Sphere Gaps and Peak Voltmeters
Chapter 15 ended on a warning: insulation breaks down on the peak, so for testing it is the peak you must measure, not the mean or the RMS. This chapter is devoted to the peak. It opens with the oldest and most trusted peak instrument of all — two polished spheres whose breakdown voltage is tabulated to the kilovolt — and then turns to the electrical peak voltmeters that read the same quantity from a current or a stored charge.
- Why the peak is the value that matters for breakdown, and why a device that breaks down is a natural peak meter.
- The sphere gap — its near-uniform field, and why that makes its breakdown voltage reproducible enough to be a primary standard.
- The S ≤ 0.5D spacing rule, where it comes from, and how accuracy and polarity behave when it is broken.
- The air-density factor \(\delta\), plus humidity and irradiation, and the 50% breakdown procedure good to about ±3%.
- The Chubb–Fortescue peak voltmeter, which rectifies the capacitive current to give \(V_{\text{pk}} = I_{\text{mean}}/(2fC)\).
- The rectifier–capacitor peak voltmeter, which stores the peak of a capacitance-divider tap on a capacitor and reads it.
Why Measure the Peak?
The last chapter left us with three different numbers for one alternating voltage — the mean, the RMS and the peak — and the observation that they part company the moment the waveform stops being a perfect sinusoid. For most of high-voltage testing the peak is the one that counts, because breakdown is a threshold phenomenon: an insulator flashes over the instant the voltage across it touches a critical value, and that instant is the crest of the wave. A withstand test, a flashover test, the rating of a bushing — all are statements about the peak.
That fact suggests a beautifully direct way to measure the peak: build a controlled gap that breaks down at a known voltage, raise the unknown voltage until the gap just sparks over, and read the peak from a table. The breakdown is the measurement. The sphere gap does exactly this, and does it so reproducibly that it serves not merely as a meter but as a primary standard against which other instruments are calibrated. The electrical peak voltmeters that follow are quicker and non-destructive, but the sphere gap is the reference they answer to.
The Sphere Gap
A measuring sphere gap is a pair of identical polished metal spheres of standard diameter \(D\) — 2, 5, 10, 25, 50, up to 200 cm — mounted with their nearest points separated by an adjustable spacing \(S\). One sphere is energised and the other earthed (or the pair is driven symmetrically), and the voltage is raised until a spark bridges the gap. International standards (IEC 60052, and IS 1876) tabulate the peak breakdown voltage against spacing \(S\) for each diameter \(D\), at reference air conditions. Read the spacing, look up the table, apply a small correction, and you have the peak — of AC, of DC, or of an impulse.
Two practical details make the reading trustworthy. The spheres must be smooth, clean and free of pits or dust, because a single sharp defect concentrates the field and triggers an early, false breakdown. And a generous clearance must be kept around the spheres — earthed walls, the floor and the framework distort the field if they crowd too close, so the standards specify minimum clearances in terms of the diameter. With those observed, the sphere gap is the most reliable absolute peak meter in the laboratory.
Field Uniformity and the 0.5D Rule
Why spheres, and not the rod gaps or point gaps that also break down? The answer is field uniformity. When the spacing is small compared with the diameter, the region between the two facing surfaces is very nearly a uniform field, like a parallel-plate gap. In a uniform field breakdown is sharp, well-defined and almost independent of polarity, and there is no corona to bleed the voltage away before the spark — exactly the qualities a measuring standard needs. As the gap is opened wider, the field grows progressively non-uniform, the spheres start to behave like points, corona sets in, polarity effects appear, and the breakdown scatters.
The standards capture this in a single working rule: the sphere gap is accurate to about ±3% provided the spacing does not exceed half the sphere diameter,
This is why a laboratory keeps several sizes of sphere. To measure a higher voltage you need a wider gap, and to keep that wider gap inside the \(S \le 0.5D\) window you must use larger spheres. A \(200~\mathrm{kV}\) peak might be measured comfortably with \(25~\mathrm{cm}\) spheres; a megavolt needs the largest spheres in the hall.
Correction Factors
The tabulated breakdown voltages hold at reference air conditions — conventionally \(20~^\circ\mathrm{C}\) and \(101.3~\mathrm{kPa}\) (760 mmHg). Real air is rarely at the reference, and since breakdown is a collision process in a gas, it depends on how many molecules sit in the gap — that is, on the air density. The correction is carried by the relative air density \(\delta\):
Denser air (higher \(\delta\)) means more collisions and a higher breakdown voltage; thinner air lowers it. The actual peak that breaks the gap is the table value scaled by a factor \(k(\delta)\) which, for conditions near the reference, is close to \(\delta\) itself:
Two smaller effects complete the picture. Humidity raises the breakdown voltage slightly, because water vapour mops up free electrons; but in the uniform field of a sphere gap the effect is small, and the standards usually omit a humidity correction within the recommended range — one of the quiet advantages of choosing a uniform-field device. Irradiation matters at the low end: below roughly \(50~\mathrm{kV}\) the gap may need a dose of ultraviolet light or a radioactive source to supply the initiatory electrons that start the avalanche, otherwise the breakdown is erratic and the statistical time lag scatters the readings.
