Analysis of output transformer frequency behaviour, December 2008.

This page analyzes the frequency performance of output transformers I have for sale. The general style of these is
different to my own general design and winding style.
Therefore I would like to explain the technical performance differences between the two styles.

I could sum up by saying that the transformers for sale have *extremely* low leakage inductance but have a lot higher shunt capacitance than my own designs which
have more leakage inductance but far less capacitance. Without any applied negative feedback to try to flatten an amplifier's response, the OPT is often the main item which determines the amplifier response and the phase shift between input and output signals.

Readers will need to be able to understand the basics about second order LC filters and interpreting equivalent models of LCR circuits. Wherever you have an audio frequency isolation transformer driven by a source resistance with separate primary and secondary, you will have a low pass filter with R plus L in series with an output point and a capacitance shunting the output to 0V.

I should familiarize readers with the theory I have explained in 2006 at  http://www.turneraudio.com.au/output-trans-theory.html
Fig 1

Fig 1  is an equivalent model of a PP output stage with a pair of class A UL 6550 tubes.  The two output tubes and OPT without its center tap may be considered like an SE stage with one generator and one Ra value to replace the  two generators and two series Ra that occur in a PP output stage. We are only interested in the  signal frequency behaviour. So the OPT primary with CT can be considered here as one winding with one end grounded. The Ra of 4k for each tube is summed to make Ra-a = 8k and this output anode resistance of the2 tubes is shown in series between OPT primary winding input and the output of an imaginary voltage generator. One may wonder why engineers exploit a silly idea like an imaginary generator but its a great way to explain how electronic devices perform at the basic level in terms of gain and dynamic output resistance. The model generator has an imaginary output = µ x grid input voltage which in this case is the summed grid to grid signal input voltage of 51.85Vrms. The µ of the output tubes = 20 which is for UltraLinear connected 6550 with approximately 40% screen taps so that the UL µ is between about 165 for pure beam tetrode and about 6 for pure triode connection. The generator produces the imaginary Vout = 1,037Vrms.

From measuring a working sample circuit we know there is 500Vrms across the the primary load anode to anode and the load is 7.8k, so we have 64mA of load current and this flows around the circuit through the generator, Ra-a and through OPT leakage inductance which appears as a series inductance, then through the winding resistance which also acts as a series resistance and finally through the transformed secondary load which appears here as 7.8k at the primary winding input.
The actual two winding transformer is considered perfect with no losses, but the imperfections are added  as they behave in the real world and are depicted as leakage inductance, LL, = 4mH, total Primary + Secondary winding Resistance, Rw, = 390 ohms. At the point of load input there is primary shunt inductance = 100H, and shunt capacitance = 500pF.

The shunt capacitance may be calculated approximately with a method shown in the page on 'PP Output Transformer Calculations'.

With most OPTs, there are numerous P and S winding sections and a far more complex number of C and L values exist if one were to draw a full equivalent model for the OPT. It would have numerous L&C sections in cascade, and with shunt capacitances from primary input to the active secondary output. The real picture is impossible for anyone to quantify perfectly.
 Predicting the actual real performance outcome of a real OPT based on dimensioning the known winding details has not
 yet been successfully attempted on a computer.  Since output transformers are not now mainstream engineering practice nobody has bothered to produce a program which will reliably simulate the use of a known OPT design 100% accurately.  But all audio transformers will behave with a response that can be explained at least as if there is a basic second order LC filter. If you measure the performance of any transformer used with audio frequencies, this is what you will find.

The presence of the leakage L and shunt C give undulations in the response above the audio band but as long as the leakage L and shunt C are both kept low, the "queer HF response" curves will not occur until above 75kHz, and the gain and phase shift of the amplifier can be tailored so the amplifier is made stable without  affecting the performance within the audio band below 20kHz and without causing audible problems. The above model of an output stage will do for a basic understanding. At 1kHz, the "reactive" elements of LL, Csh, Lp will have virtually no effect on load and gain, and the equivalent circuit could be drawn without them present. But at very low F, the Lp becomes a low impedance which shunts the signal, and at very high F the LL begins to become a high impedance in series with the load and the shunt C begins to become a low impedance to shunt the load voltage. So a tube amp operates as a bandpass filter.
But all other types of amplifiers also operate as bandpass filters. We just need to ensure the bandwidth is wide enough.

Fig 2.

Fig 2 shows the Equivalent Model Push Pull output stage in better detail and  perhaps more easily understood if anyone was utterly bamboozled with Fig1. above.
In class A with both V1 and V2 loaded equally with a 4k ohm load, the two anode signals will be subject to the attenuation of LF signals by Lp shunting
the load and the attenuation of HF due to the second order filter formed by series leakage inductance and shunt capacitances.

At LF, there is  a R + L first order filter. If Lp = 100H over all, then from one anode to the CT it is 25H.
The resistance its shunts consists of 1/2 RLa-a = 4,000 ohms in parallel with Ra for one tube, also of 4,000 ohms which gives 2,000 ohms. The equivalent series  winding resistance of primary and reflected secondary winding is so small in value we may ignore it on this occasion.

