Transformers and
Phase Shifts

note: formulas used in table calculations -
phase angle equals arc tan of the quantify R divided by X in the examples above R is the resistance of 16,000 ohms and X is equal to 2 times pi times f times L.

Now if we increase the primary inductance
then the following obtains:

DISCUSSION AND FINDINGS:

1) Sometimes it is argued that the roll off caused by a relatively small amount of inductance does not do much harm or that it is only relevant to the bass frequencies. But the numbers above should clearly show that depending on how much or how little inductance and, hence, inductive reactance the transformer has the resulting phase shifts do extend quite far up into what is conventionally referred to as the middle or lower middle frequencies.

440 hertz was put in as it is the middle A on the keyboard. With only 27 henries of primary L the phase shift at 440 hertz is 12.1 degrees. With 127 henries the phase shift at 440 hertz is reduced to 2.61 degrees.

2) Notice that with 127 henries of L the frequency that shows a 45 degree phase shift is two and a half or more octaves lower than if the transformer has only 27 henries.

3)If we consider the generation of harmonics from the phase shifted frequencies (say even on middle A on the keyboard where the 27 henry inductance yielded a phase shift of 12.1 degrees) then the first overtone at 880 hertz will have been produced by a fundamental that already had 12.1 degrees of phase shift. The third harmonic *at 1760 hertz) will also be effected and so on. So phase shifts *might* matter in the middle frequencies as well.

4)This data shows and the formula itself (upon careful study) for calculating phase shifts illustrates that the inductive impedance must be many multiples greater than the nominal impedance of the output transformer in or to minimize the magnitude of phase shift in the audio band.

5) from the formula for calculating the phase shift on the transformer:

phase angle equals arc tan of the quantity R divided by X

it is clear that to decrease or minimize phase shift we must either increase X or lower the value of R. It is the ratio of R to X that is important in this case and solely determinative of the magnitude of phase shifts whose origins are within the transformer itself and due solely to the factors in the above formula.

That is not to say that other branches, components, or circuit elements cannot have, or themselves be, a generator of phase shifts. This analysis confines the object of study to the transformer proper.

6) last point. And I specifically disclaim any extensive expertise in this consideration. But it has been mentioned and discussion has evolved around the impact of this transformer generated phase shift and what impact it has on the use, interpretation, and behavior of plate curves as supplied by tube manufacturers.

My understanding (and I, again, don't claim complete understanding of this) is that plate curves are generated with the use of resistors serving as the "load" for the tube. Resistors are inherently linear and non-reactive. Transformers are reactive... their impedance changes with frequency. At low frequencies their impedance is principally determined by the amount of primary L which they exhibit.

If this primary L is low enough then there will be phase shifts introduced. These phase shifts, as I have understood from posts on this board and other venues, change the nature and slope of the plate curves and introduce ellipsis that load the tube reactively and cause the plate generated distortion of the tube to increase. This *may* be yet another reason why we should be aware of the role of primary inductance and its resultant impedances.

Here are some other interesting facts about phase shifts: If you want or desire a phase shift of the following magnitude, then your inductive reactance must be the number of times specified greater than your nominal primary impedance.

Here is an example of how to use the above multipliers. Say you have a 5,000 ohm nominal primary impedance and you desire only 15 degrees of phase shift at 100 hertz.

So at 100 hertz we would want (by referencing the table above) an inductive impedance of 3.73 times 5,000 ohms.

Solve for the following;

18,650 equals 6.2834 times 100 times L

L is the unknown variable. In this case L must be 29.6814 henries.

The practical use of this chart is that anyone could select the frequency of interest and the amount of phase shift that they would allow and then calculate how much inductance would be required to satisfy the desired phase shift allowed.

This illustrates that the lower the phase shift you desire the greater the inductive reactance must be at any given frequency which you select.

I welcome any and all comments, questions, suggestions, criticisms. Anyone wishing to share any comments, ideas, suggestions on how I could make this paper better are also welcome to send me e-mail.

Thanks in advance, I hope this is somewhat useful or helpful to folks in understanding transformers a bit better.


MSL