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Variable-resistance conductors (VRC) for power-line de-icing

Victor F. Petrenko, Charles R. Sullivan ⁎, Valeri KozlyukThayer School of Engineering, Dartmouth College, NH 03755, USA

a b s t r a c ta r t i c l e i n f o

Article history:Received 6 October 2009Accepted 17 June 2010

Keywords:Power transmission linesThermal de-icingAnti-icing

Ice storms can result in accumulation of ice on structures, including overhead power transmission anddistribution lines and associated poles and towers; this ice may reach thicknesses of many tens ofmillimeters. Icing can cause catastrophic damage which disrupts power transmission and is expensive torepair.Normal operation of a power transmission or distribution line entails Joule heating of the conductor ascurrent flows through it. Lines are normally designed to have a constant, low resistance, so as to avoidexcessive power losses and avoid excessive operating temperatures. Because the normal heating is low (bydesign), it is of limited value in preventing or recovering from an icing event. This paper describes powerconductors that can switch their electrical resistance from a very low value, to transmit electric energy, to amuch higher value, for de-icing. The switching in between two conductor resistances does not disturb themain conductor function, which is to provide a customer with uninterrupted electric power.A variable-resistance conductor (VRC) is built of N strands (or groups of strands) insulated from each other,where N is any odd integer greater than one. For instance, N=3, 5 or 7, etc. In normal energy-transmissionoperations all the conductor strands (or strand groups) are connected in parallel, whereas in de-icing modethey all are connected in series. Switching from parallel to series connection increases the line resistance by alarge factor of N2, making the resistance sufficiently high for heating the line above the ice melting point. Oneimportant advantage of the method is that it uses low-voltage and, thus, low-cost switches.The design of a VRC de-icing system is described, including considerations for switches, conductors, controland deployment strategy. We also describe safety devices to return the line to normal operation if theelectronics get damaged. Laboratory and full-scale prototypes have both successfully demonstrated thecapability of VRC de-icing.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Ice storms can result in an accumulation of ice on structures,including overhead power transmission and distribution lines andassociatedpoles and towers; this icemay reach thicknessesofmany tensof millimeters. Although any one line typically encounters suchconditions only a few times per year, icing can cause catastrophicdamage which disrupts power transmission and is expensive to repair.

The usual mechanism of ice damage is through the weight of theice imposing addedmechanical stress on cables and towers: a cylinderof solid ice, 50 mm in radius and 1 km long, weighs 7.2 tonne; ice thisthick can add more than 8 t of weight per km of a single cable, as wellas increasing wind-induced stress on the lines. Such an ice cylinderwould increase the cable tension by a number that depends on thelength of a cable span and cable sag, but typically exceeds the ice-cylinder weight. Accumulated ice regularly causes power transmis-sion lines and poles to break, and towers to collapse. Dramatic

examples include the 2008 storm in the Guangdong province of China(Liu et al., 2008) and the 1998 storm in the northeast of North America(Gyakum and Roebber, 2001), but smaller ice storms cause damageevery year. Ice accretion can also contribute to conductor galloping inhigh winds, sometimes leading to short circuits between adjacentphases and thus to outages. In addition to disrupting powertransmission damage to power lines can cause serious risk to peopleand property on the surface.

Methods of de-icing in use today have many shortcomings, asdiscussed in the surveys by Laforte et al. (1998), Ryerson (2008), andFarzaneh et al. (2008). The primary categories of methods in use aremechanical and electrothermal. Mechanical methods in use generallyrequire access to the line. One exception is the use of a pulse of currentfrom a deliberate short circuit to create electromagnetic forces thatmay knock ice off a line (Landry et al., 2000). Electrically, this maycreate significant disruptions in power quality and may even threatengrid stability (Landry et al., 2000). Other mechanical methods arereviewed by Farzaneh et al. (2008); in addition to other difficulties,most mechanical methods require adding additional stress to asystem that may already be near the breaking point, and thus canprecipitate the failure they are intended to prevent.

Cold Regions Science and Technology 65 (2011) 23–28

⁎ Corresponding author.E-mail addresses: [email protected] (V.F. Petrenko),

[email protected] (C.R. Sullivan).

