Electr Eng (2007) 89:519–524DOI 10.1007/s00202-006-0036-0

ORIGINAL PAPER

Study on corona characteristics under nonstandardlightning impulses

Xiao Qing Zhang

Received: 31 March 2006 / Accepted: 7 July 2006 / Published online: 10 August 2006© Springer-Verlag 2006

Abstract The q − u curves under damped oscilla-tory impulses are observed experimentally in a coronacage. An analysis for the experimental results is madewith an emphasis on the difference between the q − ucurves under damped oscillatory and double exponen-tial impulses. Based on the experimental investigation,a corona model is proposed to calculate the q−u curvesunder damped oscillatory impulses. Charge, electric fieldand voltage are related macroscopically in the model.The model parameters can be determined by the mea-sured data under double exponential impulses. The cal-culated results are compared with the experimental onesand a reasonable agreement is shown between them.

Keywords Nonstandard lightning · Dampedoscillatory impulse · Double exponential impulse ·Corona · q − u curves

1 Introduction

Lightning protection and insulation coordination oftransmission lines and substations require an accurateknowledge of the magnitudes and waveforms of light-ning overvoltages. A significant phenomenon that causesthe attenuation and distortion of lightning surges propa-gating along transmission lines is impulse corona. In theanalysis of lightning overvoltages, the q − u curves, i.e.the relationship between charge and voltage, are usuallyemployed to take account of the corona effect on theovervoltages. Experimental investigations on the q − u

X. Q. Zhang (B)School of Electrical Engineering, Beijing Jiaotong University,Beijing 100044, Chinae-mail: zxqiong@hotmail.com

curves have been reported in the literature [1–3], whichwere performed mainly under double exponential im-pulses. A number of corona models have been devel-oped for simulating lightning transients on transmissionlines [4–6] and very few of them are available to pre-dict the q − u curves under nonstandard lightning im-pulses [4]. In fact, most of lightning intruding surgesinto substations take the damped oscillatory impulsewaveforms because of the refraction and reflection oflightning surge waves [7]. For lack of the outcome ofresearch on the q − u curves under damped oscilla-tory impulses, the attenuation and distortion of lightningintruding surges, regardless of what their waveforms are,are always analyzed according to the q − u curves underdouble exponential impulses. It is obvious that the exist-ing way is problematic and difficult to consider accu-rately the impulse corona effect on lightning intrudingsurges. Therefore, the practical need exists for investi-gating the q − u curves under damped oscillatory im-pulses. An attempt is made in the present paper to carryout a systematic study on the q − u curves under theimpulse waveforms of this type. The exploratory exper-iments are described to inquire into the effect of thewaveform parameters on the q−u curves. Furthermore,a corona model is proposed for calculating the q − ucurves, which is especially adaptable for damped oscil-latory impulse waveforms. In order to check the validityof the proposed model, a comparison is made betweencalculated and measured results.

2 Experimental investigation

The damped oscillatory impulse voltages can be pro-duced by a modified impulse generator, whose

520 Electr Eng (2007) 89:519–524

RfLf

C1

C2 u(t)Rt

U0

Fig. 1 Equivalent circuit of modified impulse generator

equivalent circuit is shown as Fig. 1. The frequency andoscillatory magnitude are controlled by adjusting thewavefront inductance Lf(0.025–1.3 mH) and resistanceRf (0–110 �).

Since the wave tail resistance Rt has a large value, itcan be neglected. As a result, the output voltage is givenby

u(t) = C1

C1 + C2U0

[1 − ω0

ωe−αt sin (ωt + θ)

](1)

where

α = Rf

2Lf, ω0 =

√C1 + C2

LfC1C2, ω =

√ω2

0 − α2,

θ = tg−1 ω

α

The voltage waveform corresponding to Eq. (1) issketched in Fig. 2, where Ug = C1U0/(C1 + C2).

