Data Centres are complex highly integrated systems and can present significant technical challenges. Some of these will be relatively simple, some will require in depth investigation, but some are going to need something more. These are the unforeseen challenges that need a deep dive into the science, and push practical understanding to its limits. At JDA we pride ourselves on not shying away from challenges, but finding solutions that provide value for our clients. Therefore, on occasion we will enlist support from the world of academia to fully understand the mechanisms that have led to this point.

One such academic is Ken Smith FIET SMIEE. Ken is an expert in electrical power and induced current in cables. We have collaborated with Ken to bring you this blog, which describes one particular challenge that we overcame together. This is intended to provide interest and discussion but is not an industry text or design guide and might not be applicable to every situation (there are always unknown factors after all!). We hope that you enjoy this “In-depth” technical discussion.

#### Issues with single-core cables in short three-phase cable runs

Single-core cables in general comprise of a central circular core conductor, surrounded by an annular insulation layer, which may be surrounded by other annular conductor and insulation layers.

Depending on the application, the additional layers, may include one or more of; a metallic sheath, a water blocking impervious sheath, as well as an armour to provide mechanical protection. In general the materials used for the core and metallic sheath are either copper or aluminium, the impervious layer is lead, and the armour can be stranded steel or aluminium. In some cases the geometry of the single-core cable can be complex.

In the following discussion we will limit ourselves to the simpler case of a cable with a core and a single annular conductor which we shall call a sheath.

#### The problem

Relatively short cable runs, for example between a step-down transformer and the switchboard it supplies, can on occasion cause unexpected issues due to the voltage induced in the metallic sheath that surrounds the central core conductor.

Where multiple single-core cables are applied per phase, the sharing of the currents between the parallel cables may not always be balanced, i.e. some individual cores carry larger currents than others – even although the total phase current summed across each of the individual phase conductors equals the load current in that phase, and all three supplied load currents form a balanced three-phase set.

In this case as the load currents are balanced, i.e. equal current magnitude and each phase displaced by 120 degrees from the other two phases, the unbalance in the cable currents is not due to the characteristics of the supplied load, instead it is due to the physical arrangement of the cables.

#### Single-core cables

For a single cable in isolation the flow of an alternating load current in the core conductor creates a time varying magnetic field external to the core. The annular sheath conductor is cut by this time varying magnetic field and consequently due to Faraday’s Law of induction a longitudinal voltage is induced along the length of the sheath. If single point bonding is applied to the sheath, which means that it is earthed only at one end, this induced voltage will appear as an open circuit voltage at the other end.

When carrying a large fault current which may be many times the normal maximum load current this open circuit voltage can be very large, which necessitates the use of special cross-bonding arrangements in long single-core cable three-phase circuits as found in transmission systems.

For shorter cable runs and at lower voltages, one way to avoid this open circuit voltage is to adopt solid bonding, which means that the annular sheath is earthed at both ends. The voltage induced by the alternating load current in the core will now drive a current through the loop formed by the sheath and the earth. The magnitude of this loop current is limited by the ac resistance of the sheath, and as this resistance approaches zero (due to a larger cross-sectional area and/or lower material resistivity), the magnitude of the current in the sheath conductor approaches that of the core.

To satisfy Lenz’s Law the polarity of the current flow in the sheath is in the opposite direction to the core, and from Ampere’s Law the magnetic field external to the annular sheath is zero in the theoretical case where the ac resistance of the sheath is zero. In this situation, there is Ampere-turn balance between the core and sheath; the core looks like the primary winding of an idealised single turn transformer, while the sheath looks like a single turn transformer secondary winding which is short-circuited.

The general theory of transformers can therefore be used to determine the induced voltage or current in the annular sheath. At its simplest level a transformer is two coils linked by a common magnetic flux which defines the mutual inductance Lm between the coils. If the transformer operates at 50 Hz this mutual inductance can be expressed as a reactance Xm = 2π∙50∙Lm.

