PUSH-PULL OUTPUT TRANSFORMER
CALCULATIONS, PAGE 1.
Edited 2011.

FOR  ALL OPT WITH TWO BALANCED DC FLOWS
OR WHERE THERE IS NO NET DC FLOW IN ONE DIRECTION.

For calculations for Single Ended Output Transformers
go to 'SE OPT calculations'

THERE ARE 5 SEPARATE PAGES COVERING
THIS SUBJECT, AND YOU ARE AT PP OPT calcs Page 1.
Pages 2 to 5 are listed below with links.
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Brief reference to Radiotron Designer's Handbook,  4th Ed, 1955.
Some general notes.

Design steps for 2 x 6550/KT88/KT90 tetrodes.

1.  Define the aim of this design project.
OPT-1A is for up to 75 Watts of audio power between 14Hz and
65kHz using 2 x 6550, KT88, KT90, K120 beam tetrodes.
Pda at idle should be 25Watts for each of these output tube types.

2.  LOADLINE ANALYSIS for PP BEAM TETRODE (not including triode.)
Fig 1. Ea vs Ia, Ra curves for 1 x 6550 beam tetrode.
Fig 2. Loadlines for pure class A1, and minimum class B load.
2A. Plotting the Class A loadline, Fig 2.
Fig 2. repeated.
Fig 3. Ia waveform for one 6550 in pure class A.
Fig 4. Class A and AB Ia waveforms in tube amplifier.
Fig 5. Class AB Ia waveform in tube amplifier.
2B. Class AB1 loadlines.
Plotting the minimum class AB1 RLa load.
Fig 2. repeated.
3.  Check the anode dissipation for sine wave operation.
Formula for Pda for Class AB amp.
4.  Calculate idling Ia bias current.
5.  CALCULATIONS FOR PP BEAM TETRODES,
without load line analysis :-

6.  Calculate anode to anode load for maximum pure
class A1 power.

7.  Calculate maximum Class A1 power output.
8.  Calculate minimum Class AB1 RLa-a load which will
give maximum class AB1power.

Fig 2. repeated.
Fig 6. Ra curves for 6550 with 43% UL connection, and
with loadlines for two minimum RLa values.

9.  Calculate maximum AB1 power into minimum RLa-a.
Two methods, 9A, 9B.
10. Calculate Middle RLa-a for intermediate power
and for ideal Class AB1 for Hi-Fi use.

11. Calculate PO for middle RLa-a.

PO max for middle RLa-a,
class A1 PO for middle RLa-a
.
Fig 7. Graph for PO Vs RLa-a for 6550 beam-tet, UL or CFB.
13.   List the possible ranges of primary to secondary
Load matches, list use possibilities.

Table for wanted load match possibilities.
Some Notes about use of OPT-1A.
OPT Calculator Program?
About Metric Wire size table.
Fig 32.  Blank Sheet 6550 tetrode, basic Ra curve for Eg1 = 0V.
Fig 33.  Blank Sheet with All Ra curves for 6550 tetrode.
Fig 34.  OPT No 1 Schematic from 2002.
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PP OPT Calc Page 2 :-
Design of OPT-1A continued....
14.  Calculate minimum centre leg cross sectional area, Afe.
15.  Calculate the core tongue dimension, T.
Fig 8.
Wasteless E&I lamination details.
Fig 9.
C-core details.
17.  Confirm sizes for core.
18.  Calculate the theoretical primary turns, thNp.
19.  Calculate theoretical Primary wire dia, thPdia.
20.  Find nearest suitable overall dia wire size from
the wire size table.

Table 1. Available Wire Sizes.
21.  Calculate maximum safe working Idc.
22.  Calculate the bobbin winding traverse width.
23.  Calculate no of theoretical P turns per layer.
24.  Calculate theoretical  number of primary layers.
25.  Calculate actual Np.
26.  Calculate average turn length, TL.
27.  Calculate primary winding resistance, Rwp.
28.  Calculate pri winding loss % with minimum RLa-a,
29.  Is the winding loss more than 3.0%?

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PP OPT Calc Page 3:-

Design of OPT-1A continued....
30.  Choose the interleaving pattern.
Fig 10. Cross section through a hypothetical transformer with
concentric layered windings. Basics explained.
Tables 2, 3, 4, 5, showing interleaving pattern possibilities for PP OPTs.
31.  Choose insulation thicknesses.
Table 6.
Insulation Thickness Vs Voltage.
32.  List layers of insulation which are to be used.
Fig 11.
OPT-1A bobbin winding diagram for CFB use.
33.  Calculate height of Primary layers and all insulation.
34.  Calculate the max theoretical oa dia of secondary wire.
35.  Find nearest Sec oa dia wire size.
36.  Calculate the theoretical Sec turns per layer.

Table for ZR, TR, conclusions.
37.  Choose Secondary sub-section pattern.
Fig 12, Sec = 2L,
Fig 13, Sec = 3L,
Fig 14, Sec = 4L,
Fig 15, Sec = 5L,
Fig 16, Sec = 6L.
Fig 17,
Sec sub-section pattern
4A explained.
Table for TR, ZR and loads. Conclusions.
Fig 18, Sec sub-section pattern 4C explained.

Table for TR, ZR and loads. Conclusions.
Fig 19, Link pattern for 4C sub-sections.
Fig 20, Link pattern for 4A sub-sections.
Alternative Single Simple termination.
38.  Calculate secondary winding loss %.
39.  Calculate total winding losses.
40.  Calculate total height of bobbin contents.
41.  Draw sketches of bobbin details.
Fig 21,
OPT-1A, Ultralinear config.
Fig 22, OPT-1A, CFB config.
Fig 23, OPT-1A,  Schematic of windings.
42.  Calculate Fsat with Middle RLa-a.
43.  Calculate minimum required Lp.
Fig 24,
E&I core Lp + µ vs Bac + Vac.
Many notes and calcs.
44.  Partial air gap for PP OPTs.
Fig 25,
Gapping effects.
45.  Calculate leakage inductance.

