There were two fundamental methods of connecting valve heaters in old radios.
First there is parallel connection where every valve has the same heater voltage and are all connected across the same winding of a transformer. In this circuit each valve may have a different heater current.
Secondly there is series connection: here every valve has the same heater current rating although voltage ratings may be completely different.
Of course there may be slight variations. In parallel connected sets it is often found that the HT rectifier valve has its own heater supply winding on the transformer. This is a practical consideration as it is then possible to use a cheaper form of rectifier having a directly heated filament. Here, filament is synonomous with cathode and as such carries the rectified HT voltage of the set.
With series connected heaters it is normal for the rectifier heater to be part of the series chain and therefore must be isolated from its cathode.
Dial illumination can be a problem with series connected sets as if the lamp is also wired in series its failure will open circuit the valve heater circuit and disable the set. It is therefore not uncommon to find a dial lamp having a smaller filament current rating and for the filament to be connected in parrallel with a small resistor. In the event of dial lamp failure, and until the lamp can be replaced the set will continue to work albeit with slightly reduced valve heater voltages. Unfortunately most series connected valve chains also include the dial lamps without the extra resistor and thus carry the penalty of a dead set with the failure of one of the lamps.
Some sets, particularly some of US origin, having series connected valve heaters employ what is called a "line cord". The reason for the use of a line cord which is merely a wirewound resistor built into the mains lead is to get rid of surplus mains voltage so that the valves operate correctly. Valves designed for series connection often have higher than average heater voltages. Examples are the 50L6 and 35Z4 and the reason for this is to match the total heater requirement to the mains voltage. The closer the match the less the wasted power dissipated in getting rid of excess voltage. For example if the mains supply is 115 volts and five valves require 50, 35 and three at 6.3volts, the total requirement is 104 volts. Fine for US mains where the excess voltage is a mere 12 and can be accommodated by a couple of 6 volt dial lamps, but awkward for UK mains where the nominal voltage is now 240. In this case the excess voltage is 136. If the heater chain requires 300mA then the power dissipated in the line cord is 136 times 0.3 or about 41 watts. In fact it is not uncommon for a line cord to also carry 70mA or so of HT current as well. This accounts for an extra 9 watts. The advantage of the line cord is that these 50 watts are spread out over a couple of yards and the heating effect is relatively small. Compared with a ballast resistor whose heat loss is concentrated in a small volume the line cord wins hands down. Unfortunately line cords are hard to come by, especially if the one you need has a fancy specification including partial HT drop and some dial lamp requirement as well as the main heater chain.
What if a line cord fails?
There are a number of obvious solutions: replace the cord with a ballast resistor; fit a transformer to match the total heater voltage or perhaps fit alternative valves and use a standard low voltage transformer. Unfortunately set designers may have taken advantage of the absence of a mains transformer and kept the chassis size to an absolute minimum. In this case fitting a mains transformer and keeping the overall look of the set is impossible. If however there's just enough space then before tackling the job, consider the effect on the loudspeaker. You don't want to spend an age modifying a set only to discover when you've finished that the speaker produces a loud hum from being magnetically coupled to the new mains transformer.
At the end of the day getting the set going will be some sort of compromise. If you wish to preserve as much of the original design as possible then you will not want to change any valves. For example one could remove the rectifier and fit a silicon diode. But first consider the result of doing this. The excess voltage will now be increased by 35 or 50 volts. Best keep the valve as a lot of extra watts will need to be dissipated. You could change the valve lineup. For example an early Pilot Little Maestro which uses the lineup 35Z4/50L6/6Q7/6K7/6A8 could be revalved 6L6/6Q7/6K7/6A8 with a silicon diode rectifier and the heaters supplied from a small 6.5 volt transformer. True the heaters would need rewiring unless you used a 24 volt transformer. These changes could be said to change the identity of the set somewhat and for the purists another solution would be more acceptable.
Is there another solution? Yes, one can use a silicon diode to modify the mains power from full wave to half wave. Chopping off half the AC with a diode can drastically reduce the amount of waste heat in a ballast resistor. Heaters only need "heating" after all. Unfortunately one cannot readily specify a working voltage when using half wave AC. A meter would probably not tell you the truth about the voltages as most meters are calibrated for RMS voltage. The problem isn't by any means new. Before TV tube heaters were driven from DC they were supplied at a frequency of many KHz from a small winding on the line output transformer. Trying to measure the voltage with an AVO meter was a waste of time.
Think of the power in an AC waveform as being related to the area under the curve of the sine-wave. Remove half the area and you remove half the power. Our valve heater requirement was 104 volts at 0.3 amps. The power requirement is therefore 31 watts. With half wave AC we require the same amount of power which is now delivered at half the volt rate. So I guess we now need 208 volts. The ballast resistor of 87 ohms now has to sink only 32 volts. The waste power is therefore dramatically reduced and would stand at about 12 watts if we had full wave AC but in fact only 6 watts as we are now using half wave! What sort of diode do we need? Well the forward voltage drop is a mere 0.6 and the current 0.37Amp and it's half wave so a quarter watt device is more than adequate. The voltage rating will be equal to the peak of the incoming mains minus the total heater voltage. However as the heater chain will have a low "cold" value to be safe I'd use a 400 PIV device and to prevent a surge from damaging the valves I'd consider using a temperature dependent resistor having a negative co-efficient (ie. it starts high and drops to a low resistance) in series with the ballast resistor.