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Electric charge flow in linear circuits
Electric Power Systems Research 170 (2019) 57–63
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Electric charge flow in linear circuits a
b
c,⁎
T
José Claudio de Oliveira e Silva , Antônio Roberto Panicali , Celio Fonseca Barbosa , Carlos Ermídio Ferreira Caetanoc, José Osvaldo Saldanha Paulinoc a b c
APTEMC, São José dos Campos, Brazil PROELCO, Campinas, Brazil Graduate Program in Electrical Engineering, Federal University of Minas Gerais, Belo Horizonte, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Linear circuits Electric charge Surge protective device
This paper deals with a fundamental property of circuits that can simplify certain analysis, bring new methods of measurements and possibly other applications, which has apparently passed unnoticed until now. It shows that the net charge injected by an arbitrary source into a linear circuit is divided among the circuit branches (including transmission lines) in the same proportion as the charge would divide among the same branches, if the circuit was replaced by its equivalent circuit at direct current (DC). The DC equivalent circuit alone will govern the exact sharing of the net charge, despite current oscillations caused by reactive elements or propagation effects of transmission lines. In this paper, this property is demonstrated analytically and proved by computer simulations and laboratory experiments. The paper also exemplifies its application in the calculation of the net charge distribution among the conductors of an installation struck by lightning. Finally, the paper forecasts other possible applications for the new circuit property.
1. Introduction In recent years, the charge flow through a circuit branch due to impulsive excitations has gained attention, as it is related to the rating of surge protective devices (SPDs) used for lightning protection [1–5]. Similarly, the international standards on SPDs [6–8] include the drained charge as one of the key parameters for specifying an SPD. During the investigation of the charge distribution among the conductors of a structure struck by lightning, the authors came across an interesting property of electric circuits. It was found that, although reactive components (e.g., inductors and capacitors) and transmission lines can significantly alter the amplitudes and waveshapes of the currents flowing in the circuit branches, the net charge delivered through each branch depends only on the circuit resistances. This is a fundamental circuit property that can be shortly explained by recalling that a linear circuit comprising any complex association of elements responds to a signal decomposed in frequency domain according to the impedance of the circuit at each frequency, and that at 0 Hz, i.e. for the DC component of the signal, the average current is proportional to the total flow of electric charge. To the best of the authors’ knowledge, such particular property of circuits has not been explored neither presented in the literature and it is applicable to each and every circuit, provided it is linear, time-
⁎
invariant and reciprocal. An extensive search in the literature was made by the authors, but no mention to such property has been found. This search included the database of technical journals and classic textbooks. For instance, Clarke [9] provides a detailed analysis of basic circuit properties, but it is restricted to alternating currents (AC). Greenwood [10] presents a comprehensive analysis of electric circuit transients, but he does not investigate the net charge distribution among circuit branches. Hallén [11] presents several circuit theorems, but none of them is similar to the circuit property described in this paper. Carson [12] derives several circuit properties based on Kirchhoff’s laws, including its transient behavior. These circuit properties include the reciprocity theorem and the equation of activity, which is similar to the Tellegen’s theorem [13]. However, Carson does not address the net charge distribution in circuits. A preliminary description of this property was presented in Ref. [14], which was restricted to a theoretical analysis of the subject. The present paper significantly enlarges the treatment of the new circuit property, by validating it with detailed computer simulations and controlled laboratory experiments. Moreover, a comparison of its results with those obtained by the direct solution of Maxwell’s equations (published in the literature) is included in order to highlight the power of this new property in dealing with complex situations. The paper is
Corresponding author. E-mail address: [email protected] (C.F. Barbosa).
https://doi.org/10.1016/j.epsr.2019.01.014 Received 14 October 2018; Received in revised form 16 December 2018; Accepted 15 January 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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Z12(s): mutual-impedance between Port 1 and Port 2, where Z12(s) = V1(s) / I2(s), for I1(s) = 0; Z21(s): mutual-impedance between Port 2 and Port 1, where Z21(s) = V2(s) / I1(s), for I2(s) = 0; where V1(s) and V2(s) are the voltages at Port 1 and Port 2, respectively. Note that, as the circuit is reciprocal, then Z12 = Z21. Solving the circuit for I2 (s) results:
I2 (s ) = I1 (s )
Z12 (s ) , Z22 (s ) + R
(1)
and the net charge through R up to an instant t is: t
q2 (t ) =
∫ i2 (t ) dt. 0
Fig. 1. Two-port network [N–R] that will be excited through port P1; resistor R is brought out from network [N] and defines output port P2.
(2)
Based on Laplace transform for integrals:
Q2 (s ) = organized as follows. Section 2 presents a theoretical demonstration of the circuit property and Section 3 describes how to circumvent some singularities that may arise when using perfect components. Sections 4 and 5 present its validation using computer simulations and laboratory experiments, respectively. Section 6 compares the results obtained using the new circuit property with those obtained by the full solution of Maxwell's equations. Finally, Section 7 draws the main conclusions.
I2 (s ) , s
(3)
and the total net charge passing through R is: ∞
q2 =
∫ i2 (t ) dt. 0
(4)
Based on the Final Value Theorem [15]:
q2 = limt →∞q2 (t ) = lims → 0 [s⋅Q2 (s )].
2. Theoretical analysis
(5)
Replacing (3) in (5):
q2 = lims → 0 [I2 (s )].
When impulsive sources are applied to linear circuits, the resulting currents flowing can assume complex time variations. However, the net charges flowing through the circuit branches depend only on the circuit resistances. This property is investigated theoretically in the following. Referring to Fig. 1, let P1 and P2 denote the ports of an arbitrary linear two-port network [N–R], that is necessarily reciprocal and timeinvariant. The two ports are selected so that P1 is the input port to which an arbitrary impulsive current source i1 (t) is connected and P2 is the output port where the resulting current i2 (t) through R is wanted. The circuit is at rest at t = 0 (i.e., no energy is stored in its reactive components) and the impulsive source delivers a finite net charge to the circuit. Let q1 be the net charge injected into the network by i1 (t) and q2 the net charge passing through the resistor R, while q1 (t) and q2 (t) are the referred charges calculated up to an instant t. Finally, let I1 (s), I2 (s), Q1 (s) and Q2 (s) denote the Laplace transforms of i1 (t), i2 (t), q1 (t) and q2 (t), respectively, where s is the complex frequency (jω). Given that [N–R] is a linear system, the transfer function between I1 (s) and I2 (s) can be derived from the equivalent T-network shown in Fig. 2. In this figure, the impedances are defined as:
(6)
Finally, applying (1) into (6):
q2 = I1 (0)
Z12 (0) . Z22 (0) + R
(7)
Following the same steps for q1 from (2) to (6):
q1 = I1 (0)
(8)
which results in:
q2 = q1
Z12 (0) . Z22 (0) + R
(9)
Eq. (9) demonstrates that the charge ratio q2 /q1 is given by a ratio of impedances at 0 Hz, namely Z12(0), Z22(0), and R, as expected for DC current division. Clearly, Z12(0) and Z22(0) must come from the association of resistors in the [N― R] network, after inductors have been replaced by short-circuits and capacitors by open-circuits. 3. Ideal reactive components Singularities may arise when computing the net charge in circuits with ideal reactive components (e.g., inductors without series resistances and capacitors without shunt resistances) in the absence of other circuit components that can naturally suppress the singularities. This situation is illustrated in Fig. 3, where the net charge conducted by the circuit branches cannot be determined by the DC equivalent circuit.
