Smoothing and Filtering Coefficients of an Active Compensation Filter in the Load Circuit of a Single-Phase Full-Wave Rectifier

E. I. Goldstein

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

In [1] the principle of operation and the results of an experimental study of a smoothing compensation filter were considered. In the present article, calculation expressions are derived for determining the smoothing and filtering coefficients of a tunable filter. In doing so, we neglect the internal resistance of the rectifier and represent its pulsating voltage as the sum of three components—a constant component \(U_0\), an alternating component \(U_{2\text{в}}\) of frequency \(2f_c\), and an alternating component \(U_{4\text{в}}\) of frequency \(4f_c\), where \(f_c\) is the frequency in the supply network. The corresponding equivalent circuit is shown in Fig. 1. We shall consider the output of the doubler as

Fig. 1

a connection of two generators of frequencies \(2f_c\) and \(4f_c\) with no-load voltages \(U_{2\text{у}}\) and \(U_{4\text{у}}\). In doing so, we neglect all higher harmonics in comparison with the second and fourth.

\(X_{\text{н}}\) is the internal (inductive) resistance of the doubler.

\(R_{\text{у}}\) and \(X_{\text{у}}\) are, respectively, the active and inductive (leakage) resistances of the magnetizing windings, which simultaneously perform the role of load windings.

\(R_{\text{н}}\) is the load resistance.

The principal characteristics of the filter are the smoothing coefficient \(q_c\) and the filtering coefficient \(k_f\).

\[ q_c=\frac{p}{p_f}, \tag{1} \]

\[ k_f=\frac{U_{\mathrm{p.in}}}{U_{\mathrm{p.out}}}, \tag{2} \]

\[ p=\frac{U_{\mathrm{p.in}}}{U_0}, \tag{3} \]

\[ p_f=\frac{U_{\mathrm{p.out}}}{U'_0}. \tag{4} \]

In expressions (1)--(4):

\(p\) is the ripple coefficient at the load in the absence of a filter;
\(p_f\) is the ripple coefficient at the load with the filter present;
\(U_{\mathrm{p.in}}\) is the ripple voltage at the filter input;
\(U_{\mathrm{p.out}}\) is the ripple voltage at the load;
\(U'_0\) is the rectified voltage at the load with the filter connected.

In determining \(U_{\mathrm{p.in}}\), we shall assume that in the operating mode (with compensation) the ratio between the voltages \(U_{2v}\) and \(U_{4v}\) remains the same as under no-load operation of the rectifier. But for no-load operation, using the expansion of the rectified-voltage curve \(U=U_m\sin\omega t\) in a Fourier series, we obtain:

\[ U_0=2\frac{U_m}{\pi}, \tag{5} \]

\[ U_{2v}=\frac{4}{\pi\cdot 3\cdot \sqrt{2}}\,U_m, \tag{6} \]

\[ U_{4v}=U_m\frac{4}{\pi\cdot 3\cdot 5\cdot \sqrt{2}}, \tag{7} \]

where \(U_m\) is the maximum value of the rectified voltage.

The rms value of the ripple voltage in the absence of a filter may be determined as:

\[ U_{\mathrm{p.in}}=\sqrt{U_{2v}^{2}+U_{4v}^{2}}. \tag{8} \]

And, taking expressions (6) and (7) into account,

\[ U_{\mathrm{p.in}}=1.02U_{2v}. \tag{9} \]

Whence

\[ p=\frac{U_{2v}}{U_0}\,1.02. \tag{10} \]

In the operating mode, the voltages \(U_{2v}\) and \(U_{2u}\) completely compensate each other; therefore, only the fourth-harmonic current will flow in the load circuit. Since the doubler at the second harmonic operates in no-load mode, it may, with sufficient approximation, be assumed that the phase-shift angle between the fundamental and all higher harmonics is zero [2]. We shall also take into account that the phase-shift angle between the second and fourth harmonics of the rectified voltage is equal to one quarter of the period of the fourth harmonic. Therefore, the phase shift between the fourth harmonics of the doubler and rectifier voltages is also taken as equal to \(90^\circ\).

