## Armour Currents in Single Core Cables - Generalised Method and Workbook

- Details
- Written by Wojciech Nodzyński

In 2004 a worksheet calculating armour currents in single core cables was created and placed on the PEC website. This worksheet referred to in this paper as the Original Worksheet, deals with up to 10 parallel cables arranged in a plane, but it was not fully tested.

I am studying engineering at University of Cambridge and during my 6-week summer work placement I conducted further tests under supervision. I was also given the task of improving the worksheet , so it can work for any arrangement of parallel cables, not only coplanar arrangements.

### 1) The theory behind the coplanar worksheet, Original worksheet

Worksheet is based on the theory of electromagnetism and it first calculates magnetic fluxes produced by the currents flowing in conductors. That is done by applying the Ampere`s law and integrating the magnetic flux density B with respect to distance from one conductor to the centre of the other cable. These fluxes linkages per ampere cause emf in both armour and screen circuits.

When the armour and screen are connected together at the open end, then the induced currents in both screen and armour are calculated from Ohm`s law. The theory was applied in hand calculations which were then compared with the actual results displayed by the Original Worksheet. This kind of testing was conducted for many different coplanar arrangements of cables. The conducted tests included analysis of the following arrangements:

Arrangement |
Original Worksheet |
Hand Calculations |

2 cables with suppressed earth | 51.52<-121° A | 50.61-122.965° A |

2 cables with suppressed earth and all armour suppressed | 4.51<-92.585° A | 4.51<-92.585° A |

2 cables with suppressed earth and all screen suppressed | 51.5<-120.997° A | 51.5<-120.997° A |

**Table 1.1** Testing of the coplanar worksheet

*NB: calculation of error was based on the value of amplitude

It was proved that values of mesh voltages printed by the worksheet were entirely consistent with the theory, however there were discrepancies between values of induced currents up to 1.8%. There were no errors when armour or screens were suppressed. The above results were displayed for conductor currents of 100 Amps. This error is quite small but still is showing that there is a shortcoming in either the Original Worksheet or in the hand calculations or both. Rather then focussing on the Original Worksheet, a new spreadsheet was prepared from scratch based on the flux linkages described above. The hand calculations are limited in practice to 3 conductors because of the difficulty of evaluating the mathematics. The program called the Generalised Worksheet was written for 10 conductors as for the Original Worksheet.

The formulation is all in terms of 9x9 matrices Ma and Ms . These contain elements which are the flux linkages as described above. As a consequence, expressions for induced currents in a screen and armour were deduced from scratch and written in the form of matrices in Excel. That was done separately for 2 cables +earth, 3 cables +earth and finally, 9 cables +earth:

**R _{a}I_{a} = **j

**ωM**

_{a}(I_{c}– I_{s}– I_{a})**R _{s}Is = **j

**ωM**

_{s}(I_{c}– I_{s}– I_{a})Ra - resistance of armour, Ia –armour current, Ic - conductor current , Is - screen current, Rs- resistance of screen

These expression assume that an armour current has an influence on the screen current and vice versa. The solutions of these linear equations are:

**I _{s} = ((R_{s} + **j

**M**j

_{s}) + M_{s}(R_{a}+**M**j

_{a})^{-1}M_{a})^{-1 }(**M**j

_{s}+ M_{s}(R_{a}+**M**

_{a})^{-1}M_{a}) I_{c}**Ia = ((R _{a} + **j

**M**j

_{a})^{-1 }**M**j

_{a}– (R_{a}+**M**j

_{a})^{-1 }**M**j

_{a}((R_{s}+**M**j

_{s}) + M_{s}(R_{a}+**M**j

_{a})^{-1 }M_{a})^{-1 }(**M**j

_{s}+ M_{s}(R_{a}+**M**

_{a})^{-1 }M_{a})) I_{c}The results given by these two equations were then compared with the results displayed by the coplanar worksheet for 3 cables +suppressed earth:^{1}

For 3 cables + suppressed earth, 100 Amps in phases RYB |
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Results displayed by the coplanar worksheet |
Results based on expressions 1 and 2: Hand calculations |
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Armour currents | Screen Currents | Armour currents | Screen Currents |

60.51<-121.95° | 4.35<-117.35° | 60.29<-121.82° | 4.32<-122.95° |

45.99<0.36° | 3.47<5.82° | 45.86<0.46° | 3.45<0.46° |

52.93<105.3° | 3.80<112.52° | 52.78<105.46° | 3.76<106.94° |

For 9 cables + suppressed earth, 100 Amps in phases RYBRYBRYB |
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Results displayed by the coplanar worksheet |
Results based on expressions 1 and 2: Hand calculations |
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Armour currents | Screen Currents | Armour currents | Screen Currents |

