IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. 32. NO. 2, MAY 1990 87

Power-Line Impedance and the Origin of the Low-Frequency Oscillatory Transients

MAURO M. FORTI AND LUIGI M. MILLANTA

Absfrucf- Oscillatory transients represent a large portion of power- line disturbances, and among these, lower frequency oscillations (from about 1 to more than 20 kHz) appear particularly harmful to sensitive electronic equipment. This paper addresses three main issues: a) the origin and characterization of low-frequency transients, b) the determi- nation of the impedance of power systems by deliberate switching of capacitors, and c) suggestions for remedies against these transients.

The origin of low-frequency oscillatory transients is here attributed to the connection of large capacitances across live wires of the distri- bution systems. Measurements based on the connection of capacitors of known capacitance across the line are demonstrated to be a natural, accurate, and reproducible means of probing the power-line impedance. This turns out to be constituted by resistance and inductance in the frequency range explored. A physically significant equivalent circuit can be derived, based on the consideration that the three-wire distribution system (phase, neutral, safety wire) consists of three mutually coupled loops.

The measurement technique (V-terminal data) permits derivation of both common-mode and differential-mode components. Several conclu- sions are drawn on the basis of the results of the time-domain transient analysis that allow the derivation of essential quantities of the phe- nomenon (peak voltage, duration, spectral content) and the suggestion of corrective measures.

I. INTRODUCTION

LARGE portion of the transient disturbances observed on A power ac supply mains is constituted by damped oscilla- tions. These low-frequency oscillations are suspected of caus- ing malfunctions to susceptible equipment that is connected to the power mains. In particular, these malfunctions happen to digital equipment, which is different from analog equipment, where a transient event can cause persistent failures (logic upset or operation disruption).

In this context, frequencies are currently referred to as “low” frequencies in the range between 1 and several tens of kilohertz, i.e., for transient durations of from a fraction of a cycle of the normal power-mains waveform up to durations not short enough to essentially involve propagation phenom- ena (delay, line resonances, radiation). The potential for harm from the low-frequency transients arises from the small atten- uation capabilities of the commercial power-line filters in the lower frequency range and from the high energy content of the long-duration disturbances. In fact, existing tentative tolerance curves for voltage disturbances suggest more stringent limits for longer-lasting disturbances [ 11-[3].

Manuscript received May 26, 1989; revised October 9, 1989. The authors are with the Electronic Engineering Department. University

IEEE Log Number 9034192. of Florence, Florence, Italy.

Our experience and the analysis and experiments described in this paper show that low- frequency oscillatory transients are originated by the connection of large capacitances to the mains. Connection or disconnection of resistive or in- ductive loads results in high-frequency oscillations, which are sustained by the line inductance and the distributed capaci- tance of the line and loads. The connection/disconnection of inductive loads has been carefully analyzed in [4], including the phenomena for which the (nonideal) mechanical switches are responsible. It is demonstrated in [4] that although large low-frequency oscillations are excited on the load side of the switch, only high-frequency transients can be observed on the mains (above 1 MHz and up to 1 GHz). This was confirmed by our experiments (not reported here).

The connection of a large capacitor to the line is a com- monly occurring event because of the presence of the power- factor correction capacitors, either as a single element across a specific inductive load or as a large capacitor bank for a plant or substation. In addition, relatively large differential-mode capacitors in the interference-protection filters can produce low-frequency oscillations. The transient occurs irrespectively of whether the capacitor is actually switched across the live wires, as in power-factor-correction actions, or is permanently connected in parallel with the load to be corrected, as in, for instance, fluorescent lamps.

We have performed experiments based on the deliberate switching of known capacitors across the live wires of the power distribution system and on the measurement of the re- sulting transient in the phase-to-ground and neutral-to-ground voltages. These measurements and their theoretical analysis achieve the following three main results: a) The origin and the characteristics of the low-frequency oscillatory transients are clearly identified; b) the equivalent circuit of the power distribution system is derived and related to the physical struc- ture of the system, while at the same time, a simple method is suggested for measuring the parameters of the equivalent circuit; c) remedies to reduce the impact of these transients are suggested.

The simple connection of a capacitor across the live wires of the power-distribution system is used here as an accurate and reproducible experiment that permits the determination of the impedance of the power mains at an outlet. The resulting disturbance corresponds essentially to the step response of the system, the probes of which that we have used (notch- filter networks [5]) permit the extraction of the transient part by filtering out the permanent response (the steady-state mains voltage).

OO18-9375/90/0500-0087$01 .OO 0 1990 IEEE

8 8 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 32, NO. 2 , MAY 1990

The analysis that follows involves observing the transients not only in the two-wire system constituted by the phaseheutral circuit but also as phase-to-ground and neutral- to-ground voltages in the three-wire system phase-neutral- safety earth wire. The added consideration of the earth wire allows otherwise unobtainable information to be obtained- essentially, the common or differential-mode na- ture of the voltage perturbation and a simple and physically significant equivalent (low-frequency) circuit of the three-wire power-mains system. This circuit, for which the phaseheutral mesh is concerned, corresponds to a commonly accepted re- sistivehnductive model appearing in the literature or in the technical standards (see references below, Section IV). The three-wire analysis gives additional insight into the physics of the phenomenon.

