OPTIMIZATION OF A T-SHAPED SMOOTHING \(LCL\) FILTER FOR A RECTIFIER

E. I. GOLDSTEIN, A. K. MAYER, M. S. OSTROAST

(Submitted by the scientific-technical seminar of the Department of Instruments and Devices of Automatic-Control Systems)

In the literature known to us there is no method for calculating a T-shaped smoothing \(LCL\) filter (Fig. 1) that is optimal with respect to a specified techno-economic index (volume of active materials, overall volume, weight, cost).

Such a filter can be calculated by specifying various values of \(L_1\), \(L_2\), \(C\), which ensure the prescribed smoothing coefficient, and by computing in each case the techno-economic index of interest to us; after several calculations one may stop at the most acceptable one. However, this method of calculation is rather lengthy (the method of successive approximations), and, moreover, there will never be confidence that the result of the last approximation is actually optimal. Let us pose the problem of obtaining an analytical expression that makes it possible to choose the optimal ratio among \(L_1\), \(L_2\), and \(C\), for which a specified techno-economic index (in particular, the overall volume of the filter) is minimal. The smoothing coefficient of the filter (Fig. 1) is determined by the equation

Fig. 1. T-shaped smoothing \(LCL\) filter.

\[ K_{\mathrm{sm}} = K_{\mathrm{sm}L1C}\cdot K_{\mathrm{sm}L2}. \tag{1} \]

For

\[ \omega L \gg R_{\mathrm{dr}}, \qquad K_{\mathrm{sm}L1C}=\omega^2 L_1 C; \tag{2} \]

\[ K_{\mathrm{sm}L2}=\frac{\omega L_2}{R_{\mathrm{dr}2}+R_{\mathrm{n}}}. \tag{3} \]

Substituting from (2) and (3) into (1), we obtain

\[ K_{\mathrm{sm}}=\frac{\omega^3 L_1 L_2 \cdot C}{R_{\mathrm{n}}+R_{\mathrm{dr}2}}. \tag{4} \]

Let us denote

\[ \Phi=L_1\cdot L_2\cdot C=\frac{K_{\mathrm{sm}}(R_{\mathrm{n}}+R_{\mathrm{dr}2})}{\omega^3}; \tag{5} \]

\[ X=\frac{L_1L_2}{C}; \qquad Y=\frac{L_1}{L_2}; \tag{6} \]

then

\[ L_1L_2=X\sqrt{\frac{\Phi}{X}}=\sqrt{\Phi\cdot X}; \tag{7} \]

\[ L_1=\sqrt[4]{\Phi\cdot X\cdot Y^2}; \tag{8} \]

\[ L_2=\frac{L_1}{Y}=\sqrt[4]{\frac{\Phi\cdot X}{Y^2}}; \tag{9} \]

\[ C=\sqrt{\frac{\Phi}{X}}. \tag{10} \]

The overall volume of the filter (Fig. 1) consists of the overall volumes of the chokes \(L_1\) and \(L_2\) and the capacitor \(C\).

\[ V_\Gamma=V_{\Gamma L1}+V_{\Gamma L2}+V_{\Gamma C}. \tag{11} \]

Using the expression for overall volume from [1], we write the overall volume of the choke \(L_1\)

\[ V_{\Gamma L1}=\left(\frac{N}{n_\Gamma}\right)^{0.6}\cdot 2K_{V\Gamma} =L_1^{1.2}\cdot K_{L1V\Gamma}, \tag{12} \]

where

\[ K_{L1V\Gamma}=\left(\frac{N'}{n_\Gamma}\right)^{0.6}\cdot 2K_{V\Gamma}; \tag{13} \]

\[ N'=\frac{I_0^2\cdot \rho\cdot K_{\mathrm{n}}}{R_{\mathrm{dr}}\cdot B_0^2\cdot K_c^2\cdot K_0}. \tag{14} \]

\(K_{\mathrm{n}}=1+\alpha_{\mathrm{t}}(t_{\mathrm{n}}-20^\circ)\) is the coefficient allowing for the increase in choke resistance under heating; \(\alpha_{\mathrm{t}}\) is the temperature coefficient of copper; \(t_{\mathrm{n}}\) is the choke heating temperature; \(n_\Gamma\) is the choke geometry coefficient; \(K_{V\Gamma}\) is the coefficient of the choke overall volume; \(B_0\) is the dc component of induction in the steel; \(K_0\) and \(K_c\) are the filling factors of the magnetic core and the window.

