Tube Amplifier Theory



On this page I will publish some tube amplifier theory. Output transformer impedance calculation is a bit tricky before you get the hang of it. Some useful formulas for calculation of output power, distortion and impedances will also be published. The idea to make this page came up when I bought the Namo Web Editor, which is capable of writing maths (one of the reasons why I bought it).

Transformer Impedance Calculation

A kind of transformer

Primary current in at the dot gives secondary current in at the dot.


Law of induction (Faraday-Henry law):
 

Induced emf in a solenoid:

From this formula we get (magnetic flow is the same through both windings):

For the transformer above (1:1.33 or 3:4) this means:


The power on the primary and secondary is equal, which means:



Now the tricky stuff, the impedance calculation:

If we connect a 100R resistor on the secondary and apply 1 Volt on the primary we get:






The consequence of this is:



This leaves us with the formula for impedance calculation:



Windings in parallel does not change the number of turns, windings in series adds the number of turns. This is a bit difficult to understand, normally paralleled equal resistances gives half the resistance, but not in the transformer world, and to make it even worse equal windings in series increases the resistance with a factor of 4!

In order to understand this we rewrite the formula for induced emf in a solenoid:



From this formula we see that the magnetic flow is proportional to the voltage per winding-turn. If we use parallel windings, no turns are added (acts as if the wires were connected in parallel). In series connection the turns are added (we need more voltage when the number of turns is increased to get the same magnetic flow).

LL1623 connection alternatives
Example:

Above you see the data for the Lundahl LL1620,1623,1627 output transformers. We are going to calculate some connection alternatives as a practice (LL1623):

Primary: 4x13.4
Secondary: 8x1

All primaries in parallel and all secondaries in parallel gives the transformer ratio 13.4:1, the primary impedance with 8 Ohm on the secondary is:



Two primaries in series, paralleled and all secondaries in parallel gives the transformer ratio 26.8:1, the primary impedance with 8 Ohm on the secondary is:



Four primaries in series and two secondaries in series, paralleled gives the transformer ratio 13.4:1, the primary impedance with 8 Ohm on the secondary is:



With all four primary windings in series, the number of secondaries used for a primary load of approximately 3 kOhm at 4/8/16 Ohm secondary is listed below together with the resulting load:
 

Number of secondaries (connection alternative)

Resulting load impedance

2 ((1+1)//(1+1)//(1+1)//(1+1))

2873 Ohm @ 4Ohm

3 (((1//1)+1+1)//((1//1)+1+1))

2554 Ohm @ 8 Ohm

4 ((1+1+1+1)//(1+1+1+1))

2873 Ohm @ 16 Ohm


Add the primary resistance of 164 Ohm to the figures above and you have the correct load.
 

Now the push-pull alternative, which is a bit difficult to understand. We use two primary windings for each tube in the push-pull output stage, and two windings in series on the secondary as in the example above. The resistance from anode to anode is in this case 5745 Ohm, and this is the figure used by transformer manufacturers. We are interested in the load on ONE tube, which uses 2 windings in series:



This looks suspicious, but the output tubes are in reality paralleled (with inverse phase connections), and this means that the load for each tube is 2872 Ohm (half the anode to anode load). The same applies for paralleled single end, if you use two tubes the transformer load shall be 50% of the value that is used for a single tube, and with four tubes the load shall be 25% (in this case Ohms law can be used).

From Sowter I 'borrowed' this picture showing connection alternatives for their transformers using 8 secondary windings.

Sowter output connections

The same formulas are used for interstage transformers, but these are often specified as for example 2k:4k. I think this means that it is designed for a source impedance of 2k, and the transformer output impedance is 4k. The transformer ratio is then 1:1.4 (remember the square?), and the primary inductance probably around 20-30H (see formulas below).

This concludes the transformer impedance calculation section. If anything is unclear or wrong, please contact me.

Other Important Formulas For Transformer Calculation

The output result from the transformer depends also on the frequency response, and the low cut frequency is calculated from the Primary inductance. In these calculations it is the output impedance of the driver tube that shall be used.

The -3dB low frequency limit is calculated at the frequency where the output impedance of the tube equals the input impedance of the transformer (at low frequencies this depends upon the inductance):





The LL1623SE 90mA has a primary inductance of 30H, and with a single anode loaded 300B (output impedance approximately 700 Ohm) this makes the low cut (-3dB) frequency:


Lundahl also specifies the -3dB power output point (maximum power output reduced with 50%) as the point where the primary load impedance equals the primary inductance, and with a 3 kOhm load this makes:



This is not as important as it looks, you will not find music signals with full amplitude at such low frequencies.

Power Output and Distortion Calculation 

300B example

Above is the loadline of a 300B into 5.1 kOhm. I will use this as an example how to calculate distortion and power from a curve set. The slope of the loadline is calculated from the load impedance, and with a 5.1 kOhm load the current (I=U/R) per 100V is 19.6mA. Below is the result that the Glass Ware SE Amp CAD has calculated for this loadline. Check the 'Tube CAD' link for free 'True curves', each month a new tube.

SE Amp CAD result

Below are the values at the three interesting points for power and distortion calculation (marked with green and red lines in the curve set).

SE Amp CAD min readingsSE Amp CAD middle readingsSE Amp CAD max readings

Apart from the voltage and current readings, it is also interesting to see the changes in mu and Rp. These can also be derived by graphical methods from the curve set. I will describe this in the 'Other Formulas' part.

The formula used for output power calculation is:



With the values from the loadline above we get:



The SE Amp CAD got 8.35W, the difference is because of not perfect readings above.

The formula used for calculation of the 2nd harmonic is:



With the figures from our example we get:

 

The result in dB is:

-20 log(0.026) = -31.7dB

For the 3rd harmonic calculation we need 2 more points from the curve set. These points are called I_x and I_y, and are the point at the loadline where the grid voltage is maximum positive 'grid swing' divided with the square root of 2 (I_x), and the maximum negative 'grid swing' divided with the square root of 2 (I_y). These readings are found below.

3rd harmonic curve point ix
3rd harmonic curve point iy

The formula used for calculation of the 3rd harmonic is:



With the numeric values from our example this is:



The result in dB is:

-20 log(0.0042) = -47.5dB

This calculation seems a lot more uncertain. The difference to the SE Amp Cad result is 6dB, but it will at least give some guidance.

The THD is calculated as the mean square root of the 2nd and 3rd harmonics:




Some Other Useful Formulas

I will write this a little later.


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