The same model is applicable to cables and a mutual reactance can be defined between the core and annular sheath. If the core carries an ac current of I Amperes, the voltage induced in the sheath Vs is simply Vs = I x Lm.

In the idealised case when the resistance of the annular sheath is zero, the absence of an alternating magnetic flux external to this sheath means that any additional external annular conductors (if present) or other cables cannot have any voltage induced in them; i.e. the first annular sheath can be considered to provide some magnetic screening, but in practice this magnetic screening is not fully effective since the sheath must always have a finite resistance.

Considering the finite resistance of conductors, it is apparent that with a single-core cable; if single point bonded, a magnetic field will exist external to the cable (as there is no current in the sheath providing magnetic screening), while if solidly bonded due to the presence of the current in the sheath the magnetic field external to the cable is reduced but not completely eliminated.

This has important consequences when three such single-core cables are laid together to form a three-phase circuit as the external magnetic field of one cable links with the conductors of the other two cables. This means that there is a mutual reactance between every possible pair of individual conductors of the three single-core cables that form the three-phase cable circuit.

In general, these mutual reactances are inversely proportional to the logarithm of the inverse of the distance between the geometric centres of the conductors. This means that increasing the spacing between cable centres decreases the mutual reactance or magnetic coupling between cables.

#### Three-phase circuit with three single-core cables

When laying three single-core cables to form a three-phase circuit the designer has a number of options, which include the horizontal (or vertical) flat, triangle, and trefoil arrangements.

In the trefoil arrangement where the cables centres are at the vertices of an equilateral triangle the distance between the centres of each pair of cables is identical. This equality is not present in either the flat or triangular formations. This means that the cable to cable mutual reactances are equal (or balanced) with in the trefoil arrangement, but different (or unbalanced) in both the horizontal flat and triangular configurations.

The balancing of the cable to cable mutual reactances in the trefoil arrangements results in balanced three-phase voltages being induced in the annular screen conductors surrounding the cable cores. If the cables are solidly bonded so that currents flow in the sheaths, these currents will be balanced in the trefoil case.

For both the horizontal and triangular formations, the absence of balanced cable to cable mutual reactances has little impact on the core currents, but it has a significant impact on the currents that can flow in the annular screens if solidly bonded. In this case these screen currents may be significantly larger in some phases compared to the balanced trefoil case. This can result in excessive heating of the cables carrying these higher currents. In this case, increasing the spacing between the cable’s centres can increase the currents in the screens as although the mutual reactance cable to cable is reduced, the degree of unbalance in the formation can be made worse, further increasing the screen current unbalance.

In all of these cases, as there are three cables each with a single screen, the calculation of the currents in all of the conductors requires the solution of 12 simultaneous phasor equations.

In an idealised but unpractical case where the three cables are sufficiently far apart that there is negligible cable to cable mutual coupling, the currents in the screens become identical in magnitude in each cable, and their magnitude is significantly greater than that in the trefoil case! This is due to their being no phasor addition of three magnetic fluxes of approximately equal amplitude to sum vectorially close to zero.

For this reason distribution cables which are normally installed with solidly bonded sheaths, in order to minimise the currents circulating in the sheaths of single-core cables produced by the magnetic flux linking the conductors and the sheaths, are best laid in a close touching trefoil formation.

#### Three-phase circuit with multiple single-core cables per phase

In general, for high current applications when multiple single-core cables per phase are used to supply a load, equal current division among the conductors of each phase will not occur unless specific steps are taken to balance the cable to cable mutual reactances.

The actual division of current between conductors is independent of the magnitude of the load current, the power factor of the load, the source voltage, and cable length. Therefore unbalanced currents will be observed in relatively short cable runs if the cables are not arranged correctly to balance the mutual reactance effects.

If the cables are not physically arranged so that the mutual reactances of the core conductor to the other phase core conductors are equal to that of its companion core conductor in the same phase, then a non-symmetrical configuration will exist. The load current then will not divide evenly, resulting in a current unbalance among the phase core conductors.