By analogy with (12), the overall volume of the choke \(L_2\) can be written as

\[ V_{\Gamma L2}=L_2^{1.2}\cdot K_{L2V\Gamma}. \tag{15} \]

The overall volume of the capacitor can be written as

\[ V_{\Gamma C}=\gamma_{V\Gamma}\cdot C, \tag{16} \]

where \(\gamma_{V\Gamma}\) is the specific overall volume of the capacitor.

Solving (11), (12), (15), (16), (8), (9), and (10) simultaneously, we obtain

\[ V_\Gamma = Y^{0.6}\cdot X^{0.3}\cdot \Phi^{0.3}\cdot K_{L1V\Gamma} + Y^{-0.6}\cdot X^{0.3}\cdot \Phi^{0.3}\cdot K_{L2V\Gamma} + X^{0.5}\cdot \Phi^{0.5}\cdot \gamma_{V\Gamma}. \tag{17} \]

To find the extremal values of \(X\) and \(Y\), it is necessary to take the partial derivatives with respect to \(X\) and \(Y\) of function (17), set them equal to zero, and solve the resulting system of two equations for the two unknowns.

\[ \frac{\partial V_{\Gamma}}{\partial X} =0.3\,Y^{0.6}\cdot \Phi^{0.3}\cdot K_{L1V\Gamma}\cdot X^{-0.7} +0.3\,Y^{-0.6}\cdot \Phi^{0.3}\cdot K_{L2V\Gamma}\cdot X^{-0.7} - \]
\[ -0.5\cdot \Phi^{0.5}\cdot \gamma_{V\Gamma}\cdot X^{-1.5}=0; \tag{18} \]

\[ \frac{\partial V_{\Gamma}}{\partial Y} =0.6\cdot \Phi^{0.3}\cdot X^{0.3}\cdot K_{L1V\Gamma}\cdot Y^{-0.4} -0.6\Phi^{0.3}\cdot X^{0.3}\cdot K_{L2V\Gamma}\cdot Y^{-1.6}=0. \tag{19} \]

\[ Y^{1.2}= \frac{0.6\cdot \Phi^{0.3}\cdot X^{0.3}\cdot K_{L1V\Gamma}} {0.6\cdot \Phi^{0.3}\cdot X^{0.3}\cdot K_{L2V\Gamma}}. \tag{20} \]

From the standpoint of the dielectric strength of the choke windings, it is advisable to distribute the voltage drop across the filter (Fig. 1) uniformly between the two chokes; i.e., the active resistances of the chokes must be equal. If it is also assumed that the designs of chokes \(L_1\) and \(L_2\) are identical, then

\[ K_{L1V\Gamma}=K_{L2V\Gamma}=K_{LV\Gamma}. \tag{21} \]

Then

\[ Y=1. \tag{22} \]

Substituting (21) and (22) into (19), we obtain

\[ X^{0.8}= \frac{0.5\cdot \gamma_{V\Gamma}\cdot \Phi^{0.5}} {\Phi^{0.3}\cdot K_{LV\Gamma}\cdot 0.3(1^{0.6}+1^{-0.6})} = 0.834\cdot \Phi^{0.2}\frac{\gamma_{V\Gamma}}{K_{LV\Gamma}}; \]

\[ X=\left[0.834\cdot \Phi^{0.2}\cdot \frac{\gamma_{V\Gamma}}{K_{LV\Gamma}}\right]^{1.25}. \tag{23} \]

Additional investigations show that, for \(L_1\), \(L_2\), and \(C\) determined by (15), (16), (17), in which \(Y\) and \(X\) are calculated from (22) and (23), the overall volume of the filter (Fig. 1) is indeed the minimum possible.

Thus, in order to obtain the minimum overall volume of the filter (Fig. 1), two conditions must be satisfied:

1. Chokes \(L_1\) and \(L_2\) must be identical, i.e., \(L_1=L_2\).
2. The values of the inductances and capacitance must be determined with \(X\) calculated from (23).