Is LL low enough? 2 methods.
46.  Shunt capacitance of an OPT.
12 Steps
to determine C, many notes, calcs.
------------------------------------------------------------------------------------------TAPPED SECONDARY WINDINGS.
47.  Nominate OPT Design name and its purpose. OPT-1ATS.
48.  Nominate Wanted Load matches.
Fig 26,
Graph for PO Vs RL for 3 different Ns.
49.  Calculate available height for layers of secondary.
50.  Calculate the max theoretical oa dia of secondary wire.

Determine no of layers per Sec section.
Calculate winding heights in bobbin.
Fig 27. Bobbin details for alternative OPT-1BTS.
Conclusion, 4 options to optimize design are explored.
Fig 28. OPT-1ATS bobbin details.
51.  Calculate Total winding losses, Middle RLa-a.
52.  Compare winding losses, Tapped and Wasteless Secs.
52.  Compare LL, Tapped Secs to Wasteless Secs.
53.  Shunt Capacitance with tapped Secondaries.
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FOR PP TRIODE CLASS A1 AND AB1, OPT-2A
55.  Understanding Ra curves for triodes.
Fig 29.  Ra curves for 6550 and 300B.
56.  Understanding Ra curves for triodes.
Fig 30.  Ra curves for 6550 in triode.
57.  Calculate the minimum PP Triode RLa-a for
maximum class AB1 power for OPT-2A.
Fig 31.  Graph for Po vs RLa-a 6550 PP triodes.
58.  Calculate maximum AB1 power for minimum RLa-a.
59.  Calculate RLa-a for maximum pure class A1 power.
60.  Calculate maximum class A1 PP triode power output.
61.  Calculate the Middle RLa-a for triode PP operation.
62.  Calculate PO for Middle RLa-a,
63.  Conclusions about PP triode OPT design.

14T.  Calculate minimum centre leg cross sectional area, Afe, triode PP amp.
15T.  Calculate the core tongue dimension, T.
16T.  Calculate theoretical Stack height.
17T.  Confirm sizes for core.
18T.  Calculate the theoretical primary turns, thNp.
19T.  Calculate theoretical Primary wire dia, thPdia.
20T.  Find nearest suitable overall dia wire size from wire tables.
21T.  Calculate the bobbin winding traverse width.
22T.  Calculate no of theoretical P turns per layer.
23T.  Calculate theoretical  number of primary layers.
24T.  Calculate actual Np.
25T.  Calculate average turn length, TL.
26T.  Calculate primary winding resistance, Rwp.
27T.  Calculate pri winding loss % with MIDDLE RLa-a.
28T.  Is the winding loss more than 3.0%?
29T.  Choose the interleaving pattern.

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End of list of contents for PP OPT calcs.
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General information.
The Radiotron Designer's Handbook, 4th Edition, 1955,
contains a lot of good design advice about OPT design.

The trouble with "RDH4" as it is known is that beginners who are good
craftsmen are baffled by the mathematics and electronic behaviors
described in this great book. Nevertheless, although my basic mathematics
and physics education level only extended to high school, I can comprehend
the mathematical relationships between items I encountered, eg, R = V / I,
which is Ohm's Law. I found RDH4 extremely useful and I gradually began
to understand the reason why the book was written.
many say it was because RCA wanted more people to buy vacuum tubes
at a time when most electronic things were unreliable and very expensive and
mostly non-essential, unless the use was for military, medical or scientific
establishments. To me the book comes across as a rare exercise in corporate
altruism which enabled so many ordinary people to access cutting edge
electronics information for free. If you have no copy on hand, there is a
CD you can acquire and you may then print out pages to make a book if
you want.

Chapter 5 from page 199 to page 253 should be read repeatedly until the
message sinks in. It is not easy to understand if you have no idea about basic
tube operation and other basic electronic behaviors, so as soon as you find
you don't understand something, you must find out about it from somewhere
else in RDH4 or from some other source. I ended up with a few shelves of
books with overlapping information, much of which has never been published
online, and never will be, because it is now the digital era.

Unfortunately, the associated reference material listed on pages 252 and 253
has mainly been lost or thrown out of many library archives to make way
for  the huge mountain of more modern knowledge. There is very little new
information or better information about output transformers written after
1960 because mainstream development for tube OPTs stopped in about 1959
when tube operated electronics was being dumped in favor of transistors.

Not many folks will have RDH4 in their library nor will there be a nearby
technical library with easy browsing access so I shall try to unfold the design
method I have evolved based on information in the RDH or other sources I
have collected since 1994. To gain real understanding, I began to design OPT
then wind them myself after building a simple winding lathe. I then learnt how
to test them, and prove to myself the mathematics for design were indeed
correct.

After having wound many very fine PP output transformers with bandwidth
from 14Hz to 270kHz, I feel well qualified to speak from experience. The
list of logic steps involved in producing the best possible OPT is based on
designing for low winding resistance losses, core saturation at full power at
14Hz, and adequate interleaving to extend the HF response up to at least
50kHz without reliance on negative feedback. This meant keeping both the
leakage inductance and shunt capacitances to low quantities. The end result
gives a well filled winding window with several impedance matches possible
without having wasted sections of any secondary winding so the winding
losses and response is the same for any of the chosen load matches. The
only disadvantage of always using all available turns of a secondary is the
difficulty of changing the load match which may require technical skill an
amp owner does not possess, and there is a high likelihood of a mistake

The trend today is to provide 3 terminals at the rear of an amp, Common,
4 ohms, 8 ohms, and this allows some choice in load matching. Just how to
configure secondaries for easily selectable loads is covered below. The
disadvantage of this method is the the likelihood that a non technically aware
owner will plug the speaker cables into the wrong terminals.

In my page on 'Output Transformer Theory' , I show the recipe for OPT No1,
now shown as Fig15 at the bottom of this page. It was in my 2006 pages.
This page of 2011 now shows a similar design example recipe for OPT-1A.

DESIGN OF PUSH PULL OPT-1A EXAMPLE.
For 2 x 6550/KT88/KT90 tetrodes.
Steps 1 to 13:-

1. Define the aim of this design project.

OPT-1A is to be designed for use to allow up to 75 Watts of audio power over
a frequency range between 14Hz and 65kHz using 2 x 6550, KT88, KT90, K120
BEAM TETRODES. Pda at idle should be 25Watts for each of these output tube
types.

4 x 6L6GC, 5881, KT66, EL34, 6CA7 may also be used with Pda at idle for
each output tube < 16W for long tube life.