Z11(s): self-impedance at Port 1, where Z11(s) = V1(s) / I1(s), for I2(s) = 0; Z22(s): self-impedance at Port 2, where Z22(s) = V2(s) / I2(s), for I1(s) = 0;
Fig. 2. Two-port T-network representing the arbitrary linear network shown in Fig. 1 in complex frequency domain.
Fig. 3. Examples of circuits with ideal reactive components. 58
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Indeed, by replacing L1 and L2 by short-circuits, it is not possible to assess the net charge that flows through each inductor in Fig. 3(a). Similarly, by replacing the capacitors C1 and C2 by open-circuits in Fig. 3(b), it is not possible to assess the net charge that flows through each capacitor. In order to circumvent these singularities, it is sufficient to replace each ideal inductor by a resistor RL according to (10)
RL = kL, and replace each ideal capacitor by a resistor RC according to
RC =
1 , kC
(11)
then take the limit for k → 0. Considering that a charge q0 is injected in the circuits of Fig. 3, the net charge through the inductor L1 and the capacitor C1 are given by (13) and (14), respectively
qL1 = q0 limk → 0
qC1 = q0 limk → 0
(kL1)−1 L2 = q0 L1 + L2 (kL1)−1 + (kL2)−1 + R1−1
C1 C1 + C2
(
1 + kR1 C1 1 + kR2 C2
)
= q0
C1 . C1 + C2
(12)
(13)
4. Validation by computer simulations This section presents an example intended to validate the referred circuit property using the ATP/EMTP [16] simulation software. The results from computer simulations are compared to those calculated from the DC equivalent circuit. Consider the following 6-branch circuit given in Fig. 4. The resistors subscripts n (Rn) indicate branch n. The current source I produces a unidirectional waveform according to Heidler’s time function [17], whose peak value (34.95 kA) was adjusted (through a numerical iterative process) to inject 10.00 C charge into the circuit. Its waveshape is 1/200 μs. Fig. 5 shows the current waves on the circuit of Fig. 4, i.e., the total applied current I and the currents in all six branches. The resulting applied voltage (25 kV peak) follows the same waveshape as the current on branch 1 (Fig. 5(a)), since it is represented by 1 Ω resistor. All other current waveshapes are affected by the reactive elements and by the transmission line (TL) propagation effects on branch 6. The current on branch 6 is computed at the end of the line, i.e., at resistor R6. Note in Fig. 5(c) that the first current step is around 100 A, which comes from the applied voltage (25 kV) divided by the characteristic impedance of the line (500 Ω) and multiplied by 2, since the line is practically short-circuited. The current wave goes bouncing back and forth while growing slowly, resulting in a very long wave with about 1 ms duration. As expected, currents amplitudes and waveshapes are strongly influenced by reactive elements and TL effects. The net charge per branch, however, can be easily determined by means of the circuit
Fig. 5. Currents in the circuit of Fig. 4; (a) all currents; (b) zoom showing currents on branches 2–6; (c) currents on branches 6 and 3 (current amplitude on branch 6 is multiplied by 10).
Fig. 6. DC equivalent circuit for the circuit shown in Fig. 4. Inductors and TLs are replaced by short-circuits and capacitors are replaced by open-circuits.
property presented in this paper. To this aim, the DC equivalent circuit (of Fig. 4) is shown in Fig. 6, where inductors and lossless TLs are replaced by short-circuits and capacitors are replaced by open-circuits. The charge sharing among branches 1–6 of the DC equivalent circuit (Fig. 6) is given in Table 1, together with the EMTP calculated net charges on the complete circuit (Fig. 4). It can be seen an excellent agreement between the net charges calculated by the two methods, which is in line with the referred circuit property. It is worth to mention that the calculation of the net charge for the DC equivalent circuit is rather simple, i.e., it is given by: Fig. 4. Example of circuit including resistors, inductors, capacitors and a lossless transmission line (TL). 59
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Table 1 Net charge distribution: simulated with EMTP (Fig. 4); calculated with the DC equivalent circuit (Fig. 6). Branch
R (Ω)
Simulated (C)
Calculated (C)
Source 1 2 3 4a 4b 5 6
– 1 2 Open 4 Short 5 10
10.0 4.88 2.44 ˜0a 1.22 ˜0a 0.976 0.488
10.0 (given) 4.88 2.44 0 1.22 0 0.976 0.488
a
Table 2 Values of Fig. 7 components.
Asymptotically decaying, reaching about 10–15 C at t = 50 ms.
qi = q0
Ri−1 , n ∑ j = 1 R−j 1
Component
Value
Unit
R1 R2 R3 R4a R4b R5 RL2 L1 L2 C1 C2
213.9 ± 0.1 268.0 ± 0.1 330.5 ± 0.1 149.6 ± 0.1 150.0 ± 0.1 470.2 ± 0.1 0.39 ± 0.04 18.0 ± 0.3 130 ± 2 18.5 ± 0.3 1.60 ± 0.02
Ω Ω Ω Ω Ω Ω Ω μH μH nF nF
(14)
where q0 is the total injected charge, qi is the net charge in the branch i, Ri−1 and R−j 1 are the conductances of the branches i and j, respectively, and n is the number of circuit branches. 5. Experimental validation In order to further validate the new circuit property, this section presents some experimental results. In all cases, the currents were measured by Pearson current monitors model 4100, having bandwidth from 140 Hz to 35 MHz. The current monitors were connected to a Rhode & Schwarz RTB2004 oscilloscope, with 10 bits resolution. The resistors values were measured with a precision HP 34401A meter, with 61/2 digits resolution. The uncertainties of the measurements were combined according with the procedure described in Ref. [18], whereas the values expressed in this paper are the expanded uncertainties obtained with coverage factor equal to 2, which gives a confidence level of 95.5%.
Fig. 8. Current applied to the circuit of Fig. 7.
5.1. Circuit with lumped components The first circuit considered is composed only by lumped components, as shown in Fig. 7. The resistor RL2 represents the inherent series resistance of the inductor L2, whereas the inherent resistance of the inductor L1 was added to the resistance of the resistor R2. Table 2 shows the measured values of the circuit components. The circuit of Fig. 7 was subjected to an impulsive source that delivered the current shown in Fig. 8. The currents measured in the circuit of Fig. 7 are shown in Fig. 9, where can be seen that the currents through R1, R2, and R5 are unidirectional and the currents through R3 and R4b are bidirectional, due to the series capacitor and shunt inductor, respectively. Fig. 10 shows the charges through the branches of Fig. 7, where can be seen that the charges through R1, R2, and R5 tend to a final positive value, whereas those through R3 and R4b approach zero.
Fig. 9. Currents in the circuit components of Fig. 7.