The pole for the -3dB LF cut off is at 12.7Hz. If triodes were used instead of the UL connection Ra // RL = approximately 860 ohms so the -3dB LF
point would be at 5.5 Hz. Thus the low triode Ra of approximately 1,100 ohms would be what mainly determines the -3dB pole. With tubes in beam tetrode mode, the Ra = 18,000 ohms and the resultant R = 3,270 ohms so its the resistance within the load which mainly determines the -3dB pole at 20Hz. Real loads used are actually speakers, and below 70Hz they become anything but resistive in nature and are like a parallel tuned circuits with high peaks in their impedance so negative feedback must be used to reduce the effective anode resistance and flatten the bass response and to stop bass from becoming boomy and loose sounding.

At HF on each side of the PP circuit , there is a filter formed with Ra = 4k (source resistance) in series with 2mH LL then with 4,000 ohm load. 1,000pF shunts the load.

The LL and Csh have a resonance at 113kHz, and at this F, XC = XL = 1.41k ohms. For critical damping and a non peaked response at the load,
the R required to reduce the Q should be about 2k, which is what we have, and so the response should have a -3dB pole at 113kHz and just roll off
at 12dB/octave. 

If someone were to connect a 2uF load across the OPT secondary winding instead of the R load, it appears as approximately 2,000pF across the the primary RL of 8k. This is because the OPT has an impedance ratio of 1,000:1, with a turn ratio of 31.6:1. This "reflected" or transformed C adds to the Csh shown which is 500pF anode to anode to become 2,500F. The two amounts of 2mH add to make 4mH around the whole loop and you get a series resonant F at 50kHz. The XL and XC become 1.3k, and lower than the series source R of 4,000 ohms and with just this C load we don't have any R load so the response will become "over damped", and a HF pole will occur at 16kHz and well below 50kHz, and 6dB/octave for the first octave above 16kHz. But as F rises the ultimate attenuation will be 12dB/octave as the effect of the LL becomes  factor.
Using triodes or loop NFB will effectively reduce the source resistance or Ra, and below the critical damping resistance required for an unpeaked response and so the response will become peaked because of the high current flow in the series resonant circuit formed by LL and the C load. Triodes or NFB can also cause a peaked response at 113kHz because of the transformer shunt C and LL. The actual real behaviour may be slightly different to the theoretical and you may see a dip in the response before 113kHz depending on whether the interleaving pattern in the OPT is PSPSPSPSP or SPSPSPSPS. Its not always easy to predict exact resonant behaviours. But having a lot of shunt C across the primary load while Ra is high tends to shunt the resonant impedances at higher frequencies.

With global NFB the phase shift caused by both the LL and the Csh is fed back to the input NFB terminal. With a pure resistance load the use of phase shift and gain reduction networks in early stages of the amplifier and compensation capacitance across feedback R networks work to stop the feedback at HF becoming positive and this keeps an amplifier stable. But with the right value of pure capacitance as the sole secondary load such as 0.22uF or 0.47uF, the resonance and phase shift caused my the added C with the leakage L can cause violent HF oscillations. So the "critical damping" R&C networks applied within an amplifier should be just effective enough to stop oscillations if there is no R present as a load at the output and if any value of capacitance load is connected.
Fortunately, such critical damping measures are easy to apply, and should be applied even though a pure C load is unlikely. If the leakage inductance is reduced to negligible values of say 1/10 of those in the model, the frequency of resonance between a given C and LL will rise by a factor of 3.16 times.
And the onset of phase shift cause by LL will also rise that much. To get such low LL in a given OPT means using much more interleaving and its done at the expense of increasing shunt C which tends to cause a shunt loading effect closer to the audio band, and with a 90 degree phase shift at a lower F pole.

Keen observers will see in Fig3 below that the total shunt C at each anode of OPT1 from my website design pages has Csh at 720pF, not the 1,000pF as mentioned above. The 720pF is a calculated figure and we should allow a bit more for the real world, especially after varnishing because air voids become filled with varnish which increase the C due to the varnish dielectric constant. Design of the OPT with regard to non-exact and slightly unpredictably HF response is by the simplest empirical method available, ie, to consider the amplifier as an active bandpass filter with second order LC parts enclosed by the feedback path.

Fig 3.

Fig3 shows the OPT No1 with its Csh appearing at each anode. The secondary is connected to 0V at one end and has negligible
signal voltage compared to the primary voltages so we may regard the secondary windings as earthy wound screens connected to 0V.
At each interface between P and S layers, some 570pF may exist. But the sum of the C appearing at the anode is the sum of the transformed values of the
numerous 570pF amounts present to give a total of only 720pF at each anode.

Fig 4.