Using the Sphere Gap
Because breakdown is a statistical event, a single spark is not a measurement. The standard procedure is to find the 50% breakdown voltage — the level at which the gap sparks on about half of the applications — by raising the voltage in small steps and averaging several breakdowns at a fixed spacing. That average, corrected for air density, is the peak. The whole sequence is: set the spacing within \(S\le 0.5D\); irradiate if below \(50~\mathrm{kV}\); record several breakdowns; compute \(\delta\) from the barometer and thermometer; and report \(V_{\text{actual}} = k(\delta)V_{\text{table}}\).
The sphere gap's limitations are the mirror of its strengths. It is destructive in the sense that it must spark to read, so it cannot monitor a voltage continuously; each measurement interrupts the test. It is comparatively slow, and the spark stresses the test object. For these reasons it is used chiefly to calibrate the convenient electrical voltmeters and to make occasional absolute checks, while the day-to-day reading is done by the peak voltmeters we turn to now.
The Chubb–Fortescue Peak Voltmeter
The first electrical peak meter exploits the capacitive current we met in Chapter 15. A capacitor \(C\) across the voltage carries a current \(i = C\,\dfrac{dv}{dt}\) — largest where the voltage rises fastest, and crucially, its time integral over the rising quarter of the wave is the charge \(C\) needs to move from the trough to the crest. Chubb and Fortescue's idea was to rectify that current with a pair of diodes and pass it through an averaging (moving-coil) meter. The charge delivered each cycle is fixed by the peak-to-peak swing, so the mean current reports the peak directly.
The bookkeeping is short. In each cycle the rectified branch carries a charge equal to \(C\) times the full swing from the negative peak to the positive peak; for a symmetrical wave that swing is \(2V_{\text{max}}\). Multiplying the per-cycle charge by the frequency gives the mean current, which inverts to the peak:
A moving-coil meter reading the average rectified current, scaled by the fixed factor \(2fC\), reads the true peak — and it does so for any waveform with a single maximum and minimum per cycle, which is the method's great virtue. Its one demand is an accurately known frequency.
The Rectifier–Capacitor Peak Voltmeter
The second electrical peak meter is the passive peak detector, applied to the low-voltage tap of a capacitance divider. The divider (the \(C_1\)–\(C_2\) pair from Chapter 15) brings the voltage down to a safe level; a diode then charges a storage capacitor \(C_s\) up to the peak of that tapped voltage and holds it there, where a high-impedance electrostatic or electronic voltmeter reads it at leisure. A large resistance \(R\) bleeds the storage capacitor slowly, so the reading can follow a falling envelope without sagging within a single cycle.
The reading is the peak of the divider output, lifted back to the line by the known divider ratio:
Its errors are the errors of any peak detector: the diode's forward voltage drop is subtracted from the stored peak, leakage and the bleed resistor let the charge sag between crests as ripple, and the time constant \(R\,C_s\) must be chosen long against the period (to hold the peak) yet short against the envelope (to track changes) — the same compromise that governs every envelope detector. Well-designed versions reach a fraction of a percent. Compared with the sphere gap, both electrical meters are continuous, non-destructive and fast; compared with each other, the Chubb–Fortescue is the more absolute (it needs only \(f\) and \(C\)), while the rectifier–capacitor meter is simpler and reads on a steady scale.
Choosing a Peak Method
All three instruments answer the same question — what is the crest of this voltage? — but they earn their places differently. The sphere gap is the absolute reference, slow and destructive but answerable to nothing but geometry and air. The two electrical voltmeters are the working instruments, fast and continuous, and are themselves calibrated against the gap.
| Instrument | Reads | Key relation | Continuous? | Role |
|---|---|---|---|---|
| Sphere gap | peak (AC, DC, impulse) | \(V_{\text{actual}} = k(\delta)\,V_{\text{table}}\) | no (must spark) | primary standard, calibration |
| Chubb–Fortescue | peak (AC) | \(V_{\text{max}} = I_{\text{mean}}/(2fC)\) | yes | absolute electrical peak meter |
| Rectifier–capacitor | peak (AC) | \(\hat V = \hat V_2\,(C_1{+}C_2)/C_1\) | yes | simple working peak meter |
Worked Examples
Problem. You must measure a peak of about \(250~\mathrm{kV}\). A \(25~\mathrm{cm}\) sphere gap needs a spacing of roughly \(7.5~\mathrm{cm}\) for that voltage. Is the measurement within the accurate range?