The total charge delivered by the source and the charge through one component were obtained simultaneously by integrating the current recorded at each surge application. Each pair of measurements was repeated 10 times, in order to better characterize the uncertainties related to these measurements. The average value recorded for the total charge was 64.32 μC. Table 3 shows the net charge measured in the relevant components of Fig. 7, alongside with their uncertainties. The same table shows the values of the net charges calculated according to (14), i.e., considering
Fig. 7. Circuit with lumped components for the experimental validation. 60
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Fig. 11. Circuit with distributed parameters for the experimental validation. Table 4 Values of Fig. 11 components. Component
Description
Value (Ω)
R1 R2
Resistor near the source R2 = Z0 R2 < Z0 R2 > Z0 Inner coaxial conductor resistance Outer coaxial conductor resistance
100.68 ± 0.05 50.09 ± 0.05 21.96 ± 0.05 98.80 ± 0.05 2.50 ± 0.01 1.72 ± 0.01
RCI RCO
with a circuit-simulation software (PSpice), which agree almost exactly with those from the DC equivalent circuit.
Fig. 10. Charges in the circuit components of Fig. 7.
the DC equivalent circuit. It can be seen in Table 3 that the calculated values agree very well with the measured ones, whereas the small differences are within the uncertainties of the measurements. This result is an experimental validation of the circuit property described in this paper, considering lumped components. The table also shows the charge values obtained with a circuit-simulation software (PSpice), which agree almost exactly with those from the DC equivalent circuit.
6. Example of application A recent paper from Heidler and Camara [19] presented an analysis of the lightning current distribution among the conductors of the lightning protection system (LPS) and the internal electric lines of a building. Several parameters were calculated and, as highlighted by the authors, "The most important parameter is the maximum transferred charge", which was calculated with a computer code based on the Method of Moments (MoM) [20] that solves the full Maxwell equations in the frequency domain. The time-domain solutions were obtained from the inverse Fourier transform, whereas the skin effect was automatically treated by the software. In order to compare the results from [19] with those calculated using the circuit property described in this paper, it is chosen the Configurations (a) and (b) of the "single family house", as shown in Fig. 13. The LPS is represented by the thin lines installed along the corners of the roof and walls, whereas the electric line, one for each configuration, is represented by bold lines. In both configurations, it is assumed that the electric lines and the LPS are bonded at the striking point by a copper bonding conductor 0.3 m long and having 16 mm2 cross-section area. Of course, the electric line shall be bonded through an SPD, but its effect was neglected in the simulations presented in Ref. [19]. The ground plane was supposed as perfectly conducting. To apply the referred circuit property to this structure, it is necessary to build its DC equivalent circuit. Due to symmetry, the two halves of the LPS present the same resistance from the striking point to ground. Moreover, the horizontal LPS sections along the top of the walls do not contribute to the resistance to ground. Therefore, the equivalent LPS
5.2. Circuit with distributed parameters This section considers a distributed parameters circuit, as shown in Fig. 11. The transmission line (TL) is made by 60 m long coaxial cable with 50 Ω nominal characteristic impedance (Z0). The currents i1 and i2 were simultaneously measured (see Fig. 11). The components values are shown in Table 4, where three different values were considered for R2. In addition to these values, it was also considered R2 = 0 and R2 = ∞. Fig. 12 shows the currents i1 and i2 for the conditions R2 = Z0, R2 < Z0, and R2 > Z0, where the propagation effects on the TL can be clearly seen. The net charges obtained for the cases considered are shown in Table 5, alongside with those calculated using (14), i.e., with the DC equivalent circuit for Fig. 11. In the DC equivalent circuit, the resistances of the coaxial cable conductors (RCI and RCO) are in series with R2. Table 5 shows that the calculated values agree very well with the measured ones, whereas the small differences are within the uncertainties of the measurements. This result is another experimental validation of the circuit property described in this paper, considering propagating waves. The table also shows the charge values obtained
Table 3 Net charges in the circuit of Fig. 7: calculated with the DC equivalent circuit; simulated with PSpice. Component
Calculated (μC)
Measured (μC)
Diff. (%)
Simulated (μC)
Diff. (%)
R1 R2 R3 R4b R5
17.49 13.95 0.00 0.071 7.95
17.43 ± 0.25 13.87 ± 0.20 0.01 ± 0.05 0.07 ± 0.05 7.87 ± 0.11
−0.34 −0.57 – −1.4 −1.0
17.48 13.95 0.01 0.07 7.95
−0.06 0.00 – −1.4 0.00
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Fig. 13. LPS representing a single family house and two electric line configurations. Adapted from Ref. [19].
Fig. 14. DC equivalent circuit of the LPS (RLPS) and electric line (REL) of Fig. 13.
The DC equivalent circuit is shown in Fig. 14, where RLPS and REL are the LPS and the electric line resistances from the striking point to ground, respectively, which are calculated as Fig. 12. Currents recorded in the circuit of Fig. 11. (a) R2 = Z0; (b) R2 < Z0; (c) R2 > Z0.
RLPS =
length from the striking point to ground is given by:
REL =
LLPS =
1 6 5 ⎛7.5 + + ⎞ = 6.5 m 2⎝ 2 2⎠
ρLPS LLPS ALPS
,
(15)
ρEL LEL , n AEL
(16)
where ALPS and AEL are the cross-section areas of the LPS conductor and of a single electric line conductor, respectively, and n is the number of conductors of the electric line. The electric line is considered as made of copper, with resistivity ρEL = 17.8 nΩ m. For the LPS, it is considered here two materials: steel, with resistivity ρLPS = 120 nΩ m and stainless steel, with resistivity ρLPS = 700 nΩ m. The net charge qEL that flows through REL in Fig. 14 is:
The length of the electric line from the striking point to ground (LEL) is 17.2 m and 9.17 m for the Configurations (a) and (b), respectively. The LPS is made of round conductors with 8 mm diameter (nominal 50 mm2 area). For the electric line, two options were considered: singlephase circuit made of 3 conductors, i.e., phase, neutral, and protective earth (PE) with 1.4 mm diameter each (nominal 1.5 mm2 area); and three-phase circuit made of 5 conductors with 1.8 mm diameter each (nominal 2.5 mm2 area).
qEL = Q
RLPS , REL + RLPS
(17)
Table 5 Net charges in the circuit of Fig. 11: calculated with the DC equivalent circuit; simulated with PSpice. Condition
Comp.
Calculated (nC)
Measured (nC)
Diff. (%)
Simulated (nC)
Diff. (%)
R2 = Z0
R1 R2 R1 R2 R1 R2 R1 R2 R1 R2
44.20 81.93 31.32 120.43 53.89 52.67 7.97 190.22 71.50 0
44.06 ± 0.40 82.07 ± 0.60 31.34 ± 0.40 120.41 ± 0.80 53.83 ± 0.50 52.73 ± 0.50 8.12 ± 0.30 190.07 ± 1.20 71.47 ± 0.60 0.029 ± 0.200
−0.32 0.17 0.06 −0.02 −0.11 0.11 1.9 −0.08 −0.04 –
44.20 81.93 31.32 120.43 53.89 52.67 7.97 190.22 71.45 0.046
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 −0.07 –
R2 < Z0 R2 > Z0 R2 = 0 R2 = ∞
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This property may apply to the dimensioning of SPDs, especially when the SPDs are subjected to the flow of part of the lightning current (see Section 6). Note that the property fits well in the cases of unidirectional impulses like those specified by IEC 62305-1 [21] and when the SPD equivalent resistance (though non-linear) is much smaller than the other resistances in the same circuit branch. This is particularly true for spark-type SPDs, whereas the arc voltage drop is usually negligible. However, if the SPD voltage is not negligible (e.g., clamping type SPD) or if the current through the SPD is oscillatory, then the net charge calculation may not be appropriate for SPD rating. Another possible application concerns the measurement of earthing resistances, to which this circuit property may give rise to a new method with some advantages in comparison with the traditional methods. In fact, preliminary measurements by the authors in actual earthing systems have already produced excellent results [22]. Moreover, it is expected that this property may be helpful in the dealing with electrical behavior of complex circuitry, particularly in the field of instrumentation. These and other applications are under study by the authors.
