Liquid Resistive HV divider
Introduction
The measurement of pulsed voltages generated by a Tesla Coil (TC) is usually problematic for the
following reasons:
- the voltage generated is in the 100 kV to 1 MV range. Too high for most
of the traditional hand-held HV probes
- The TC output impedance is pretty high, in the range of some hundreds of kiloohms. Its output
voltage and resonance frequency are very sensible to resistive and capacitive loading. A
resistance higher than one megaohm and a capacitance less than 10 pF are the minimum
requirements for a decent accuracy. These requirements make unsuitable most of the
commercially available resistive and capacitive dividers.
- TCs require a flashover distance much larger than the one dictated by their peak output voltage. A
TC with an output voltage of "only" 500 kV can easily require a clearance of 3 meters.
Contactless voltage meters like field mills, capacitive pickups, etc. don't load the TC but require the
contribution of conduction current to be taken into account and separated from the one of the displacement
current. This typically translates into bandwidth limitations, need of prohibitive screening or of recalibration
after each change in the measuremnt setup.
The liquid resistive divider described here is of simple construction and of low cost. It exhibits a high resistance
together with a low capacitance, yet featuring a bandwidth of DC to 10 MHz.
A divider balanced by design
A voltage divider can be easily constructed as a tube filled with a liquid resistive solution, with
a top and bottom electrode (Fig.1). An intermediate tap (a grid, plate, etc.) connects to the liquid solution
and experiences only a fraction of the full voltage applied. The distance between the tap and the
bottom electrodes sets the dividing ratio. The liquid column between tap and ground is called lower arm,
while the one between tap and HV electrode is called upper arm of the divider.
A voltage divider has got a broadband frequency response when it is compensated (balanced).
For a divider implemented as a vertical column this means that, along all of its length,
the product of the resistance and the capacitance between any two points (neglecting the inductance) is
constant. A construction like the one proposed here is therefore quite attractive. Resistance
variations due to temperature or concentration changes are reflected identically in lower and upper arm.
The division ratio is thus unchanged.
The resistive and capacitive parts of both arms are also balanced
by their geometrical symmetry (i.e. by their inverse dependance from section and length). This
means that even changes in the tube diameter along its vertical axis won't influence the
divider balance.
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Fig.1 Simple liquid resistive divider. |
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Tackling the stray capacitances
Any object located in space has got a stray capacitance to ground and a voltage divider has got
one too. If the divider total resistance (top to bottom) measures, say, 5 kohms, 10 pF of
stray capacitance result in an RC constant of 50 ns. But if the resistance is 1 Mohm (as we
wish for our TC), the RC constant becomes 10 µs. This accounts for a usable bandwidth of
0.36 / 10 µs = 36 kHz. Not very good...
There are many ways to describe this problem in order to understand it, but a useful one
is the following. If we apply a voltage step to our divider, the electric field along it will
set in zero time according to the electrostatic field distribution, i.e. the one existing
between its HV and ground electrodes if the resistive column wasn't there. After a certain amount
of time the stray capacitances will get charged and the electric field will reach the
distribution dictated by the resistance of column. The time needed to reach steady state is
the divider response time.
Now, it is obvious that if we could arrange the two field distributions (with and without
divider column) to resemble each other, the response time would turn to zero. This technique
is employed in commercial dividers, but usually the divider resistance is linear
(along its axis) and the surrounding electric field is "linearized" by using one or more extra
electrodes. This method goes under the name of field grading and actually modifies the existing
field to follow a linear distribution.
The approach used in the divider I have built is, instead, to modify the resistive
distribution of the divider in order to follow the one of the original electric field.
As the position of the divider during measuremnts is fixed and always the same, this doesn't constitute a
problem.
Divider design
Dimensional design
The divider has been designed to be placed on the top of the TC, hanging upside down from
the ceiling. For flashover distance requirements its total length has been fixed to 2.7 m and
a top (grounded) toroid has been used to ensure a certain predictability of the field distribution
above the TC.
A FEM (Finite Element Method) solver program (Bela V1.0) has been used to model the electrical field around the TC.