The connection mode may be for pure
Beam Tetrode with fixed screen +Vdc
supply,
Beam Tetrode with CFB windings and fixed screen +Vdc supply,
aka Acoustical, Ultralinear with screen taps, Ultralinear with or without CFB
windings, AND triode class AB1, or AB2.

The OPT-1A is to have secondary winding layers sub-divided into enough
separate windings to allow a number of winding arrangements made up with
seriesed and or paralleled windings useful range of speaker loads to suit
optimal tube operation, usually about 3 matches between 1.5 ohms and 16 ohms.

The range of anode load values to be considered are nominated as :-
Minimum RLa-a for maximum safe AB1 power,
RLa-a for maximum pure class A1 power,
Middle RLa-a for intermediate power between max AB1 and pure A1 powers.

THERE ARE SEPARATE OPT CALCULATIONS FOR TRIODE AB1
OPERATION OF BEAM TETRODES OR PENTODES BELOW AT
TRIODE PP OUTPUT TRANS CALCULATIONS.

2.  LOADLINE ANALYSIS for PP BEAM TETRODE
modes, but not including triode.

Fig 1.

Fig 1 shows my tidied up copy of New Sensor Corp's Ra curves for pure
beam tetrode operation of Russian made 6550EH with Eg2 = +250Vdc.
Before moving on to Loadline Analysis, everyone might like to become
familiar with the basic Ea/Ia Ra curves for the 6550. Higher values of Eg2
will show the Ra line for Eg = 0V giving a "knee" at a slightly higher Ia.

The slope of Ra curves between 0V and Ea = 50V shows a near straight line
limiting line for Ea swing. The "slope" of the line has a resistance value
which may be calculated as Ea / Ia swing below Ea = 66V.
Ia swing = 300mA for the Ea swing of 50V, so the Ra limiting line = 166 ohms.

I have found the limiting line ohm value for many Russian made output tubes
to be a higher ohm value than indicated in the Fig 1 data curves. In other words,
the anode voltage cannot swing as low in recently made Russian tubes compared
to the best NOS samples ever made, from which many of the curve sheets were
drawn.
Nearly all beam tetrodes and pentodes will never be used in pure tetrode or
pentode mode with a stable Vdc as Eg2, and the cathode bypassed to 0V,
but in fact most amps will employ UL taps or CFB windings or both.
Where this is the case the Ra limiting line for Eg1 = 0V may be assumed
to be 280 ohms or more.
When such multigrid output tubes are strapped in triode mode their Ra limiting
line for Eg1 = 0V is simply the Ra value given for the tube, or the triode curve
shown in data sheets.
It is usually impossible to swing Ea to the left side of the Ra limiting line even
with class A2 or AB2 operation with pentodes or tetrodes. But with a large
number of real triodes or triode strapped beam tetrodes the Ea swing can be
forced to the left of the triode Ra curve for Eg1 = 0V, but only by means
of class A2 or AB2 operation.

I have added a table for Ra and Gm for a range of bias conditions which I
found approximately correct within +/-10% for Russian samples of 6550EH,
KT88EH, all 'Sovtek' labeled 6550 and KT88, and "winged C" Svetlana 6550
and KT88.

Fig 2.

Fig 2 above provides the most basic load line analysis for ONE 6550 in a PP
amp to help determine the range of anode load values.
It saves everyone the
trouble of going to another page on load line analysis.

To draw all required loadlines, all that is needed is the curve of Ra for Eg1 = 0V,
for the correct value of Eg2, and the curve for maximum Pda.

The slope of the Ra curve between Ea = 0V and 100V is drawn as a straight
line with resistance value = 280 ohms. This part of the Ra curve is also called
the "diode line", and it indicates the extent of possible Ea load swing.
The 280 ohms is approximately correct for all output beam tetrodes and
pentodes
where they are used of hi-fi amps which most probably will have
UL screen taps and/or cathode feedback windings.

I have chosen commonly used idle conditions for 6550 with Ea = +500Vdc,
Ia dc = 60mA, Eg2 = +350V, which could be between +300V and +400V
without changing the design outcome.
The "knee" of the Ra curve where the Ra changes rapidly between a very
low Ra below Ea = 100V to a high Ra of thousands of ohms will be at
approximately point D where Ea = 100V and Ia = 350mA.

Pda at idle = 25W, therefore
Iadc = Pda / Ea chosen = 50mA.
There is no point considering the use of idle Ea above +520Vdc for most
power output beam tetrodes or pentodes in Hi-Fi amps because the higher
Ea will mean Ia will be low and Pda at idle will be low so there is less initial
class A power and most most power made in class AB which means higher
distortion. The higher Ea means less reliable operation, and a bigger
dependance on fixed bias. The higher the Ea, the higher the RLa-a becomes
so the OPT needs more turns per volt to keep the frequency of of saturation
low. However, the higher the RLa-a, the higher the tube gain so the use of
CFB or UL taps is more effective. All things considered, the following range
of Ea should be used for class AB PP hi-fi amps :-

+350V to +520V for 6550, KT88, KT90,
+300V to +430V for KT66, 6L6GC, 807, 5881, EL34,
+250V to +330V for 6V6, EL84.

Eg2 for all may equal Ea for UL connection.
For CFB, the fixed Eg2 may be
+300V to +400V for 6550, KT88, KT90,
+250V to +350V for KT66, 6L6GC, 807, 5881, EL34,
+200V to +300V for 6V6, EL84.

NOTE. If no 6550 data curves are available, there is a blank sheet at the
bottom of this page for anyone to print and use for amps where 6550, KT88,
KT90 are proposed.

NOTE. There is no need to consider harmonic distortion during the load
line analysis for load calculations. My methods steer people away from
coupling tubes with loads which would generate excessive distortion.
In class AB1 amps, there is considerable distortion in tube *currents* because
tube current is switched on and off during each AB wave cycle.
But in the overall *voltage* waveforms at transformer terminals there is
usually less than 4% THD at just under clipping and it is reduced to low levels
with negative feedback. THD during the first few watts of pure class A is only
marginally more than than for the same few watts from a totally pure class A
amplifier, and often only 0.03%.

2A. Plotting the Class A loadline, Fig 2.

