EXTENDING THE OPERATING RANGE OF INDUCTIVE DIVIDERS

A. I. Kramnyuk, M. S. Roitman

(Submitted by the scientific seminar of the Department of Radio Engineering)

In [1] it is shown that the frequency error of inductive dividers (ID) increases sharply as the frequency is raised. Although this error is systematic in character and can be allowed for, the inconvenience of such a possibility is obvious.

The present work is devoted to consideration of one of the possibilities for extending the operating frequency range while retaining high accuracy of division by an ID.

In [1] it is shown that the appearance of frequency error is associated with the presence of reactances, in particular, distributed capacitances and leakage inductances. For the multifilar winding used in an ID, the values of the leakage inductances and active resistances of the individual windings may be regarded as approximately equal, but the effect of the distributed capacitances on the individual windings will be different.

To clarify the influence of distributed capacitances, let us represent the equivalent circuit of the divider, replacing in it the distributed elements by lumped ones. As an example, consider a divider with a division ratio of four. In [1] the values of the resistances in the equivalent circuit of the divider are determined as

\[ (n-1)\dot{Z}_1=\frac{a_0}{\left(1+\frac{n}{6}\Delta\right)2}\simeq\frac{a_0}{2}; \tag{1} \]

\[ \dot{Z}_2=\frac{b_0}{1+\frac{n}{6}\Delta}\simeq b_0. \tag{2} \]

Taking (1), (2) into account, the divider circuit may be represented as in Fig. 1, where

\[ a_0=r_0+j\omega L'_{s0}\quad\text{is the equivalent resistance of one winding;} \tag{3} \]

\[ b_0=\frac{1}{j\omega C_{0g}}\quad\text{is the resistance determined by the value} \tag{4} \]

of the distributed capacitance between windings [1].

We shall regard the circuit formed by the resistances \(a_0\) as the main one, and the circuits formed by the resistances \(b_0\) as auxiliary ones.

Let us represent the auxiliary circuits by the sum of as many equal resistances as each of them encompasses of the elements of the main circuit (Fig. 2). For example, the auxiliary circuits connected between points 2—4, 4—6, 6—8 are replaced by two series-connected resistances equal to \(b_0/2\); the circuits 4—8, 2—6 by four resistances, and so on. Taking into account that the conductances of the auxiliary circuits are considerably smaller than the conductance of the main one, the potentials of points \(3, 3^{\mathrm{I}}, 3^{\mathrm{II}}, 3^{\mathrm{III}}\) may be regarded as approximately equal to one another. The same may also be noted with respect to points \(4, 4^{\mathrm{I}}, 4^{\mathrm{II}}, 4^{\mathrm{III}}, 4^{\mathrm{IV}}\), etc.

Fig. 1.

Fig. 2.

Let us connect the points having identical potentials. In this case, in parallel with the resistances of the main branch there are connected the resistances of the auxiliary branch divided by the corresponding coefficient.

Taking into account that \(b_0 = \dfrac{1}{j\omega C_{0g}}\), and summing all the capacitances connected to each of the branches, we obtain the equivalent circuit (Fig. 3a). The inconvenience of this circuit is that the equivalent capacitances are connected to halves of the windings.

If one half of a winding is shunted by a capacitance equal to \(mC_0\), and the other half by \(nC_0\), then the value of the equivalent capacitance, as is easily shown, will be equal to

\[ C_{\mathrm{eq}}=\frac{m+n}{4}C_{0g}. \tag{5} \]

Taking (5) into account, we obtain the equivalent circuit of the divider (Fig. 3b).

As is evident from the analysis presented and from Fig. 3b, the values of the equivalent capacitances referred to the various windings are not the same,

which is also the cause of the appearance of frequency error. To reduce it, it is necessary to equalize the capacitance values at each of the windings. For this purpose, additional correcting capacitances equal to \(4C_{0g}\) must be connected in parallel with the end windings of the divider (Fig. 3б).

Fig. 3.

Fig. 4.

The conversion coefficients for finding the equivalent capacitances of a divider with other division ratios are given in Fig. 4; they were calculated for the static value of the distributed capacitance between windings [1].

Experimental verification showed that, with the aid of correcting capacitances, the operating range of an inductive divider can be extended to frequencies of \(20 \div 50\) kHz with a frequency error of no more than 0.05%. By careful selection of the correcting capacitances, the value of the frequency error can be reduced.

Conclusions

The calculated values of the equivalent capacitances connected to the individual windings make it possible to reduce significantly the frequency error of inductive dividers and thereby extend the operating range.

References

1. A. I. Krainyuk, M. S. Roitman. A method for analyzing the frequency error of an inductive divider. This collection.