;
\(\alpha,\beta\) are approximation coefficients of the basic magnetization curve;
\(l\) is the length of the mean line of force.
Analysis of the ACF frequency doubler in the regime of sinusoidal variation of voltage (induction)
In the steady-state no-load regime, the magnetic state of the cores, with the adopted method of approximating the basic magnetization curve and with input voltage \(U_{1y}=U_{1ym}\cos\omega t\), is described by the following equations:
\[ \left. \begin{aligned} h_y+h_0&=\operatorname{sh}\Theta_A,\\ h_y-h_0&=\operatorname{sh}\Theta_B, \end{aligned} \right\} \tag{3} \]
where \(h_y,h_0\) are the instantaneous values of the primary field intensity and the biasing field intensity,
\(\Theta_A,\Theta_B\) are the inductions in cores A and B.
\[ \left. \begin{aligned} \Theta_A&=\Theta_1\sin\omega t+\Theta_0+\sum_{n=1}^{\infty}\Theta_{2n}\cos 2n\omega t,\\ \Theta_B&=\Theta_1\sin\omega t-\Theta_0-\sum_{n=1}^{\infty}\Theta_{2n}\cos 2n\omega t . \end{aligned} \right\} \tag{4} \]
Solving system (3) with allowance for (4),
\[ \left. \begin{aligned} h_y&=\operatorname{sh}(\Theta_1\sin\omega t)\operatorname{ch}\!\left[\Theta_0+\sum_{n=1}^{\infty}\Theta_{2n}\cos 2n\omega t\right],\\ h_0&=\operatorname{ch}(\Theta_1\sin\omega t)\operatorname{sh}\!\left[\Theta_0+\sum_{n=1}^{\infty}\Theta_{2n}\cos 2n\omega t\right]. \end{aligned} \right\} \tag{5} \]
After a simple transformation of equations (3)—(5), we obtain expressions for the instantaneous value of the excitation field intensity and for the values of the even harmonics of the induction of the frequency doubler:
\[ h_y=\operatorname{sh}(\Theta_1\sin\omega t)\operatorname{ch}\!\left[\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(\Theta_1\sin\omega t)}\right], \tag{6} \]
\[ \left[\Theta_0+\sum_{n=1}^{\infty}\Theta_{2n}\cos 2n\omega t\right] = \operatorname{Arsh}\frac{h_0}{\operatorname{ch}(\Theta_1\sin\omega t)}. \tag{7} \]
The function \(f(t)=h_y(t)\) is a periodic function with period \(T=2\pi\) and is equal to zero at the ends of the interval. Let us represent it in the form
\[ h_y(\omega t)=h_{1m}\sin\omega t+h_{3m}\sin 3\omega t+h_{5m}\sin 5\omega t+\cdots . \tag{8} \]
The harmonic components are found by the method of trigonometric interpolation [2]
\[ h_{km}=\frac{2}{n}\sum_{\alpha=1}^{n-1} y_{\alpha}\sin k\alpha\frac{\pi}{n}, \tag{9} \]
where \(y_{\alpha}\) are the values of the function at the interpolation nodes
\[ \begin{aligned} y_1&=\operatorname{sh}(0.5\Theta_1)\operatorname{ch}\left[\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(0.5\Theta_1)}\right],\\ y_2&=\operatorname{sh}(0.866\Theta_1)\operatorname{ch}\left[\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(0.866\Theta_1)}\right],\\ y_3&=\operatorname{sh}(\Theta_1)\operatorname{ch}\left[\operatorname{Arsh}\frac{h_0}{\operatorname{ch}\Theta_1}\right]. \end{aligned} \tag{10} \]
From (9), taking (10) into account, we obtain
\[ \begin{aligned} h_{1m}&=\frac{1}{3}(y_1+1.73y_2+y_3),\\ h_{3m}&=\frac{1}{3}(2y_1-y_3),\\ h_{5m}&=\frac{1}{3}(y_1-1.73y_2+y_3). \end{aligned} \tag{11} \]
Similarly, we determine the harmonic components of the secondary induction from (7):
\[ \Theta(\omega t)=\Theta_0+\Theta_2\cos 2\omega t+\Theta_4\cos 4\omega t+\Theta_6\cos 6\omega t+\cdots, \tag{12} \]
\[ \Theta_{2k}=\frac{2}{n}\sum_{\alpha=0}^{n}y_{\alpha}\cos k\alpha\frac{\pi}{n}, \tag{13} \]