Fig 4 shows the bobbin winding details for one of the larger output transformers I have for sale, numbered OP1, produced by Mr Bruce Flanagan.
Its properties need to be explained.
It uses C-cores which have Afe = 2,700sq.mm and Np = 1,496turns. It has P wire of 0.45mm dia and is OK for use with 6 x 6550 output tubes.
600Vrms across the primary loaded with 2,800 ohms will give 128W. The frequency of saturation at 600Vrms is at 20Hz.
My OPT-No1 has Afe = 2,200sq.mm and Np = 2,320turns, with wire = 0.35mm dia, OK for a pair of 6550, but with 600Vrms a-a the power is only 45W into a load of 8,000 ohms. My transformer Fsat is at 16Hz at 600Vrms. There is not a huge difference in the Fsat of either transformer. Most users of amps with 45W or 135W will not use very different sound levels in normal use, perhaps 10W maximum when 282Vrms is needed across 8k in the 45W amp and 167Vrms is needed across the 2,800 ohms of the 135W amp. At 10W, the Fsat for the 45W amp is at 7.5Hz and for the 135W amp its 5.6Hz, so the larger amp just wins the battle of who saturates first  during normal use. Therefore, when used to its natural ability optimum, the Flanagan LF performance is very good indeed.
One must consider other makers whose product saturates at a higher F than either mine or Mr Flanagan's. Such makers *NEVER* disclose where their transformers saturate, and *NEVER* honestly disclose the details within their transformers. I do.

At high frequencies, there is a major difference between my designs and those by Mr Flanagan.
Because the P and S windings have been interleaved maximally, ie, there are nearly an equal number of P and S sections, and a high number of both,
we can calculate the LL across the whole transformer be 0.24mH, a tiny amount indeed.

So at each anode there is 0.12mH in series with 1/2 the anode to anode load during class A operation.

The Csh is much higher though and I calculated about 3,600pF which is more than what I would have when using a 2uF load at the secondary of
my design. I measured the transformer to confirm what I have calculated. In fact the Flanagan transformer has 5 times the effective shunt C of my design.

However, it is meant to be used with 3 times the number of output tubes, so the HF loading effect of the Csh per tube used is only 1.7 times greater than in an amp using my OPT No1.

The resonant F between 0.12mH LL and 0.0036uF Csh is at  265kHz, which is at twice the Fo of my design. So effects of resonance will be easy to deal with because they occur at such a high F. If a 2 uF load is connected across the secondary, the impedance ratio changes its value to appear anode to anode as
0.0035uF which adds to the Ca-a of 0.0018 I calculated to make a total of 0.0053uF anode to anode This this will be resonant with 0.24mH LL at 140kHz,
so C loads won't worry an amp with this OPT because the AF band only goes to 20kHz.

Because the Csh at the anodes with the Flanagan OPT is so high, we can calculate the HF -3dB pole caused by this C and the R formed with RL in parallel with  Ra. Considering that we will use 6 x 6550 with the Flanagan transformer, Ra each side is 1,333ohms for UL operation and load is 1,400ohms so the parallel load each side is about 700 ohms. Anode to anode this becomes 1,400 ohms and we have an RC filter with transformer shunt C = 0.0036uF, so the pole is at 31kHz. This is lower than with my OPT but in not much of a loading problem at 20kHz. There is hardly any attenuation or phase shift caused by the leakage inductance. The use of triodes would raise the HF pole to over 55kHz.
The roll off is at 6dB/octave until the F approaches the Fo due to the leakage LL and Csh resonance and beyond.
Therefore a UL amp using my OPTNo1 design has what is called an inductive output impedance character, but it is a low amount of L at 4uH at the secondary,
shunted by the transformed Ra-a = 4 ohms and Csh = 0.5uF.
With the Flanagan transformer the output impedance is more capacitive in nature at HF and would be about 2uF in shunt with the Ra-a at the sec = 4.6 ohms. The leakage inductance would be only 0.42uH at the secondary.

Therefore the leakage inductance of both my designs and those of Mr Flanagan appear to be quite low enough.

Larger amounts of leakage inductance will always give more HF attenuation when the amp is used with with low loads and ESL speakers.
With larger amounts of shunt C, there is less HF attenuation as the load becomes lower.

I can conclude that the higher capacitance of the Flanagan transformers should not adversely affect the music. I would defy anyone who thought they could detect a difference in AB trials between Mr Flanagan's transformers and my own.

In my 300 watt amps, I did use more interleaving than in OPTNo1, and I had 6S x 5P sections instead of the 5S x 4S of OPTNo1. Mr Elson Silva who ordered all the transformers from Mr Flanagan began to use mostly my designs after 2002. Even Mr Flanagan had to admit that slightly less interleaving and hence less capacitance was a better way to make an OPT suitable for say a pair of output tubes which have a higher Ra-a than a six pack have.
Mr Silva is keeping all his spare OPTs made to my design for the future.

I can guarantee that the transformers I do have for sale will certainly handle music well. 

Back to output transformers for sale.