0165-232X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.coldregions.2010.06.003

Contents lists available at ScienceDirect

Cold Regions Science and Technology

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Normal operation of a power transmission or distribution lineentails Joule heating of the conductor as current flows through it.Lines are normally designed to have a constant, low resistance, so as toavoid excessive power losses and avoid excessive operating tempera-tures. As wire reaches high temperatures, due to electrical self-heating and high ambient temperatures, it tends to lengthen andweaken. This lengthening can cause the lines to sag between poles ortowers, possibly causing hazard to persons or property on the surface.Because the normal heating is low (by design), it is of limited value inpreventing or recovering from an icing event. Various approacheshave been proposed or tested to enable increased heating forpreventing ice formation or removing ice.

One option is to simply change power system operations to increasethe current flowing through a particular line. Although Huneault et al.(2005a,b) andMerrill andFeltes (2006) show that this can be effective insome circumstances, it requires a network with adequate configurationoptions, and theheatingpower is limited.Another option is todisconnectthe line from service, short-circuit it, and apply ac or dc power to it. Dchas the advantage that the current is not limited by the reactance of theline. Applying dc requires expensive rectification equipment, although itmay be possible to configure this equipment to be useful for otherpurposeswhen it is not used for de-icing (Horwill et al., 2006). The shortcircuit method is widely used (Farzaneh et al., 2008), even thoughremoving a line from service is problematic, particularly during anemergency. To provide significant, controlled heatingwithout removinga line from service, it is possible to increase the level of current in anetwork through the use of phase-shifting transformers and capacitors(Cloutier et al., 2007). Evenwith this proposedenhancement, theheatingavailable is limited, and there may be negative effects on grid stability.

It is also possible to keep a line in service, and add additionalpower dissipation by superimposing a higher frequency current in theform of a sinusoid McCurdy et al. (2001) or pulse (Peter et al., 2008).However, these approaches require additional expensive equipment,as well as raising issues with electromagnetic interference ordielectric breakdown.

Another approach to keeping a line in service but increasing powerdissipation is to make some modification to the line itself. Forexample, if there are parallel conductors, some of them can beswitched out of the circuit, as suggested by Couture (2004) and Pierce(1954). This increases the resistance, and the heat dissipated, by afactor up to the number of parallel conductors (if each has equalresistance). In this paper, we introduce a similar method that canprovide a larger increase of resistance and can simultaneously heat allparallel conductors in a bundle.

2. The variable-resistance conductor method

2.1. Introduction

This paper describes power conductors that can switch theirelectrical resistance from a very low value, to transmit electric energy,to a much higher value, for de-icing (Petrenko and Sullivan, 2008).The switching in between two conductor resistances does not disturbthe main conductor function, which is to provide a customer withuninterrupted electric power.

A variable-resistance conductor (VRC) is built of N strands (orgroups of strands) insulated from each other, where N is any oddinteger greater than one. For instance, N=3, 5 or 7, etc. In normalenergy-transmission operations all the conductor strands (or strandgroups) are connected in parallel, whereas in de-icing mode they allare connected in series. Three- and five-strand versions are shown inFigs. 1, 2 and 3. Switching from parallel to series connection increasesthe line resistance by a large factor of N2, making the resistancesufficiently high for heating the line above the ice melting point. Oneimportant advantage of the method is that it uses low-voltage and,

thus, low-cost switches. We also describe safety devices to return theline to normal operation if the electronics are damaged.

2.2. Power requirements

The heating power required for anti-icing is primarily the powerrequired to maintain the line at an elevated temperature (e.g., by 10 °C)by balancing convective losses. For de-icing, additional power is requiredto raise the temperatureof the lineandthe ice, and tomelt the ice.A smallamount of additional power can then eventually, slowly melt the ice.However, as discussed inmore detail by Petrenko et al. (2003), Petrenkoand Sullivan (2003) and Peter et al. (2008), and in a companion article inthis issue (Petrenko et al., 2010), higher power heating can reduce theenergy requirement for de-icing. This is because of reduced energy loss toconvection through a reduced time at an elevated temperature, andbecause, at sufficiently high power, the interface between the ice and theline can bemelted before thewhole body of the ice is brought up to 0 °C.In the worst case, the ice may encircle the line, and may thus need to bemelted through to be released. If sufficient heating power is available,this can easily be accomplished when necessary.

