Low-Frequency Response

At low frequencies, the leakage reactances are negligibly small. Resistance R_P may then be combined with Z_g to form R₁ for a pure resistance source, and R_s with Z_L to form R₂ for a resistance load. At low frequencies both source and load are pure resistance, and the circuit may be simplified to that of Fig. 107(b). Here the a² has been dropped; in other words, a transformer with a 1:1 ratio is shown, referred to the primary side. X_N is the primary open-circuit reactance, or 2πf times the primary open-circuit inductance (OCL) as measured at low frequencies.

If shunt resistance R_N is included in load resistance R₂, the circuit becomes like that of Fig. 107 (c). Winding resistances are small compared with source and load resistances in well-designed transformers. Likewise, R_N is high compared with load resistance, especially if core material of good quality is used.

Therefore, to a good approximation, in Fig. 107 (c), R₁ may represent the source impedance and R₂ the load impedance. On a 1:1 turns ratio basis, the voltages E₂ and E₁ are proportional to the impedances across which they appear or

[61]

The scalar value of this ratio is found by taking the square root of the sum of quadrature terms:

[62]

Equation 62 holds for any values of R₁, R₂, and X_N whatsoever, but there are three cases that deserve particular attention: (a) R₂ = R₁; (b) R₂ = 2R₁; and (c) R₂ = ∞. Of these, (a) corresponds to the usual line-matching transformer with the source and load impedances equal; (b) is often recommended for maximum undistorted output of triodes; (c) is realized practically when the load is the grid of a class A amplifier. For these cases, equation 62 becomes

[62a]

[62b]

[62c]

These three equations are plotted in Fig. 108 to show low-frequency response as "db down" from median.

Fig. 108. Transformer characteristics at low frequencies.

The median frequency in an audio transformer is the geometric mean of the audio range; for other transformers it is a frequency at which the ratio X_N/R₁ is very large. At median frequency the circuit is properly represented by Fig. 103(b).

The equivalent voltage ratio E₂/E₁ has maxima of 0.5, 0.667, and 1.0 for cases (a), (b), and (c), respectively, at the median frequency, or for X_N/R₁ = ∞ in Fig. 108. The higher OCL, the nearer the transformer voltage ratio approaches median-frequency value. The lower the value of loading resistance R₂, the lower the equivalent voltage ratio is. The factors 0.5, 0.667, and 1.0 multiplied by the turns ratio, a, give the actual voltage ratio at median frequency. At lower frequencies, the factors diminish.

The transformer loaded by the lowest resistance has the best low-frequency characteristic. A transformer having an open-circuit secondary has twice the voltage ratio and gives the same response at twice the "low end" frequency of a line-matching transformer of the same turns ratio.

Figure 108 is of direct use in determining the proper value of primary OCL. Permissible response deviation at the lowest operating frequency fixes X_n/R₁ and therefore X_N. At the corresponding frequency, this represents a certain value of primary OCL. As this inductance determines the size and weight of the transformer, the importance of Fig. 108 is evident.

If the primary and equivalent (1:1) secondary winding resistance each are 5 per cent of R_1; the total effect will be a decrease of 10 per cent in the median-frequency voltage ratio, in the case of the line-matching transformer, with corresponding decreases at lower frequencies. On the other hand, the primary resistance of an open secondary transformer has no effect upon the median-frequency voltage ratio but has some effect at lower frequencies, whereas the secondary resistance has no effect either at median or at lower frequencies. Hence it is important in the open secondary case, for the sake of low-frequency response, to keep the primary winding resistance low, but the secondary winding resistance may be any value. The maximum number of secondary turns may be determined by the smallest practicable wire size rather than by winding resistance.

As the frequency increases, the primary inductive reactance X_N also increases until it has almost no effect upon frequency response. This is true for median frequency in Fig. 108. It is also true for higher frequencies; in other words, the OCL has an influence only on the low-frequency end of the frequency response curve. The ratio of R₂ to R₁ still limits the voltage ratio, however. If the amplifier works at one frequency only, OCL is determined by the deficiency in voltage gain that can be tolerated in the amplifier design. This can be found in Fig. 108.

In an amplifier with a band of operating frequencies, e.g., the audio band, a well-designed transformer has uniform voltage ratio for a frequency range extending from the frequency at which X_N ceases to exert any appreciable influence, upward to a zone designated as the high-frequency end of the transformer frequency range.