Table 6 Net charges in the electric line conductors. LPS
Steel
Conf.
a b
Stainless steel
a b
Circuit
Calculation method DC circuit
MoM [20]
Dif. (%)
1 3 1 3
phase phases phase phases
3.22 3.98 5.17 5.54
3.31 4.02 5.28 5.48
−2.7 −1.0 −2.1 1.1
1 3 1 3
phase phases phase phases
9.83 8.07 12.2 8.94
9.91 8.04 12.1 8.80
−0.81 0.37 0.83 1.6
where Q is the total charge injected in the circuit. For the calculation of the charge through the power conductors, Heidler and Camara [19] considered the positive first return stroke defined in IEC 62305-1 [21], which has 100 kA peak value and 10/350 μs waveshape. The total charge delivered by this stroke current is obtained by integrating the current time function as per Ref. [17], which gives Q = 51.2 C. Inserting these values in (17) yields the net charge through the power conductors. The net charge through a single conductor can be obtained by dividing qEL from (17) by the number of conductors (n). The calculated values according to the equivalent DC circuit depicted in Fig. 14 are shown in Table 6, alongside with those computed from the complete electromagnetic equations applied to the conductor configurations of Fig. 13 using the MoM and presented in Ref. [19]. It can be seen a very good agreement between the two sets of data, as the differences range from −2.6% to +1.6%. These small differences could be justified by small errors in the representation of the structure by the DC equivalent circuit. It shall be highlighted that the results obtained with a sophisticated software are replicated by the application of the simple circuit property described in this paper. It is worth mentioning that in Ref. [19] there are also results for aluminum and copper LPS, but the charge values reported for these cases do not match those calculated with the DC equivalent circuit. A personal communication with the authors of Ref. [19] revealed that the current waveforms for these cases were bidirectional, whereas the current waveforms for steel and stainless steel LPS were unidirectional. As the charge values reported in Ref. [19] are the integral of the absolute value of the current and the net charge is defined as the integral of the current, the values are not comparable for this case. Of course, if the net charge values were calculated for the aluminum and copper LPS, they should match those provided by the equivalent DC circuit.
Acknowledgments This work was supported in part by the Brazilian National Council for Scientific and Technological Development (CNPq) and by the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES). References [1] J. Birkl, C.F. Barbosa, Modeling the current through the power conductors of an installation struck by lightning, Proceedings of the 11th International Symposium on Lightning Protection - SIPDA (2011). [2] T. Kisielewicz, C. Mazzetti, G.B. Lo Piparo, F. Fiamingo, Stress to surge protective devices system due to direct flashes to low voltage lines, Electr. Power Syst. Res. 129 (Dec) (2015) 44–50. [3] T. Kisielewicz, G.B. Lo Piparo, C. Mazzetti, Surge protective devices efficiency for apparatus protection in front of direct flashes to overhead low voltage lines, Electr. Power Syst. Res. 134 (May) (2016) 88–96. [4] W. Bassi, J.M. Janiszewski, Evaluation of currents and charges in low-voltage surge arresters due to lightning strikes, IEEE Trans. Power Deliv. 18 (January (1)) (2003) 90–94. [5] T. Kisielewicz, G.B. Lo Piparo, C. Mazzetti, A. Rousseau, Dimensioning of SPD for the protection against surges due to lightning to LV overhead lines, Proceedings of the 32nd International Conference on Lightning Protection (ICLP) (2014). [6] IEC 61643-11, Low-voltage Surge Protective Devices — Part 11: Surge Protective Devices Connected to Low-voltage Power Systems — Requirements and Test Methods, (2011). [7] ITU-T K.77, Characteristics of Metal Oxide Varistors for the Protection of Telecommunication Installations, (2011). [8] IEC 61643-22, Low-voltage Surge Protective Devices — Part 22: Surge Protective Devices Connected to Telecommunications and Signalling Networks — Selection and Application Principles, (2015). [9] E. Clark, Circuit Analysis of AC Power Systems vol. 1, John Wiley & Sons, 1948, pp. 38–41. [10] A. Greenwood, Electrical Transients in Power Systems, John Wiley & Sons, 1971, pp. 32–70. [11] E. Hallén, Electromagnetic Theory, John Wiley & Sons, 1962 (Chapter 24). [12] J.R. Carson, Electric Circuit Theory and the Operational Calculus, Chelsea Pub. Co., 1953, pp. 1–12. [13] P. Penfield Jr, R. Spence, S. Duinker, A generalized form of Tellegen’s theorem, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. ct-17 (3) (1970). [14] J.C.O. Silva, A.R. Panicali, C.F. Barbosa, Electric charge flow in linear circuits, Proceedings of the 14th International Symposium on Lightning Protection (SIPDA) (2017). [15] M.R. Spiegel, Mathematical Handbook of Formulas and Tables, McGraw-Hill Book Co., 1968. [16] ATP/EMTP, Electromagnetic Transient Program, http://eeug-test.hostingkunde.de/. [17] F. Heidler, Analitsche blitzstromfunktion zur LEMP-berechnung, Proceedings of the 18th International Conference on Lightning Protection (ICLP) (1985). [18] Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement, (2008) JCGM 100:2008, Sept.. [19] F. Heidler, A. Camara, Currents on electric installation lines in case of equipotential bonding at roof level, Proceedings of the 34th International Conference on Lightning Protection (ICLP) (2018). [20] H. Singer, H.-D. Bruens, A. Freiberg, CONCEPT II — Manual of the Program System, University Hamburg, Harburg, Germany, 2005 Apud Heidler and Camara [19]. [21] IEC 62305-1, Protection Against Lightning — Part 1: General Principles, (2010). [22] C.E.F. Caetano, J.O.S. Paulino, C.F. Barbosa, J.C.O. Silva, A.R. Panicali, A new method for grounding resistance measurement based on the drained net charge, IEEE Trans. Power Deliv., in press, https://doi.org/10.1109/TPWRD.2018.2879838.
7. Summary and conclusion This paper presented a fundamental circuit property that governs the distribution of the net charge in a linear circuit, due to an impulsive excitation. If written in the form of a theorem, it may read as: “For a linear, reciprocal and time-invariant circuit excited by an arbitrary time limited source, the net charge carried by each circuit branch is the same as the one computed for the DC equivalent circuit, where inductors and lossless transmission lines are replaced by short-circuits and capacitors are replaced by open-circuits.” For the application of this property, the following conditions must be fulfilled:
• The circuit shall be at rest at t = 0 (i.e., no energy is stored in its reactive components); • The source shall deliver a finite charge to the circuit; • The currents shall be computed until the circuit comes back to its state of rest.