Fig 2 repeated.

Plot point Q at Ea = 500V and at Ia = 50mA.
Plot point A at twice Ia on Ra curve.
Draw straight line from A through Q and onwards through the Ea axis at
point B.

Plot point C at Ea = 500V, Ia = 0.0mA, on Ea axis.

The distance between A and Q should be exactly equal to between Q and B.

AQB is the Class A load line for one 6550 in the pair of PP tubes.

The Ea minimum voltage at B is read off the graph from vertically below
B = 28V.
Ea change = load voltage change = Ea Q - Ea min = 500 - 28 = 472Vpk.
The Ia change = Ia at Q = 50mApk.
RLa for each 6550 = Ea change / Ia change = 472V / 0.05A = 9,440 ohms.
( This is Ohm's law being applied. )

The anode signal voltage = 0.707 x Pk swing voltage = 0.707 x 472V = 333.7Vrms.

The Ia at idle reduces to zero as Ea swings positively. The point B
on the graph should appear at Ea = IaQ x RLa, which is at 972V.
With two tubes working together with oppositely phased voltage and currents,
when one tube anode swings up to 972V, the other swings down to 28V,
and this voltage is applied across the whole primary, and transformed to high
current and low voltage required to drive a speaker.
The anode to anode signal voltage, Vaa, measured from one anode to the other
is twice the anode voltage at each anode = 2 x 333.7Vrms = 667Vrms.
The Class A RLa-a anode to anode primary load is effectively the sum of the
loads of each tube in series, so the RLa-a = twice the class A RLa for one tube.

Calculate PP Class A RLa-a load for maximum pure class A
= 2 x Class A RLa for one tube = 2 x 9,440 ohms = 18,880 ohms.

This formula is true for all class A loads for PP output stages.

The above class A load produces the maximum possible pure class A power.
If the RLa was a lower value, pure class A power is limited to where Ia change
never more than the idle IaQ value. So where the RLa becomes a lower
value than calculated so far, the class A power reduces and maximum output
power rises because the tubes begin to work in class AB.

Pure class A1 power = 0.5 x RLa-a x Ia squared, where Ia is Iadc at idle.
This example, Class A PO = 0.5 x 18,880 x 0.05 x 0.05 = 23.6Watts.

Power may also be calculated as Vaa squared / RLa-a, with Va-a in Vrms.
This example, Vaa = 2 x 0.707 x 472Vpk = 667Vrms,
Power = 667 x 667 / 18,880 = 23.7Watts.

The class A load line has less slope than any class AB load lines so the class A
loadline crosses through the least number of Ra curves for the various Eg1
voltage values.
In other words, when the class A loadline is drawn on a full set of Ra curves,
the Ea change requires little Eg1 change, ie, the anode voltage gain is high.
This means that a given negative feedback network will be most effective
to reduce the distortion and effective anode resistance during class A operation.

Tetrodes and Pentodes are not very linear for the whole possible Ea swing
in class A. Single ended tetrodes pentodes at clipping might produce 13% THD
without any NFB.
In PP, with "cancellation" of even numbered H because of the OPT the THD might
be 5% because the odd numbered H are not cancelled by PP action.
The use of the UL screen taps or Cathode Feedback windings may
reduce the THD to less than PP triode class A operation.

Fig 3.

Fig 4.

Fig 5.

2B. Class AB1 loadlines.

What is a class AB loadline?
The class AB loadline is really two load lines.
Class AB load line ohm values are always less
than the RLa-a for pure class A.
The class AB loadline has a Class A loadline and a Class B loadline.

During class AB operation in a full wave cycle up to clipping, the loading
for the first few watts for each tubes is pure class A where each tube sees a
load = 1/2 RLa-a. During this operation, +/- Ia change in each tube is limited
+/- peak current change equal to the idle bias current.
Above the first few watts the operation moves to class AB
and the load each tube sees becomes 1/4 RLa-a which is the same loading one
would see if the amp was set up to operate in class B without any bias current
at idle. The AB amp operation changes to partial class B operation because one
tube's Ia becomes completely cut off during 1/2 its wave cycle, while the other
powers the load alone through 1/2 the OPT primary and with a larger peak Ia
change than the Idle Ia.
If the tubes were biased to have no idle current and could only increase with
a positive going grid signal the tubes would be said to operate in class B,
and each tube could only provide power for each half wave cycle.
The load line for class B action in ONE tube is all that is needed to
determine possible class B or AB output power and anode voltage swings.

Class AB operation exploits the ability of an output tubes to produce much
more peak current change than for pure class A where Ia change is limited to
+/- Ia dc at idle. Therefore much more power is possible from a pair of tubes
working in class AB than for the same pair working in class A.

Plotting the minimum class AB1 RLa load.
Fig 2. ( repeated here the same as above ).

For all PP hi-fi amps considered at this website, only class A1 or AB1 will
be considered suitable for hi-fi because there are  no net benefits of class
A2 or AB2 which must include more complicated methods of low impedance
drive to output tube grids to cope with grid current. For any hi-fi amp, it is
better to use a quad of 6550 in class AB1 than have a pair of 6550 being
flogged to death in class AB2 to extract the same PO as the Quad.

To draw the load for minimum Class AB RLa-a the minimum Class B
load for a single tube of the pair must be plotted.

Examine Fig 2. This shows the Ra limiting line = 280 ohms.
The Knee  of this curve is shown at Ea = 100V and Ia = 350mA.

Plot point D at the Knee.

Point D is usually below the Pda curve and if D appears above Pda limit,
then plot point D where Ra curve intersects Pda curve, so that D appears
at the lower of the two possible positions.

Draw a straight line from Point C on Ea axis at 500V through D and on to
Ia axis and plot Point E.

The line E D C is the minimum class B load for the tube, ie, 6550.

The RLa B load value = Ea at C / Ia at E = 500V / 450mA =  1,111 ohms.

The Class A load line for the AB loading conditions may be plotted :-

Plot Point F on line EDC where Ia = 2 x IaQ.
Point F is at Ia = 2 x 50mA = 100mA.
Draw straight line from F through Q and on to Ea axis and plot Point G.
The distance between F and Q should be equal to between Q and G.

The line FQG is the class A loadline for each tube of the PP pair for AB
operation.