To determine the power required for anti-icing operation weestimate heat losses from a line held above 0 °C in an ambient below0 °C. These may include wind-driven convection, radiation andnatural convection. For worst-case design, we consider the case of astrong wind, in which case wind-driven convection dominates overradiation and natural convection, and we can consider only theconvective loss, which may be estimated by (Admirat, 2008):

Q = 14:2ðvdÞ0:61ΔT ð1Þ

where Q is heat loss (forced convection only) in watts per meter lengthof a cable, v is wind velocity in m/s, d is diameter of the cable in m, andΔT is the difference between the cable and the ambient temperatures.For example, Eq. (1) predicts that holding a 10 mmdiameter cable 10 °Cabove ambient (e.g., at 1 °C in a−9 °C ambient), would require asmuchas 35W per meter of cable at high wind speeds (10 m/s), as shown inFig. 4. Higher power is needed for de-icing, both because the additionalsurface area of the ice increases convective loss, and because energy isneeded to melt ice. More detailed models of power requirements arediscussed in Merrill and Feltes (2006) and Personne and Gayet (1988).

To compare the power requirement to the power that can beproduced in a typical VRC conductor, we now consider an examplecable comprising 5 strands of 2.3 mm diameter aluminum with0.7 mm insulation, such that each strand has a diameter of 3.7 mmincluding insulation. The overall diameter is 10 mm, and the totalcross-sectional area of aluminum available for conducting current is21 mm2. A 1.6-mmdiameter steel wire could optionally be included inthe center for mechanical support without affecting the overalldimensions. Although such a cable could have an ampacity over135 A (CME Wire and Cable, Inc., 2006), we conservatively assume anominal current of 100 A rms. In the normal conductor configuration,such a cable would have a resistance of 1.35 mΩ per meter, and woulddissipate 13.5 W/m at full 100 A rms current. If we switch it into de-icing mode with the 5 strands in series, the resistance and powerdissipation are increased by a factor of 25, for a dissipation of 338 W/mat full current, or 84 W/m at half current. Comparing this to the anti-

Fig. 1. Variable-resistance conductor system using three parallel strands of conductor,represented by resistors. With the switches all off the resistance from the left terminalto the right terminal is 32=9 times the resistance when the switches are on and all theconductors are connected in parallel.

24 V.F. Petrenko et al. / Cold Regions Science and Technology 65 (2011) 23–28


icing requirements calculated above, we see that even at half ofnominal current, the power available is more than twice that requiredfor maintaining a temperature above freezing in anti-icing operation,with the balance (50 W/m) available for the additional convectionload present with an ice coating, and for melting the ice.

2.3. Switch design

A power transmission line is typically operated at an ac potentialhundreds of kilovolts above ground. Equipment designed to supportthese voltages is large and expensive. However, the voltage across aswitch in a VRC de-icing system is only the extra voltage drop acrossthe segment it switches. This voltage may be hundreds to a fewthousands of volts, within the range of standard low-cost semicon-ductors or electromechanical contactors. The voltages with respect toground could still pose difficulties, except that the whole switchingapparatus, including its housing, can be maintained at the potential ofthe line, and can be mounted suspended from the towers, similarly toequipment proposed in Divan and Johal (2007) for reactive powercompensation.

When more than three strands are used, the switches may beconfigured in two different ways—series, as shown in Fig. 2, or parallelas shown in Fig. 3. The series configuration has a lower maximumvoltage across any switch, whereas the parallel configuration has alower maximum current through any switch. The total volt–ampere(VA) requirement for the switches is equal in the two configurations,as shown in Appendix A, so the choice of switch configurationdepends more on the practical limitations of the particular switchesused, and how those compare to the voltage and current require-ments, which depend on the distance between switching units.

For example, for a long distance between switching unitsthyristors are a good choice for the switches, and a parallelconfiguration (Fig. 3) is likely to be preferred, because the switchassembly will have lower total on-state voltage drop, and the highervoltage requirement is not a problem. On the other hand for a shorterdistance, MOSFET switches are likely to be preferable, and a seriesconfiguration can reduce the voltage requirement for the MOSFETs,and thus allow the use of MOSFETs with lower on-state resistance.Particularly with thyristors, a hybrid switch combining semiconduc-tors with a parallel electromechanical relay may be desirable to

reduce on-state power losses and heating (Atmadji and Sloot, 1998;Polman et al., 2001). An electromechanical relay may also be usedwithout a parallel semiconductor device.

For the example parameters discussed above, the voltage drop atfull current in one strand is 0.54 V rms per meter at full current. Thus,the peak voltage across a switch in the series switch configuration(Fig. 2) is 1.53 V per meter of distance between switch boxes atnormal maximum current. This is compatible with MOSFETs for asingle span, or with thyristors for up to several kilometers.

