High-Frequency Response

The factors that influence the high-frequency response of a transformer are leakage inductance, winding capacitance, source impedance, and load impedance. Hence a new equivalent diagram, Fig. 107(d), is necessary for the high-frequency end. Winding resistances are omitted or combined as in Fig. 107(b). Winding capacitances are shown across the windings. If primary and secondary leakage inductances and capacitances are combined, X_N is omitted as if it were infinitely large, and a² is dropped as before, the circuit becomes that shown in Fig. 107(e). X_L is the leakage reactance of both windings, X_c the capacitive reactance of both windings, and R₂ the load resistance, all referred to the primary side on a 1:1 turns ratio basis.

At any frequency, the equivalent voltage ratio in the circuit of Fig. 107(e) can be found by the ratio of impedances, as for the low-frequency response. The scalar value is

[63]

In equation 63 the term X_L/X_C may be written 4π²f²LC = f²/f_r², where 1/(2π√(LC)) = f_r , the resonance frequency of the leakage inductance and winding capacitance, considered as lumped and without resistance. Also X_L/R₂ = X_Cf²/R₂f_r². Assign to the ratio X_c/R₁ a value B at frequency f_r. Then at any frequency f, X_c/R₁ = Bf_r/f. In the three cases considered at the low frequencies,

R₂ = R₁,

[63a]

R₂ = 2R₁,

[63b]

R₂ = ∞,

[63c]

Equations 63a, b, and c are plotted in Figs. 109, 110, and 111. If X_c/R₁ has certain values at frequency f_r, the frequency characteristic is relatively flat up to frequencies approaching f_r. In particular, performance is good at B = 1.0 in all three figures.

Fig. 109. Transformer characteristics at high frequencies (line matching).

When leakage inductance and winding capacitance are regarded as "lumped" quantities, current distribution in the windings is assumed to be uniform throughout the range of frequencies considered. As shown in R-F Chokes, this assumption is valid up to the resonance frequency f_r. At frequencies higher than f_r, there may be appreciable error in Figs. 109, 110, and 111. But good frequency characteristics lie mainly below the frequency f_r, where the curves are correct within the assumed limits.

Fig. 110. Transformer characteristics at high frequencies (triode output).

To use these curves in design work, choose the most desirable

characteristic curve and, from a knowledge of the source impedance, find the proper value of capacitive reactance X_c at frequency f_r. The value of f_r should be such that the highest frequency to be covered lies on the flat part of the curve. X_o and f_r determine the values of winding capacitance and leakage inductance that must not be exceeded in order to give the required performance.

In Fig. 107(e) the capacitance is shown across the load. This is correct if the main body of capacitance is greater on the secondary than on the primary side. Normally this is true if the secondary winding has the greater number of turns. Figures 109, 110, and 111 are thus plotted specifically for step-up transformers.

Fig. 111. Transformer characteristics at high frequencies (class A grid).

Modifications are necessary for step-down transformers, the equivalent circuit for which is shown in Fig. 113. Analysis shows the scalar voltage ratio to be

[64]

Notice the similarity to equation 63. In fact, if R₁ = R₂, equation 64 reduces to equation 63; for this case the response is the same for step-down and step-up transformers, and is given by Fig. 109.

For R₂ = 2R₁ ; equation 64 becomes, after substitution in terms of frequency,

[65]

which is plotted in Fig. 113. Non-uniform response comes at somewhat lower frequency than in Fig. 110.

The case of R₂ = ∞ for step-down transformers is not important. By inspection it can be seen to be the response of R₁ and X_c in series, because X_L carries no current. This case rarely occurs in practice.