\[ \begin{aligned} y_0&=\operatorname{Arsh}h_0,\\ y_1&=\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(0.259\Theta_1)},\\ y_2&=\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(0.5\Theta_1)},\\ y_3&=\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(0.707\Theta_1)},\\ y_4&=\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(0.866\Theta_1)},\\ y_5&=\operatorname{Arsh}\frac{h_0}{\operatorname{ch}(0.966\Theta_1)},\\ y_6&=\operatorname{Arsh}\frac{h_0}{\operatorname{ch}\Theta_1}. \end{aligned} \tag{14} \]
\[ \left. \begin{aligned} \Theta_0 &= \frac{1}{3}\left(\frac{1}{2}y_0+y_1+y_2+y_3+y_4+y_5+\frac{1}{2}y_6\right),\\ \Theta_2 &= \frac{1}{3}\left(\frac{1}{2}y_0+0.866y_1+0.5y_2-0.5y_4-0.866y_5-\frac{1}{2}y_6\right),\\ \Theta_4 &= \frac{1}{3}\left(\frac{1}{2}y_0+0.5y_1-0.5y_2-y_3-0.5y_4+0.5y_5+\frac{1}{2}y_6\right),\\ \Theta_6 &= \frac{1}{3}\left(\frac{1}{2}y_0-y_2+y_4-\frac{1}{2}y_6\right). \end{aligned} \right\} \tag{15} \]
To obtain an analytical expression for the output voltage of the doubler, we shall use the known relation
\[ e_y=\frac{2}{\beta}W_2S\frac{d}{dt}\left(\Theta_{2k}\cos 2k\omega t\right). \tag{16} \]
In this case
\[ \left. \begin{aligned} e_{2\mathrm{gr}}&=\frac{2}{\beta}W_2S\frac{d}{dt}\left(\Theta_{2k}\cos 2\omega t\right)=U_{2m}\sin 2\omega t,\\ e_{4\mathrm{gr}}&=\frac{2}{\beta}W_2S\frac{d}{dt}\left(\Theta_{4k}\cos 4\omega t\right)=U_{4m}\sin 2\omega t,\\ e_{6\mathrm{gr}}&=\frac{2}{\beta}W_2S\frac{d}{dt}\left(\Theta_{6k}\cos 6\omega t\right)=U_{6m}\sin 6\omega t, \end{aligned} \right\} \tag{17} \]
\[ U_{2y}=\sqrt{\frac{U_{2m}^{2}+U_{4m}^{2}+U_{6m}^{2}}{2}}. \tag{18} \]
The harmonics of the primary current are found from the corresponding voltages
\[ i_y=\frac{h_y\alpha l}{W_1}, \tag{19} \]
\[ I_y=\sqrt{\frac{I_{1m}^{2}+I_{3m}^{2}+I_{5m}^{2}}{2}}, \tag{20} \]
\[ H_y=\frac{I_yW_1}{\alpha l}, \tag{21} \]
where \(H_y\) is the rms value of the excitation intensity in relative units.
Analysis of the ACPF doubler in the regime of sinusoidal variation of current (voltage)
For the case
\[ \left. \begin{aligned} I_y&=I_{ym}\sin\omega t,\\ h_y&=h_{ym}\sin\omega t \end{aligned} \right\} \tag{22} \]
the systems of equations (3) and (4) are written in the form
\[ \left. \begin{aligned} h_{ym}\sin\omega t+h_0&=\operatorname{sh}\Theta_A,\\ h_{ym}\sin\omega t-h_0&=\operatorname{sh}\Theta_B, \end{aligned} \right\} \tag{23} \]
\[ \left. \begin{aligned} \Theta_A&=\operatorname{Arsh}\left(h_{ym}\sin\omega t+h_0\right),\\ \Theta_B&=\operatorname{Arsh}\left(h_{ym}\sin\omega t-h_0\right). \end{aligned} \right\} \tag{24} \]
To determine the values of the primary and secondary voltages, we use the known relations:
\[ e_1=\frac{2}{\beta} W_1 S \frac{d}{dt}\left(\Theta_{1k}\sin k\omega t\right); \]
\[ e_2=\frac{2}{\beta} W_2 S \frac{d}{dt}\left(\Theta_{2k}\cos 2k\omega t\right), \tag{25} \]
where
\[ \left. \begin{aligned} \Theta_{1y}=(\Theta_A+\Theta_B) &=\operatorname{Ar\,sh}(h_{ym}\sin\omega t+h_0) +\operatorname{Ar\,sh}(h_{ym}\sin\omega t-h_0),\\ \Theta_{2y}=(\Theta_A-\Theta_B) &=\operatorname{Ar\,sh}(h_{ym}\sin\omega t+h_0) -\operatorname{Ar\,sh}(h_{ym}\sin\omega t-h_0). \end{aligned} \right\} \tag{26} \]
But, in turn,
\[ \Theta_{1y}=\sum_{k=1}^{\infty}\Theta_{1k}\sin k\omega t,\qquad \Theta_{2y}=\Theta_0+\sum_{k=1}^{\infty}\Theta_{2k}\cos 2k\omega t. \tag{27} \]
We determine the harmonic components of the induction
\[ \Theta_{1y}=\Theta_1\sin\omega t+\Theta_3\sin 3\omega t+\Theta_5\sin 5\omega t+\cdots, \tag{28} \]
\[ \Theta_k=\frac{2}{n}\sum_{\alpha=1}^{n-1} y_\alpha \sin k\alpha\frac{\pi}{n}, \tag{29} \]