2.4. Cables

The VRC de-icer requires an odd number of conductor strands orgroups, insulated from each other. Fortunately, the voltage differencebetween the strands is small compared to the line voltage, and simple,thin, polymer coating as is used for low-voltage insulation or forcovering on some distribution lines is adequate. The maximumvoltage between strands is the product of the number of strandsN andthe voltage drop per strand. For the example discussed above, for 7strands, and a 100 m distance between switch boxes, the maximumvoltage is 378 V rms. The addition of the polymer may affect theampacity of the conductor. Heat dissipated in the conductor duringnormal operation must be conducted through the polymer beforetransferring to the ambient by convection and radiation. Theconduction path adds thermal resistance, whereas the slight increasein diameter increases convective area. The emissivity of the coatingalso affects heat loss by radiation and solar heat gain. In practice, thenet result tends to be a slight increase of ampacity as reflected inmanufacturers' ampacity ratings (CME Wire and Cable, Inc., 2006).

It is also possible to use bare bundled conductors separated byinsulated spacers. These are commonly used in transmission lines ingroups of 4; for VRC an odd number (e.g., 5) can be used instead.

2.5. Deployment strategies

For several reasons, it is desirable to have short distances betweenswitch boxes suspended on a line. This allows finer granularity in thechoice of which sections are heated. It also allows smaller, lighterswitch boxes, which are easier to suspend from a line. Finally, thedisturbance to the power network when a segment is turned on or offwill be very small.

Fig. 2. Variable-resistance conductor system using five parallel strands of conductor,represented by resistors and a series switch arrangement. With the switches all off theresistance from the left terminal to the right terminal is 52=25 times the resistancewhen the switches are on and all the conductors are connected in parallel.

Fig. 3. Variable-resistance conductor system using five parallel strands of conductor,represented by resistors and a parallel switch arrangement. Compared to thearrangement in Fig. 2, this is functionally equivalent but changes the requiredcapabilities of the switches by reducing the current through some switches, butincreasing the voltage across them.

Fig. 4. Calculated convective heat loss from a 10 mmdiameter cable maintained at 10 °Cabove ambient as a function of wind speed. At high wind speeds, this loss dominatesover other power losses and is approximately equal to the power required to maintainthe cable at this temperature.

25V.F. Petrenko et al. / Cold Regions Science and Technology 65 (2011) 23–28


The disturbance can be further minimized by coordinatingswitching times such that when one segment is turned off, the nextsegment is simultaneously turned on, such that overall dissipationand voltage drop in the line are constant.

When a section of the line is uniformly coated with ice, high-powerheating can be appliedwithout overheating the line, as the icemaintainsthe temperature near 0 °C until the ice is cleared, at which point theheating can be turned off. If the segment has sections that are coveredwith ice and bare sections, the bare sectionswill be heated above 0 °C inthe time it takes to clear the iced sections. Rated operating temperaturesof polyethylene insulation are 75 °C or 90 °C (CMEWire and Cable, Inc.,2006), leaving adequate headroom in ambient temperatures below0 °C.However, it is desirable for this reason, and for anti-icingoperation, to beable to adjust the heating power.

Although it would be possible to use switch configurations otherthan all on and all off to get different heating powers, the heatingwould then be non-uniform, stronger in some strands than in others.Another better approach to modulating heating power is to cycle theheating on and off and vary the duty cycle to vary the power. Withsynchronized switching of many segments, such pulse-width modu-lation can be used to achieve many different average power levelswhile maintaining constant total power dissipation and voltage drop.

2.6. Practical implementation

Although full documentation of a practical implementation is notof interest here, we briefly address some practical strategies forimplementation, including powering the switch boxes, control andcommunications, and failure modes.

A switch box equipped with line- and air-temperature sensors, icesensors, and a microcontroller could autonomously decide when toactivate de-icing. However, wireless communications systems can beeasily implemented, for example using cellular networks such as GSM(Global System for Mobile Communications), and offer many advan-tages. Central control allows coordination of the activation of differentsegments, and allows activationdecisions tobemadewith knowledgeofweather forecasts. It also allows coordination with power networkoperation. Bidirectional communication allows data from sensors to bereturned to a control center. This data is not only valuable formonitoringand addressing an ice storm, but also for monitoring line temperaturesduring other situations, such as a hot summer day.