FQG is *always* = 2 x Class B RLa.  ie, = 2,222 ohms, this example.

The class AB RLa-a is always considered 4 x Class B RLa, or 2 x
Class A RLa.

OPT-1A, Minimum Class AB RLa-a = 4 x 1,111 = 4,444 ohms.

The peak anode voltage swing at each anode = EaQ - Ea minimum.
= 500V - 100V = 400Vpk. Ea minimum occurs at point D, and it
may be read off the graph looking vertically below D to Ea axis.

The Va-a = 0.707 x 2 x Ea pk swing at each anode
= 0.707 x 2 x 400Vpk = 565.6Vrms.

Class AB output power for plotted AB load = Vaa squared / RLa-a.
OPT-1A, PO = 565.6 x 565.6 / 4,444 = 72 Watts.

Class A power for any class AB RLa-a less than the RLa-a for maximum
pure class A = 0.5 x RLa-a x IaQ squared.

OPT-1A, RLa-a = 4,444 ohms, Ia dc per tube at idle = 50mA,
PO = 0.5 x 4,444 x 0.05 x 0.05 = 5.6 Watts.

This means that the power up to 5.6 Watts is pure class A, but all the rest
of the power is produced by class B action.

3. Check the anode dissipation for sine wave operation.
The tubes will dissipate varying amounts of heat for varying audio signal
levels and varying load values at different frequencies. The anode
dissipation, Pda, is the product of sustained voltages x currents flowing
in the tube. If a 6550 has Ea = 500Vdc and Ia = 50mA dc, at idle, then its
Pda = 500v x 0.05A = 25 Watts. The dc Pda plus signal caused Pda should
not be allowed to exceed the limits quoted in data for the tube for more than
a very brief time.

If the amplifier is designed by using design methods shown at this website it
is unlikely that the load range suitable for the OPT will ever cause tubes to
overheat. The 2 main reasons are :-

1. The RLa-a load value is chosen so the class B load line for one tube of the
PP pair will intersect the Ra limiting line at Ia lower than where the Pda
maximum intersects the Ra limiting line, see point D in Fig2 above.

2. Ea voltages chosen are NOT the highest which could be used. Although
Ea could be +800Vdc for KT88/6550 etc, and although 140 Watts is available
in virtually class B conditions, such high Ea bring reliability problems and high
THD. If for any reason the RLa-a is slightly lower than optimal, the tubes
may all too easily overheat. Adjusting bias with high Ea can be tricky.

As a general rule, l could suggest the proposed maximum AB power out
never be more than 0.7 x Pda limit of all tubes in an output stage.
For example, 2 x 6550, Pda limit for two tubes = 84Watts, so do not design for
working loads to give more than 0.7 x 84Watts, ie, 58 Watts of audio power.
2 x EL34, Pda limit = 56Watts, power output < 39Watts.
Even though higher maximum Class AB output power is available if RLa-a
is reduced on ohm value, just don't, because it leads to overheating, and
increases THD at all levels, reduces class A, increases Rout, and ruins
music.

I have enormous reluctance to use Ea higher than :-
+500Vdc for any octal tubes such as KT88, 6550, KT90, KT120, or for
non octal industrial tubes like 13E1.
+450Vdc for EL34, 6L6GC, 807, KT66, 5881.
+350Vdc for EL84, 6V6, 6CM5, 6GW8.
+250V for EL86, 6BM8, EL95.
+1,000V for 813.

The same table could be used for most of the above when used in triode
or UL connected with some exceptions, ie, +375Vdc for 13E1, and
probably lower also for 813. One must be very careful not to use Eg2 too
high lest Pg2 exceed limits or cause Eg1 bias to be excessively negative and
unable to properly control Ia bias current.

For real triodes, Ea should not exceed :-
+300Vdc for 2A3,
+450Vdc for 300B,
+1,200Vdc for 845, 211, 805, GM70.

Whenever the tubes operate in pure class A1, the anode dissipation will
never be higher than the idle Pda, Max Pda = EaQ x IaQ, and in Fig 2 with 6550,
Pda max = 500V x 0.05A = 25 Watts. With a load giving only pure class A the
Pda will reduce when any audio power is produced, and Pda in minimum
at clipping. In fact, with tetrodes and pentodes in pure class A the anode
efficiency is highest at about 45% so that if there are two 6550 with total Pda
= 50Watts, then Maximum pure class A power = 22.5Watts.
At this maximum Class A, Pda = Pin from PSU - PO = 50W - 22.5W = 27.5Watts
for both tubes, so Pda per tube = 13.75Watts. But in a class A or class AB
amp used for hi-fi the average PO level is rarely above 1/10 of the maximum
possible intended PO, so Pda is always nearly constant and temperature is
stable unless faults are present or loads are too low.

But with class AB where the amount of possible initial class A power is
less than 1/4 of the total AB power possible for the load value, then the
Pda will always rise after the first few watts are produced.
------------------------------------------------------------------------------------------------------
Calculating Pda for output tubes for class AB RLa-a.
For where RLa-a is low enough to cause maximum peak Ia to be more than
3 x Pda rating / Ea, the Pda max will not differ much from a pure class B amp,
and will be at a maximum at less than the clipping level.
The lower the RLa-a load value, the higher the max Pda will be.

All class AB amps meant for hi-fi performance will have substantial Ia at idle
which will have little effect on Pda max for the lowest RLa-a likely to be used.

For much more info on Pda and how to calculate it go to my page
Anode-dissipation+waveforms

Formula for calculating Class AB Pda :-

OPT-1A.  Calculate Pda for Minimum Class AB RLa-a.
RLa-a = 4,444 ohms, Va-a max = 566Vrms, Ia at idle = 50mA.
Maximum class AB PO = 72Watts.

Pda maximum will occur with beam tetrodes or pentodes at
approximately 0.67 x maximum class AB PO = 0.67 x 72W = 48 Watts.
Vaa at 48W = 461Vrms.

Pda =
500 [[ ( 0.364 x 0.05 ) + ( 1.8 x 461 ) + (    0.364 x 0.05 x 0.05    )  ]]  - 48
4,444              [ ( 2.83 x 461 ) - Ia ]
4,444
= 500 [[  0.0182 + 0.1867 + 0.0037 ]] - 48

= 56.32 Watts for the two x 6550.
Pda per tube = 28.16 Watts.