First, empirically, the size of the divider toroid has been selected in order to strighten a little bit the field
above the TC. Then the liquid resistive column has been introduced and its diameter along its length
has been changed in order to end up with a field distribution (see Fig.3) approximating the original one.
Six different tube sections with different diameters have been employed.
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Fig.2 Electric field distribution and equipotential lines around the divider. |
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Fig.3 Electric field distribution along the divider axis. |
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Divider drawings and construction
As a picture is better than a hundred words, please find here below hyperlinks to detailed drawings of
my liquid resistive divider
The divider is made by 10 sections of acrylic pipe, glued together with epoxy, filled with 20 l of deionized tap water.
If deionized water is a luxury, distilled water can be used too. In that case, purging of the air dissolved into the
water will take a little bit longer but it is still feasible.
Note that pipe section n.3 is used only to stiffen the whole structure.
A BNC connector is used to separate the tap and ground electrode by 10 mm (actually reduced to 5 mm). Two copper pipes are
used for water filling and air venting. The cavity originated behind the ground electrode is used to
collect the air bubbles possibly forming. Corrugated HVAC aluminum pipe is used to form the top toroid.
The divider arm has got two plexiglass collars, one secured directly to the PVC table
and the other through three PVC rods. In the top cap detail drawing, the cap is built using separate sections
of plastic glued together. In my actual divider both upper and lower caps are built from solid copper.
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Fig.4 The built divider placed on the top of Thor. |
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Fig.5 The additional attenuator with ratio 98:1. |
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Division ratio setup
The resistance between the divider's HV electrode and ground electrode has been measured using AC voltage (50 Hz) and
accounted for 1.618 MΩ. The resistance between the tap and ground measured 2.32 kΩ. Initial tests with a step
generator indicated that the best response could be obtained by adding 1.7 kΩ between the tap and ground. The
resulting division ratio was therefore:
The above division ratio has been incremented by adding an additional attenuator (Fig.5) build inside a shielded
case and using non-inductive discrete resistors. The attenuator is located right on the upper cap and provides an
attenuation ratio as 98:1. The overall estimated division ratio is thus:
The attenuator uses low inductance carbon resistors and is assembled in a shielded case with two BNC panel
connectors. It has been tested to have a flat frequency response. Its input impedance is 1.74 kΩ which is just suitable for
providing the required compensation to the divider.
Current buffer
The divider has been equipped with a battery powered broadband current buffer (Fig.6) having an amplification of 1 when
driving a 50 Ω load. This was mandatory in order to drive several meters of coaxial cable and thus provide the required
clearance between Thor and the measuring instruments.
Also the current buffer is assembled in a shielded case with two BNC connectors for signal input and output. A compensation
resistor of 1.7 kΩ can be optionally connected in parallel to its input to compensate correctly the divider when the
attenuator is not used (e.g. during calibration).
The current buffer accepts an input voltage of ±3.6 V, which corresponds roughly to ±5.8 kV applied to the divider
or to ±573 kV when the additional attenuator is also used.
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Fig.6 The battery-powered current buffer. |
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Characterization of the divider
Estimation of the frequency response
The transfer characteristic of the divider and its current buffer can be described in a comprehensive way by its frequency response.
Instead of performing a frequency sweep over the desired frequency range, a much more handy method is to stimulate the divider
with a known input waveform and record its output waveform. A step generator was used to apply a voltage changing from approx. 250 V to
0 V with a fall time of about 1 ns. The attenuator was removed and replaced by the 1.7 kΩ resistor internal to the current buffer.
Thor's upper toroid was disconnected from the secondary coil and connected to the step generator through a 390 Ω damping resistor.
Averaging 200 readings most of the random noise could be removed. The measured fall time at the current buffer output was about 40 ns.
The ringing of the response tail (Fig.7) is due to the resonant circuit formed by the stray capacitances of Thor's toroid to its
secondary coil and of the coil to ground. This was easily verified by grounding the secondary top and observing a marked frequency
shift of the ringing.
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Fig.7 Divider original step response. |
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Fig.8 Divider frequency response (modulus). |
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Fig.9 Divider frequency response (phase). |
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The waveform of an ideal step was calculated by averaging the initial values and the tail ringing values. A ramp extending
linearly from the first to the last sample was subtracted from both the measured and ideal waveform in order to obtain two
"periodic" signals. A Fast Fourier Transform (FFT) was computed for both waveforms and the divider response was found by
dividing the measured FFT by the FFT of the ideal step. The processing was performed using MATLAB1) and the source
code used for it can be found HERE.
