PVC Single Layer Tesla Coils

In spring 2011, the students in my AC class at ACC built simple single layer solenoids using hardware store PVC pipe sections as forms, and magnet wire from Radio Shack. They then measured the resistance and inductance of their coils, and more fun, found self-resonant frequency of these coils using neon bulbs and fluorescent bulbs as high voltage detectors.

Family of Coils

The Top Coilers in Reverse Alphabetical Order are

The students used 1" PVC pipe (1.3" OD), cut to about 8 inch lengths for the initial forms. They drilled two small 0.060 holes at the base of the PVC as a starting wire holder, then patiently built a close wound coil using their entire chosen spool of Radio Shack magnet wire.

Radio Shack magnet wire - 278-1345B - has three spools of different gauge wire. The red spool has 200 feet of 30 gauge wire. The green spool has 75 feet of 26 gauge wire, while the bronze spool has only 40 feet of 22 gauge wire. Tesla coils are a transmission line in disguise, operated at a frequency corresponding to a 1/4 wavelength. f_r = (c/(4*lambda)). Consequently, the longest length wire has the lowest frequency resonance. The red coils, while the most tedious to wind, were easily sent into resonance with our lab equipment. The green coils, using a third harmonic of a square wave could likewise be excited. However, as built, the bronze coils were unable to be be driven at resonance without capacitive loading.

Radio Shack 278-1345 Wire Properties
Wire Color
Quarter Wavelength
Frequency in Space
30 g
200 ft
11 mil
103 Ohm/ft
1.23 MHz
26 g
75 ft
19 mil
40.8 Ohms/ft
3.28 MHz
22 g
40 ft
30 mil
16.14 Ohms/ft
6.15 MHz

Winding the red forms took about 45 minutes. The green and bronze forms took correspondingly less.

After the coils were wound, we drilled small holes at the form terminus to hold the finished wiring, then sprayed a coat of polyurethane varish over the coil which we left to dry overnight.

The next class period, the students removed varnish from the ends of the wire using sandpaper or jeweler files, and measured the resistance and inductance using a DVM and LCR meter.

Red Coil Measurements

1 inch PVC, wood base
R measured 23.3 ohms. We expected 26 ohms, so okay. (A bit short.)
Winding length 6.4375 inches. Expect number of turns to be about N=6.5375/0.011 = 585
L measured = 2.21 mH using Elenco LCR-1801 LCR meter. (calculate 2.1 mH using Wheeler's long/narrow formula)
Measured resonance at 1932 kHz using neon indicator at end of coil

Green Coil Measurements

R measured 3.9 Ohms. Expected 75'*40.8O/1000' = 3.0 Ohms. (More resistive, or longer than expected.)
Number of turns from length/(pi*D) = 75'*12"/(3.14*1.315) = 217.
L measured = 482 uH  (calculated 483 uH using N=217)
Measured resonance at 4130 kHz using neon indicator at end of coil

Bronze Coil Measurements

R measured 1.3 Ohms. Expected 40'*16.14/1000' = 0.65 Ohms. (This should be remeasured on a four wire Keithley.)
Winding length 3.0625 inches. Counted turns - N = 112
Number of turns from length/(pi*D) = 40'*12/(3.14*1.315) = 116. Use N = 112.
L measured 156 uH (calculated 148 uH)
Could not light neon using sine or square waves.

Capacitive Loading

Two items are noteworth about these Tesla coils. First, the actual resonance is above the straight line, free-space expected resonance. There is substantial `short cutting' (57% overspeed for the red, versus 25% overspeed for the green) being done by the signal capacitively cutting across adjacent windings, rather than staying strictly on the wire.

The actual capacitances of these circuit is natively in 3 pF range. These circuits are this affected by human presense, scope probe capacitance, and nearby metal and ionized gasses.

With the oscilloscopes, student can easily see 1kV on the coil output with the 10X probes. The frequency, however, is much lower than open coil resonance as found with a neon due to the 10pF 10X probe capacitance. Likewise, using a reduced signal with a 1X probe (100 pf) give lower resonance still. One technique to find the unloaded resonance frequency of the coil is to suspend the coil between two small four inch, 25 turn coils which act as transmitter and reciever. At the intermediate coil resonance, a remarkable increase in coupling is found.

For simple testing, the neon bulb sensor technique works well. The NE-2 bulbs ionize around 60V/mm field strength. The victim student  is encouraged to hold the neon leads, with the glass end adjacent to the top of the coil. (To increase emotional impact, it is better to call out the field strength as 60kV/m, rather than 60V/mm). At resonances, there is usually enough field to light the bulb, and the bulb can be moved down the length of the winding to provide a qualitative indication of the voltage distribution down the length of the coil. With the red coils, we can easily see the quarter wavelength, open far end voltage distribution, as well as the half-wavelength, 'jump rope' distribution where you drive both ends of the coil with the same driving signal, and see high voltage in the middle of the coil.