Measurements of common-mode or differential-mode volt- ages in the power mains have been in use for a long time (see, e.g., the Vee or Delta artificial-mains networks in [6]). Lim- ited results, however, appear in the literature, although the importance of CM/DM analyses and measurements has been recently stated, with special emphasis on the application to a more effective design of the power-line filters [7]. The effect of the switching of capacitive loads and the evidence of DM phenomena has recently appeared in [8]. It is believed that the results of the present work, in particular the physical descrip- tion of the mains and its lumped-element equivalent circuit, will help improve the protection criteria of the apparatus es- pecially in the frequency range below 0.1 MHz that has so far received limited attention and where, as has already been noted, high-energy transients are common, and the powerline filters are almost ineffective. In addition, our results show a high DM content in the transients; therefore, isolation trans- formers are expected to be almost ineffective for protection as well (see [9] and some manufacturer’s specifications).

The essential aspects of the time-domain transient behav- ior were studied using the equivalent circuit, and several in- teresting results are outlined. First, two contributions to the power-mains impedance can be differentiated: one belonging to the line itself and the other to the sending-end or generator impedance. Then, the maximum values of the peak DM volt- age resulting from the disturbance are derived, and it turns out that they can almost reach twice the amplitude of the mains voltage. Bouncing in the switches can make things much worse in some highly unlikely although possible cases, resulting in the above peak voltage being doubled(or more). Finally, sug- gestions on the reduction of the amplitude of the transients to low or even negligible values can be derived. For instance, the insertion of important capacitor banks could be made at the zero crossings of the mains voltage waveform, or the capacitor bank could be subdivided into sections that are sequentially switched into the network. A small-value resistance could be inserted permanently or momentarily in series with the ca- pacitor to reduce (or eliminate) the overshoot. Special care should be taken to control bouncing in the switches.

11. ORIGIN AND DESCRIPTION OF THE LOW-FREQUENCY OSCILLATORY TR4NSIENT

When a capacitor (initially discharged) is suddenly con- nected across the live (phase = P , neutral = N) distribution

Fig. 1 Oscillatory transient at the connection of a capacitor (C = 47.5 pF) across P and N. Voltage versus time is represented. Horizontal scale: 0.5 ms/div. Vertical scales adjusted for equal overall scale ratio for the two channels, i.e., 100 V/div.

conductors, a transient is excited as is described in Fig. 1.’ The 220-V, 50-Hz voltage on the P conductor is shown and is seen to correspond to a quarter cycle (5 ms) on the complete horizontal trace. The capacitor is 47.5 p F (nom.). The 50-Hz N voltage is almost zero (the minor zero offset shown in the Fig. 1 is a display artifact and can be disregarded.) Both volt- ages are measured between live conductors and safety ground (i.e., green wire G). When the connection is made, the P and N voltages are seen to go instantaneously together, i.e., they meet each other at an intermediate value between the in- stantaneous P voltage value of the 50-Hz waveshape and the zero steady-state value of the N conductor. This comes from the capacitor imposing V = 0 at the instant of connection. In the example shown, the 50-Hz P voltage was in the pos- itive cycle, and therefore, a negative variation occurs on P, whereas one observes a positive variation on N. The reverse obviously occurs when the connection happens in the negative 50-Hz half cycle. A somewhat larger instantaneous variation is observed on P than on N. This is caused by the different impedances from P to G and from N to G , as is discussed below. A damped oscillation then follows the instant of con- nection with a frequency that is dependent on the capacitance value and the mains network inductance and with a damping depending on the mains resistance (below). This measurement was performed with high-impedance compensated resistive di- viders.

111. MEASIJREMENT

In Fig. 1, the transient is shown together with the 50-Hz waveshape to show their relative amplitude and phase rela- tions. The actual amplitude-time function of the transient is best measured after removing the 50-Hz mains waveform. This, in our experiment, was done using notch-filter networks with broadband frequency response and very low distortion at low frequencies [5] . The circuits used had a - 3-dB low- frequency limit of 400 Hz, a flat frequency response up to 30 MHz, and a coupling ratio of 200: 1. Low phase distortion is essential in these experiments if quantitative results are to be derived. One filter network was connected between P and G

‘ A distribution using an N line is assumed here, but the general consider- ations also apply to the distribution system where both wires are “hot,” such as the black and red wires of the USA NEC.

FORTI AND MILLANTA: POWER-LINE IMPEDANCE AND ORIGIN O F TRANSIENTS 89

+ 1 i

neutral I . 1

Fig. 2. Example of voltage oscillatory transient subsequent to the connection of a capacitor (12.5 pF) between P and ,V. Frequency 5.5 kHz. Voltages measured with respect to safety (ground) wire.

as was another one having the same characteristics between N and G . A capacitor of known value was connected between P and N, and the transient was recorded with a digital sampling oscilloscope (20 Ms/s maximum sampling rate with simul- taneous two-channel sampling, 1024 samples per trace). An example is presented in Fig. 2, corresponding to a capacitance value of 12.5 pF (nominal). The two oscillations appear op- posite in phase and have similar amplitudes (predominantly differential mode). The frequency is 5.5 kHz.