NOTE. This result is consistent with the Pda graph for RLa-a = 4k0 at
my page at
Anode-dissipation+waveforms

NOTE. In class AB PP amps, Pda max should not be more than
0.67 x Pda limit for the tube.
This allows for some tube heating effects of Eg2 x average Ig2 which
may become considerable, and for the speaker loading of the tubes to be
less ohms than a nominal load, ie, there is some room for error.

Conclusion. RLa-a should not be less than 4k4 lest Pda rise too high.

4.  Calculate idling Ia bias current.
For PP class AB1 6550 for all connection modes, Ultralinear,
Beam Tetrode, Acoustical with CFB, or Triode.

Number of output tubes = 2.
Choose Ea = +500Vdc. Confirm Pda limit from data sheets = 42Watts.
For class AB1, choose Pda = 0.6 x Pda limit = 0.6 x 42 = 25Watts.
Calculate Iadc at idle, each tube = Idle Pda / Ea = 25W / 500V
Calculate Idle Pda, all output tubes = No tubes x Pda for one
tube = 2 x 25W = 50W.

For output stages where only pure class A is intended, Pda at idle may be
up to 0.75 x Pda maximum allowed. In the case of 6550, Pda for pure
class A only = 0.76 x 42 = 31.5Watts. For Ea = 500V, Ia = Pda / Ea
= 31.5 / 500 = 63mA.

But longest tube life, Pda at idle should never exceed 0.6 x Pda max.

Instead of using 2 x 6550 for class AB1, it is possibly better to use
4 x 6L6GC or 4 x EL34, with Ea at +400Vdc max, and Pda at idle
= 0.6 x Pda max for the tubes. For 6L6GC, Pda = 22 Watts so
at idle Pda = 0.6 x 22 = 13.2 Watts. These smaller and cheaper tubes
have Ea = 400V so Ia = 13.2 / 400 = 330mA. Two 6L6 will have nearly
the same Ia on each side of the PP circuit as for 1 x 6550. The amount
of power produced by a quad of 6L6 working with very easy conditions
will be similar to 2 x 6550, and sound just as good, and despite the hi-fi
cognoscenti prejudice against the 6L6 based on its major use in guitar
amps which are designed to have high distortion.

------------------------------------------------------------------------

5.  CALCULATIONS FOR PP BEAM TETRODES,
without load line analysis :-

( Triodes are separately considered below. )

6. Calculate anode to anode load for maximum pure
class A1 power.
RLa-a for 2 tubes =  1.8 x Ea / Iadc where Ea & Idc are for one tube at idle.

OPT 1A, Class A1 RL a-a load = 1.8 x 500 / 0.05 = 18,000 ohms.

For more than one pair of OP tubes, divide the above by the number
of pairs of tubes.

7.  Calculate maximum Class A1 power.

Max class A1 PO = 45% x Pda for all tubes at idle.
OPT 1A,  PO A1 max  = 0.45 x 50W = 22.5Watts.

Calculate maximum Va-a swing = Square Root ( Power x RL ),
OPT 1A, Va-a max = sq.rt ( 22.5 x 18,000 ) = 636Vrms.

8.  Calculate minimum Class AB1 RLa-a load for
maximum class AB1power.

To calculate the minimum class AB1 RLa-a, there must be some inspection
of the Ea vs Ia anode curves for the tube which show Ra curves for Eg1
and overall value of Eg2. The minimum class B RLa must be calculated.

Fig 2. ( repeated again )

The knee of the Ra curve for Eg1 = 0.0V is at point D, where
Ea minimum = 100V, Ia maximum = 360mA,
The minumum class B RLa = ( Ea - Ea min ) / Ia max
= ( 500V - 100V ) / 0.36A = 1,111 ohms.

RLa-a = 4 x RLa = 4 x 1,111 = 4,444 ohms.

Examine a set of curves for 43% UL connected 6550 :-

Fig 6.

Fig 6 shows curves for a 6550 with 43% UL taps. The point D is
not recognized as an easy to see knee of any curve but it can be
established at a convenient point on the Ra curve for Eg1 = 0.0V
and below where the Pda curve intersects the Ra curve.
In the above Fig 6 the point D is at Ea = 87V and Ia = 345mA.
For Ea = 450V, RLa minimum = ( 450V - 87V ) / 0.34 A = 1,052 ohms
or approximately 1,000 ohms giving RLa-a = 4,000 ohms.

Point D may be constant for all values of Ea between +350V and +500V,
thus giving a range of RLa minimum from 686ohms  to 1,100 ohms and
range of RLa-a = 2k7 to 4k4. Notice that the Ia at idle does not affect the
RLa min calculation.

The Output Transformer should give the widest number of load matches
depending on the Ea chosen.

Regardless of how useful the Fig 2 and Fig 6 graphs may appear to be,
They only suit 6550, KT88, KT90 and KT120. For all other tube types
the calculations will involve different Pda, Idle Ea, Ea minimums, and
Ia maximum and you MUST examine and understand what you are looking at.

9. Calculate maximum AB1 power into minimum RLa-a.

9A. Calculate max class AB1 PO from the data gained in Step 8.
PO at clipping = 2 x ( Ea - Ea minimum ) squared / RLa-a
For Ea = 500V, Ea min at point D = 100V, RLa-a = 4k4,
OPT-1A, Max AB1 PO = 2 x ( 500 - 100 ) x  ( 500 - 100 ) / 4,444
= 72.72 Watts.

9B. Assume the value of limiting Ra line = 280ohms for all power tetrodes
and pentodes :-

Maximum safe class AB1 Power,

PO =   0.125 x        RLa-a x Ea squared
( [ RLa-a / 4 ] + 280 ) squared

OPT 1A, RLa-a minimum = 4,444 ohms, Ea = 500V,
Max class AB PO =  0.125 x          4,444 x 500 squared

( [ 4,444/4 ]+ 280 ) squared

= 0.125 x 4,444 x 250,000 / ( 1,391 x 1,391 ) = 71 Watts.

NOTE.  For any other RLa-a between the minimum RLa-a, and up to
the RLa-a for pure class A, the same formula may be used for where the limiting
Ra line slope = 280 ohms.

