OPTIMAL GEOMETRY OF A HIGH-VOLTAGE SMOOTHING CHOKE ON PL-TYPE CORES

E. I. GOLDSHTEIN, A. I. ANOKHIN, A. K. MAIER, V. K. SKRIPKO

(Presented by the seminar of the Department of TOE, TIRiET.)

High-voltage smoothing chokes (HVSCh) are intended for smoothing pulsations in the supply circuit of special radio-engineering devices. As a rule, HVSCh are made on PLV-type cores with an enlarged window [1], since, as the operating voltage increases, the winding insulation is reinforced. In the literature known to us, questions of optimization of HVSCh have not been considered, although the latter have considerable weight and dimensions. Let us formulate the problem of investigating an HVSCh in order to determine its geometry ensuring the minimum possible weight or overall volume. To simplify the analysis, we do not take into account the weight and dimensions of the insulating material (compound), fastening parts, and insulators. Then the weight and overall volume can be expressed by the following equations [2]

\[ G = a^{3}\cdot K_{0}\cdot \gamma_{0}(\beta\cdot K_{Vc}+K_{Vo}), \tag{1} \]

\[ V_{\mathrm{r}}=a^{3}\cdot 2\cdot K_{Vr}, \tag{2} \]

where

\(a\) is the principal standard size of the choke (Fig. 1),

\[ \beta=\frac{K_{c}\cdot \gamma_{c}}{K_{0}\cdot \gamma_{0}}, \]

\(K_{0}, K_{c}\) are the coefficients of filling the window with conductor material and the magnetic circuit with steel,

\(\gamma_{0}, \gamma_{c}\) are the specific weights of copper and steel,

\(K_{Vc}, K_{Vo}, K_{Vr}\) are dimensionless coefficients of the geometry of the HVSCh, determined by the relative dimensions \(x=\frac{b}{a}\); \(y=\frac{c}{a}\); \(z=\frac{h}{a}\) (Fig. 1). The values of the coefficients \(K_{Vc}, K_{Vo}, K_{Vr}\) for PL-type magnetic circuits are given in Table 1.

In engineering practice, two calculation cases for HVSCh are encountered most often:

I c.c. — calculation for a specified voltage drop across the choke,

II c.c. — calculation for a specified overheating.

Fig. 1

In accordance with this, the basic standard size can be determined from the following expressions [2]:

\[ a_{\mathrm{I}}=\left(\frac{N_{\mathrm{I}}}{n_{\mathrm{r}}}\right)^{0.2}, \]

\[ a_{\mathrm{II}}=\left(\frac{N_{\mathrm{II}}}{n_{\mathrm{r}}\cdot K_{\mathrm{gp}}}\right)^{1/7}, \tag{3} \]

where

\[ N_{\mathrm{I}}=\frac{(LI_0)^2\cdot \rho \cdot K_{\mathrm{nI}}}{R_{\mathrm{gp}}\cdot B_0^2\cdot K_c^2\cdot K_0}; \]

\[ N_{\mathrm{II}}=\frac{(LI_0)^2\cdot \rho \cdot K_{\mathrm{nII}}}{2(B_0)^2\cdot K_c^2\cdot K_0\cdot \sigma \cdot \tau}; \]

\(L\) — inductance of the THB,
\(I_0\) — direct-current component,
\(R_{\mathrm{gp}}\) — permissible active resistance of the THB,
\(\rho\) — resistivity of copper,
\(B_0\) — direct-current component of induction,
\(K_{\mathrm{nI}} = 1 + \alpha_{\mathrm{t}}(t_{\mathrm{okr}} - 20^\circ)\),
\(K_{\mathrm{nII}} = 1 + \alpha_{\mathrm{t}}(t_{\mathrm{okr}} + \tau - 20^\circ)\),
\(\alpha_{\mathrm{t}}\) — temperature coefficient of copper,
\(t_{\mathrm{okr}}\) — ambient temperature,
\(\tau\) — overheating temperature of the THB,
\(\sigma\) — heat-transfer coefficient,
\(n_{\mathrm{r}}\) — geometry coefficient, the expressions for which are given in Table 1,
\[ K_{\mathrm{gp}}=\frac{S_{\mathrm{ok.n}}}{2a^2} \]
— cooling-surface coefficient of the THB, the expressions for which are given in Table 1.

Substituting (3) into (2) and (1) and analyzing the expressions obtained, one may conclude that it is possible (when finding the optimum geometry) to operate with the following specific techno-economic indicators:

\[ G'_{\mathrm{I}}=(n_{\mathrm{r}})^{-0.6}(\beta K_{Vc}+K_{Vo}); \]

\[ G'_{\mathrm{II}}=(n_{\mathrm{r}}\cdot K_{\mathrm{gp}})^{-3/7}(\beta K_{Vc}+K_{Vo}); \tag{4} \]

\[ V'_{\mathrm{rI}}=(n_{\mathrm{r}})^{-0.6}\cdot K_{Vr}; \]

\[ V'_{\mathrm{rII}}=(n_{\mathrm{r}}\cdot K_{\mathrm{dp}})^{-3/7}\cdot K_{Vr}. \]

Table 1

To minimize expressions (4), a special program was compiled for a digital computer to search for the optimal geometry parameters that ensure the minimum of the corresponding techno-economic indicator. The search limits were taken as:

\[ x=\frac{в}{a}=1\div 5; \]

\[ y=\frac{c}{a}=1\div 10; \]

\[ z=\frac{h}{a}=1\div 10. \]

The minimization was carried out at values of the coefficient \(\beta=\dfrac{\gamma_c\cdot K_c}{\gamma_0\cdot K_0}\), calculated for \(K_c=0{,}85\), \(\gamma_c=7{,}65\ \text{g}/\text{cm}^3\), \(\gamma_0=8{,}8\ \text{g}/\text{cm}^3\), and \(K_0=0{,}246;\ 0{,}182;\ 0{,}146;\ 0{,}109;\ 0{,}082\). The window fill factors were determined experimentally for DC machines produced in series. The search step in \(x, y\), and \(z\) was taken to be the same: \(\Delta x=\Delta y=\Delta z=0{,}1\). The results of the minimization are given in Table 2.

Table 2

To clarify the question of the expediency and effectiveness, in some cases, of switching from the standard geometry, i.e., from the geometry parameters of the chokes of the “Henry P” series [1], to the optimal geometry, we shall compare the specific technical-and-economic indicators (4) of the chokes of the “Henry P” series and of the optimal geometry. The comparison is carried out under identical conditions. In doing so, we assume, to a first approximation, that the geometry does not affect the heat-transfer coefficient, the magnetic regime, or the filling factors. The comparison results are given in Table 3. The estimate of the gain in weight and in overall volume was calculated respectively as \(\dfrac{G'_{\mathrm{st}}}{G'_{\mathrm{opt}}}\cdot 100\%\) and \(\dfrac{V'_{\mathrm{g.st}}}{V'_{\mathrm{g.opt}}}\cdot 100\%\), with the specific indicators of the optimal chokes taken as 100%.

Table 3

Conclusion

As shown by the analysis of the comparison results (Table 3), the use of a transformer of optimal geometry makes it possible to reduce its overall volume by a factor of 1.6–3.5 and to obtain a weight saving of up to 80%.

References

1. Radio-Industry Standard (draft). High-voltage filter chokes. Technical specifications. Revision 2-67.
2. E. I. Goldshtein. Some questions in the design of optimal smoothing chokes. Elektromekhanika, 1964, No. 4.