Several trials were obtained by varying the capacitance value, and for each case, the oscillation frequency and damp- ing was derived. A plot of frequency versus capacitance is shown in Fig. 3 with logarithmic scales. The measurements along the upper dashed line in Fig. 3 were taken at an outlet within a large building. Three different series of measure- ments were taken on different days at different times of the day to observe a possible effect of variable network loadings and to check for the stability of the results (dots, dotted cir- cles, triangles). All points are seen to lie on a straight line with negligible scattering over a wide range. The measure- ments along the lower dashed line in Fig. 3 (X's) were taken within the same building but at a wall outlet that was more distant than the previous one from the service entrance. This plot also corresponds to a straight line, and both have a slope corresponding to a variation of frequency with the inverse root of C. Thus, the results indicate a series R-L-C tran- sient below the critical damping and can be further analyzed using the standard results for the elementary circuit in Fig. 4. It is obvious that frequency and damping, as derived from V-terminal data such as those in our experiments, remain the same in the pure DM circuit of Fig. 4. The source e ( t ) rep- resents the mains voltage. Let us assume for the moment that the transient duration is much less than the power-line pe- riod of the mains waveform. The excitation of the circuit then corresponds essentially to a voltage step having an amplitude Ej equal to the line voltage at the instant when the switch is closed. The step response can be described as

Wn w,'

where s is the Laplace variable, D(s ) = s2 +s (R /L)+ 1 / ( L C )

1 3 10 30 100 C(,,F)

Fig. 3 . Oscillation frequency plotted versus the capacitance values. Upper plot: three series of measurements (dots, dotted circles, triangles) at an outlet. Lower plot (X's): one series of measurements at a distant outlet.

I I

connected to the capacitor C . Fig. 4. Elementary circuit representation of the power line (R, L , e ( t ) )

and

The frequency of oscillation is measured on the transient- voltage plots (e.g., Fig. 2) and is

and the ratio of two successive peaks is

(3)

From these, the natural frequency f ,, U, /2a and damping factor 6 are derived, and finally, the impedance parameters R and L are obtained. Note that fn and fo turn out to be practi- cally coincident throughout the range of Fig. 3, including the lower frequency values where p is larger and the discrepancy is more evident (but still less than 5%). For ease of reference, the expression of the complete transient is written down here:

(4)

The constant term 1 represents the amplitude E; of the step at the closing of the switch and is removed by the notch filter. From the data of Fig. 3, an L value of 66 pH is derived for the upper plot, and a value of 183 pH is derived for the

90 IEEE TRANSAC

~

low-frequency approximation

I I

1 high-frequency approximation I

0 46 -,--- 0 5 10 15 20 f (kHz)

Fig. 5 . Behavior of the line resistance versus frequency. Measured values: dotted circles. Continuous line: best-fitting curve according to theory of round wires. Dashed line: region where low-frequency approximation is no longer valid. Dot-dash line: region where high-frequency approximation is no longer valid.

lower plot. Both L values have a low dispersion, which can be described by a standard deviation of 6%.

The resistance as derived by the procedure outlined above is, as expected, frequency dependent (skin effect). The re- sults, corresponding to the series of measurements of Fig. 3 (upper plot) are shown in Fig. 5 (dotted circles). The other series of measurements in Fig. 3 shows higher values with a presumably similar behavior. The resistance variation versus frequency is discussed in the next section.

IV. POWER-LINE EQUIVALENT CIRCUIT PARAMETERS

We have so far seen that the impedance at the outlet, at least in the range explored (from 1 to more than 20 kHz), corre- sponds to a frequency-dependent resistance plus a (constant) inductance:

z = R ( f ) t s L . (5 )

It is desirable to be able to link Ihe above-derived quantities to the characteristics of the power distribution system so that a physical, rather than a phenomenological, description can be obtained. We first observe that the constant value of L was identified by the experiment described. We also note that the L values derived from the results of Fig. 3 are consistent with the orders of magnitude existing in the literature. For instance, the 66 p H from Fig. 3 (upper plot) can be compared with Fig. 17 in 191, which shows 65 pH, or with the 50 pH specified for the CISPR artificial-mains networks [lo]. The result also compares favorably with the average values quoted in [ 1 11 for the European distribution and with orders of magnitude that can be derived from [12]. A much higher inductance value is derived from the lower plot of Fig. 3 , i.e., 183 pH. This can be accommodated by the maximum values deducible from [ 1 11. In addition, a very high inductance can be derived from the artificial network specified in Fig. 9(b) in [13], i.e., 796 pH between P and N.

The resistance values shown in Fig. 5 are again of the same order of magnitude as those quoted in the literature, e.g., [9], [12]. The frequency behavior can be compared with the theo- retical dependence of R versus f for a round conductor [ 141. A best-fitting high-frequency approximation of the measured

‘TIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 32, NO 2. MAY 1990

points was obtained as

R ( f ) = A f i + B (6)

where A and B are the parameters obtained in the best-fitting procedure, with A = 7 x lop3, and B = 0.148. The behavior of (6) is the high-frequency approximation of the exact for- mula involving Bessel functions of a complex argument (see also [ 151). The low-frequency behavior is approximately de- rived, as it was in Sect. 5.17 Eq. 5.17-4) in [14] and is a dependence of the type

R ( f ) = C f 2 +D. (7)

The quantities C and D, however, are not directly relatable to the experimental points of Fig. 5 but are instead derived from the best fitting high frequency such that the two ap- proximations are continuous in the region of their crossover. The two approximations should theoretically join for r E 26, where r is the wire radius, and 6 is the skin depth, i.e., for f 4/npuar2 where p is magnetic permeability, and U is con- ductivity. The quantitative results are acceptable in comparison with the values mentioned in the previously quoted literature. The Etting process, however, is more complex and thus less reliable than that for the line inductance. This is particularly true for extrapolations such as the dc value ( R d c 2i 0.46 0). The power-network configuration was complicated and only roughly known. The resistance values, however, agree with the order of magnitude predictable for a reasonable wiring.’