10. Calculate Middle RLa-a for intermediate power
and for ideal Class AB1 for Hi-Fi use.

Calculate Middle RLa-a =

Minimum RLa-a x square root of ( Class A RLa-a / RLa-a minimum ).

OPT-1A, Minimum RLa-a = 4,444 ohms from Step 2B,
Class A RLa-a = 18,800 ohms from Step 2A.

Middle Beam Tetrode RLa-a
= 4,400  x sq root ( 18,800 / 4,444 )
= 4,444 x sq root 4.23 = 9,140 ohms, ( say 9k0.)

11. Calculate PO for middle RLa-a.

PO = 0.125 x      RLa-a x Ea squared
( [RLa-a / 4] + 280 )squared

OPT-1A, PO, RLa-a = 9,140 ohms, Ea = 500V,
PO = 0.125 x   9,140 x 250,000 / ( 2355 x 2355 )
= 51.5 Watts.

10. Calculate pure class A1 portion of power within class AB1 total
for middle RLa-a.
Class A1 PO for any RLa-a between RLa-a min and pure class A RLa-a for
maximum PO = 0.5 x Ia squared x RLa-a, where Ia is Ia dc at idle.

OPT-1A, Class A PO into 9,140 ohms
= 0.5 x 0.05 x 0.05 x 9,140 = 11.4 Watts.

Maximum AB PO for middle RLa-a will be between maximum class
AB1 PO for lowest RLa-a and maximum class A1 PO.

Fig 7.

Fig 7 shows the clipping power levels for one pair of 6550 in beam tetrode, UL, or
Acoustical with CFB and for Ea = 500V and for various RLa-a values.
The PO curves for limit of class A1 and for total AB1 are only valid for
the shown Ea and Ia at idle.

NOTE. The ideal load value chosen by anyone for an OPT should suit
everyone, because during any amplifier's life the owner may change and the
speaker impedance value may change. I therefore I believe is is stupid to
design any OPT for only ONE ideal load value, say 8 ohms.

There should be two or more load values which may be used depending on
how multiple secondary windings are configured or on what tapping points are
chosen along a single secondary winding. Plenty of dynamic dome&cone
speakers which have a nominal Z = "4 ohms" have in fact Z varying between
say 2.5 and 25 ohms with the 2.5 ohm dip between the crossover between
bass and midrange at say 250hz where there is maximum musical energy.
So the amplifier needs to be comfortable driving a low 2.5 ohms, and able to
generate high current at low voltage and at low THD. The load of 25 ohms
is no trouble to any amp, as current is low, and voltage high. Some full range
electrostatic speakers have high Z below 1 kHz, but falling Z above 1kH,
maybe only 1.5 ohms by 15kHz. Most amps can cope OK with this because
very little musical energy exists above 7kHz.

Therefore the amplifier which may be set for "4 ohms" should be arranged
so the RLa-a minimum load seen by the tubes is never less than RLa-a
minimum as I have described in previous steps. Many amplifier makers
arrange their amps so the absolute maximum PO occurs when the tubes
have RLa-a equal to one and only load which gives this absolute maximum
PO. It is done to increase sales figures, and to make their products look
better than they really are.

If you look at the Fig 7 graph you will see that the RLa-a = 3,500 ohms
gives PO = 76 Watts. If the OPT has an impedance ratio of 3k5 : 4 ohms,
or 875:1, and a "4 ohm" speaker has a dip to 2.5 ohms, then the RLa-a
becomes about 2k2, and such a load might overheat the tubes is a high
level is used. Old AR9 speakers were classic amp killers, because their
bass sensitivity and Z were both so poor.
The low load reduces the output tube gain by half, reduces the amount
of class A1 PO, reduces the damping factor while very much increasing
THD and IMD all all levels, and all that ruins music.

To avoid overheating tubes, it is better to have the OPT ratio of 4k5 : 2.5
ohms, ie, 1,800:1, or about TWICE what most amp makers provide today.
If an amp has 3 terminals on the rear panel and labelled COM, 4 and 8,
The use of an "8 ohm" speaker plugged between COM to 4 will give a
far better load match and better music than the COM to 8 terminals.

99% of listeners should find a pair of 6550 or KT88 will give excellent
sound *if the load-matching* is optimal.

Therefore, for a pair of 6550 in beam tetrode or UL or acoustical mode
should have the following ratios :-
RLa-a MIDDLE VALUE : 2.5ohms, 5ohms and 10 ohms.

The steps above have calculated the Middle Value RLa-a at 9k0 approx,
and this gives a healthy maximum PO = 45 Watts class AB1, with pure
class A max at about 11 Watts.

What is ideally wanted are 3 secondary load matches for between
2.5 and 10 ohms to give RLa-a = Middle RLa-a value.

13.  List the possible ranges of primary to secondary
load matches, list use possibilities.

OPT-1A. Middle RLa-a = 9k0.
Sec = 2.5 ohms, ZR = 3,600:1, TR = 60.0:1,
Sec = 5.0 ohms, ZR = 1,800:1, TR = 42.4:1,
Sec = 10.0 ohms, ZR = 900:1, TR = 30.0:1.

Table for wanted load match possibilities.
 ZR TR RLa-a ohms Sec RL ohms Nominal RL PO 3,600:1 60:1 4k5 to 9k0 1.25 to 2.5 2.0 High, class AB 9k0 to 18k0 2.5 to 5.0 4.0 Medium, class AB 18k0 to 36k0 5.0 to 10 8.0 Low, pure class A 1,800:1 42.4:1 4k5 to 9k0 2.5 to 5.0 4.0 High, class AB 9k0 to 18k0 5.0 to 10.0 8.0 Medium, class AB 18k0 to 36k0 10.0 to 20.0 16.0 Low, class A 900:1 30:1 4k5 to 9k0 5.0 to 10.0 8.0 High, class AB 9k0 to 18k0 10.0 to 20.0 16.0 Medium, class AB 18k0 to 36k0 20.0 to 40.0 32.0 Low, class A

NOTE. In Steps below, turn ratios required for load matches for Middle RLa-a to
approximate loads of 4, 8 and 16 will be calculated. All easy available matches
should evaluated. For example, If a match of 4k5 : 1.0 ohms is found after
calculation of secondary turns it indicates that other load matches may be
possible, ie, 9k0 : 2.0 ohms and 18k0 : 4.0 ohms.

