PROCEEDINGS
OF THE TOMSK ORDER OF THE OCTOBER REVOLUTION AND ORDER
OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. Kirov

Vol. 228 1974

TOWARD THE DEVELOPMENT OF AN ENGINEERING METHOD FOR CALCULATING SMOOTHING CHOKES FOR A SPECIFIED VOLTAGE DROP

E. I. Goldshtein, A. K. Maier

(Presented by the scientific and technical seminar of the Department
of Instruments and Devices of Automation Systems)

Existing methods for calculating smoothing chokes (s. ch.) have a number of shortcomings. Thus, in [4] only a method is given for calculating a s. ch. for a specified overheating. However, in engineering practice the problem quite often arises of calculating a s. ch. for a specified efficiency or for a specified voltage drop. Such a problem can be solved using the method for calculating a s. ch. [5]. But this path is rather laborious, since the problem, in essence, is solved by the method of successive approximations.

Let us set the task of developing an engineering method for calculating a s. ch. for a specified voltage drop.

For calculating a s. ch. we shall use formulas obtained earlier by one of the authors [1, 2, 3].

a) The basic dimension \(a\) is determined by formula (10) from [1]:

\[ a=\left(\frac{N}{n_{\Gamma}}\right)^{0.2}, \tag{1} \]

where

\[ N=\frac{L^{2} I_{0}^{2}\cdot \rho \cdot K_{\mathrm{n}}}{R_{\mathrm{dop}}\cdot B_{0}^{2}\cdot K_{\mathrm{c}}^{2}\cdot K_{0}} =\frac{Q_{\mathrm{c}}^{2}\cdot Q_{0}}{l_{0}}; \tag{2} \]

\(K_{\mathrm{n}}=1+\alpha_{\mathrm{T}}(t_{\mathrm{n}}-20^\circ)\) is the coefficient accounting for the increase in the active resistance of copper during heating;
\(\alpha_{\mathrm{T}}\) is the temperature coefficient of copper;
\(t_{\mathrm{n}}\) is the heating temperature of the choke,
\(L\) is the inductance of the choke,
\(I_{0}\) is the dc component of the rectified current,
\(\rho\) is the specific electrical resistivity of copper,
\(R_{\mathrm{dop}}\) is the permissible resistance of the choke in operating mode,
\(B_{0}\) is the dc component of induction,
\(K_{\mathrm{c}}, K_{0}\) are the fill factors of the magnetic core and window,
\(n_{\Gamma}\) is the geometry coefficient,
\(Q_{\mathrm{c}}\) is the cross section of the magnetic core,
\(Q_{0}\) is the area of the magnetic-core window,
\(l_{0}\) is the mean length of a turn.

When calculating chokes made on normalized core...

in the windings, the standard size can be selected according to the value \(N\) calculated from equation (2). In this case, for the selected standard size the quantity

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

must be no less than the calculated value.

For standardized cores of the ShL, PL, and OL types, tables with values of \(N\) were calculated, making it possible to select the necessary standard size that provides the required electromagnetic parameters (see the article in this same collection).

b) When calculating the permissible active resistance of the choke, one can use the well-known expression

\[ R_{\mathrm{perm}}=W^2\cdot \rho\cdot K_{\mathrm{n}}\cdot \frac{l_0}{Q_0\cdot K_0}. \tag{3} \]

From the last expression one can determine the necessary number of turns

\[ W=\sqrt{\frac{R_{\mathrm{perm}}\cdot Q_0\cdot K_0}{\rho\cdot l_0\cdot K_{\mathrm{n}}}}. \tag{4} \]

c) When determining the magnitude of the nonmagnetic gap, one may use the method of [2, 7].

1. Determine

\[ \frac{\delta'}{K_{\phi}}=K_{\delta}'\cdot IW_0, \tag{5} \]

where

\[ \delta'=\frac{\delta}{l_c} \]
is the relative gap,

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

\(\delta\) is the magnitude of the gap,

\[ IW_0=\frac{I_0\cdot W}{l_c} \]
is the specific ampere-turns,

\(K_{\phi}\) is the fictitious-gap coefficient;

\(K_{\delta}'\) is a coefficient whose values are given in Table 1.

Table 1

1. From the dependences \(\dfrac{\delta'}{K_{\phi}}=f(\delta')\), determine the relative gap \(\delta'\).

2. Calculate the magnitude of the nonmagnetic gap

\[ \delta=\delta'\cdot l_c. \tag{6} \]

When calculating a smoothing choke for a specified voltage drop, the following must be specified: the inductance of the smoothing choke \(L\), the constant component of the rectified current \(I_0\), the maximum voltage drop across the choke \(\Delta U\), the ripple frequency \(f\), and the maximum heating temperature of the smoothing choke \(t\).

Calculation Procedure

1. For design, technological, or dimensional considerations, the type of magnetic core is selected.

2. The material for the choke core and winding is selected.

3. The permissible resistance of the choke at the maximum heating temperature is determined:

\[ R_{\text{perm}}=\frac{\Delta U_{\phi}}{I_0}. \]

1. The power expended on heating the smoothing choke is calculated:

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

1. From the tables in [4], the values of the coefficients \(K_0\) and \(K_c\), and the current density \(\Delta\), are selected.