Under these assumptions we obtain the following expression for determining the ripple voltage at the load in the presence of the filter:

\[ U_{\text{p.out}}=\sqrt{U_{\text{p.v}}^{2}+U_{\text{p.u}}^{2}} . \tag{11} \]

The ripple voltage dependent on the voltage of the fourth harmonic at the output of the frequency doubler is

\[ U_{\text{p.u}}= \frac{U_{4\text{u}}R_{\text{n}}} {\sqrt{(R_{\text{n}}+R_{\text{u}})^{2}+(X_{\text{u}}+X'_{\text{in}})}}}. \tag{12} \]

Here \(X'_{\text{in}}\) is the internal resistance of the frequency doubler, determined from the no-load and short-circuit parameters [3],

\[ X'_{\text{in}}=\frac{U_{4\text{u}}}{I_{4\text{k}}}. \tag{13} \]

The short-circuit current \(I_{4\text{k}}\) is determined with the aid of Bessel functions of zero and fourth order with a purely imaginary argument [3]:

\[ I_{4\text{k}}=\sqrt{2}\, I_{0}\, \frac{\tau_{4}(\varphi\Theta_{1})}{\tau_{0}(\varphi\Theta_{1})}. \tag{14} \]

In expression (14):
\(I_{0}\) is the constant component of the rectified current;
\(\Theta_{1}\) is the magnitude of the first harmonic of the induction, expressed in relative units,

\[ \Theta_{1}=\beta B_{1}, \tag{15} \]

where \(\alpha\) and \(\beta\) are coefficients of approximation of the magnetization characteristic \(H=\alpha \operatorname{sh}\Theta\).

The ripple voltage dependent on the current of the fourth harmonic of the rectifier is

\[ U_{\text{p.v}}= \frac{U_{4\text{v}}R_{\text{n}}} {\sqrt{(R_{\text{n}}+R_{\text{u}})^{2}+(X_{\text{u}}+X''_{\text{in}})^{2}}}}. \tag{16} \]

The resistance \(X''_{\text{in}}\) depends on the parameters of the frequency doubler and can be determined with sufficient rigor only from experimental characteristics of simultaneous magnetization that take into account the magnetic operating condition of the frequency-doubler cores.

As a first approximation, as shown by additional analysis and by the results of studies of experimental samples of the active compensating filter, one may take

\[ X''_{\text{in}}=K_{x}X'_{\text{in}}, \tag{17} \]

where \(K_{x}\) is the proportionality coefficient.

For cores made of steel E-41:

\[ K_{x}=0.3\div 0.5. \tag{18} \]

The dc voltage at the load with the filter switched on will be less than the rectifier voltage because of the voltage drop across the filter resistance:

\[ U'_{0}=\frac{U_{0}}{R_{\text{u}}+R_{\text{n}}}\cdot R_{\text{n}}. \tag{19} \]

It is noted in [1] that the use of the filter under consideration is promising mainly only with a low-resistance load. In this cas-

where \((R_{\mathrm{n}}+R_{\mathrm{y}})\) is considerably smaller than \((X_{\mathrm{in}}+X_{\mathrm{y}})\). Under the stated assumption, the active resistance may be neglected in comparison with the inductive resistance. Then, from expressions (4), (11), (12), (16), and (19), we obtain:

\[ p_{\mathrm{f}}= \sqrt{ \frac{U_{4\mathrm{v}}^{2}}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime\prime})^{2}} + \frac{U_{4\mathrm{y}}^{2}}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime})^{2}} } \cdot \frac{R_{\mathrm{n}}+R_{\mathrm{y}}}{U_{0}} . \tag{20} \]

Similarly, from expressions (2), (9), (11), (12), and (16),

\[ K_{\mathrm{f}}= \frac{1.02\,U_{2\mathrm{v}}}{R_{\mathrm{n}}} \cdot \frac{1}{ \sqrt{ \frac{U_{4\mathrm{v}}^{2}}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime\prime})^{2}} + \frac{U_{4\mathrm{y}}^{2}}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime})^{2}} } } . \tag{21} \]

Accordingly, from expressions (1), (10), (20),

\[ q_{\mathrm{c}}= \frac{1.02\,U_{2\mathrm{v}}}{(R_{\mathrm{n}}+R_{\mathrm{y}})} \cdot \frac{1}{ \sqrt{ \frac{U_{4\mathrm{v}}^{2}}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime\prime})^{2}} + \frac{U_{4\mathrm{y}}^{2}}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime})^{2}} } } . \tag{22} \]

To establish the relationship between the voltages \(U_{2\mathrm{v}}\), \(U_{4\mathrm{v}}\), and \(U_{4\mathrm{y}}\), we use the obvious relations

\[ \frac{U_{2\mathrm{y}}}{U_{4\mathrm{y}}}=K_{\mathrm{y}}; \tag{23} \]

\[ U_{2\mathrm{v}}=U_{2\mathrm{y}}, \tag{24} \]

\[ \frac{U_{2\mathrm{v}}}{U_{4\mathrm{v}}}=5, \tag{25} \]

where \(K_{\mathrm{y}}\) is a calculated coefficient depending on the operating mode of the frequency doubler.