V. THREE-TERMINAL EQUIVALENT CIRCUIT AND ANALYSIS

We have seen that the impedance looking into the P and N terminals at an outlet corresponds to a resistance in series with an inductance. An additional important observation is derived from the experiment (see Fig. 2). The voltages shown there are measured with respect to safety ground or green wire. It is seen that the P-G voltage is opposite in phase with respect to the N-G voltage with unequal amplitudes ( P larger than N). The analysis requires a model taking into account a three- wire situation, which can be represented with a Y network, as is shown in Fig. 6. Points P, N , and G represent the phase, neutral, and ground terminals at the outlet, respectively. The physical justification of this network is not obvious, and a discussion is presented in Appendix I. From the previous re- sults, Z s appears to be essentially inductive and is usually small with respect to the wiring impedance (see Appendix I and below).

The analysis of the transient in Section I11 was confined within the P-N mesh, and a step excitation was assumed.

’From the outlet to the MVILV transformer, we had about 16 m of line with a 6-mm2 cross section (each conductor). We then had about 30 m of line with one conductor (phase) of 26 mm2 and the other (neutral) of 16 mm2 and finally 57.5 m of line with a 240-mm? cross section (each conductor). The minimum theoretical dc resistance would amount to 0.14 R. The measured value conceptually includes some contribution from the power distribution network (12-kV lines, HV transmission lines, transformers, etc.) as well as various possible causes of increased resistance (contacts, proximity effects, i.e., increased current-distribution nonuniformity that adds to skin effect, etc.). However, as previously stated, the extrapolation process is suspected of being responsible of most of the disagreement.

FORTI AND MILLANTA: POWER-LINE IMPEDANCE AND ORIGIN OF TRANSIENTS 91

Fig. 6 . Equivalent circuit of the power distribution system as seen from the outlet terminals P, N, and G , including the sending-end phase-to-neutral impedance Z S , the line voltage generator e ( t ) , and the transient-generating capacitor C. Wiring impedance Z W = Rw + j w L ~ in the case of equal and symmetric wires.

The phenomenon is now analyzed in greater detail by consid- ering the voltages with respect to ground. We again assume for the following a step excitation, which is adequate for a basic understanding. A more general analysis, which includes the variation with time of the power-line voltage during the transient, is also interesting. This is shown in Appendix 11.

Let us now decompose U, (U, = U p N ) into the two ex- perimentally observed voltages up(; and U N G . This permits a derivation of the unknown circuit quantities, i .e., R w , L w , and Zs = Rs + SLS. The previousiy considered quantities correspond to L = LW + Ls and R = R w + Rs. Using the Laplace transform Vpc and V N G , from the circuit of Fig. 6:

1 2 L D(s )

Lw +2Ls s + 1 / T P G V p G = --E; ~-

where r p G = ( i L w + L s ) / ( f R w S R s ) , and TNG = L w / R w . A step excitation E; is assumed.

More generally

1 z w + 2 z s 2 z w t z s

V P G = --(E - V,)- (9)

Here, general impedances for each wire ( Z W ) and termination (Zs) are assumed, and a general (Laplace transform) E is assumed for the excitation, thus including time variation of the input voltage e( f).

The ratio V P G / ~ / N G is given as

The presence of a Z S # 0 causes an unbalance of the two voltages in the otherwise symmetrical three-wire system. An essentially differential-mode voltage distribution ( V p G N -

V N G ) requires that lZsl << IZwl. The discontinuity at t = 0 permits derivation of Ls and L w . Using the initial-value theorem on (8)

1 L w t 2 L s L limupG(t) = lim s V p ~ ( s ) := --E;

t-0 S i C c 2

1 Lw limuNG(t) = lim ~ V N G ( S ) = -Ei-. 1-0 S - o i 2 L

The difference of these two quantities is obviously -E; , i .e., the total initial discontinuity E; is subdivided between phase and neutral as is clearly shown in Fig. 1 . The ratio of the two quantities is

( 1 1 )

which is intuitively correct (only the inductances are seen at the closing of the switch, t = 0 +). When the above ratio is measured together with total L measured as in Section 111, the wiring and the source inductances can be determined. If we examine the time behavior after t = 0 + , antitransforming (8), we see that u p G ( t ) and U N G ( ~ ) correspond to two damped oscillations that differ both in amplitude and phase (same fre- quency and same damping). The phase is given by

IUfC(O>I - - $w + L s /UNG(O)) i L W

The two oscillations are therefore not in phase opposition in the general case, i.e., either Zs = 0 is required (in which case the amplitudes are also equal) or equal time constants are required ( L s /Rs = L w / R w , with unequal amplitudes).

Examples referring to a simulated situation are shown in Figs. 7 and 8 . In those figures, the transients are excited and observed at the input of a dummy line constituted by a symmetrical three-wire power cord (length 11.1 m, wire diameter N 1 mm, center-to-center wire spacing N 2.2 mm, Rdc = 0.56 R, and LW = 7 p H as measured at 0.4 MHz with a HP 4193A vector impedance meter). The transient was excited with a dc-charged capacitor (2-pF nominal). In Fig. 7 , the three wires of the line are short circuited at the far end. The oscillation frequency, corrected for the damping, turns out to be f n = w,/2.rr = 40.2 kHz. This compares satisfacto- rily with the expected value 112%- = 42.7 kHz, which is obtained for LW = 7 pH. In fact, the discrepancy (6%) is moderate and reduces to a negligible value if one takes into account the variation of the internal inductance of the wires with frequency [14]. Stated differently: the correct value of LW is indeed that measured using the ringing frequency, and the value measured independently (with the impedance me- ter) at a higher frequency is lower, as accounted for by the internal-inductance reduction. In Fig. 8 an inductance Ls is connected between the P and the N wires, where the N wire is short circuited to the G wire (actually, the three wires are all equal and interchangeable). In this case, LS = 8.65 pH (measured at 0.4 MHz). The dc resistance was 0.12 R. We see that the transient is no longer symmetrical. The ringing frequency corrected for the damping turns out to be 27 kHz and corresponds with that predictable from 1 / 2 a m (again, we observe an acceptable discrepancy, which is accounted for by the variation of the internal inductance). We now compare the peak amplitude ( 1 = 0) of the transient on P with that