63.91<-125.04° | 4.54<-120.24° | 63.7<-124.87° | 4.51<-125.87° |

38.51<-2.77° | 3.00<4.26° | 38.44<-2.74° | 2.97<-0.986° |

49.49<123.92° | 3.71<128.4° | 49.31<124.03° | 3.68<123° |

47.09<-122.7° | 3.53<-117° | 46.97<-122.58° | 3.50<-122.19° |

46.45<0.46° | 35.03<5.85° | 46.31<0.56° | 3.48<0.48° |

46.22<118.3° | 3.48<124.12° | 46.1<118.391° | 3.45<118.73° |

44.86<-116.24° | 3.42<-111.33° | 44.71<-116.18° | 3.40<-116.66° |

54.46<1.63° | 4.00<6.11° | 54.26<1.77° | 3.97<0.64° |

50.98<98.86° | 3.63<107.3° | 50.88<99.04° | 3.59<101.7° |

For 9 cables + earth, 100 Amps in phases RYBRYBRYB |
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Results displayed by the coplanar worksheet |
Results based on expressions 1 and 2: Hand calculations |
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Armour currents | Screen Currents | Armour currents | Screen Currents |

63.60<-125.05° | 4.67<-119.75° | 63.7<-124.87° | 4.51<-125.87° |

38.45<-2.76° | 2.96<1.88° | 38.44<-2.74° | 2.97<-0.986° |

49.54<123.97° | 3.63<130° | 49.3<124.03° | 3.68<123° |

47.11<-122.78° | 3.65<-116.33° | 46.98<-122.58° | 3.49<-122.186° |

46.37<0.45° | 3.46<3.81° | 46.32<0.56° | 3.48<0.48° |

46.21<118.44° | 3.39<125.79° | 46.1<118.39° | 3.45<118.73° |

45.06<-116.26° | 3.56<-111.00° | 44.7<-116.18° | 3.4<-116.66° |

54.41<1.23° | 3.96<3.99° | 54.26<1.77° | 3.97<0.64° |

50.18<98.627° | 3.47<108.254° | 50.88<99.04° | 3.59<101.7° |

1.39<80.63° | 1.25<78.544° | 1.36<78.36° | 1.23<75° |

**Table 1.2** Comparing the results displayed by the worksheet with results based on expressions 1 and 2 ; for 3 cables + suppressed earth. Cable OD=27mm, CSA armour=76mm^2 (Cu) ,CSA screen=4.68mm^2(Cu), CSA conductor = 50mm^2 (Cu),cables separation=100mm

After doing some other tests on the expressions 1 and 2 like:

- 1 cable+earth
- 3 cables + earth
- 9 cables + earth

were done.

### 2) Generalised Worksheet

The analysis of magnetic fluxes produced by the cables in different arrangements and positions showed that for the general case only matrix defining distances between the cables needed to be introduced. The worksheet determines these distances according to x&y coordinates of the centres of cables typed in by the user. After determining values of magnetic fluxes worksheet proceeds as in the coplanar version.

In order to make the worksheet more robust as well as easier to test, structure of the worksheet was changed too. It should be noted at this point that the coplanar version of worksheet used different matrices for different numbers of cables, ie. 3x3 matrices were used to calculate the induced currents for 3 cables + earth, whilst 6x6 matrices were used for 6 cables+earth. In order to change it, resistance matrices were modified, so that worksheet always uses 9x9 matrices. For example, when user enters data for 7 cables + earth only, worksheet sets the resistances of two cables to a very high value (10^6 Ohms) and proceeds just as in case when all 10 cables are present.

This generalised version of worksheet was tested in many different ways:

- comparing the generalised version with the coplanar one – co-ordinates of cables could be chosen to lie in one plane . That allowed for comparing results with well tested , coplanar worksheet. See Table 2.1 & Table 2.2 below
- comparing the generalised worksheet with the theory – some simple cases were considered and the results displayed by the worksheet were compared with hand calculations. All results were consistent with the theory
- analysis of cables arranged in the trefoil – all results were consistent with the theory

Results displayed by the coplanar worksheet |
Results displayed by the generalised worksheet |
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Armour currents |
Screen Currents |
Armour currents |
Screen Currents |