Switch boxes may derive power from the line current throughcurrent transformers or small photovoltaic panels with battery backup.Given the inverse correlation between icing events and solar poweravailability, current transformers are a preferred approach, althoughphotovoltaics would be viable with adequate batteries.

The purpose of the VRC de-icing system is to make the powernetwork more reliable as well as to avoid expensive damage. If theVRC de-icing system were to fail and disrupt the power network,reliability would be hurt rather than helped. Thus, in addition toengineering the switch boxes for high reliability, it is important toensure that a failure restores the power line to normal operation,rather than defaulting to the high-resistance state, which could lead tooverheating and failure of the cable. Scenarios that must beconsidered include failure of the insulation between strands or failureof power or control components in the switch box resulting from anycause including a lightning strike. Multiple levels of fail-safeprotection can be used to ensure that this is the case.

The first level of protection is an electronic cable temperaturesensor and logic that shuts of the system in the case of cableoverheating. With an electromechanical relay as the switch, or acomponent of the switch, the next level of protection can be anormally-closed relay that reverts to the low-resistance state when ithas no control signal. This ensures that in the case of a failure of theelectronics or the power supply, the system reverts to the low-resistance state. For designs that use only semiconductor switches, a

failure of the drive electronics would, with most types of semicon-ductors, lead to the switch turning off and the VRC stuck in the high-resistance state. Although redundant drive systems could be used tomitigate this hazard, an emerging option is the use of SiC JFETswitches, which typically are on in the absence of a control signal. SiCJFETs offer very low resistance when on and very high voltagecapability (Cooper and Agarwal, 2002). In typical applications, theirnormally-on characteristic is a problem, but in this application it is anadvantage. If a third level of protection is desired, a purely mechanicalbackupmechanism can be triggered by amechanical fusible link usingmetal or polymer that melts at a moderate temperature (e.g. 120 °C)to close a set of contacts and revert to low-resistance mode. Testingunder simulated lightning strikes should be used to verify that thefailure mode is normal operation even under extreme circumstances.

Failure of insulation between strands, for example due tomechanical impact or lightning, will have no effect on operation inthe normal mode, as the potential difference between strands is zero.However, in some scenarios, an insulation failure could lead to non-uniform heating in de-icing mode, potentially leading to damagingtemperatures at some points on the line before the sensedtemperature exceeds 0 °C. Although thermal damage is unlikelywhen the ambient temperature is low enough to lead to icing, aprotective measure for such a fault would be to sense the voltageacross pairs of series strands and compare this to expected valuesbased on line current, length, and temperature, and revert to normaloperation if a significant discrepancy is detected.

2.7. Applicability

The VRC method is widely applicable. Because the switches seeonly the voltage drop along the length of the line, not the voltage fromthe line to ground, their rating is independent of the transmissionvoltage. Thus, the method is applicable to high-voltage transmission(hundreds of kilovolts) as well as medium-voltage distribution (tensof kilovolts). For higher voltage, the main additional consideration iscorona, and thus possible degradation of insulation material. Wherethis is a problem, spacing conductor strands apart with insulatedspacers may be preferable to using polymer insulation.

Although the incremental cost of adding polymer coatings toconductors is low, the cost of replacing conductors in an existinginstallation is high. Thus, the VRC method is least expensive either innew installations or on existing lines when replacement of conductorsis being done for other reasons. However, in situations where icing is asevere or recurrent problem, or in situations where extremely highreliability is desired, the cost of new conductors may be justified.

3. Test results

Although the VRC de-icing system is based on well establishedphysical principals and proven technology, laboratory and fieldtests have been conducted to verify its performance.

3.1. Laboratory tests

A power cable with a steel reinforced core was stretched betweentwo insulator posts set 3.5 m apart, as shown in Fig. 5. The cablecomprises seven insulated aluminum strands, each 2.6 mm diameter(AWG 10), and a steel reinforcement cable in the center. The aluminumstrands were connected through switch boxes at the ends of the span.The cables for this and other tests are shown in detail in Fig. 6. Theswitches were controlled by a radio remote control. The line wasdesigned to carry current of 70 A rms in normal operation, with aresistance of 2 mΩ for the length of the test span. The resistance isincreased to 110 mΩ in the de-icing mode. At 70 A rms, the powerdissipation is 539W, or 154W/m of cable length, in de-icing mode,whereas it is under 5 W/m in normal operation (low-resistance) mode.



