Capacitance Evaluation

In high-frequency transformers, the capacitances differ from those in audio transformers in that the windings are usually single layers, whose turn-to-turn capacitance is negligible compared to capacitance between windings and to the core. For example, in the transformer in Fig. 166, the primary and secondary are each wound in a single layer concentrically, in the same rotational direction, and in the same traverse direction (right to left).

Fig. 166. Single-layer windings.

It will be assumed that the right ends of both windings are connected to ground (or core) through large capacitances, as shown dotted, so that the right ends are at substantially the same a-c potential. Primary capacitance C₁ is composed of many small incremental capacitances C_p, and secondary C₂ of many small incremental capacitances C_S , each of which has a different voltage across it. Likewise, many small incremental capacitances C_a exist between primary and secondary, which have different potentials across them. If the transformer is step-up,

[97]

If the transformer is step-down,

[98]

If the ratio is 1:1,

[99]

For transformers with opposite angular rotations of primary and secondary windings, or with opposite traverse directions (but not both), minus signs in the factors (N_p - N_S)²/N_P² and (N_s - N_p)²/N_s² in equations 97 and 98 become positive; there is no other change. For transformers with both angular rotations and traverse directions opposite there is no change at all in these equations. If there is a shield between primary and secondary, omit terms containing C_a in these equations, and make ∑C_S and ∑C_P include the capacitance of secondary and primary to shield, respectively. ∑C_S is the measurable capacitance of the short-circuited secondary to core, and ∑C_P that of the short-circuited primary to core.

In push-pull amplifier transformers, the secondary winding is interleaved between two primary halves. The rotational directions of winding and traverse are important, as they affect not only effective capacitance but also plate-to-plate coupling. It is usually best to have all windings with the same rotational direction and traverse, and to connect the primary halves externally.

Winding resistance increases with frequency because of eddy currents in the larger wire sizes, and copper loss increases proportionately. Formulas for single-layer coils are given in handbooks.⁽¹⁾ Eddy-current resistance of layer-wound coils in deep open slots is plotted in Fig. 167 as a function of conductor thickness, frequency, and number of coil layers; it approximates the increase of winding resistance in a transformer.⁽²⁾

Fig. 167. Increase in coil resistance at high frequencies.

(1)	See Natl. Bur. Standards Circ. 74, p. 304.
(2)	See "Eddy-Current Resistance of Multilayer Coils," by T. H. Long, Trans. AIEE, 64, 716 (October, 1945).