63
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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Electric charge flow in linear circuits a
b
c,⁎
T
José Claudio de Oliveira e Silva , Antônio Roberto Panicali , Celio Fonseca Barbosa , Carlos Ermídio Ferreira Caetanoc, José Osvaldo Saldanha Paulinoc a b c
APTEMC, São José dos Campos, Brazil PROELCO, Campinas, Brazil Graduate Program in Electrical Engineering, Federal University of Minas Gerais, Belo Horizonte, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Linear circuits Electric charge Surge protective device
This paper deals with a fundamental property of circuits that can simplify certain analysis, bring new methods of measurements and possibly other applications, which has apparently passed unnoticed until now. It shows that the net charge injected by an arbitrary source into a linear circuit is divided among the circuit branches (including transmission lines) in the same proportion as the charge would divide among the same branches, if the circuit was replaced by its equivalent circuit at direct current (DC). The DC equivalent circuit alone will govern the exact sharing of the net charge, despite current oscillations caused by reactive elements or propagation effects of transmission lines. In this paper, this property is demonstrated analytically and proved by computer simulations and laboratory experiments. The paper also exemplifies its application in the calculation of the net charge distribution among the conductors of an installation struck by lightning. Finally, the paper forecasts other possible applications for the new circuit property.
1. Introduction In recent years, the charge flow through a circuit branch due to impulsive excitations has gained attention, as it is related to the rating of surge protective devices (SPDs) used for lightning protection [1–5]. Similarly, the international standards on SPDs [6–8] include the drained charge as one of the key parameters for specifying an SPD. During the investigation of the charge distribution among the conductors of a structure struck by lightning, the authors came across an interesting property of electric circuits. It was found that, although reactive components (e.g., inductors and capacitors) and transmission lines can significantly alter the amplitudes and waveshapes of the currents flowing in the circuit branches, the net charge delivered through each branch depends only on the circuit resistances. This is a fundamental circuit property that can be shortly explained by recalling that a linear circuit comprising any complex association of elements responds to a signal decomposed in frequency domain according to the impedance of the circuit at each frequency, and that at 0 Hz, i.e. for the DC component of the signal, the average current is proportional to the total flow of electric charge. To the best of the authors’ knowledge, such particular property of circuits has not been explored neither presented in the literature and it is applicable to each and every circuit, provided it is linear, time-
⁎
invariant and reciprocal. An extensive search in the literature was made by the authors, but no mention to such property has been found. This search included the database of technical journals and classic textbooks. For instance, Clarke [9] provides a detailed analysis of basic circuit properties, but it is restricted to alternating currents (AC). Greenwood [10] presents a comprehensive analysis of electric circuit transients, but he does not investigate the net charge distribution among circuit branches. Hallén [11] presents several circuit theorems, but none of them is similar to the circuit property described in this paper. Carson [12] derives several circuit properties based on Kirchhoff’s laws, including its transient behavior. These circuit properties include the reciprocity theorem and the equation of activity, which is similar to the Tellegen’s theorem [13]. However, Carson does not address the net charge distribution in circuits. A preliminary description of this property was presented in Ref. [14], which was restricted to a theoretical analysis of the subject. The present paper significantly enlarges the treatment of the new circuit property, by validating it with detailed computer simulations and controlled laboratory experiments. Moreover, a comparison of its results with those obtained by the direct solution of Maxwell’s equations (published in the literature) is included in order to highlight the power of this new property in dealing with complex situations. The paper is
Corresponding author. E-mail address: [email protected] (C.F. Barbosa).
https://doi.org/10.1016/j.epsr.2019.01.014 Received 14 October 2018; Received in revised form 16 December 2018; Accepted 15 January 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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Z12(s): mutual-impedance between Port 1 and Port 2, where Z12(s) = V1(s) / I2(s), for I1(s) = 0; Z21(s): mutual-impedance between Port 2 and Port 1, where Z21(s) = V2(s) / I1(s), for I2(s) = 0; where V1(s) and V2(s) are the voltages at Port 1 and Port 2, respectively. Note that, as the circuit is reciprocal, then Z12 = Z21. Solving the circuit for I2 (s) results:
I2 (s ) = I1 (s )
Z12 (s ) , Z22 (s ) + R
(1)
and the net charge through R up to an instant t is: t
q2 (t ) =
∫ i2 (t ) dt. 0
Fig. 1. Two-port network [N–R] that will be excited through port P1; resistor R is brought out from network [N] and defines output port P2.
(2)
Based on Laplace transform for integrals:
Q2 (s ) = organized as follows. Section 2 presents a theoretical demonstration of the circuit property and Section 3 describes how to circumvent some singularities that may arise when using perfect components. Sections 4 and 5 present its validation using computer simulations and laboratory experiments, respectively. Section 6 compares the results obtained using the new circuit property with those obtained by the full solution of Maxwell's equations. Finally, Section 7 draws the main conclusions.
I2 (s ) , s
(3)
and the total net charge passing through R is: ∞
q2 =
∫ i2 (t ) dt. 0
(4)
Based on the Final Value Theorem [15]:
q2 = limt →∞q2 (t ) = lims → 0 [s⋅Q2 (s )].
2. Theoretical analysis
(5)
Replacing (3) in (5):
q2 = lims → 0 [I2 (s )].
When impulsive sources are applied to linear circuits, the resulting currents flowing can assume complex time variations. However, the net charges flowing through the circuit branches depend only on the circuit resistances. This property is investigated theoretically in the following. Referring to Fig. 1, let P1 and P2 denote the ports of an arbitrary linear two-port network [N–R], that is necessarily reciprocal and timeinvariant. The two ports are selected so that P1 is the input port to which an arbitrary impulsive current source i1 (t) is connected and P2 is the output port where the resulting current i2 (t) through R is wanted. The circuit is at rest at t = 0 (i.e., no energy is stored in its reactive components) and the impulsive source delivers a finite net charge to the circuit. Let q1 be the net charge injected into the network by i1 (t) and q2 the net charge passing through the resistor R, while q1 (t) and q2 (t) are the referred charges calculated up to an instant t. Finally, let I1 (s), I2 (s), Q1 (s) and Q2 (s) denote the Laplace transforms of i1 (t), i2 (t), q1 (t) and q2 (t), respectively, where s is the complex frequency (jω). Given that [N–R] is a linear system, the transfer function between I1 (s) and I2 (s) can be derived from the equivalent T-network shown in Fig. 2. In this figure, the impedances are defined as:
(6)
Finally, applying (1) into (6):
q2 = I1 (0)
Z12 (0) . Z22 (0) + R
(7)
Following the same steps for q1 from (2) to (6):
q1 = I1 (0)
(8)
which results in:
q2 = q1
Z12 (0) . Z22 (0) + R
(9)
Eq. (9) demonstrates that the charge ratio q2 /q1 is given by a ratio of impedances at 0 Hz, namely Z12(0), Z22(0), and R, as expected for DC current division. Clearly, Z12(0) and Z22(0) must come from the association of resistors in the [N― R] network, after inductors have been replaced by short-circuits and capacitors by open-circuits. 3. Ideal reactive components Singularities may arise when computing the net charge in circuits with ideal reactive components (e.g., inductors without series resistances and capacitors without shunt resistances) in the absence of other circuit components that can naturally suppress the singularities. This situation is illustrated in Fig. 3, where the net charge conducted by the circuit branches cannot be determined by the DC equivalent circuit.
