This example looks at an unusual large planar spiral coil Tesla coil to demonstrate modelling and measurement of its fundamental resonant frequency.
The coil was built in 2007 by Bart Anderson, http://www.classictesla.com/photos/bigflat/bigflat.html. Close examination of the photos show that the inner of the winding is octagonal, the middle section has 16 sides and the outer has 32 sides. Rather than model this detail, this example will treat the coil as a planar circular coil.
The coil is operated with grounded rim and the high voltage is produced at the center. The measured resonant frequency in this configuration with no center load is 438.5 kHz.
Input file baps-rg.in contains
ground { center 0,0,0 axis 0,0 }
coil {
radius1 50.625/2 in
radius2 3.625/2 in
end1 0,0, 31.25 in
axis 0,0
length 0
turns 95
wirad 18 awg
}
and the Spice sub-circuit is generated with the command
lcng -o spice baps-rg
The correct method to measure the resonant frequency of a coil is to apply a low impedance signal generator to the end which would normally be grounded, leaving the other end, and taps if there are any, open circuit. Sometimes a coil is measured with its topload connected if the resonance of the combined system is required.
Resonance can be detected either by monitoring the current from the signal generator via a current transformer, or by detecting the electric field of the coil by a sensitive E-field probe placed not too near the coil. The probe must be far enough away so that its capacitance to the coil does not significantly affect the resonance, therefore it needs to be quite sensitive.
The Spice model for the test circuit is very simple, just an AC source connected to the outer rim terminal of the coil. File baps-rg-test.spice contains
.include baps-rg.spice * pin 1: zero of potential at infinity * pin 2: L1 end1, 0.00 turns * pin 3: L1 end2, 95.00 turns V1 1 0 DC 0 AC 1 X1 0 1 2 baps-rg .control ac dec 1000 10k 1000k setplot ac wrdata /tmp/baps-rg abs(V(2)) .endcand this is run with
ngspice -b baps-rg-test.spiceto produce a file /tmp/baps-rg.data which contains two columns: frequency and abs(V(2)) which is the center terminal voltage. Plot the data using your favourite plotting program. For gnuplot you might use the commands
set terminal x11 set style data lines set logscale y plot '/tmp/baps-rg.data' using 1:2The resonant frequency can be read off the plot by placing the cursor. Alternatively you can use a shell command to extract the record with peak value of abs(V(2)) by sorting reverse numerically on column two and selecting the top row:
sort -gr -k2 /tmp/baps-rg.data | head -1But that only works if, as in this case, there is only one resonance in the frequency range of the Spice simulation. The result is 429.5 kHz which is 2% below the measured resonant frequency. Usually lcng predictions are a bit on the high side because the model doesn't take account of stray capacitances of the surroundings. lcng predicting a lower frequency suggests the model doesn't have enough resolution with its default of 10 sections.
Inserting
sections 20into the coil description in baps-rg.in and re-running produces a resonance prediction of 439.5kHz which is more realistic with an error of +0.23%. Trying 40 sections (which is rather a lot for any coil) reports a resonance at 445.6 kHz which is +1.6% with respect to the measured value. This is more typical of the expected error of a realistic model.
The model can be improved slightly by introducing some features of the environment around the coil. In this example a wall 15 feet away is 10 foot high, and there is the E-field probe which is a 10.2 inch diameter disc level with the coil and placed 5 feet from its edge.
Putting these into the input file, we get baps-rg2.in which contains
ground { center 0,0,0 axis 0,0 }
coil {
radius1 50.625/2 in
radius2 3.625/2 in
end1 0,0, 31.25 in
axis 0,0
length 0
turns 95
wirad 18 awg
sections 40
}
electrode {
; Nearby wall 15 feet away
plate {
points { 15 ft, 30 ft, 1 in } { 15 ft, 30 ft, 10 ft }
{ 15 ft, -30 ft, 10 ft} { 15 ft, -30 ft, 1 in }
}
; The E-field probe plate 10.2 inch diameter 5 ft away at the same
; height as the coil, facing the coil
disc {
center 0, -(5 + 50.625/2 * 12) ft, 31.25 in
radius 10.2/2 in
axis 90,90
}
}
The wall and probe are effectively grounded so these are lumped together
into a single electrode.
We also need a modified Spice model baps-rg2-test.spice because the coil sub-circuit now has an extra pin:
.include baps-rg2.spice * pin 1: zero of potential at infinity * pin 2: E1 * pin 3: L1 end1, 0.00 turns * pin 4: L1 end2, 95.00 turns V1 1 0 DC 0 AC 1 X1 0 0 1 2 baps-rg2 .control ac dec 1000 10k 1000k setplot ac wrdata /tmp/baps-rg abs(V(2)) .endcThe predicted resonant frequency falls to 444.6 kHz which is +1.4% which shows that the wall and probe are not having much effect on the coil.
Is is likely that the supporting insulators of the coil are providing another couple of percent of 'stray' capacitance, but unfortunately lcng does not model dielectrics.
This result is fairly typical of coil models. When everything is working correctly the resonant frequency predictions tend to be on the high side due to un-modelled electric field effects. The opposite can occur if there are sheet conductors within the magnetic field of the coil. These will reduce the effective inductance and raise the resonant frequency. lcng accounts for these effects in the ground plane, but not those due to other electrodes.
This coil required 40 sections to model the fundamental accurately. With solenoid coils, usually only 10 are required but with rim-grounded planar coils, there is a large voltage rise across the outer few turns and the extra number of sections is needed to model this accurately enough.