92 LEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. 32. NO. 2 , MAY 1990

L -

1011s

I I R i L

Fig 7 Example of d symmetrical transient on a simulated line Three-wire symmetrical power cord short-circuited at the far end

1L

Fig. 8. Example of a nonsymmetrical transient. Same experiment as in Fig. 7 with an inductance connected between the “phase” and “neutral” wires at the far end. “Neutral” and “ground” shorted.

on N . These, when carefully read on the oscilloscope (using markers and expanding the waveshape to better dispose of the high-frequency ringing at the onset), give a ratio

The theoretical value from (11) is 3.21, and again, the dis- crepancy is well within the experimental uncertainties. Fig. 8 also clearly shows a phase shift between the two waveshapes, as is expected from the theory. This can be better discussed in conjunction with the real-world situation of the experiments on the actual power network (see below).

As was observed in Fig. 2 , an evident unbalance appears in the amplitudes, whereas the phase shift is very nearly n radi- ans, or at least, a difference from 7r is not discernible in the figure. The almost perfect phase opposition was always ob- served in the many transients recorded in different situations. In several cases, a very good amplitude balance was also ob- served (see an example in Fig. 1 of [16], where a transient of natural origin produced by the power-factor-correction capac- itor at the switching on of a fluorescent lamp is shown). All of the above behavior is compatible with the theoretical model. We know that a perfect phase opposition is only reached when either 1) IZs 1 is negligible ( / Z s I a< /ZW I) or when 2) the two time constants LW f R w and Ls fR:; are equal. This last case is perhaps uninteresting because it represents a mere coinci- dence. The observation of a significant Dhase difference (with

respect to n) is, however, unlikely in the real situations, as was confirmed by our experiments. In fact, a large difference in time constants and a significant damping is required. The case in Fig. 8 can be considered to refer to a forced situa- tion that is almost extreme. Note also that according to our observations and physical assumptions (see Appendix I), / Z s I can never be large with respect to IZwl in order to make the unbalance more conspicuous than has been shown in the previous figures.

The impedance Z s of the real power system is presently a poorly defined quantity and is not clearly relatable to the (complicate) physical structure of the system. Its presence is, however, clearly identified phenomenologically. In particular, the experiment is such that the generator inductance comes out quite evident (related to the amplitudes at the instant of connection). The experiment is instead rather insensitive to the generator resistance (related to the phase differences), but this reduced information is not likely to appreciably limit the interpretation or prediction of most phenomena (see also the comments in Appendix I).

VI. SIMPLIFIED ANALYSIS, ESSENTIAL QUANTITIES, AND THE

WORST-CASE TRANSIENT

On the basis of the previous model, we can also develop a quantitative expression for the worst-case transient that can be observed on the line. We consider the differential transient between P and N . The step response gives an initial value equal to the instantaneous power-line voltage, i.e., the maxi- mum is

where V is the rms mains voltage. Following t = 0, the backswing that is nearly in phase opposition on the P and N wires gives rise to a maximum P-N voltage difference, which corresponds to the first peak of the transient after the onset (see example in Fig. 1 where a P-N voltage difference of > 400 V is measured). The overvoltage depends on the damping and can be expressed according to (3) as

U p N ( t ) I m a x = E , ( I +e-s6/d1p6*) (13)

and the absolute maximum possible U P N corresponds to E, = V in the limit for negligible damping (6 + 0), i.e.

Absolute max. u p N ( t ) = 2 . & V. (14)

For the time duration that is concerned, the transient can be briefly quantified by the approximate half-amplitude time ration of the dip immediately following t = 0, i.e., one-half of a quarter period of the ringing

1 1 71 E - -

8 f o where fo is the frequency of the ringing. In addition, the approximate half-amplitude duration of the first backswing (overvoltage) can be given as

1 1 72 = 271 E ,I.

v 4 ~ 0

FORTI AND MILLANTA: POWER-LINE IMPEDANCE AND ORIGIN OF TRANSIENTS 9 3

which can be of use in very simple technical descriptions of the phenomenon. For instance, it can be seen that plausible values of fo and of Absolute max. u p ~ ( f ) can easily fall outside an amplitudehime tolerance curve such as that in [3].

The spectral analysis of transients permits us to obtain addi- tional simple results of practical use. The transient oscillation across the capacitor has the 6: transform given by

E; s 1 +SRC + : S ~ L C s

_ _ _ - _ 1 v; = 5

i.e., the spectrum of the complete voltage form minus E; /s (the spectrum of the step function with amplitude E;). Thus

is what is observed after removing the mains voltage. The dc component of the unilateral (real) voltage spectrum is

S v ( j 0 ) = 2E;RC. (16)

The peak spectral amplitude (approximation around the peak frequency fn fo), again in terms of unilateral spectrum, is

If we assume, for example, R = 1 R , L = 60 pH, and C = 30 p F with E; = ’ 220 V, we find

S v ( j 0 ) = 85.4 dB (pV/Hz).

This is in accordance with experimental values existing in the literature (see Fig. 7 in [17]). The peak spectral density would be

S V I M ~ ~ = 93.2 dB (pV/Hz).