Z11(s): self-impedance at Port 1, where Z11(s) = V1(s) / I1(s), for I2(s) = 0; Z22(s): self-impedance at Port 2, where Z22(s) = V2(s) / I2(s), for I1(s) = 0;
Fig. 2. Two-port T-network representing the arbitrary linear network shown in Fig. 1 in complex frequency domain.
Fig. 3. Examples of circuits with ideal reactive components. 58
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Indeed, by replacing L1 and L2 by short-circuits, it is not possible to assess the net charge that flows through each inductor in Fig. 3(a). Similarly, by replacing the capacitors C1 and C2 by open-circuits in Fig. 3(b), it is not possible to assess the net charge that flows through each capacitor. In order to circumvent these singularities, it is sufficient to replace each ideal inductor by a resistor RL according to (10)
RL = kL, and replace each ideal capacitor by a resistor RC according to
RC =
1 , kC
(11)
then take the limit for k → 0. Considering that a charge q0 is injected in the circuits of Fig. 3, the net charge through the inductor L1 and the capacitor C1 are given by (13) and (14), respectively
qL1 = q0 limk → 0
qC1 = q0 limk → 0
(kL1)−1 L2 = q0 L1 + L2 (kL1)−1 + (kL2)−1 + R1−1
C1 C1 + C2
(
1 + kR1 C1 1 + kR2 C2
)
= q0
C1 . C1 + C2
(12)
(13)
4. Validation by computer simulations This section presents an example intended to validate the referred circuit property using the ATP/EMTP [16] simulation software. The results from computer simulations are compared to those calculated from the DC equivalent circuit. Consider the following 6-branch circuit given in Fig. 4. The resistors subscripts n (Rn) indicate branch n. The current source I produces a unidirectional waveform according to Heidler’s time function [17], whose peak value (34.95 kA) was adjusted (through a numerical iterative process) to inject 10.00 C charge into the circuit. Its waveshape is 1/200 μs. Fig. 5 shows the current waves on the circuit of Fig. 4, i.e., the total applied current I and the currents in all six branches. The resulting applied voltage (25 kV peak) follows the same waveshape as the current on branch 1 (Fig. 5(a)), since it is represented by 1 Ω resistor. All other current waveshapes are affected by the reactive elements and by the transmission line (TL) propagation effects on branch 6. The current on branch 6 is computed at the end of the line, i.e., at resistor R6. Note in Fig. 5(c) that the first current step is around 100 A, which comes from the applied voltage (25 kV) divided by the characteristic impedance of the line (500 Ω) and multiplied by 2, since the line is practically short-circuited. The current wave goes bouncing back and forth while growing slowly, resulting in a very long wave with about 1 ms duration. As expected, currents amplitudes and waveshapes are strongly influenced by reactive elements and TL effects. The net charge per branch, however, can be easily determined by means of the circuit
Fig. 5. Currents in the circuit of Fig. 4; (a) all currents; (b) zoom showing currents on branches 2–6; (c) currents on branches 6 and 3 (current amplitude on branch 6 is multiplied by 10).
Fig. 6. DC equivalent circuit for the circuit shown in Fig. 4. Inductors and TLs are replaced by short-circuits and capacitors are replaced by open-circuits.
property presented in this paper. To this aim, the DC equivalent circuit (of Fig. 4) is shown in Fig. 6, where inductors and lossless TLs are replaced by short-circuits and capacitors are replaced by open-circuits. The charge sharing among branches 1–6 of the DC equivalent circuit (Fig. 6) is given in Table 1, together with the EMTP calculated net charges on the complete circuit (Fig. 4). It can be seen an excellent agreement between the net charges calculated by the two methods, which is in line with the referred circuit property. It is worth to mention that the calculation of the net charge for the DC equivalent circuit is rather simple, i.e., it is given by: Fig. 4. Example of circuit including resistors, inductors, capacitors and a lossless transmission line (TL). 59
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Table 1 Net charge distribution: simulated with EMTP (Fig. 4); calculated with the DC equivalent circuit (Fig. 6). Branch
R (Ω)
Simulated (C)
Calculated (C)
Source 1 2 3 4a 4b 5 6
– 1 2 Open 4 Short 5 10
10.0 4.88 2.44 ˜0a 1.22 ˜0a 0.976 0.488
10.0 (given) 4.88 2.44 0 1.22 0 0.976 0.488
a
Table 2 Values of Fig. 7 components.
Asymptotically decaying, reaching about 10–15 C at t = 50 ms.
qi = q0
Ri−1 , n ∑ j = 1 R−j 1
Component
Value
Unit
R1 R2 R3 R4a R4b R5 RL2 L1 L2 C1 C2
213.9 ± 0.1 268.0 ± 0.1 330.5 ± 0.1 149.6 ± 0.1 150.0 ± 0.1 470.2 ± 0.1 0.39 ± 0.04 18.0 ± 0.3 130 ± 2 18.5 ± 0.3 1.60 ± 0.02
Ω Ω Ω Ω Ω Ω Ω μH μH nF nF
(14)
where q0 is the total injected charge, qi is the net charge in the branch i, Ri−1 and R−j 1 are the conductances of the branches i and j, respectively, and n is the number of circuit branches. 5. Experimental validation In order to further validate the new circuit property, this section presents some experimental results. In all cases, the currents were measured by Pearson current monitors model 4100, having bandwidth from 140 Hz to 35 MHz. The current monitors were connected to a Rhode & Schwarz RTB2004 oscilloscope, with 10 bits resolution. The resistors values were measured with a precision HP 34401A meter, with 61/2 digits resolution. The uncertainties of the measurements were combined according with the procedure described in Ref. [18], whereas the values expressed in this paper are the expanded uncertainties obtained with coverage factor equal to 2, which gives a confidence level of 95.5%.
Fig. 8. Current applied to the circuit of Fig. 7.
5.1. Circuit with lumped components The first circuit considered is composed only by lumped components, as shown in Fig. 7. The resistor RL2 represents the inherent series resistance of the inductor L2, whereas the inherent resistance of the inductor L1 was added to the resistance of the resistor R2. Table 2 shows the measured values of the circuit components. The circuit of Fig. 7 was subjected to an impulsive source that delivered the current shown in Fig. 8. The currents measured in the circuit of Fig. 7 are shown in Fig. 9, where can be seen that the currents through R1, R2, and R5 are unidirectional and the currents through R3 and R4b are bidirectional, due to the series capacitor and shunt inductor, respectively. Fig. 10 shows the charges through the branches of Fig. 7, where can be seen that the charges through R1, R2, and R5 tend to a final positive value, whereas those through R3 and R4b approach zero.
Fig. 9. Currents in the circuit components of Fig. 7.
