There are several ways to heat a tube filament, each having it's own advantages. Here are a few methods. These are numbered. Explanations are sometime long, so read only what concerns to you!
1) Serialized filaments
The filament current has some tolerance. So sometimes a 5Volt / 1Ampere filament can draw 1.1 Ampere, or sometimes 0.9 Ampere. Only at 5 Volt, the filament temperature is correct. A problem may come if you have an unstabilized DC heater circuit, since the voltage depends on the current load. So at 1.05 Amps the voltage drops slightly below 5V, and at 0.95 Amps it gets a little above 5V.
Even larger problems can come if you serialize a 0.95 Amps tube and a 1.05 Amps tube of the same type! That is because a tube filament is a (very) unlinear resistor. One way or another, this current in the circuit will be the same for each tube. But... what will a heater do, that was produces 5Volts at 1.05 Ampere, and now it is supplied with only 1Ampere? You may expect perhaps one tube has now 4.75V and the other has 5.25V but unfortunately the difference is a lot higher. That is because the a the tube heater has quite a large positive temperature coefficient. (So the opposite of an NTC resistor). This makes the voltage difference much higher as expected. In all cases, when you serialize two volts that are 5 Volts, and put 10V on it, you will definitely not see exactly 5Volts in each tube.
Conclusion: Please use the old basic rule, heaters are to be used in series connection, when they are specified for this. Like P-Series TV tubes. Put 10 random PCC88 tubes in series, and they will all glow with the same temperature. Put them all in parallel, and they will all glow with a difference temperature, which is not what we intended. Well of course nobody will so do. However, with tubes made parallel connection, like E-Series tubes, in the same way, we have to say these should not be serialized. If you parallel 10 ECC88 tubes, they will all glow with the same temperature. Not so when you serialize them! Specially not when they are from different vendors. This is simply wrong use.
If the above is understood, it should be clear it is in general wrong, to supply a current source to a voltage specified heater.
2) Safe operated AC circuits
In one sentence: 5% maximum tolerance, no matter how the 5% adds up. So if you have 5 mains variation already, you should have no additional variation due to amplifier transformers not being fully correct.
3) AC heating
The Control Grid Voltage of a DHT (directly heated tube) when DC heated produces a DC electric field inside the tube Anodes. It is from the "left to the right" and indeed the tube is not symmetrically loaded by this. There is a misconception that AC Heating will make this effect disappear. The only way to deal with this is use tubes with best possible linearity and then it doesn't matter anyway. The EML tubes of the latest generation have a filament center tap, to ensure best possible symmetry inside the tube. Connecting this center tap to the outside world is not possible, since it requires an extra pin. However indeed in all (new generation) EML tubes this center tap is physically present inside the tube, enforcing in a mechanical way, that the filament electrical center and physical center are the same. (Even Western Electric 300B does not have this feature!). At EML we have drastically reduced AC hum of our tubes this way, and all output tubes can be used AC heated. For driver tubes or pre-amp tubes we do not recommend it.
4) Ultrasonic, or RF heating
Heating a tube Ultrasonic or RF, is a very interesting method, since any noise resulting from this will be eliminated by the output transformer. Besides, it is not audible anyway. The electronics for this is more difficult as one expects, and if anything unexpected will go wrong with the oscillator, the result is a broken filament. So do not experiment with this, unless you are an expert.
5) Safe operated DC linear circuits
This is what most people are using. HOWEVER.... there is no such a thing as a stable regulator IC, specially the low drop types are instable and tend to oscillate at the smallest mistake. Avoid low drop regulators if you are not an expert, and use normal ones. You can have an error with those unexpected and without explanation. Some modules are sold simply instable, and the very moment you make the slightest error, the output voltage can swing wildly, or add a few Volts RF signal to it. What you think is a silly simple regulator IC, is in fact a 10...30 Watt power amplifier with very high gain, very fast, and ultra high feedback. Any small error with the wiring and the feedback will become feed-forward due to inductance of the wire. This requires an RF frequency, for a piece of wire to become an inductor, but these IC's are incredible fast, and it happens before you know it. Then the manufacturer puts tricks inside to make it a low drop type. The normal (not low drop) types these cost only 30 cents in production in China, and for that price you can not expect something good.
Here are some observed problem and things to take care of:
Best is always: to set up a normal schematic for AC and then connect DC to that. If that will hum only very little with AC heating, it will sure not hum with passive DC heating, even if residual ripple is on it.
Hint: Try making a DC stabilized circuit with a suited power Zener Diode. You can't go wrong with this.
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