The above quantitative results can be modified unfavorably by the frequently occurring phenomenon of switch bouncing. Contact reopening on a bounce temporarily isolates the ca- pacitor from the line. The following contact closure places the charged capacitor across the line at an instant when the line voltage is different from the previous value. The worst case evidently corresponds to two subsequent contacts at peak values of opposite sign of the line voltage. This is an unlikely, yet possible, occurrence. The analysis can be performed as was done previously in connection with (8) and (9), consid- ering that the charged capacitor is equivalent to the same ca- pacitor (uncharged) in series with an ideal voltage generator having the voltage equal to that to which the capacitor was charged. It can be seen that the worst case corresponds to doubling the value in (14), which in a 220-V,,, network would be more than 1.2 kV. In principle, multiple bouncings could further multiply the effect with even higher voltages. An example corresponding to an unfavorable, but not worst- case, bounce is shown in Fig. 9(a). where the switch opened the contact during the 6rst backswing of the transient oscilla- tion and closed again during the subsequent half cycle. The phenomenon is better seen in Fig. 9(b), where the second os-

C h l

r^w2

(b) Fig. 9. Example of a switch bounce in two contiguous half cycles of the

power-line voltage: (a) Large overvoltage across P and N is shown; (b) shows the second contact with the time scale expanded 10: 1.

cillation is shown with an expanded time scale (1O:l). The sampling frequency is not entirely adequate to accurately rep- resent the waveshape3; however, a DM voltage of over 600 V results at the second contact. The initial voltage difference between P and N (A-A’ in Fig. 9) corresponds to the volt- age left on the capacitor at the previous disconnection ( = 430 V, negative). The bouncing rate was relatively slow with the switch in the example (a magnetothermic automatic switch of nominal current 10 A). In many cases, bounces occur closer to each other, thus presenting a less dangerous situation.

VII. CONCLUSIONS

We have shown that the important category of the low- frequency oscillatory transient disturbances originates from the connection of large (greater than 1 pF) capacitances across the live wires of the power distribution system. The tran- sient has a frequency of oscillation that is simply determined by the capacitance value and by the inductance of the mains wiring as is seen at the outlet. The switch characteristics do not come into play, except for possible bouncing phenom- ena. The disconnection is expected to produce small high- frequency disturbances. This behavior is different from the inductive case, where the transient disturbances on the line are high-frequency, are produced at the opening, and depend

’Note the slow rate of rise and the initial minor ringing at the onset due to sine interpolation of the samples.

94 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. 32. NO. 2 . MAY 1990

essentially on the switch behavior (mechanical and electrical, such as bouncing, sticking, and arcing).

A second important result is the possibility of obtaining a simple and reliable measure of the low-frequency impedance seen between phase and neutral by connecting known capac- itors across the line and determining frequency and damping of the transient thus produced. A constant inductance is ob- served in series with a frequency -dependent resistance. The values are consistent with those in the literature and appear to correspond predominantly to the resistance and inductance of the wiring.

Since the experiment measures the transient both on the hot wire (phase-to-safety wire) and on the neutral wire (neutral- to-safety wire), additional information is obtainable by the comparison of the two. The main results are as follows: 1) the voltage disturbance is essentially differential mode, and its amplitude is simply that of the line voltage at the instant of connection; the overshoot due to the ringing gives rise to a larger voltage at the peak of the first cycle of oscillation; volt- age and duration are accurately predictable by simple analysis or can be determined as orders of magnitude. 2) The dif- ference in voltage amplitude and phase between phase and neutral is due to the impedance terminating the line on the generator (utility) side. Finally, 3) a physical model (low- frequency) of the power line is presented, which lends itself to a simple equivalent circuit such that steady-state or transient analysis becomes straightforward and significantly related to the structure of the distribution system (essentially the charac- teristics of wiring). Considerable interest can be attributed to the finding of the maximum possible DM transient (see (14)), corresponding to an overvoltage equal to twice the amplitude of the mains voltage.

Several suggestions can be drawn from the above analy- sis and results. For instance, connecting the capacitive loads at the zero crossings of the power voltage wave would give very small amplitude transients. Subdividing large capacitor banks into sections and connecting the sections sequentially to the network would give reduced duration of the first tran- sient (same amplitude) and would give reduced amplitude for the subsequent sections. A resistance added in series to the capacitor (permanently or temporarily) would give low or no ringing, thus drastically reducing overshoot and duration.

The present results are confined to a frequency range (es- sentially below 50 kHz), where the high-frequency behavior of the lines is not dominant. Actually, high-frequency ringing is observable in many experiments, and this can be attributed to propagation phenomena on the lines (resonances and mul- tiple reflections). There is probably little hope that a simple and general high-frequency model of the power-distribution network can be obtained (including the transformation into common-mode disturbances). However, it would be desirable to investigate into the transition region between low-frequency and propagation ranges to determine the operating limits of the low-frequency model and connect them to the structure of the network. In addition, further investigation into the origin and characteristics of the sending-end or generator impedance would be of interest.

APPENDIX I

Y-NETWORK EQUIVALENT CIRCUIT OF THE THREEWIRE POWER DISTRIBUTION SYSTEM

A Y , or star, equivalent circuit of the wire triplet is very helpful for the quantitative treatment of the distribution sys- tem. An equivalent circuit should be connected to the physical structure of the system as closely as possible to make the re- sults and predictions more significant. We then make some greatly simplifying physical assumptions, which are deemed acceptable in the range of parameter variations of our interest. They are essentially frequency range, geometry of the cabling and wire characteristics, and impedance level of the network.