The total charge delivered by the source and the charge through one component were obtained simultaneously by integrating the current recorded at each surge application. Each pair of measurements was repeated 10 times, in order to better characterize the uncertainties related to these measurements. The average value recorded for the total charge was 64.32 μC. Table 3 shows the net charge measured in the relevant components of Fig. 7, alongside with their uncertainties. The same table shows the values of the net charges calculated according to (14), i.e., considering
Fig. 7. Circuit with lumped components for the experimental validation. 60
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Fig. 11. Circuit with distributed parameters for the experimental validation. Table 4 Values of Fig. 11 components. Component
Description
Value (Ω)
R1 R2
Resistor near the source R2 = Z0 R2 < Z0 R2 > Z0 Inner coaxial conductor resistance Outer coaxial conductor resistance
100.68 ± 0.05 50.09 ± 0.05 21.96 ± 0.05 98.80 ± 0.05 2.50 ± 0.01 1.72 ± 0.01
RCI RCO
with a circuit-simulation software (PSpice), which agree almost exactly with those from the DC equivalent circuit.
Fig. 10. Charges in the circuit components of Fig. 7.
the DC equivalent circuit. It can be seen in Table 3 that the calculated values agree very well with the measured ones, whereas the small differences are within the uncertainties of the measurements. This result is an experimental validation of the circuit property described in this paper, considering lumped components. The table also shows the charge values obtained with a circuit-simulation software (PSpice), which agree almost exactly with those from the DC equivalent circuit.
6. Example of application A recent paper from Heidler and Camara [19] presented an analysis of the lightning current distribution among the conductors of the lightning protection system (LPS) and the internal electric lines of a building. Several parameters were calculated and, as highlighted by the authors, "The most important parameter is the maximum transferred charge", which was calculated with a computer code based on the Method of Moments (MoM) [20] that solves the full Maxwell equations in the frequency domain. The time-domain solutions were obtained from the inverse Fourier transform, whereas the skin effect was automatically treated by the software. In order to compare the results from [19] with those calculated using the circuit property described in this paper, it is chosen the Configurations (a) and (b) of the "single family house", as shown in Fig. 13. The LPS is represented by the thin lines installed along the corners of the roof and walls, whereas the electric line, one for each configuration, is represented by bold lines. In both configurations, it is assumed that the electric lines and the LPS are bonded at the striking point by a copper bonding conductor 0.3 m long and having 16 mm2 cross-section area. Of course, the electric line shall be bonded through an SPD, but its effect was neglected in the simulations presented in Ref. [19]. The ground plane was supposed as perfectly conducting. To apply the referred circuit property to this structure, it is necessary to build its DC equivalent circuit. Due to symmetry, the two halves of the LPS present the same resistance from the striking point to ground. Moreover, the horizontal LPS sections along the top of the walls do not contribute to the resistance to ground. Therefore, the equivalent LPS
5.2. Circuit with distributed parameters This section considers a distributed parameters circuit, as shown in Fig. 11. The transmission line (TL) is made by 60 m long coaxial cable with 50 Ω nominal characteristic impedance (Z0). The currents i1 and i2 were simultaneously measured (see Fig. 11). The components values are shown in Table 4, where three different values were considered for R2. In addition to these values, it was also considered R2 = 0 and R2 = ∞. Fig. 12 shows the currents i1 and i2 for the conditions R2 = Z0, R2 < Z0, and R2 > Z0, where the propagation effects on the TL can be clearly seen. The net charges obtained for the cases considered are shown in Table 5, alongside with those calculated using (14), i.e., with the DC equivalent circuit for Fig. 11. In the DC equivalent circuit, the resistances of the coaxial cable conductors (RCI and RCO) are in series with R2. Table 5 shows that the calculated values agree very well with the measured ones, whereas the small differences are within the uncertainties of the measurements. This result is another experimental validation of the circuit property described in this paper, considering propagating waves. The table also shows the charge values obtained
Table 3 Net charges in the circuit of Fig. 7: calculated with the DC equivalent circuit; simulated with PSpice. Component
Calculated (μC)
Measured (μC)
Diff. (%)
Simulated (μC)
Diff. (%)
R1 R2 R3 R4b R5
17.49 13.95 0.00 0.071 7.95
17.43 ± 0.25 13.87 ± 0.20 0.01 ± 0.05 0.07 ± 0.05 7.87 ± 0.11
−0.34 −0.57 – −1.4 −1.0
17.48 13.95 0.01 0.07 7.95
−0.06 0.00 – −1.4 0.00
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Fig. 13. LPS representing a single family house and two electric line configurations. Adapted from Ref. [19].
Fig. 14. DC equivalent circuit of the LPS (RLPS) and electric line (REL) of Fig. 13.
The DC equivalent circuit is shown in Fig. 14, where RLPS and REL are the LPS and the electric line resistances from the striking point to ground, respectively, which are calculated as Fig. 12. Currents recorded in the circuit of Fig. 11. (a) R2 = Z0; (b) R2 < Z0; (c) R2 > Z0.
RLPS =
length from the striking point to ground is given by:
REL =
LLPS =
1 6 5 ⎛7.5 + + ⎞ = 6.5 m 2⎝ 2 2⎠
ρLPS LLPS ALPS
,
(15)
ρEL LEL , n AEL
(16)
where ALPS and AEL are the cross-section areas of the LPS conductor and of a single electric line conductor, respectively, and n is the number of conductors of the electric line. The electric line is considered as made of copper, with resistivity ρEL = 17.8 nΩ m. For the LPS, it is considered here two materials: steel, with resistivity ρLPS = 120 nΩ m and stainless steel, with resistivity ρLPS = 700 nΩ m. The net charge qEL that flows through REL in Fig. 14 is:
The length of the electric line from the striking point to ground (LEL) is 17.2 m and 9.17 m for the Configurations (a) and (b), respectively. The LPS is made of round conductors with 8 mm diameter (nominal 50 mm2 area). For the electric line, two options were considered: singlephase circuit made of 3 conductors, i.e., phase, neutral, and protective earth (PE) with 1.4 mm diameter each (nominal 1.5 mm2 area); and three-phase circuit made of 5 conductors with 1.8 mm diameter each (nominal 2.5 mm2 area).
qEL = Q
RLPS , REL + RLPS
(17)
Table 5 Net charges in the circuit of Fig. 11: calculated with the DC equivalent circuit; simulated with PSpice. Condition
Comp.
Calculated (nC)
Measured (nC)
Diff. (%)
Simulated (nC)
Diff. (%)
R2 = Z0
R1 R2 R1 R2 R1 R2 R1 R2 R1 R2
44.20 81.93 31.32 120.43 53.89 52.67 7.97 190.22 71.50 0
44.06 ± 0.40 82.07 ± 0.60 31.34 ± 0.40 120.41 ± 0.80 53.83 ± 0.50 52.73 ± 0.50 8.12 ± 0.30 190.07 ± 1.20 71.47 ± 0.60 0.029 ± 0.200
−0.32 0.17 0.06 −0.02 −0.11 0.11 1.9 −0.08 −0.04 –
44.20 81.93 31.32 120.43 53.89 52.67 7.97 190.22 71.45 0.046
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 −0.07 –
R2 < Z0 R2 > Z0 R2 = 0 R2 = ∞
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This property may apply to the dimensioning of SPDs, especially when the SPDs are subjected to the flow of part of the lightning current (see Section 6). Note that the property fits well in the cases of unidirectional impulses like those specified by IEC 62305-1 [21] and when the SPD equivalent resistance (though non-linear) is much smaller than the other resistances in the same circuit branch. This is particularly true for spark-type SPDs, whereas the arc voltage drop is usually negligible. However, if the SPD voltage is not negligible (e.g., clamping type SPD) or if the current through the SPD is oscillatory, then the net charge calculation may not be appropriate for SPD rating. Another possible application concerns the measurement of earthing resistances, to which this circuit property may give rise to a new method with some advantages in comparison with the traditional methods. In fact, preliminary measurements by the authors in actual earthing systems have already produced excellent results [22]. Moreover, it is expected that this property may be helpful in the dealing with electrical behavior of complex circuitry, particularly in the field of instrumentation. These and other applications are under study by the authors.