Solution. The accurate range requires \(S \le 0.5D\):
Since \(S < 0.5D\), the field is near-uniform and the reading is good to about ±3%. Smaller spheres would push \(S\) past the limit and lose accuracy.
Problem. A gap whose table value is \(40~\mathrm{kV}\) (peak) at reference conditions is used where the barometer reads \(700~\mathrm{mmHg}\) and the temperature is \(35~^\circ\mathrm{C}\). What peak will actually break the gap?
Solution. Find \(\delta\), then \(V_{\text{actual}} = k(\delta)V_{\text{table}}\approx\delta\,V_{\text{table}}\):
The thinner, warmer air breaks down at about 35 kV instead of 40 — a 12% shift that the density correction is there to catch.
Problem. A Chubb–Fortescue voltmeter uses \(C = 100~\mathrm{pF}\) on a \(50~\mathrm{Hz}\) supply, and the averaging meter reads \(I_{\text{mean}} = 1~\mathrm{mA}\). Find the peak voltage.
Solution. Apply \(V_{\text{max}} = I_{\text{mean}}/(2fC)\):
A 1 mA mean current corresponds to a 100 kV peak — and the result holds whatever the waveform's shape, provided it has one crest and one trough per cycle.
Problem. The supply in Example 3 is actually \(51~\mathrm{Hz}\), but the scale assumes \(50~\mathrm{Hz}\). What is the percentage error in the indicated peak?
Solution. Since \(V_{\text{max}}\propto 1/f\), assuming too low a frequency over-reads in proportion:
The indicated peak is 2% high — which is exactly why the method demands an accurately known, stable frequency.
Problem. A rectifier–capacitor peak voltmeter reads the tap of a divider with \(C_1 = 100~\mathrm{pF}\) and \(C_2 = 9900~\mathrm{pF}\). The stored peak is \(\hat V_2 = 2~\mathrm{kV}\). Find the line peak (ignore the diode drop).
Solution. Lift the tap peak by the divider ratio \((C_1+C_2)/C_1\):
The stored \(2~\mathrm{kV}\) peak corresponds to a 200 kV line peak. A real instrument would add back the diode's forward drop, a fixed offset of a volt or two at the tap.
Chapter Summary
Breakdown is a threshold at the crest, so testing needs the peak. A calibrated breakdown gap turns the crest into a tabulated number.
Polished spheres of diameter \(D\) at spacing \(S\); the standards tabulate peak breakdown vs \(S\). It is a primary standard.
Near-uniform field — and ±3% accuracy — only while \(S\le 0.5D\). Wider gaps go non-uniform, with corona and polarity effects.
\(V_{\text{actual}}=k(\delta)V_{\text{table}}\) with \(\delta=\frac{b}{760}\frac{293}{273+t}\); humidity minor, irradiation for sub-50 kV gaps.
Rectify the capacitive current and average it: \(V_{\text{max}}=I_{\text{mean}}/(2fC)\). Waveform-independent; needs accurate \(f\).
A diode stores a divider tap's peak on \(C_s\); read it and scale by \((C_1{+}C_2)/C_1\). Watch diode drop, ripple and \(RC_s\).
Problems
For each item, first identify what it tests — the reason for peak measurement, the uniformity rule, the density correction, or one of the two peak-voltmeter relations — then apply it. Difficulty rises down the list.
- Explain why high-voltage testing usually calls for the peak of a voltage rather than its mean or RMS, and why a breakdown gap is a natural peak meter.
- State the reason spheres are preferred over rods or points for a measuring gap.
- A sphere gap uses \(50~\mathrm{cm}\) spheres. Find the largest spacing that keeps it in the accurate (±3%) range.
- Compute the relative air density \(\delta\) for \(b = 750~\mathrm{mmHg}\) and \(t = 25~^\circ\mathrm{C}\), and say whether the breakdown voltage is raised or lowered relative to reference.
- A table value of \(60~\mathrm{kV}\) peak applies at reference conditions. Find the actual breakdown peak where \(\delta = 0.90\).
- Explain why a humidity correction is normally unnecessary for a sphere gap but important for a rod gap.
- Why must a sphere gap below about \(50~\mathrm{kV}\) sometimes be irradiated, and what does the irradiation supply?
- A Chubb–Fortescue voltmeter has \(C = 200~\mathrm{pF}\) at \(50~\mathrm{Hz}\) and reads \(I_{\text{mean}} = 2~\mathrm{mA}\). Find the peak voltage.
- Derive the Chubb–Fortescue relation \(V_{\text{max}} = I_{\text{mean}}/(2fC)\) from \(i = C\,dv/dt\), stating the assumption about the waveform.
- A rectifier–capacitor peak voltmeter feeds from a \(500{:}1\) capacitance divider and stores \(\hat V_2 = 0.45~\mathrm{kV}\). Find the line peak, then explain how a diode drop of \(0.7~\mathrm{V}\) at the tap would shift the result, and which way.