64.306<-125.242° | 4.996<-121.552° | 64.091<-125.057° | 4.967<-126.996° |

38.115<-2.745° | 2.669<-2.070° | 38.069<-2.730° | 2.654<-7.549° |

49.785<124.225° | 3.688<135.413° | 49.581<124.320° | 3.649<130.018° |

47.248<123.188° | 3.958<-118.956° | 47.125<-123.047° | 3.932<-124.137° |

45.903<0.472° | 3.150<0.651° | 45.789<0.551° | 3.137<-4.923° |

46.408<119.232° | 3.429<132.076° | 46.271<119.306° | 3.388<126.689° |

45.929<-117.054° | 3.898<-114.408° | 45.764<-116.968° | 3.876<-119.56° |

53.092<-0.235° | 3.593<-0.064° | 52.914<-0.071° | 3.578<-5.689° |

46.702<100.602° | 3.208<116.456° | 46.660<100.809° | 3.156<110.926° |

9.302<49.408° | 4.539<47.166° | 9.197<50.149° | 4.523<43.374° |

** Table 2.1** Comparison of the results displayed by the coplanar worksheet with the generalised one for current equal to 100Amps for 9 cables+earth, cable separation = 100mm, currents: 100 Amps in phases RYBRYBRYB, CSA armour=76mm^2, CSA screen=4.68mm^2, CSA conductor= 50mm^2,outer diameter =27mm

Results displayed by the coplanar worksheet |
Results displayed by the generalised worksheet |
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Armour currents |
Screen Currents |
Armour currents |
Screen Currents |

54.009<-130.757° | 4.251<-129.532° | 53.875<-130.570° | 4.226<-134.879° |

40.569<73.011° | 3.055<85.477° | 40.402<73.014° | 3.025<80.154° |

42.110<-120.465° | 3.524<-121.617° | 42.026<-120.337° | 3.508<-126.692° |

41.807<62.531° | 3.019<76.637° | 41.693<62.584° | 2.982<71.227° |

41.303<-110.776° | 3.459<-114.527° | 41.162<-110.707° | 3.451<-119.568° |

49.541<51.159° | 3.342<64.781° | 49.448<51.327° | 3.299<59.15° |

9.152<8.452° | 3.650<6.608° | 9.037<9.195° | 3.616<2.738° |

**Table 2.2** Comparison of the results displayed by the coplanar worksheet with the generalised one for current equal to 100 Amps in phases 0 for 6 cables + earth, cable separation = 100mm, currents: 100 Amps in phases RYBRYBRYB, CSA armour=76mm^2, CSA screen=4.68mm^2, CSA conductor= 50mm^2,outer diameter =27mm

**Fig 2.1** Graphical summary of testing

### 3) Decription of the generalised worksheet

Generalised worksheet calculates induced currents in armour and screen for any arrangement of up to 10 parallel single core cables, for example for cables arranged in trefoil. The input table is very similar to the Original Worksheet. User is still asked to key in values of outer diameters of cables, their length, cross sectional area of screen , armour and conductor, values of conductor currents, phase angles and a frequency. There are two versions of the generalised worksheet, one of them includes macros and the other does not. Generalised Worksheet with Macros allows the user to change the position of cable by using special buttons. In the other version the same thing can be done “manually” by changing the values of co-ordinates in the boxes . User can also decide whether the earth should be shown in the diagram or not.

As in the coplanar version of worksheet, it is suggested that the last cable is reserved for an earth. An electrical earth connection is always present and induced currents will flow in this path, as they will in the armour of the power cables. Specifying an earth path also enables armour voltages relative to earth to be calculated.

### Summary

The coplanar version of worksheet was tested by comparing the results displayed by the worksheet and hand calculations for different arrangements of cables. As a result new expressions for induced armour and screen currents were implemented and tested. Generalised Worksheet was also developed, so it can calculate currents for any arrangement of cables, not only coplanar one. The user is asked to choose coordinates of cables ,which can be seen on the graph. The generalised version was tested by comparing it with the coplanar one as well as by comparing the printed results with hand calculations.

*1. For currents equal to 100 Amps and phases 0, 120 and 240 degrees. Outer diameter of cable : 27mm, cross sectional area of screen: 4.68 mm^2, armour: 76 mm^2 and conductor: 50mm^2, frequency: 60Hz, length of cable:100m, distances between cables: 100mm in every case. Resistance of earth was set to a very high value*

* *

### DISCLAIMER

PLEASE NOTE: The information and worksheet on this page are preliminary. Consultation and worksheet testing continues. The results obtained from the worksheet should not be relied upon.

Any persons using this spreadsheet, or results obtained from this spreadsheet, do so at their own risk. PEC cannot accept responsibility for any consequences

**Right click the links below to download the workbooks.**

GENERALISED INDUCED CURRENTS SPREADSHEET - WITHOUT MACROS