We first assume that the network is small with respect to the shortest wavelength of interest so that lumped- circuit analysis can be adopted. For example, with dimensions I X/10 and a maximum frequency of 50 kHz, the system is

“lumped” up to 600 m. Propagation effects and radiation can be neglected (actually, with intentional experiments, we have also observed propagation delay with the initial discontinu- ity of the ringing waveform, but this does not invalidate the lumped-circuit assumption in the frequency range of interest).

We assume that the three wires are connected with low impedances towards the power-entrance side or even short circuited (such as N and G at the service entrance). The com- parison impedance level is the characteristic impedance of the parallel-wire line constituted by each couple of wires, and this can be assumed to be relatively invariable from case to case and can also be assumed to be of about 100 12 or not much less ( Z , = 120 cosh-‘ ( s / d ) / a f , where s = spacing, d = diameter of the conductors, teff = effective rel- ative dielectric constant, which for wires running very close to each other is not much lower than the E of the insulating mate- rial, of the order of 3 ) . With this assumption, admittances in parallel across each couple of wires are not needed, and the series impedances are adequate to model the line behavior. In high-voltage networks instead, the impedance levels are high, and the spacing of the conductors is large with respect to the conductor cross dimensions and is not small with respect to the distance to the earth ground. Thus, parallel admittances (capacitances) would be required across the conductors and to the earth ground [18]. Further discussion of the impedances connected across the P and N wires is needed and is deferred after the presentation of the equivalent circuit. In addition to the equivalent-circuit simplification, our assumption also per- mits a simpler physical representation and interpretation of the phenomena; the distribution system is essentially a closed system, the interaction with the surroundings (earth ground, large conductors, etc.) being negligible, and the wires being only coupled by mutual inductance and wire resistance, with magnetic fields rapidly falling down away from the three-wire system.

The simplest model corresponds to the maximum geometri- cal symmetry: three equal wires at equal distances from each other. The hypothesis of symmetry will be later removed for greater generality, but it is well justified in our present analysis since the distribution lines are usually composed of three equal

FORTI AND MILLANTA: POWER-LINE IMPEDANCE AND ORIGIN OF TRANSIENTS 95

Fig. IO. Simplest physical representation of a three-wire power distribu- tion line: symmetric lines, P , N, and (2 are terminals at the outlet. An impedance-measuring generator V and impedance Z.S are also shown. PI , NI, and GI are the opposite extremes of the wires (power-generator side) with NI and GI shorted.

wires run in the conduit close to each other and symmetrically placed, at least in statistical sense, with respect to each other and to the surroundings. Pieces of three-wire power cord are also used and can be more rightfully assumed to be symmetri- cal. The system corresponds to three mutually coupled loops (P-PI-NI-N, P-P’-G’-G, and N-NI-GI-G) (see Fig. 10). The inductance contributed by each of the loops as observed across any couple of input terminals (i.e., P , N , and G) is given by

(18)

where i and j are the two terminals considered, k is the third terminal, and M ; , is the mutual inductance between the two loops i-to-k and j-to-k. All inductances are equal; therefore

L . . - L . + L . . - 2 M . . IJ - i k jk

Mij = Mik = Mjk = L w / 2 .

If we apply voltage V across P and N , we observe the current IPN in the mesh P-PI-NI-N

I/ SLW + z s

IPN =

and induced EMF’S arise (assume M > 0):

Lw V VNG = -SMIPN = --.s- 2 SLW + z s

and

VPG = ~ M I P N + ZSIP,Y Lw v V

- Z S - - s- 2 s L w + Z s s L w + Z s ’

( VNG is due to the induced EMF into N-N’-G’-G, whereas VPG is due to voltage induced into P-PI-GI-G plus voltage drop in Z s . Induced EMF into P’-N’-G’ is zero or negligi- ble.) We see here V P N = VPG +- VGN = VPG - V N G = V as it should be, and we see that the two voltages differ be- cause Z S is present. When Zs = 0, the system becomes com- pletely symmetrical: Vpc = - V N ~ ; = V / 2 . Wire resistance is unessential in the above reasoning and will be added later.

For the lumped-element equivalent circuit, let us consider a

(b) outlet

(a) Fig. 11. (a) Impedances of the line equivalent circuit in relation to

impedances Z P N , Zpc , and Z N C observed across each couple of outlet ter- minals with Z s = 0; (b) simplest case of purely inductive and symmetrical impedances Z ~ N = Zpc = ZNC = j w L w .

star circuit representing the line with Zs short-circuited as in Fig. 11. The equivalent impedances are defined on the basis of the impedances seen across each couple of terminals (third terminal open) i.e.

ZPN = z P + z N

which identifies Z p , Z N , and Z G when the terminal impedances are identified. If we assume in the simplest case Z P N = Zpc = ZNG = j o L w , we have Z p = ZG = Z N = jwLw/2 . Total inductance L w / 2 + L w / 2 represents the in- ductance of each loop connected across each terminal pair, and L w / 2 represents the mutual coupling between the two loops sharing the same conductor (it is not to be interpreted as the inductance of each wire, or worse, half the inductance of a loop).

If we now introduce the resistance of the wires, the circuit results turn out to be the same as in Fig. l l (b) with R w / 2 added in each branch, where Rw is the resistance of the com- plete loop between each couple of terminals (this time R w / 2 actually represents the resistance of each wire). The resistance R w / 2 now contributes to mutual coupling.