From expressions (21)--(25) we obtain the calculated expressions for the filtering and smoothing coefficients:

\[ K_{\mathrm{f}}= \frac{1.02}{R_{\mathrm{n}}} \cdot \frac{1}{ \sqrt{ \frac{0.04}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime\prime})^{2}} + \frac{1}{K_{\mathrm{y}}^{2}(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime})^{2}} } } , \tag{26} \]

\[ q_{\mathrm{c}}= \frac{1.02}{(R_{\mathrm{n}}+R_{\mathrm{y}})} \cdot \frac{1}{ \sqrt{ \frac{0.04}{(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime\prime})^{2}} + \frac{1}{K_{\mathrm{y}}^{2}(X_{\mathrm{y}}+X_{\mathrm{in}}^{\prime})^{2}} } } . \tag{27} \]

To determine \(K_{\mathrm{y}}\), we use the method described in [2], where, for the no-load mode, the equations obtained are:

\[ \Theta_{2} = 0.5\left( \operatorname{arsh} h_{0} - \operatorname{arsh}\frac{h_{0}}{\operatorname{ch}\Theta_{1}} \right), \tag{28} \]

\[ \Theta_{4} = 0.25\left( \operatorname{arsh} h_{0} + \operatorname{arsh}\frac{h_{0}}{\operatorname{ch}\Theta_{1}} - 2\operatorname{arsh}\frac{h_{0}}{\operatorname{ch}\frac{\Theta_{1}}{\sqrt{2}}} \right). \tag{29} \]

In deriving equations (28)--(29):

a) losses due to hysteresis and eddy currents were not taken into account, and leakage was not taken into account;

b) the field strength was taken into account as the relative quantity \(h_{0}\):

\[ h_{0}=\frac{H_{0}}{a}=\frac{I_{0}W_{0}}{a l_{\mathrm{c}}}, \tag{30} \]

where \(W_0\) is the number of turns of the load winding, which serves as the bias winding;

\(l_c\) is the length of the mean line of force.

In [2] an analysis was carried out and the dependences \(\dfrac{\Theta_4}{\Theta_2} = \varphi(\Theta_1)\) were plotted for \(h_0 = \mathrm{const}\) in the range \(4 \leqslant \Theta_1 \leqslant 8\). Of greater interest to us is the dependence \(K_y = \varphi(\Theta_1)\) in the range \(1 \leqslant \Theta_1 \leqslant 3\), since for \(\Theta_1 > 3\), \(K_y < 3.8\) and the smoothing and filtering coefficients are small.

Using equations (28), (29), the values of \(\Theta_2\) and \(\Theta_4\) were calculated. But

\[ K_y = \frac{B_2}{2B_4} = \frac{\Theta_2}{2\Theta_4}, \tag{31} \]

where \(B_1\), \(B_2\) are the maximum values of the induction of the corresponding harmonic.

Figure 2 shows the obtained dependences \(\Theta_2 = \varphi(\Theta_1)\) and \(K_y(\Theta_1)\).

Fig. 2

For the selected range of variation of \(\Theta_1\), variation of the bias in the range \(50 \leqslant h_0 \leqslant 500\) has practically no effect on the value of \(K_y\), which simplifies the calculation and adjustment of the filter under consideration.

To determine the induction \(\Theta_1\), we use relation (32):

\[ \Theta_1=\frac{U_1\cdot 10^{4}\cdot \beta}{8.88 f_c Q_c K_c W_1}, \tag{32} \]

where \(W_1\) is the number of turns of the magnetizing winding connected to the AC source;

\[ \begin{aligned} U_1&\text{ is the voltage applied to the magnetizing windings;}\\ Q_c&\text{ is the geometrical cross section of the core, } \mathrm{cm}^2;\\ K_c&\text{ is the steel filling factor.} \end{aligned} \]

From the induction \(\Theta_1\), all calculated quantities needed to find the resistance \(X'_{\mathrm{vn}}\) from expression (13) can easily be determined.

Indeed,

\[ U_{4y}=8.88\cdot 4 f_c\cdot Q_c\cdot K_c\cdot W_0\cdot 10^{-4}\cdot B_4, \tag{33} \]

\[ B_4=\frac{\Theta_4}{\beta}, \tag{34} \]

\[ I_{4k}=I_0 A, \tag{35} \]

\[ A=\frac{\tau_4(\varphi\Theta_1)}{\tau_0(\varphi\Theta_1)}\sqrt{2}, \tag{36} \]

\[ X'_{\mathrm{vn}}=8.88\cdot 4 f_c Q_c K_c W_0\cdot 10^{-4}\frac{C}{\beta I_0}, \tag{37} \]

\[ C=\frac{\Theta_4}{A}. \tag{38} \]

Figure 2 gives the dependence \(C=\varphi(\Theta_1)\), used in calculating the resistance \(X'_{\mathrm{n}}\).