2. From Table 1, the value \(B_0\) is selected. As a first approximation, for electrical steels of type Э-310 one may take \(B_0=0.8\) T.

3. The design coefficient \(N\) is calculated by formula (2).

4. According to the calculated value \(N\), from Tables 1, 2, 3 (see the same collection) the standard size of the smoothing choke is selected for which the value of the coefficient

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

is not less than the calculated value.

1. The number of turns of the winding is determined from expression (4).

2. The design cross section of the conductor is determined,

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

and, from the tables in [6], a conductor is selected from the condition \(S_{\text{pr}}\leq S_{\text{pr.r}}\).

1. The actual current density is determined,

\[ \Delta=\frac{I_0}{S_{\text{pr}}}, \]

and compared with that recommended in [4].

1. The value of the optimum nonmagnetic gap is determined:

a) the value

\[ \frac{\delta'_{\text{opt}}}{K_{\phi}} \]

is calculated from expression (5);

b) from the dependences

\[ \frac{\delta'_{\text{opt}}}{K_{\phi}}=f(\delta) \]

\(\delta'_{\text{opt}}\) is determined;

c) the value of the optimum nonmagnetic gap is determined from expression (6).

1. The technical and economic indicators of the smoothing choke are calculated—the weight, the volume of active materials, the overall volume, and the cost of the active materials:

\[ G=a^3\left(K_{Vc}\cdot \gamma_c\cdot K_c+K_{V0}\cdot \gamma_0\cdot K_0\right), \]

\[ V=a^3\left(K_{Vc}+K_{V0}\right), \]

\[ V_{\Gamma}=a^3\cdot 2K_{V\Gamma}, \]

\[ Ц=a^3\left(K_{Vc}\cdot \alpha_c+K_{V0}\cdot \alpha_0\right), \]

where

\(K_{Vc},\ K_{V0},\ K_{V\Gamma}\) are dimensionless volume coefficients, the expressions for which are given in the table of work [1];

\(\gamma_0,\ \gamma_c\) are the specific weights of copper and steel;

\(\alpha_0,\ \alpha_c\) are the specific costs of copper and steel.

When calculating a smoothing choke designed on an arbitrary (optimal) core, in addition to the listed initial data, there must be specified—

the goal of minimization (weight, volume of active materials, overall volume, cost of active materials).

A special feature of calculating the smoothing choke in this case is the use of chokes of optimal geometry. The optimal relative dimensions of the choke, \(x=\dfrac{b}{a}\), \(y=\dfrac{c}{a}\), \(z=\dfrac{h}{a}\), are determined from the tables in [1] for preliminarily calculated coefficients indicating the optimization objective:

\[ \beta_G=\frac{\gamma_c\cdot K_c}{\gamma_0\cdot K_0}; \qquad \beta_V=1; \qquad \beta_{\mathrm{c}}=\frac{\gamma_c\cdot K_c\cdot \alpha_c}{\gamma_0\cdot K_0\cdot \alpha_0}. \]

The principal standard size \(a\) is determined from expression (1).

Using the proposed method, a smoothing choke was calculated on a standardized core (example No. 1) and on an arbitrary (optimal) core (example No. 2), ensuring the minimum overall volume of the choke.

The calculation results are summarized in Table 2.

Table 2

In all examples the following were adopted:

\[ t_{\mathrm{n}}=105^\circ; \quad K_{\mathrm{n}}=1.334; \quad \Delta U_{\mathrm{f}}=0.5\ \mathrm{V}; \quad \rho=1.72\cdot 10^{-8}\ \Omega\cdot \mathrm{m}; \]

\[ \gamma_c=7.65\ \mathrm{g/cm^3}; \quad \gamma_0=8.8\ \mathrm{g/cm^3}; \quad \alpha_c=0.0032\ \mathrm{rub/cm^3}; \]

\[ \alpha_0=0.01\ \mathrm{rub/cm^3}; \quad B_0=1\ \mathrm{T}; \quad \Delta=3.5\ \mathrm{A/mm^2}; \]

magnetic-core material: steel E-340.

Conclusions

1. An engineering method has been developed for calculating a smoothing choke for a specified voltage drop, taking into account heating of the choke.

2. Examples of calculating smoothing chokes show that, in order to reduce technical and economic indicators, it is reasonable to use arbitrary (optimal) geometry.

References

1. E. I. Goldshtein. Some Problems in Designing Optimal Smoothing Chokes. Proceedings of Higher Educational Institutions. Electromechanics, No. 4, 1964.

2. E. I. Goldshtein, L. V. Traut. On the Engineering Calculation of the Main Parameters of Electromagnetic Devices on Standardized Shell-Type Cores with Gaps. Proceedings of Higher Educational Institutions. Electromechanics, No. 4, 1968.

3. E. I. Goldshtein. Some Problems in the Design and Calculation of Smoothing Chokes. Dissertation, Tomsk, 1964.

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

1. A. M. Bamdas, Yu. A. Savinovsky. Filter Chokes for Radio Equipment, “Sov. radio,” 1962.

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

3. L. V. Traut, E. I. Goldshtein. Toward the engineering calculation of the basic parameters of electromagnetic devices on standardized rod cores with gaps. Izv. TPI, vol. 179, 1969.