When designing a filter (Fig. 1) optimized with respect to overall volume, the following sequence may be followed:

1. The magnetic-core design is selected for chokes \(L_1\) and \(L_2\), and mean values of the relative dimensions are specified: \(x=\frac{b}{a}\); \(y=\frac{c}{a}\); \(z=\frac{h}{a}\). When chokes of arbitrary (optimal) geometry are used, the values of the relative dimensions for which the overall volume of the choke is minimal are selected from table [1].
2. The magnetic-core material is selected.
3. The type of capacitor is selected according to the permissible temperature, permissible ripple voltage, frequency, dimensions, and other parameters.
4. The dimensional specific volume of the selected capacitor type of the nominal rating used is determined:

\[ \gamma_{V\Gamma\mathrm{nom}}=\frac{V_{\Gamma\mathrm{nom}}}{C_{\mathrm{nom}}}. \]

Table 1

NOTE: *) a bronze choke with a reduced high-frequency window according to NO 666000.

Table 2

Taking into account the reduction of capacitance due to a decrease in operating temperature and an increase in frequency, the specific overall volume of the capacitor will increase.

\[ \gamma_{V\Gamma}=\frac{\gamma_{V\Gamma\text{ nom}}}{\dfrac{C_{\text{fact}}}{C_{\text{nom}}}} =\frac{\gamma_{V\Gamma\text{ nom}}}{K}. \]

1. The power of the smoothing choke is calculated (with respect to heating)

\[ P=\Delta U_{\text{dr}}\cdot I_0=I_0^2\cdot R_{\text{dr}}. \]

1. From the tables [2] the coefficients \(K_c\) and \(K_0\) are selected, and from [3] the value \(B_0\). As a first approximation for steel of type E-310, \(B_0=0.8\ \text{T}\) may be used.

2. The coefficient \(N'\) is calculated from (4).

3. For the selected magnetic-core design, the coefficients \(n_\Gamma\) and \(K_{V\Gamma}\) are calculated [1].

4. The coefficient \(K_{LV\Gamma}\) is calculated from (13).

5. The quantity \(\Phi\) is calculated from (5).

6. The quantity \(X\) is calculated from (23).

7. The optimal value of the capacitance is determined from (17).

Taking into account the decrease in capacitance with decreasing temperature and at elevated frequency, it is necessary to use a capacitance larger than the calculated value.

\[ C_{\text{fact}}=\frac{C_{\text{calc}}}{K}, \]

where \(K<1\) (determined by the type of capacitor, the operating temperature, and the frequency).

The obtained actual value of the capacitance is rounded to an integer value realizable by the adopted capacitor rating.

1. The optimal values of the inductance at the lowest operating temperature are determined.

\[ L=L_1=L_2=\sqrt{\frac{\Phi}{C_{\text{fact}}\cdot K}} =\sqrt{C_{\text{fact}}\cdot K\cdot X}. \]

1. The smoothing coefficient provided by the \(L_1C\)-filter at the highest operating temperature is determined.

\[ K_{\text{sm}\,L1C}=\omega^2 L_1C. \]

1. The amplitude of the ripple on the capacitance is determined,

\[ U_{C\sim}=\frac{U_{\sim \text{in}}}{K_{\text{sm}\,L1C}}, \]

and compared with the permissible value.

1. The coefficient \(N\) is calculated:

\[ N=\frac{Q_c^2\cdot Q_0}{l_0}=N'\cdot L^2, \]

where

\[ \begin{aligned} Q_c&\text{ — cross section of the magnetic-core steel;}\\ Q_0&\text{ — window area of the magnetic core;}\\ l_0&\text{ — mean length of a turn.} \end{aligned} \]

1. A magnetic core is selected for which

\[ N=\frac{Q_c^2\cdot Q_0}{l_Q} \]

is not less than the calculated value.

1. Determine the number of turns of the choke:

\[ W=\sqrt{\frac{R_{\mathrm{pr}}\cdot Q_0\cdot K_0}{\rho\cdot l_0\cdot K_{\mathrm{n}}}}. \]

1. Determine the calculated wire cross section

\[ S_{\mathrm{pr.r}}=\frac{Q_0\cdot K_0}{W} \]

and, from tables [4], select the wire from the condition \(S_{\mathrm{pr}}<S_{\mathrm{pr.r}}\).