It is basically of the type of (1–sin ω t) with the oscilla-tory component decaying with the inverse of the damp-ing factor α that is associated with Rf and Lf. The dampedoscillatory impulse voltage is applied to the inner con-ductor of a corona cage to produce the corona dis-charge. The corona cage is a coaxial cylindrical elec-trode system, as shown in Fig. 3, in which the electricfield geometry approximates to that of a transmissionline. Two guard rings are set at both sides of the main

Fig. 2 Damped oscillatory impulse voltage

Φ0.

002m

voltage divider

qu

Shielded room

Φ1m

ComputerDigital Memorizer

Fig. 3 Corona cage and digital measurement system

cylindrical electrode and grounded for reducing the endeffect on electric field. The waveforms of the voltageu and charge q are recorded respectively by a digitalmeasurement system and the q − u curves are obtainedthrough data processing. A set of typical measured q−ucurves under damped oscillatory and double exponen-tial impulses are shown in Figs. 4 and 5 for both polari-ties, where the wavefront time and amplitude of the twotypes of impulses are adjusted to approximately equalvalues for comparison. These measured results revealthat the q−u curves under damped oscillatory impulsesare mainly coincident with those under double expo-nential impulses before their respective charges reachthe maximum values. They differ from each other inthe parts after the maximum charges. In these parts,the q − u curves under double exponential impulses godown monotonically while those under damped oscil-latory impulses display narrow helical appearance. Theformation mechanism of the helical trajectory might beinterpreted schematically in Fig. 6.

A descending section AB has a slope larger than thegeometric capacitance C0 of the corona cage as the volt-age u decreases to a certain extent. Such an increase inthe slope makes the section AB unable to coincide withthe next rising section BC. This phenomenon might beascribed to the occurrence of the opposite polar corona.With the voltage u decreasing and approaching to itswave trough point, the electric field near the surfaceof the inner conductor could be reversed. After the re-versed field strength exceeds a critical value, the oppo-site polar space charge might be produced due to thedevelopment of the opposite polar corona. Therefore,the total charge q decreases more rapidly. This is mainreason why the helical trajectories appear on the q − ucurves when the damped oscillatory impulse voltages

Electr Eng (2007) 89:519–524 521

Fig. 4 q − u curves underdamped oscillatory anddouble exponential impulseswith negative polarity

Fig. 5 q − u curves underdamped oscillatory anddouble exponential impulseswith positive polarity

522 Electr Eng (2007) 89:519–524

q

uO

A

C

B

D

EF

C0

C0

C0

Fig. 6 Formation mechanism of the helical trajectory

are applied. As seen from Figs. 4 and 5, the relativedifference between the q−u curves under double expo-nential and damped oscillatory impulses is more pro-nounced for negative polarity than for positive polarity.Also, the helical area for negative polarity is larger thanthat for positive polarity. These features are importantto lightning protection design, since most of lightningsurges have negative polarity.

3 Corona model

The impulse corona occurring in the corona cage is char-acterized by the ionization processes that are of thestreamer nature. It is assumed that the corona geom-etry is symmetrical, as shown in Fig. 7. The ionizationzone is a cylinder surrounding the inner conductor andits boundary radius is denoted by Rib. The exact distri-bution characteristic of electric field in the ionizationzone has not been found in the literature. Based on thedistribution characteristics proposed in Refs. [4–6], weassume that the electric field in the ionization zone cannot exceed the critical value

Ecr = a√

b + r2

r(2)

where a and b are constants. They are identified ina(+), b(+) and a(−), b(−) corresponding to Ecr(+) andEcr(−) for positive and negative polarities, respectively.By fitting the data points selected from the rising partsof the q − u curves, a and b can be determined. Accord-ing to the experimental investigation stated above, theinitial rising part of the q−u curve under damped oscilla-tory impulse is roughly identical with that under doubleexponential impulse if both the applied voltages haveapproximately equal wavefront time. So far the mea-sured q − u curves under double exponential impulseshave been accumulated a lot [1–3]. This means that it

r0

uRib

R0

r

Fig. 7 Corona geometry

r0

C2

C1

Ri b R0

r

dr

Fig. 8 Configuration of the corona model

O – – – – –+ + + + + + + + + + + + + + + + + + +

rECr(—)

P

E

rmr0

Fig. 9 Occurrence of opposite polar space charge

is convenient to use the measured data under doubleexponential impulse to determine a and b for calculat-ing the q − u curves under damped oscillatory impulses.An optimistic fitting procedure has been given in Ref.[8].