Analysis of expressions (26), (27), (31), (37), and (38) shows that the accuracy of determining the smoothing and filtering coefficients is largely determined by the correct choice of induction. The point is that expression (32) was obtained under the assumption that the entire supply voltage is applied to winding \(W_1\), whereas in practice, owing to leakage, the calculated primary voltage \(U_1\) is \(0.85 \div 0.65\) of the applied voltage [4, 5]. As a first approximation, we use the following relation:

\[ U_{1p}=(0.8\div 0.85)U_1. \tag{39} \]

The inductive reactance of the load windings of the doubler can be determined from the design expressions for the leakage inductance \(X_p\) of a two-winding transformer [6]:

\[ X_y=2X_p=\frac{2\cdot 4\cdot 4 f_c\cdot W_0^2\delta_{12}l_0}{h_d}\,10^{-8}, \tag{40} \]

where \(4f_c=200\ \mathrm{Hz}\);

\(\delta_{12}\) is the reduced width of the leakage-flux channel, mm;

\[ \delta_{12}=a_{12}+\frac{a_1+a_2}{2}; \tag{41} \]

\(a_1\) and \(a_2\) are the widths, respectively, of the load and magnetizing windings;

\(a_{12}\) is the width of the leakage-flux channel;

\(l_0\) — length of the mean turn of the load winding, mm;
\(h_d\) — coil height, mm;

\[ h_d=(h-1)-2\Delta_{\text{ins}}; \tag{42} \]

\(h\) — height of the winding window;
\(\Delta_{\text{ins}}\) — thickness of the frame cheek.

To verify the calculated expressions obtained, the filtering coefficient was calculated for the active filter described in [1]. The following data were used:

\(W_0=300;\ a_1=5\ \text{mm};\ a_2=10.0\ \text{mm};\ a_{12}=0.5\ \text{mm};\ h=44\ \text{mm};\ l_0=138\ \text{mm}\) (the load winding is wound over the magnetizing winding); \(f_c=50\ \text{Hz};\ \Delta_{\text{ins}}=1.0\ \text{mm}\).

From expression (41) it was obtained that \(\delta_{12}=8.0\ \text{mm}\); from expression (42), \(h_d=41\ \text{mm}\); from expression (40), \(X_y=39\ \Omega\).

\(\Theta_1\) was calculated from expression (32), taking expression (39) into account, for

\[ W_1=3000;\qquad K_c=0.88; \]

\[ Q_c=3.2\ \text{cm}^2;\quad \beta=6.28\ \frac{1}{\text{mm}}; \]

\[ a=38.1\ \text{A/m}. \]

\(X'_{\text{in}}\) was calculated from expression (37) for the corresponding values of \(I_0\).

\(k_\phi\) was calculated from expression (25).

The main stages in calculating the filtering coefficient are reflected in Table 1. For comparison, the corresponding experimental results from [1] are also given here—the values \(k_{\text{exp}}\).

Table 1

Conclusions

1. The possibility has been shown of making an approximate assessment of the capabilities of an active compensation filter by means of the calculated expressions obtained for the smoothing and filtering coefficients.

2. It is advisable to carry out a separate study of questions concerning the influence of leakage on the filter characteristics and the method for determining the inductive reactance of the fourth-harmonic doubler of the rectified current.

References

1. E. I. Goldshtein, A. I. Kutyavin. Active compensation filter for smoothing pulsations in the load circuit of a single-phase full-wave rectifier. Izvestiya TPI, vol. 155, 1966.

2. V. A. Rayushkin. Calculation of the no-load mode of a frequency doubler. Elektrichestvo, No. 5, 1959.

3. V. A. Rayushkin. Analytical calculation of the no-load and short-circuit modes of a frequency doubler, taking higher harmonics into account. Proceedings of the Kharkov Polytechnic Institute, vol. XXVII, Electrical Engineering Series, issue 1, 1959.

4. L. L. Rozhanskii. Static Electromagnetic Frequency Converters. Gosenergoizdat, 1959.

5. V. G. Tikhomirov. Study of the possibility of creating a frequency multiplier in the range 50–10000 Hz for induction heating, and a method for its calculation. Dissertation, LETI, 1953.

6. I. I. Belopol’skii, L. G. Pikalova. Calculation of Low-Power Transformers and Chokes. Gosenergoizdat, 1963.