-----------------------------------------------------------------------------------------------

Some notes about the use of OPT-1A :-

NOTE.  OPT-1A, When RLa-a = 9k0, there is an initial 10 Watts of pure
class A, with the remainder of 34 Watts produced by class B action. 10 Watts
is enough to produce an SPL = 99dB with speakers rated for 89dB/W/M.
The total of 44 Watts will generate SPL = 105dB, and for both channels the
SPL = 108dB at clipping. The levels heard while standing in the midst of a
busy 50 member orchestra may exceed what your speakers could accurately
produce, but the recording engineer has probably set a level limiter or signal
compressor to reduce excessive dynamic range before 108 dB SPL is reached.
99% of people I know are happy with one pair of 6550 in each channel.

NOTE.   It is ALWAYS WRONG to connect a 4 ohm speaker to a
winding meant for a higher number of ohms. For example, if 4 ohms is used
at a winding meant for 8 ohms, the RLa-a anode load is halved, and may
become lower than the safe allowable RLa-a. The result may damage
tubes by causing thermal runaway, but at low levels THD and IMD will be
at least doubled, and damping factor halved.

NOTE.   It is NOT WRONG to use a 4 ohm speaker connected to a winding
meant for a lower number of ohms. For example, if an 8 ohm speaker is used
at a winding meant for 4 ohms, the RLa-a anode load is doubled and the amp
produces a higher amount of class A1 power with a reduced amount of total
AB1 power. Providing there is still enough power without clipping, THD/IMD
is lower, and damping factor higher, so music will sound better.

NOTE.   Because most 16 ohm speakers will be old models made before 1970,
and because they are often more sensitive than speakers made after 1970, they
require little power to give good results; A pair of 16 ohm 1969 dual concentric
15" Tannoy drivers in 180 Litre ported reflex boxes will make magnificent hi-fi
with amps capable of only 10 Watts. Therefore if an amp is capable of 50 Watts
with 8 ohms there will be more than enough output voltage for any 16 ohm
speaker so there is no need to have a load match for anything above 8 ohms.

NOTE.    Quad-II amps have a load match choice for 8 ohms or 16 ohms.
I find that the ESL57 which has quite high Z below 500Hz is better driven
with the OPT terminals strapped for 8 ohms. Unfortunately, Quad-II do not
have a recommended way of making the OPT match 4 ohm speakers. If one
uses 4 ohm speakers with Quad-II amps set for 8 ohms, the RLa-a falls below
a safe RLa-a minimum. The use of 6550, KT88, KT90 in Quad-II will give
better results than KT66, 6L6GC, 5881, EL34.
Many old amps like Quad-II and many modern amps do not like low impedance
speakers at all and they offer very poor performance because not enough load
matches are are available. With Quad-II OPT, they may indeed be removed
from the amp chassis, gently heated to melt out the the potting compound
which is saved for re-use. The OPT may then have the existing wiring from
secondaries slightly altered and two added terminals fitted to allow low loss
configuration of available secs to give a load match of 4k0 : 4 ohms, with losses
being the same as for 4k0 : 16ohms. When 8 ohms is used at the 4 ohm setting,
the RLa-a becomes 8k0, and these venerable old ancient amps are then optimized
to meet modern expectations for hi-fi.

NOTE. With OPT-1A, and when using the 60t secondary winding configuration,
the load match is 10k5 : 8 ohms. Now if someone connects 4 ohms, then
RLa-a is halved to become 5k3. But this is still above the safe minimum RLa-a
and the tubes will not overheat and fidelity will remain passable. Doing things
my way will give you many years of trouble free hi-fi listening.

-------------------------------------------------------------------------------------------------

OPT Calculator program?

As far as I know there is no available "Output Transformer Calculator"
program online or available anywhere.  My design logic flow could be
used to construct a computer program where one would enter the design
requirements such as power, secondary load, tube Ra, core dimensions,
then with a click on a "calculate" button, out would come a specification
sheet listing core size, winding wire details and there would be a detailed
diagram showing the layers of wire built up in the winding bobbin, so that
a paper copy may be given to a manufacturer who could then proceed to
wind the OPT.

Alas, I am not a computer expert, but I can give you the flow of logic plus
background notes needed to understand what you may try to achieve
when you design your OPT.

Besides a PC with a printer, you will need some tools :-
Exercise book, pencil,
pocket calculator, scale ruler, and open mind, persistent attitude.

I invite anyone interested to prepare a PC program to include all diagrams
and details of the windings using a range of common wire sizes and core
types. It is October 2011 and so far since 2000, 4 gentlemen have tried
to make a program but none have completed the work.

--------------------------------------------------------------------------------------------

ABOUT THE METRIC WINDING WIRE SIZE CHART

The metric winding wire sizes were kindly given to me by a local Sydney
wire and transformer parts supplier. The original chart contained the same
copper sizes as shown for grade 1 with less enamel thickness and grade 3
with more enamel thickness.
I only use grade 2 which is the only grade shown in the chart below.
Grade 2 is the only grade stocked by my supplier because it is the industry
norm for 99% of high temperature rated winding wire for electric motors and
stressful industrial applications. The range of sizes shown are not all obtainable
off the shelf, and to get some sizes a wait for an order is involved, so I
sometimes have to design around the wire size available, which adds to the
challenge. Anyone not used to measuring in millimetres better start getting
used to metric because here the diameter measurement matters more than
the wire gauge, and there are AWG, SWG, BS, all very confusing, and I
don't have conversion charts so if you work in gauges and inches and feet,
provide your own solutions.

Before winding anything, make sure you have an accurate micrometer to
confirm that the size is correct. Wire should be measured with enamel
coating and with enamel carefully removed.
-----------------------------------------------------------------------------------------------

Fig 32. Blank sheet for printing out for plotting load lines.

Fig 33. Blank sheet for drawing 6550 beam tetrode load lines.
.

Fig 34. For Reference.

Fig 35. Blank sheet for drawing 6550 triode load lines.

Fig 36. Blank sheet for drawing
6550 UL load lines.

Forward to PP OPT Calc page 2.

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