The analysis and results of the more general case when the three-wire line is asymmetrical do not differ substan- tially from the symmetrical case. Again, since our exper- imental results show resistivehnductive impedances, we set Z P N = R ~ N + jwLpN, etc. The equivalent circuit of Fig. 12 results. This more general asymmetrical equivalent cir- cuit does not appear to be presently required to interpret the experimental data gathered so far, and it is much more com- plicated to mentally follow than is the symmetrical case. It can, however, be interesting in analyzing the consequences of minor asymmetries or for partial asymmetries, such as P and N running close together with the safety G wire somewhat separated.

A few comments are needed regarding the sending-end impedance Z s . The origin of this quantity is presently lit- tle understood, but its existence and inductive nature appear from our experiments and are confirmed by results in the literature [ 121. It can be thought of as the complex combina- tion of the impedance seen looking towards the public utility at the service entrance (leakage inductance of the M V I L V

96 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 32, NO. 2. MAY 1990

ideal step. For a given transient duration (e.g., = 1 ms in Fig. l) , this is acceptable near the peaks of the power-line voltage, whereas around the zero crossings, the excitation is more closely represented by a linear ramp. The more general situation is analyzed in the following. With reference to the circuit of Fig. 4, we assume that the line voltage is represented by

e ( t ) = EM sin (ant + p) (20)

where w p is the power angular frequency, and the capacitor is connected at t = 0 (when the instantaneous phase is p). Expanding e ( t ) in series of Mac Laurin one obtains

Fig. 12. Equivalent circuit of the distribution system in the nonsymmetrical case.

transformer?) in parallel with the several loads fed by the distribution system within the premises together with the as- sociated wiring. The overall combination of several indistinct impedances corresponds to a complicated result that appears to behave as a simple inductivehesistive impedance of low value. It is to be stressed that Iupc;I always turned out in our experiments to be larger than ~ U ~ G 1 (or l u p ~ I = ( U N G 1 in some cases [16]), and this appeared to be a constant be- havior of the distribution system, complicated as it would be. We note also that the harmonic level observed between P and G has always been larger than that between N and G. It is also very interesting to note that the unbalance UPG /UNG has always been moderate (1.5 or less). The experiment (obser- vation of the initial discontinuity at t = 0 + ) is capable of making evident the presence of a small Ls (1 1). For instance U P G / U N G = 1.5 corresponds to L,s = Lw/4. The contribu- tion of 2 s to the total mesh impedance Z p + Z N + Z s is thus modest, amounting theoretically to 20% of the total for the maximum U ~ G / V N G observed. Small 2 s and zw are required anyway in the normal operation of the power network (small

1 e ( t ) = EM sin p + ( E M W ~ cos p)t - - (EMU; 2! sin q)t2 + . . .

(21)

The first term is a step excitation, the second is a linear ramp, etc.

The Laplace transform is

1 1 E(s) = EM sin p- S + (EMwp cos 9)- S2

1 2 -(EMU; sin p)T + . . 2! S

(22)

With the transfer function (LC .D(s))-' (see (l)), we ob- tain V,(s), and after antitransforming, we have

u c ( t ) = E ~ s i n p u , ~ ( t ) + E M W ~ C O S ~ u , , ( ~ ) d ~ I' voltage-drop from no load to full load) so that finding low values of 2 s up to the highest frequencies examined here is compatible with the expected behavior.

A final important remark regards the concept per se of impedance of the power network; we must assume that when

The first term ucl corresponds to the step response of (4), and the following terms correspond to successive integrations of the same time function.

Performing the successive integrations in (23) one can write

looking into a power outlet, we see essentially a reasonably long section of the distribution line. Any 2 s should be distant from the observation point so that the behavior is not en- tirely dominated by 2 s itself but rather that the parameters of the three-wire line are essentially contributing to the phe- nomenon. Stated differently, we exclude that a large load is connected in close proximity to (or at) the outlet being mon- itored, since in that case, it is evident that the measured impedance would be that of the load itself in parallel with the line impedance, where the latter is expected to be much lower due to the limitation of the no-loadhll-load voltage drop. The case of large nearby loads is considered among others in [ 121.

APPENDIX I1

TRANSIENT ANALYSIS TAKING THE POWER-MAINS VOLTAGE VARIATION INTO ACCOUNT

When the transient duration is very short compared with one power-frequency cycle, it can be assumed that the ex- citation (generator e ( t ) in Figs. 4 and 6) corresponds to an

w epar + EM cos p 2 ~ sin (wot + 27) + . . .(24) J1-s2 where y = a r c t a n ( d m / & ) , and CY = 6wn. Here, the first three terms represent the permanent forced response (which represents merely the expansion of the mains voltage and is filtered out in the experiment), and the remaining terms repre- sent the transient forced response, which is what is observed in our experiment. Each time an integration is performed, a term w p /Un appears to multiply the result; thus, the weight of the higher order terms is progressively reduced. The conver- gence of the series is fast, where wp/w,, is of the order of at least a few tens.

FORTI AND MILLANTA: POWER-LINE IMPEDANCE AND ORIGIN OF TRANSIENTS

The limiting cases for what phase is concerned are 1) p = r / 2 and ip = 0. The first case gives a solution such as that of (4) corrected by the third term in ( 2 3 ) , which is normally at least a few hundred times lower than the first and can be neglected. In the case p = 0, the first term is zero, and the dominant term is the second, i.e., essentially a damped sine with a weight up /an. As physically intuitive, large transients are only obtained near the peaks of the power-line voltage wave. Small amplitudes result when the transient is excited when the line voltage is zero, i.e., the maximum amplitude when cp = a12 is E M , whereas when p : 0, the maximum amplitude is of the order of EM . up /an.

ACKNOWLEDGMENT

S. Lazzerini, M. Polignano, and M. Monticelli contributed valuable assistance in collecting experimental data.

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