Grounded Neon Used as Voltage Probe

In the picture above, the ground lead substitutes for the experimenter's hand. The high voltage connection is capacitively coupled.The single wire or no wire at all connection is a fine source of brain extension for students who have been working with low frequency circuits all semester long.

Inducance Formulas

Not covered in the class are two useful formulas for inductance of single layer solenoids. These formulas are approximations from the elliptic integral expressions, and are limited to the regime indicated by the name. (Turns out the Long/Narrow formula works well for the short coil as well.)

Wheeler's Long/Narrow Formula

L= (d^2 * n^2)/(18d+40l)


L is inductance in micro Henrys,
d is coil diameter in inches,
l  is coil length in inches, and
n is number of turns.

This formula works very well (within 5% for the coils above).

Wheeler's Short/Fat Formula

Wheeler's formula for short coils (l<a, c<a)

L = (aN^2/13.5) log_10(4.9a/(l+c))

a = average radius
l = coil height
c = winding thickness

3% when (l+c) = a, and better as (l+c)/a approaches 0
(I assume inches and uH.)

  _ _ _
 |     |  }<- c
  - - -

- - - - - -  \
              | <- a
  - - -      /
 |_ _ _|

L =  (Diameter*N^2/27) log_10(2.45*Diameter/(height+thickness))

Testing Wheeler's Long/Narrow Formula

To test the Wheeler formula, a few more coils were made which varied the form diameter, and gauge. As seen in the photo at the top, we have an acrylic form
(1.758 inch in diameter), a 3/4 inch pvc form (1.050 inch diameter) with reduced length winding, a 1/2 inch pvc form (0.855 inch diameter) with reduced length winding, and finally a black, non-magnet wire form.

Acrylic, free standing
R measured 22.9 Ohms.
Winding length 4.754 inches. Number of turns 432
l = 4.754 N = 4.754/0.011 = 432
L measured = 2.63 mH (calculation - 2.6 mH)
Measured resonance 1741 kHz

3/4 PVC form using partial spool
R measured = 9.0 Ohms. Wire length calculated as 87 ft. (9.0 Ohms/103.2 Ohms/1000ft)
Winding length l = 2.892 in. Calculate N=2.892/0.011 = 263 turns
L measured = 583 uH (calculate 888uH)
L calculated (1.05)^2*(263)^2/(18*1.05 + 40*2.892) = 567 uH.
Measured resonance 4167 kHz

1/2 PVC form using partial spool
R measured = 8.0 Ohms.Wire length calculated as 77 ft. (8.0 Ohms/103.2 Ohms/1000ft)
Winding length l = 3.280 in. N = 3.28/0.011 = 298 turns
L measured = 444 uH
L calculated (0.845)^2*(298)^2/(18*0.845 + 40*3.280) = 433 uH
Measured resonance 5232 kHz (using third harmonic of square wave technique)

Black Coil Measurements
R measured 1.6 Ohms. (Length calculated as 1.6*1000/16.14 = 99'. Deemed not reliable. Wire more resistive than expected.)
Winding length 6.625 inches. Counted turns - N = 137.
Calculated length = 137*3.14*1.315"/12 = 47 ft.
L measured 113 uH (calculate 112 uH)

Testing Wheeler's Short/Fat Formula

I wanted to test a short, squat Tesla coil, but keep similar construction technique with previous work above. For the form, I used a wholesale CD container cover (polypropylene), and wound a 200 foot coil of 30 gauge wire.

Short and squat winding

Short Aspect Ratio Coil

height = 1.7 inches
N = 1.7/0.011 = 155
ID 4.95 inches
plastic thickness = 50 mils
OD = 5.05 inches

DC resistance 23 Ohms
L = 3.83 mH
Resonance 675 kHz

Calculated inductance using Long/Narrow formula
 L_long = D^2 N^2 /(40 l + 18 D)
        = 5.05^2 155^2/(40*1.7 + 18*5.05)
        = 612697/158.9
        = 3.85 mH   (This is pretty good . . .)

Calculated using the short form formula

L = (aN^2/13.5) log_10(4.9a/(l+c))

a = average radius
l = coil height
c = winding thickness

a = 2.525 inch
N = 155
l = 1.7 inch
c = 0.011 inch

L = (2.525*155^2/13.5) log_10(4.9*2.525 /(1.7+0.011))
  = 4494 log_10(7.23)
  = 3.86 mH

Turns out either formula is fine here.