The line was covered by mixture of ice and snow with athickness of approximately 10 mm to mimic atmospheric icingconditions. Dense snow was first saturated with water and themixture set at a temperature of 0 °C. The mixture then was placedon a plastic film and wrapped around a cold wire, forming a wetsnow cylinder inside a plastic-film sheath. After the mixture wasfrozen the plastic film was removed. In a −5 °C cold room, withthe line and ice starting at ambient temperature, it took 3 to 5 minto clean the line completely of ice and snow in various tests. Thetemperature profiles from two cold-room tests, with and without2 m/s air flow, are plotted in Fig. 7.

3.2. Field tests

A full-scale VRC de-icing system was tested on a 10.5 kV powerdistribution line in Orenburg, Russia in early 2009, as shown inFig. 8. A sample of the cable used is shown at the left in Fig. 6, andused the same stranding as the cables in the laboratory tests. Theswitch boxes were powered by the line current, and controlled bya radio transmitter from the ground. Although atmospheric icingwas not encountered during the test, ice was frozen onto the line,and testing was conducted with the line in active use in the powernetwork, carrying 60 to 70 A rms. The ambient temperature wasnear 0 °C and wind speeds were 4 to 6 m/s. The system worked asexpected, heating the line and removing the ice. Fig. 9 shows asequence of ice beginning to melt and falling off the line.

4. Conclusion

The variable-resistance conductor de-icing method offerssimple, reliable de-icing for power transmission and distribution

lines, using cables similar to those already in use and low-costswitching systems that do not need to withstand the full linevoltage. Laboratory and field tests have confirmed operation asexpected.

Fig. 5. Laboratory test setup.

Fig. 6. Cables used in laboratory tests. Each cable has a steel core and seven 2.6 mm(AWG 10) insulated aluminum conductors, twisted around the steel core. The left cablewith, with an outer jacket, was used in the field test; the center unjacketed cable wasused in lab tests.

Fig. 7. Laboratory test results of cable temperature vs. time. The cable temperature risesquickly and then levels off when automatic temperature control engages, and finallydrops off when de-icing is turned off.

Fig. 8. Test installation on one span of a power line. The spheres on top of the first andlast towers contain the switches for the protected cable between them.

Fig. 9. Ice being removed from an active power line. The first frame shows ice beginningto melt; the second ice about to fall, the last shows ice falling.

27V.F. Petrenko et al. / Cold Regions Science and Technology 65 (2011) 23–28


Acknowledgement

The authors thank Ice Engineering, LLC for multi-year financialsupport of this research.

Appendix A. Switch ratings

Consider a section of cable made up of an odd number N ofinsulated strands, with a total load current flowing through it I0.Each strand has a resistance RS, such that the resistance of thesection in normal operation (low-resistance mode) is RS/N. In thismode, the current in each strand is

Is = I0 =N ð2Þ

In de-icing mode (high resistance), the total resistance is N·RS.The full current I0 flows through each strand, and thus the voltagedrop across a single strand is

Vs = I0·RS ð3Þ

For the series switch configuration (Fig. 2), the voltage across eachswitch when it is off (high-resistance mode) is 2·VS. The currentswhen the switches are on are not all equal. At each end of thesegment, there are (N−1)/2 switches. These carry currents {2· IS,4· IS, … (N−1)IS}. We can evaluate the total switch requirement bysumming the volt–ampere (VA) requirements for each switch.Because the voltage requirements are identical for each switch, wefirst sum the currents and thenmultiply by the voltage. The sum of thecurrents at each end is 2(N−1)IS, for a total of 4(N−1)IS at both ends.Multiplying by the voltage, we obtain the total switch VA requirementfor the series configuration:

VAseries = 8 N–1ð ÞVSIS ð4Þ

For the parallel switch configuration (Fig. 3), the current througheach switchwhen it is on (normal operation) is 2· IS. The voltageswhenthe switches are off are, {2·VS, 4·VS,… (N−1)VS}. To find the total VAof the requirements for each switch,we can separately sum the voltagesand then multiply by current, because the current requirements areidentical for each switch. The sumof the voltages at each end is 2(N−1)VS, for a total of 4(N−1) VS at both ends. Multiplying by the voltage, weobtain the total switch VA requirement for the parallel configuration:

VAparallel = 8 N–1ð ÞVSIS ð5Þ

We see that in terms of the sum of VA ratings of the switches, therequirements are identical.

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