The author presented this paper on audio transformers at the 122nd AES Convention in Vienna, Austria, in May 2007.

Published in audioXpress Magazine, Jan 2011, pp. 18+

ABSTRACT: At the threshold of audibility, the signal and flux density levels in an amplifier with audio transformers are very small. At those levels the relative magnetic permeability of the iron transformer core collapses and the inductance of the transformer becomes very small. The impedances connected to the transformer plus its signal level and frequency dependent inductance behave as a high pass filter [whose] corner frequencies slip into the audio bandwidth, resulting in a nonlinear signal transfer through the transformer. This research dxplains deviations in the reproduction of micro details at the threshold of audibility.

**INTRODUCTION**

The capabilities of our ears are amazing and it is surprising what we can hear. Some amplifiers and speakers reproduce sound in a fantastically open manner, others do not, and it is not always clear why this is the case. To find clues and answers to these differences between sound systems, I decided to start reverse-thinking research; that is, to start with the capabilities of ours ears and to determine which minimal specs the electronics should have.

In this article I give a simple, but very effective, demonstration of this reverse approach by using the well-known audibility threshold curves [Fletcher-Munson] of our ears. This famous research determined the lowest SPL level per frequency that the ear can notice. Only the happy few with golden ears can hear below that level, but most people have a threshold close to that measured in those days.

For instance, you can just notice audibility at 4kHz at a SPL level of -4dB, which is even less than the 0dB level of 20 microPascal at 1kHz. Now consider a loudspeaker with an efficiency of 90dB / W,m and calculate that the power which the speaker needs to reproduce such a weak sound level at 1m distance is an [amazingly tiny] 4x10^-10W. Also consider that the signal levels at the speaker terminals are extremely small.

My professional work is focused on valve amplifiers and audio transformers [1-5]. So, I decided to continue research in that area. A valve amplifier with output transformers needs to transfer such small voltages with great accuracy...

What happens inside an output transformer at such small voltage levels? The flux density in the core will be amazingly small as well, [on the order] of 10^-8 Tesla. There exist not so much handbook [nor research] data about the relative magnetic permeability inside the core at such low flux densities. This permeability is closely related to the mobility of magnetic iron domains, called the Weiss-areas.

At the indicated signal levels, the musical magnetic force on these domains will be almost negligible. They will be busy with each other, and consequently their mobility will be very small. So, when you listen to sound at such low SPL levels, the transfer of sound through the audio transformer mainly depends on the capability of the small magnetic domains to move, and it is not certain at all whether they can do this in the right manner.

I started to do measurements there and noticed that interesting deviations from linear behavior occur. And that there is a real conflict between linear signal transfer and the actual mobility of the small magnetic domains.

The easiest way to explain this conflict is to realize that a transformer and its inductance, which is directly related to the permeability, in combination with the driving and loading impedances, acts as a first-order high-pass filter. At several frequencies and belonging SPL threshold levels, it is shown that the permeability of an iron core is hardly able to linear-transfer the weak audio signal. This means that with a given iron in the transformer core, with a given amplifier topology, and a given loudspeaker efficiency, you can calculate in advance whether the transformer will be able to transfer the weak audio signal in a linear manner.

From there it will be proven that SE amplifiers with low impedance triodes have great advantage, not by miracle or belief, but just by their low impedance combined with gapped core magnetic behavior. Then you can understand why these amps are able to produce such a holographic soundstage, and why you truly can hear the micro details beacuse they are not weakened inside the output transformer. This research combines many characteristics of valve amplifiers into a holistic picture, combinining magnetism with the fantastic capabilities of the human ear; it gives objective answers to what you subjectively hear.

**CALCULATIONS**

Figure 1 shows the well-known ISO-curves of the human ear, in this case taken from the famous book of Blauert [6]. The lowest dashed line in this figure shows the threshold of audibility in a totally silent environment.

