[TCML] tank circuit power

[Jim Lux]

The question comes up of what is the RMS current in the primary tank. 
This comes up when looking at losses, and when looking at RF safety 
calculations for the magnetic field from the primary.

A few basic assumptions:

The loaded Q of a tesla coil is approximately 10.
The equation for Q = 2*pi * Energy stored/(energy lost per cycle)
or, Energy lost = Energy stored * 2 *pi/Q
For Q = 10, that's 6.28/10 or .628...
That is, the tank loses about 62.8% of its stored energy in every cycle. 
  If you look at the voltage, it goes as the square root (so each cycle 
is about 60% the amplitude of the previous)

Previous analyses have shown that one of the larger losses is in the 
spark gap (which has a  "cathode drop" of around 100V per gap. (why a 
rotary or blast gap is more efficient than a static multi gap)

 From that, we can calculate what the stored energy in the primary 
capacitor is (1/2 C *V^2) and from the stored energy in the cap, we can 
calculate the peak current in the primary (= energy in cap, to a first 
order)

the next question is "how much energy is dissipated in the resistance 
for a half sine with peak value X".  For this, I assume that there's no 
loss during the actual half cycle, so the current is perfectly 
symmetrical around the peak:

So what we want is

integral[sin(omega t)^2] for t=[0, pi/omega]

which is

(omega t)/2 - sin(2omega t)/4 +C

This is zero for t=0

(omega pi/omega)/2 - sin(2* omega pi/omega)/4
or
pi/2 - sin (2pi)/4 = pi/2

So the energy dissipated in the primary resistance, R, if the peak 
current is I, is
R *  pi/2 * I^2 * thalfcycle

Then you build up a little spreadsheet and sum things up..
( stored is how much energy is in the cap at the peak, the loss is the 
joules lost to the resistance, to sec is the energy transferred to the 
secondary)

Here's an example:
L	3.70E-05	Q	10		
C	6.80E-08	r	0.1		
f 	1.00E+05				
	V	I			
t			stored	loss	to sec
0	21000	900	14.99	0.63	7.30
0.5	14406	618	7.06	0.30	3.44
1	9883	424	3.32	0.14	1.62
1.5	6779	291	1.56	0.07	0.76
2	4651	199	0.74	0.03	0.36
2.5	3190	137	0.35	0.01	0.17
3	2189	94	0.16	0.01	0.08
3.5	1501	64	0.08	0.00	0.04
4	1030	44	0.04	0.00	0.02
4.5	707	30	0.02	0.00	
					
	Total joules		1.20	13.78


With twice the Q
L	3.70E-05	Q	20		
C	6.80E-08	r	0.1		
f 	1.00E+05				
	V	I			
Cycle			stored	loss	to sec
0	21000	900	14.99	0.63	3.70
0.5	17703	759	10.66	0.45	2.63
1	14924	640	7.57	0.32	1.87
1.5	12581	539	5.38	0.23	1.33
2	10605	455	3.82	0.16	0.94
2.5	8940	383	2.72	0.11	0.67
3	7537	323	1.93	0.08	0.48
3.5	6353	272	1.37	0.06	0.34
4	5356	230	0.98	0.04	0.24
4.5	4515	194	0.69	0.03	
					
		Total joules		2.12	12.21

with 0.4 ohms and Q=10
L	3.70E-05	Q	10		
C	6.80E-08	r	0.4		
f 	1.00E+05				
	V	I			
Cycle			stored	loss	to sec
0	21000	900	14.99	2.54	5.40
0.5	14406	618	7.06	1.19	2.54
1	9883	424	3.32	0.56	1.20
1.5	6779	291	1.56	0.26	0.56
2	4651	199	0.74	0.12	0.26
2.5	3190	137	0.35	0.06	0.12
3	2189	94	0.16	0.03	0.06
3.5	1501	64	0.08	0.01	0.03
4	1030	44	0.04	0.01	0.01
4.5	707	30	0.02	0.00	
					
		Total joules		4.79	10.19