Output transformer analysis.

Frequency behavior, December 2008.

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

I could sum up by saying that the OPTs for sale have *extremely* low
leakage inductance but have a lot higher shunt capacitance than my own
designs which have more leakage inductance which is quite low enough,
but also have far less shunt capacitance. Without any applied negative
feedback to try to flatten amplifier frequency response, the OPT is often
the main item which determines the open loop 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 source R feeding load R through LL and
a capacitance shunting the load R output to 0V.

I should familiarize readers with the theory I have explained in 2006 at

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 behavior.
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 the 2 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 ultra-linear
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,
shunt capacitance = 500pF.

The shunt capacitance may be calculated approximately with a method
shown in the pages 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, and
no online OPT calculator programs are yet available.

Predicting the actual real performance outcome of a real OPT based on
winding dimensions and geometry of the known winding details has not
yet been successfully attempted using a computer program.

Because output transformers are not now mainstream engineering practice
in 99.99% of mainstream audio engineering nobody has been willing to spend
enough time to understand all OPT operation principles and then spend months
to perfect a program which will reliably simulate the use of a known OPT
with 95% accuracy. But all audio transformers will behave with a response
that can be partially explained in terms of basic LCR network theory at least
where the OPT is a passive bandpass filter with L&R first order HPF below the
LF pole, and L&R and C&R second order LPF above the HF pole.
If you measure the properties 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 with NFB and 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 an active 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 Class A1 output stage in better detail
and perhaps more easily understood if anyone was utterly bamboozled with Fig1
There are two output tubes V1 and V2 which could be type 6550 beam tetrode and
each is drawn up consisting of their equivalent model being a low impedance voltage
generator producing an "imaginary output voltage" in series with the dynamic anode
resistance Ra, shown here as 4k0.
The Ra of each tube affects the response measured across the "anode to anode"
load, RLa-a, shown as 7k8.

In class A with both V1 and V2 loaded equally, The load of each = 1/2 RLa-a = 3k9.
In addition to the RLa load of each tube there is a series winding resistance Rw =
195, say 200 ohms, so each tube is loaded with 3k9 + 200 ohms = 4k1.
The signals across RLa-a 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 load resistance of 3k9 is in parallel with 4k0 plus series Rw of 200
ohms. The total of these two parallel R = 2,020 ohms, or say 2k0, near enough.

2k0 is shunted by 25H.

Response of the amp is down -3dB when XL = R, so L = 2k0 / ( 25 x 6.28 ) = 12.74Hz.
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. But the OPT response
is very dependent on the source resistance.

With tubes in beam tetrode mode, the Ra could be 25k0 and the resultant R = 3,450
ohms and it the load resistance within the load which mainly determines the -3dB pole
at 22Hz. 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"

At HF on each side of the PP circuit , there is a filter formed with Ra = 4k0, in series with
2mH LL then with RLa 3k9 load, 1,000pF shunts the load. Rw may be neglected.

The LL and Csh have a resonance at 113kHz, and at this Fo, 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 2k0, which is what we have with the ultralinear connection, and
so the response should have a -3dB pole at 113kHz and response across RLa-a falls
at at 12dB/octave above 113kHz. 

If someone were to connect a 2uF load across the OPT secondary winding instead of
the RLa-a load, it appears as approximately 2,000pF across the the
primary RL of 7k8. This is because the OPT has an impedance ratio of 1,000:1, with
a turn ratio of 31.6:1. 1/2 This "reflected" or transformed C load appears across each
1/2 primary as 1,000pF, so Cshunt at each OPT 1/2 primary = 2,000pF.
The 2mH LL and 2,000pF give a series resonant Fo at 80kHz.
At Fo, XL = XC = 1,000 ohms, and lower than damping resistance of 2k0, so response
will become under damped and roll off will be gradual and less sharp. 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 peaked because of the high current flow in the series LC resonant
So triodes or NFB can also cause a peaked response at 113kHz because of
the transformer shunt C and LL. The actual real behavior 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 behaviors. But having a lot of shunt
C across the primary load while Ra is high tends to shunt the resonant
impedances at higher frequencies.

It is not easy to answer to the question, "What is the OPT response?"

In fact, it is a silly question unless the signal source resistance and
output load are specified and both used for measurement tests.

But for some idea about the quality and properties of OPTs, we may measure
the OPT so that the OPT is in a pure class A amp, that Ra-a = 2 x Ra.
Ra-a = RLa-a load across the primary.
RLa-a = RL at secondary x OPT Impedance Ratio, or,
RLa-a = RL at secondary x ( Turn Ratio squared.)

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 feedback 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 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 John 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

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

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 OPT No1, 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.

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

OP3 listed at my For-sale pages was used in two 60 Watt SE monobloc
amps each with 6 x 6550 in parallel with CFB use. Sound is excellent.

Back to output transformers for sale.