Table 6 Net charges in the electric line conductors. LPS
Steel
Conf.
a b
Stainless steel
a b
Circuit
Calculation method DC circuit
MoM [20]
Dif. (%)
1 3 1 3
phase phases phase phases
3.22 3.98 5.17 5.54
3.31 4.02 5.28 5.48
−2.7 −1.0 −2.1 1.1
1 3 1 3
phase phases phase phases
9.83 8.07 12.2 8.94
9.91 8.04 12.1 8.80
−0.81 0.37 0.83 1.6
where Q is the total charge injected in the circuit. For the calculation of the charge through the power conductors, Heidler and Camara [19] considered the positive first return stroke defined in IEC 62305-1 [21], which has 100 kA peak value and 10/350 μs waveshape. The total charge delivered by this stroke current is obtained by integrating the current time function as per Ref. [17], which gives Q = 51.2 C. Inserting these values in (17) yields the net charge through the power conductors. The net charge through a single conductor can be obtained by dividing qEL from (17) by the number of conductors (n). The calculated values according to the equivalent DC circuit depicted in Fig. 14 are shown in Table 6, alongside with those computed from the complete electromagnetic equations applied to the conductor configurations of Fig. 13 using the MoM and presented in Ref. [19]. It can be seen a very good agreement between the two sets of data, as the differences range from −2.6% to +1.6%. These small differences could be justified by small errors in the representation of the structure by the DC equivalent circuit. It shall be highlighted that the results obtained with a sophisticated software are replicated by the application of the simple circuit property described in this paper. It is worth mentioning that in Ref. [19] there are also results for aluminum and copper LPS, but the charge values reported for these cases do not match those calculated with the DC equivalent circuit. A personal communication with the authors of Ref. [19] revealed that the current waveforms for these cases were bidirectional, whereas the current waveforms for steel and stainless steel LPS were unidirectional. As the charge values reported in Ref. [19] are the integral of the absolute value of the current and the net charge is defined as the integral of the current, the values are not comparable for this case. Of course, if the net charge values were calculated for the aluminum and copper LPS, they should match those provided by the equivalent DC circuit.
Acknowledgments This work was supported in part by the Brazilian National Council for Scientific and Technological Development (CNPq) and by the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES). References [1] J. Birkl, C.F. Barbosa, Modeling the current through the power conductors of an installation struck by lightning, Proceedings of the 11th International Symposium on Lightning Protection - SIPDA (2011). [2] T. Kisielewicz, C. Mazzetti, G.B. Lo Piparo, F. Fiamingo, Stress to surge protective devices system due to direct flashes to low voltage lines, Electr. Power Syst. Res. 129 (Dec) (2015) 44–50. [3] T. Kisielewicz, G.B. Lo Piparo, C. Mazzetti, Surge protective devices efficiency for apparatus protection in front of direct flashes to overhead low voltage lines, Electr. Power Syst. Res. 134 (May) (2016) 88–96. [4] W. Bassi, J.M. Janiszewski, Evaluation of currents and charges in low-voltage surge arresters due to lightning strikes, IEEE Trans. Power Deliv. 18 (January (1)) (2003) 90–94. [5] T. Kisielewicz, G.B. Lo Piparo, C. Mazzetti, A. Rousseau, Dimensioning of SPD for the protection against surges due to lightning to LV overhead lines, Proceedings of the 32nd International Conference on Lightning Protection (ICLP) (2014). [6] IEC 61643-11, Low-voltage Surge Protective Devices — Part 11: Surge Protective Devices Connected to Low-voltage Power Systems — Requirements and Test Methods, (2011). [7] ITU-T K.77, Characteristics of Metal Oxide Varistors for the Protection of Telecommunication Installations, (2011). [8] IEC 61643-22, Low-voltage Surge Protective Devices — Part 22: Surge Protective Devices Connected to Telecommunications and Signalling Networks — Selection and Application Principles, (2015). [9] E. Clark, Circuit Analysis of AC Power Systems vol. 1, John Wiley & Sons, 1948, pp. 38–41. [10] A. Greenwood, Electrical Transients in Power Systems, John Wiley & Sons, 1971, pp. 32–70. [11] E. Hallén, Electromagnetic Theory, John Wiley & Sons, 1962 (Chapter 24). [12] J.R. Carson, Electric Circuit Theory and the Operational Calculus, Chelsea Pub. Co., 1953, pp. 1–12. [13] P. Penfield Jr, R. Spence, S. Duinker, A generalized form of Tellegen’s theorem, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. ct-17 (3) (1970). [14] J.C.O. Silva, A.R. Panicali, C.F. Barbosa, Electric charge flow in linear circuits, Proceedings of the 14th International Symposium on Lightning Protection (SIPDA) (2017). [15] M.R. Spiegel, Mathematical Handbook of Formulas and Tables, McGraw-Hill Book Co., 1968. [16] ATP/EMTP, Electromagnetic Transient Program, http://eeug-test.hostingkunde.de/. [17] F. Heidler, Analitsche blitzstromfunktion zur LEMP-berechnung, Proceedings of the 18th International Conference on Lightning Protection (ICLP) (1985). [18] Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement, (2008) JCGM 100:2008, Sept.. [19] F. Heidler, A. Camara, Currents on electric installation lines in case of equipotential bonding at roof level, Proceedings of the 34th International Conference on Lightning Protection (ICLP) (2018). [20] H. Singer, H.-D. Bruens, A. Freiberg, CONCEPT II — Manual of the Program System, University Hamburg, Harburg, Germany, 2005 Apud Heidler and Camara [19]. [21] IEC 62305-1, Protection Against Lightning — Part 1: General Principles, (2010). [22] C.E.F. Caetano, J.O.S. Paulino, C.F. Barbosa, J.C.O. Silva, A.R. Panicali, A new method for grounding resistance measurement based on the drained net charge, IEEE Trans. Power Deliv., in press, https://doi.org/10.1109/TPWRD.2018.2879838.
7. Summary and conclusion This paper presented a fundamental circuit property that governs the distribution of the net charge in a linear circuit, due to an impulsive excitation. If written in the form of a theorem, it may read as: “For a linear, reciprocal and time-invariant circuit excited by an arbitrary time limited source, the net charge carried by each circuit branch is the same as the one computed for the DC equivalent circuit, where inductors and lossless transmission lines are replaced by short-circuits and capacitors are replaced by open-circuits.” For the application of this property, the following conditions must be fulfilled:
• The circuit shall be at rest at t = 0 (i.e., no energy is stored in its reactive components); • The source shall deliver a finite charge to the circuit; • The currents shall be computed until the circuit comes back to its state of rest.
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