In the ionization zone, as illustrated in Fig. 8, the spacecharge density ρ can be given by Possion’s equation incylindrical coordination

d(rE)

rdr= ρ

ε(3)

By substituting Eq. (2) into Eq. (3) [4], ρ is expressedas

ρ = εa√b + r2

(4)

Electr Eng (2007) 89:519–524 523

Fig. 10 Calculated andmeasured q–u curves underdamped oscillatory impulseswith negative polarity

Fig. 11 Calculated andmeasured q–u curves underdamped oscillatory impulseswith positive polarity

An initial value Rib is assigned to the boundary radiusof the ionization zone. Through the integration of elec-tric field E from r0 to Rib, the voltage drop in the ioni-zation zone is obtained by

ui =Rib∫

r0

Edr (5)

where E can be calculated by an algorithm proposed inRef. [8].

In the non-ionization zone, the electric field is

E = q2πεr

(R0 > r > Rib) (6)

The total charge q is calculated by

q = C0u +Rib∫

r0

ρ(1 − β)2πrdr (7)

The charge-induced coefficient β is represented as [9]

β(r) = C2

C1 + C2= ln(R0/r)

ln(R0/r0)

524 Electr Eng (2007) 89:519–524

Integrating Eq. (6) from Rib to R0, yields the voltagedrop u0 in the non-ionization zone

u0 = q2πε

lnRib

R0(8)

From Eqs. (5) and (8) the total voltage drop is givenby

ui0 = ui + u0 (9)

The total voltage drop ui0 should be equal to theapplied voltage u, or a deviation between ui0 and u cannot exceed a given tolerance δ∣∣∣∣u − ui0

u

∣∣∣∣ < δ (10)

If inequality (10) is untenable, Rib must be replaced by anew value. The calculation process from Eqs. (5), (6), (7),(8), (9) and (10) is conducted repeatedly until Eq. (10)is tenable. Then, the total charge q can be obtained fromEq. (7) for the determined value Rib.

With the voltage u oscillating and approaching to thewave trough value, the electric field E near the inner con-ductor might be reversed. If the reversed E increases tothe critical value of the other polarity, a reversal of spacecharge could occur. As shown in Fig. 9, the reversedE intersects Ecr(−) at point P, so a space charge layerwith negative polarity is assumed to replace the formerpositive charge in the range of r0 − rm. If the voltage uhas a larger oscillatory magnitude, it is possible to forma multiplayer structure of space charge with alternativepolities in the ionization zone.

4 Calculated results from the proposed model

The proposed model is applied to the corona cage shownin Fig. 3. The calculated results from the proposed modelare shown in Figs. 10 and 11, where the correspond-ing measured results are simultaneously given for com-parison. The model parameters for Fig. 10 are: a(−) =1.5 kV/mm, b(−) = 18.3, a(+) = 0.507 kV/mm, b(+) =161.0, and those for Fig. 11 are: a(+) = 0.507 kV/mm,b(+) = 167.0, a(−) = 1.5 kV/mm, b(−) = 18.3.

As seen from Figs. 10 and 11, a reasonable agreementis shown between calculated and measured results forboth polarities. This makes certain of the validity of theproposed model.

5 Conclusions

1. The experimental results obtained from the coronacage shows that the main difference between theq − u curves under double exponential and dampedoscillatory impulses appears in the parts after thepeak value of the surge. In these parts the curvesunder double exponential impulses go down monot-onously while those under damped oscillatory im-pulses display narrow helical appearance.

2. The model proposed in this paper has capability toproduce the q − u curves under damped oscillatoryimpulses in a practical manner. The model parame-ters can be determined by fitting the measured dataunder double exponential impulses, which is easierto accomplish based on a lot of the existing datafrom the traditional experiments.

3. The calculated q − u curves from the model canagree with those from the experiments reasonablyand so the validity of the model is confirmed.

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