Get Longer TC Sparks !!
TC Efficiency Theory
Much has been said about TC efficiency over the years, and a certain amount of controversy still exists.  This is an area that I've been investigating for some years now and some of my findings have been surprising and thought-provoking. I'll present some ideas in a question and answer format.

    1)  Wire thickness, secondary proportion issues, etc.

Ques.  I've heard that even for a small TC project, it's important to use thick wire on the primary and secondary, to seal the PVC secondary form before winding, to use robust connections, etc.  Is this really important?

Ans.  You are correct to question those statements.  In reality none of those things will have a noticeable effect on the spark length of a small Tesla coil.  In fact, contrary to popular belief, the use of thinner wire and more turns for both the primary and the secondary will result in longer sparks.

Ques.  Thinner wire... won't that increase the wire losses?

Ans.  Yes, it will, but the thinner wire lets more turns be used on the primary which will increase the primary surge impedance thereby reducing the gap losses.  The benefits of the reduced gap losses outweighs the negative effects of the thin wire, so the overall sparks will be longer, brighter and better.

Ques.  And this really makes a noticeable difference?

Ans.  In a comparison I did, the sparks were 10% longer for the same power input using thinner wire.  

Ques.  I looked at a lot of websites, and no one talks about this method.

Ans.  The only other person that I know of who tried this is Richie Burnett of the UK, he also gained 10% spark length when he used more turns of a thinner wire on his secondary.

Ques.  OK, but why does a higher primary surge impedance reduce the gap losses?

Ans.  Gap losses are proportional to the primary tank current, a higher impedance results in lower currents and lower losses.

Ques.  You say thin wire is good.  How thin should it be?  Should a huge coil use real thin wire?

Ans.  The wire should be proportional to the size of the coil. I like to use about 1600 turns on the secondary, and about 24 to 34 turns on the primary.  Realize though that it is not the thin wire or the many turns that is giving the benefit, rather it is the resulting many turns in the primary and higher primary inductance and Zsurge that gives the benefit.  

Ques.  Do the proportions of the secondary matter?

Ans.  It seems that wider secondaries are better, because they give a greater inductance which also helps to reduce gap losses, by requiring the use of more primary turns as explained above.  A proportion of 1:4 or 1:5 is probably quite good.
However, excellent results can be obtained from narrower secondaries, esp if one wants to build a compact coil and is willing to sacrifice some efficiency.

Ques:  Well, suppose I just build a huge secondary so I can get a high inductance but still use fewer turns of thicker wire?

Ans:  This might work, but often it is not desired to have short wimpy sparks coming off of a huge TC.  Generally the size of the coil is matched to the size of the sparks to keep a sense of proportions.  Also, if the secondary is too large, it will shade too much of the toroid's area, and decrease its effectiveness.

Ques:  Just how much loss is there in the spark gap?

Ans:  There's some controversy here, but it may be 15% to 30% or more depending on the coil design.

Ques:  Is it possible that the lower frequency that results from using more turns or a wider secondary also helps the spark length?

Ans:  Yes, it is possible that the lower frequency also helps by reducing losses or helping the quenching.  If the lower frequency is an important factor, then larger more powerful coils may benefit less than smaller ones from the use of thiner wire with many turns, because the frequency of large coils tends to be low anyway, and may be low *enough*.  I still think the larger coils should benefit from the reduced gap losses when using more turns of thinner wire though.  
      2) Break rate issues:

Ques.  I've heard that a higher break rate gives longer sparks?

Ans.  This can be true, but is just part of the story.  When all is said and done, I've found that the ultimate efficiency is obtained at a relatively low break rate (bps).  Any coil that gives a certain spark length at a high bps, can be redesigned to give an even longer spark for the same power input at a lower bps.

Ques.  What are the factors to consider then?

Ans.  If a coil uses a relatively small capacitor, it will need a high break rate to give long sparks.  A coil with a larger capacitor (or a higher input voltage) will give long sparks at a lower bps.  Neon sign transformers are so fragile, they tend to burn out if used with a non sync rotary, esp at low bps, so it is best to use a synchronous rotary spark gap with NST's.  It is possible that the sync gap is also best (efficiency-wise) even on pig or potential xfrmer powered TC's.

Ques.  What bps seems to be the best then?

Ans.  I've gotten excellent results at 120 bps.  The overall efficiency was 20% better than at 240 bps.  However, a low bps TC will require a larger taller secondary and a larger toroid to handle the larger "bang" size to prevent racing sparks on the secondary coil, and premature breakout from the toroid.
                                    More bps and other TC theory
    3) Magnifiers

Ques:  I heard that magnifiers are much more efficient than normal TC's because they use tight coupling and that they also require smaller capacitors, higher breakrates and special rotary gaps?

Ans:  Magnifiers work very similarly to normal TC's.  Although the driver coupling is tight, the overall coupling is similar to a normal TC.  In my work, magnifiers worked best using the same cap sizes and breakrates as a normal TC.  I did not see any advantage in using a special type of rotary gap.  I have never seen a magifier that outperformed a normal well built TC.
     4)  Other efficiency issues:

Ques:  How important is the size of the toroid?

Ans:  It is very important for longest sparks.  The toroid size depends on the power input.  A toroid that is too large or too small will not give the best results.  Longest sparks will be obtained when the toroid is sized so that the sparks can barely break out, and just one streamer is formed.  However often it is desired to use a little smaller toroid to give multiple streamers.  Often the sparks will reach about 3 or 4 times the toroid diameter, but only if the toroid is sized correctly.  The sparks often reach 10 times the toroid minor diameter.  For example a toroid that is 4" thick may give sparks that are 40" long or so, if the coil is properly designed.  The toroid should overlap the secondary coil and be two or three times wider in diameter.  The toroid should be raised above the coil high enough so that it gives the longest sparks, but not so high that corona begins to flair off of the top secondary windings.  A smooth toroid will give longer sparks than a rough surfaced one, for a given toroid size.

Ques:  You mentioned something about using higher primary voltages?

Ans:  This is a good way to increase the efficiency of a TC, but often suitable xfrmers are not readily available.  Also suitable clearances must be provided for the higher voltages to prevent arc-over between components in the spark gap, primary, etc.  The higher voltages reduce the spark gap losses by reducing the current in the gap.

Ques:  How long will my sparks be for a given input power?

Ans:  In a well designed TC, the spark length will follow the formula:
      
     spark length (inches) = 1.7*sqrt input power (wallplug watts)

I also have a newer more complex formula that I did not post here.