Now imagine a loudspeaker with an efficiency "n" of, say, 90dB/W,m. In order to reproduce the signal level at each given frequency at the threshold of audibility at a distance "d" between loudspeaker and ear, the loudspeaker needs a power "P" given by:

P(f) = 10 ^ [ [ SPL(f) - "n" + 20 log( "d" )] / 10 ] (eq. 1)

For this study I assume that the frequency characteristic of the loudspeaker and its impedance are absolutely constant in the frequency range of interest. Later I will say more about these aspects. As an example, the required powers at threshold are calculated for "n" = 90dB.W,m at "d" = 1m, and the results are shown in Fig. 2.

Now let's focus on my profession: valve amplifiers with output transformers. I assume that the valves used are excellent and you only need to focus on the output transformer with a primary of "Zaa".

Assuming negligible losses inside the output transformer (OPT), then the effective voltage "Vaa" delivered by the valves over the total primary winding is given by foruma [2]. Firgure 3 shows the results of this calculation for Zaa = 4,000 Ohms parimary.

Vaa = sqrt( P * Zaa ) (eq. 2)

Next, use a real-world output with Np = 2,000 primary turns and a cross-sectional core surface A = ( 10 ^ -3 ) * ( m ^ 2 ). With the primary voltages known, you can calculate the amplitude of the magnetic flux density B, using formula (eq. 3). See Fig. 4 for the results.

B = ( Vaa * sqrt( 2 ) ) / ( 2 * pi * f * Np * A ) (eq. 3)

This figure is one of the first reasons why I became concerned. Imagine, at threshold SPL, the core is working at very small flux density levels. Then it might be the case that the relative magnetic permeability of the OPT core material will be too small to give the OPT any primary inductance. Core steel manufacturer information gives almost no clues about magnetic behavior at such small flux densities. Their info is in nice graphs from 0.1 to 2 Tesla, but we are looking now at what happens [well] below 0.1 T.

I performed relative permeability measurements on two steels: GOSS (Grain Oriented Stainless Steel) and stamped lamination VM11 annealed steel. The results are shown in Fig. 5.

Below 10 ^ -3 T it is clearly visible that an initial permeability stays present, which depends also on the temperature of the core (the measurements were performed at 20deg C, while temperature rise makes the initial permeability larger, but not by that much - about 0.07% oer degree Celsius, according to measurements of L. Alberts and B. J. Shepstone).

With the flux densities known (Fig. 4) at threshold level, you can derive the permeability with Fig. 5, and calculate the inductance of "Lp" of the primary winding with formula 4. There "lc" is the mean magnetic path length, and "lg" is the width of the gap in the core.

Lp = ( u_0 * Np ^ 2 * A ) / ( lg + ( lc / u_r ) ) (eq. 4)

Using u_0 = 4 * pi * 10 ^ -7 Hm ^ -1 for the permeability of vacuum and our example output transformer has lc = 0.236m, while lg is negligible. At each threshold SPL level and belonging frequency the primary inductance Lp can be calculated. The results are shown in figure 6 for a transformer with a GOSS core.

It clearly is visible that Lp is not a constant and collapse to rather small values, especially at higher frequencies threshold SPL levels.

The next question is crucial: "Is Lp large enough to have no negative effect on the linear signal transfer at threshold level through the output transformer?" To answer the question, I used the modeling of a valve power amplifier [1-4]. There the power valves are replaced in their operating point by a single voltage source in series with their summed plate resistances Ri, eff. This voltage source drives the primary inductance Lp plus the primary impedance Zaa in parallel (you assume that the loudspeaker impedance is a constant). Figure 7 shows the situation.

In Fig. 7 it is clear that the valves plus the OPT driving a speaker "Zs" can effectively be replaced by a pure voltage source with series resistance Ri,eff in parallel with "Zaa" driving an inductance Lp. This circuit is a first-order high-pass filter, and the transfer function of this filter is given by formula t, where s = j * 2 * pi * f and Req is Ri,eff in parallel with Zaa.

Material herein added and updated constantly; presented for inspirational and educational purposes per Fair Use.

*Last modified 22 Feb 2021*

Milbert Amplifiers

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Made in USA since 1986 8051 Citation Drive #C

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The Most Musical Amplifiers

Made in USA since 1986 8051 Citation Drive #C

Gaithersburg, MD 20877 USA

phone 202-558-5502