As during normal operation Thor acts as a voltage source, the resonance peaks due to the secondary coil proximity must be
removed from the calculated frequency response. This is done using linear interpolation in the intervals 83-110 kHz and
227-250 kHz. The resulting characteristic is shown in Fig.8 and Fig.9, truncated to 12 MHz to account for noise margin.
Compensation for the non-ideal frequency response
Knowing that the divider/current buffer pair hasn't got an ideal, flat frequency response but the one depicted in Fig.8, it is
still possible to perform accurate measurements by "counterbalancing" the nonlinearities and postprocessing the measured waveforms.
The algorithm used involves deconvolution in the frequency domain and uses the overlap-add method. The measured sample sequence is
sliced into overlapping intervals that are deconvolved with the probe frequency response and added back together in the time
domain. Differences between sampling rates are taken into account using interpolation and decimation. The resulting signal is also
low-passed with a cut-off frequency of 12 MHz. The algorithm source code can be found HERE.
Fig.10 shows the measured step from Fig.7 after deconvolution postprocessing. Note that for this test the resonances caused
by the secondary coil were not removed from the frequency response.
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Fig.10 Divider step response reconstructed from Fig.7. |
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Divider calibration
The built divider has been calibrated by removing the attenuator and applying a standard 1/50 µs surge pulse with peak value
of about 4 kV. The setup was similar to the one used with the 250 V step generator but this time also a reference capacitive divider
was used.
The reference divider measured a peak value of 3780±10 V (Fig.11). The waveform acquired from the build
divider is shown in Fig.12. Thor's secondary coil is responsible for the ripple already noticed with the step response.
Fig.13 shows the same signal after deconvolution postprocessing.
The ringing residual is due to the slightly different measurement setup used for the calibration. The changed
stray capacitance of the secondary causes a frequency shift of the ringing from 96 kHz (step test) to 92 kHz (calibration). During
normal operation this is not an issue as the secondary acts as a voltage source.
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Fig.11 Test surge as measured with a reference divider. |
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Fig.12 Test surge as measured with the built divider. |
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Fig.13 Test surge measured with the built divider after postprocessing. |
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The divider ratio can be estimated by comparing the peak value measured by the reference divider Vout
with the value V ' out measured by the built divider
When the 98:1 attenuator is installed, the overall divider ratio is estimated as
The difference between the ratio rt measured at 50 Hz and r 't estimated now is due
to the non-ideal frequency response of the divider and of its current buffer. Clearly 135.4E3 has to be considered the
best approximation of the divider real ratio.
Experiments
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The built divider has been used to measure several one-shot discharges of Thor. No flashover or damages to the
divider, the attenuator or the current buffer have been detected. Fig.15 shows a bang resulting in a breakdown
to a grounded stick and Fig.16 shows its corrected and scaled version. The shape differences between the two waveforms
are minimal as Thor's resonance frequency is pretty low (66 kHz). Anyway, the deconvolution is responsible alone for an
amplification of about 17%, compensating the corresponding valley in the divider frequency response in the 66 kHz zone.
The rest of the scaling is purely due to the division ratio of 135.4E3:1.
By increasing Thor's feed voltage the divider has been tested up to a peak voltage of 418 kV.
During more than 6 months of divider "life" no sedimentation, colouring or deterioration of the divider solution
and electrodes have been detected.
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Fig.14 Experiment setup. |
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Fig.15 Voltage burst as measured from Thor with the divider. |
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Fig.16 Burst from Fig.15 after postprocessing and scaling. |
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Data files
The divider calibration data files can be can be found here: 
Divider data files
- videoamp_nognd.csv: the divider step response (Fig.7)
- videoamp_nognd_g.csv: the divider frequency response (modulus and phase)
- videoamp_nognd_g_corr.csv: the divider frequency response corrected for the secondary coil effect (Fig.8 and 9)
Trademarks
1) MATLAB is a registered trademark of The MathWorks, Inc.