Almost any problem involving paralleled cables can be represented by simultaneous phasor equations of voltage drops caused by self-impedance and mutual reactances but such equations become numerous and cumbersome with even just two parallel three-phase cables, which makes hand-calculations difficult. Also, if the cable route length is very short, then significant errors may occur in the calculated result due to the change in the relative positions of the cables as they approach terminations.

It is difficult to achieve a balanced current distribution for a cable circuit with three, five, or seven single conductor cables per phase as it is difficult to arrange a symmetrical cable tray or trench configuration for such circuits. It is possible to achieve a perfectly balanced cable current distribution for two and four cables per phase. Examples of some of the different core and sheath current distributions that can be calculated for six cables in a flat formation supplying a balanced total load current of 100 A is shown below.

These examples show that for a flat formation:

- Placing all cables of the same phase together results in a highly unbalanced circuit. In this case the currents in the cable cores range from 44.4 A to 55.6 A, while the sheath currents range from 34.8 A to 44.4 A. In general, cables of the same phase should not be grouped together.
- Adopting mirror symmetry, so that the cables are laid as two subgroups R1/Y1/B1 and B2/Y2/R2, i.e., symmetrical about the central axis, balances the core currents at 50 A. However, the sheath currents remain unbalanced and range from 28.7 A to 34.8 A. Note that the magnitude of the sheath current is reduced by introducing mirror symmetry.
- With mirror symmetry, reversing the phase rotation of the source voltages has no impact on the core currents which remain at 50 A, however the distribution of the sheath currents across the cables changes.

In all of these cases, as there are 6 cables each with a single screen, the calculation of the currents in all of the conductors requires the solution of 6 x 2 x 3 = 36 simultaneous phasor equations.

When the cables are arranged in trefoil, with phase mirror symmetry in the centre line between the two trefoil subgroups, this arrangement produces balanced core currents (as did the flat case with mirror symmetry), and with this arrangement the sheath currents are perfectly balanced and at 13.9 A are of considerably lower amplitude than any of the flat arrangements shown above.

These examples for the six single-core cable case (two cables per phase), demonstrate the general rules of thumb that minimise the effect of core current unbalance and sheath induced voltages and currents (if solidly bonded), namely:

- To achieve a balanced current distribution for a group of single-core cables, wherever possible the cables must be arranged symmetrically.

Putting all cables of the same phase together will introduce a highly unbalanced current distribution and this practice should be avoided.

Include one cable per phase in a cable subgroup and install them symmetrically. - A configuration with adjacent subgroups arranged in mirror symmetry can obtain near balanced core currents for two and four cables per phase. For three and five subgroups, applying mirror symmetry between the adjacent subgroups, provides the lowest possible core current unbalance.
- Trefoil formations are preferred over flat formations as with balanced core currents in either case, it is the trefoil formation which has the lowest and balanced sheath currents.

#### A theoretical or practical issue?

At the time of writing this article the author is aware of four installations where upon commissioning of cable circuits issues were identified with the cable arrangements.

In three of these cases large currents were found to be flowing in the armours of low and medium voltage cables which led to excessive heating at the glands in the cable termination boxes.

In the fourth case, two separate three-phase cables each comprising of three single-core cables were installed in close proximity. All attempts to operate these two circuits as dual redundant feeders failed; as soon as the last circuit breaker was closed to form a closed loop around the two cable circuits, both cables near instantaneously tripped, causing loss of power supply to the downstream loads. This was due to the unbalance in the core currents of each circuit, causing the presence of a zero-sequence current. Zero-sequence currents would also be present during earth faults on the cables, and the protection system was correctly responding to the measured zero sequence current anticipated for this fault condition.

In all of these cases, the rearrangement of the individual single-core cables following the general rules of thumb listed above and verified by supporting calculations was necessary to mitigate these issues.