1. Determine the actual current density

\[ \Delta=\frac{I_0}{S_{\mathrm{pr}}} \]

and compare it with the value recommended in [2].

1. Determine the magnitude of the nonmagnetic gap

\[ \delta=K'_{\delta}\cdot I_0\cdot W. \]

Recommended values of \(K'_{\delta}\) are given in [3]. As a first approximation, for steel of type E-310 one may use

\[ K'_{\delta}=1.6\cdot 10^{-6}\ \frac{m}{a\cdot \text{turn}}. \]

1. Check whether the filter reactance is inductive. In order for the reactance of the filter (Fig. 1) to be inductive, the inequality

\[ \omega L> \frac{ \frac{1}{\omega C}(R_{\mathrm{pr}}+R_{\mathrm{n}})^2+\frac{L}{C}\left(\omega L-\frac{1}{\omega C}\right) }{ \left(\omega L-\frac{1}{\omega C}\right)^2+(R_{\mathrm{pr}}+R_{\mathrm{n}})^2 }. \]

must be satisfied.

1. Determine the overall volume of the filter

\[ V_{\Gamma}=2a^3\cdot 2K_{V\Gamma}+C_{\mathrm{fakt}}\cdot K\cdot \gamma_{V\Gamma}. \]

Using the proposed method, a series of filters for currents of 20 and 100 A was calculated. The calculation results are summarized in Table 1. In all examples the following were assumed:

\[ f=800\ \text{Hz},\quad t_{\mathrm{n}}=105^\circ,\quad K_{\mathrm{n}}=1.334,\quad \Delta U_{\mathrm{f}}=0.5\ \text{V}, \]
\[ \rho=1.72\cdot 10^{-8}\ \Omega\cdot m;\quad B_0=1\ \text{T},\quad \Delta=3.5\ a/mm^2; \]
\[ K'_{\delta}=1.5\cdot 10^{-6}\ m/\text{ampere-turn};\quad \text{capacitor type K-50-3A}; \]

magnetic-core material—steel E-340.

In the calculations, the possibility was considered of using, for the filters, chokes with arbitrary (optimal) geometry \((x=2.7;\ y=0.5;\ z=1.5)\), providing the minimum overall volume of the choke (Example No. 3), and chokes on standard cores with a reduced window height (Example No. 2).

For comparison, a filter (Fig. 1) was calculated by the generally accepted method, when it is unknown what the ratio between \(L_1, L_2\) and \(C\) should be (Example No. 5); in this case the smoothing coefficients of the \(L_1C\)- and \(L_2\)-sections were taken to be equal.

From the calculation results it is evident that the filter calculated by the proposed method (Example No. 4) has an overall volume 2 liters smaller than the filter calculated by the generally accepted method with arbitrary choice of the ratios between the parameters of the sections (Example No. 5).

It should be noted that, for rectifiers operating into a low-resistance load, the use of the filter (Fig. 1) is more appropriate than that of an inductive filter. This is confirmed by the results of calculations for two inductive filters (Table 2), which provide the same smoothing factors as in Examples 1–5.

Conclusions

1. When chokes with a geometry that provides the minimum overall volume are used, the gain in the overall volume of the filter is only 30% compared with Example No. 1 (a standardized choke is used), since the very method for calculating a filter optimized with respect to overall volume takes into account the choice of the geometry of a standardized choke for which the overall volume of the filter is minimal.

2. When standardized chokes with a reduced window height (NO 666000) are used, the gain in the overall volume of the filter compared with Example No. 1 is 25%.

3. The method presented for calculating a filter optimized with respect to overall volume can also be applied when optimizing filters according to other techno-economic indicators (volume, weight, cost).

REFERENCES

1. E. I. Goldshtein. Some problems in the design of optimal smoothing chokes. Izv. vuzov, Elektromekhanika, No. 4, 1964.

2. I. I. Belopolsky, L. G. Pikalova. Calculation of Low-Power Transformers and Chokes. GEI, 1963.

3. E. I. Goldshtein. Dissertation, Tomsk, 1964.

4. Electrical Cables, Wires, and Cords. State Standards of the USSR, 1964.