\[ \left. \begin{aligned} y_1&=\operatorname{Ar\,sh}(0{,}5h_{ym}+h_0)+\operatorname{Ar\,sh}(0{,}5h_{ym}-h_0),\\ y_2&=\operatorname{Ar\,sh}(0{,}866h_{ym}+h_0)+\operatorname{Ar\,sh}(0{,}866h_{ym}-h_0),\\ y_3&=\operatorname{Ar\,sh}(h_{ym}+h_0)+\operatorname{Ar\,sh}(h_{ym}-h_0). \end{aligned} \right\} \tag{30} \]
\[ \left. \begin{aligned} \Theta_1&=\frac{1}{3}\left(y_1+1{,}732y_2+y_3\right),\\[4pt] \Theta_3&=\frac{1}{3}\left(2y_1-y_3\right),\\[4pt] \Theta_5&=\frac{1}{3}\left(y_1-1{,}732y_2+y_3\right). \end{aligned} \right\} \tag{31} \]
\[ \Theta_{2y}=\Theta_0+\Theta_2\cos 2\omega t+\Theta_4\cos 4\omega t+\Theta_6\cos 6\omega t, \tag{32} \]
\[ \Theta_{2k}=\frac{2}{n}\sum_{\alpha=0}^{n} y_\alpha \cos k\alpha\frac{\pi}{n}, \tag{33} \]
\[ \left. \begin{aligned} y_0&=\operatorname{Ar\,sh}h_0+\operatorname{Ar\,sh}h_0=2\operatorname{Ar\,sh}h_0,\\ y_1&=\operatorname{Ar\,sh}(0{,}259h_{ym}+h_0)-\operatorname{Ar\,sh}(0{,}259h_{ym}-h_0),\\ y_2&=\operatorname{Ar\,sh}(0{,}5h_{ym}+h_0)-\operatorname{Ar\,sh}(0{,}5h_{ym}-h_0),\\ y_3&=\operatorname{Ar\,sh}(0{,}707h_{ym}+h_0)-\operatorname{Ar\,sh}(0{,}707h_{ym}-h_0),\\ y_4&=\operatorname{Ar\,sh}(0{,}866h_{ym}+h_0)-\operatorname{Ar\,sh}(0{,}866h_{ym}-h_0),\\ y_5&=\operatorname{Ar\,sh}(0{,}966h_{ym}+h_0)-\operatorname{Ar\,sh}(0{,}966h_{ym}-h_0),\\ y_6&=\operatorname{Ar\,sh}(h_{ym}+h_0)-\operatorname{Ar\,sh}(h_{ym}-h_0). \end{aligned} \right\} \tag{34} \]
The values \(\Theta_0,\ \Theta_2,\ \Theta_4,\ \Theta_6\) are determined from (15).
The theoretical assumptions are confirmed by experimental data. Table 1 gives the results of an experimental pro-
…of checking the doubler in the regime of sinusoidal variation of the induction \(C\) with parameters \(S_{\mathrm{st}} = 4.5 \cdot 10^{-4}\ \mathrm{m}^2\), \(W_1 = 500\), \(W_0 = 100\), \(I_0 = 0.4\ \mathrm{A}\), \(U_{1y} = 70\ \mathrm{V}\), made of material E-42 (whose basic magnetization curve is approximated by a hyperbolic sine with coefficients \(\alpha = 31.4\ \mathrm{A/m}\) and \(\beta = 4.02\ \frac{1}{\mathrm{T}}\)). Satisfactory agreement was also obtained for the regime of sinusoidal variation of the voltage.
Table 1
| \(U_{2\mathrm{gr}}\) (V) | \(U_{4\mathrm{gr}}\) (V) | \(U_{6\mathrm{gr}}\) (V) | \(I_{1\mathrm{gr}}\) (A) | \(I_{3\mathrm{gr}}\) (A) | Note |
|---|---|---|---|---|---|
| 8.99 | 3.03 | 0.728 | 0.0618 | 0.00213 | determined analytically |
| 8.81 | 3.11 | 0.74 | 0.0624 | 0.0022 | determined experimentally |
Conclusion
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Analytical dependences have been obtained for instantaneous values in the regimes of sinusoidal variation of the induction and sinusoidal variation of the voltage in a system of relative units.
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The proposed analytical method for calculating harmonic components is readily amenable to tabulation and can be used for the analytical calculation of ferromagnetic devices with accuracy sufficient for practice.
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The method considered makes it possible to obtain equations for any harmonic taken into account in explicit form.
LITERATURE
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E. I. Goldstein, A. I. Kutyavin, V. V. Sorokovikov. Active Compensation Smoothing Filters. Nauka Publishing House, Moscow, 1968.
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K. Lanczos. Practical Methods of Applied Analysis. Moscow, Fizmatgiz, 1962.