Three-phase AC 50/60 Hz cable lines have magnetic fields (H) of core currents A,B,C. In normal operation, the currents A,B,C are equal to each other and have a 120° shift, so it seems that the magnetic fields A,B,C should compensate each other. However, this is not happening well, and as a result, the line has a resulting magnetic field. One of the consequences is the appearance of induced voltages (Us) and currents (Is) in the cable screens.

The screens must be grounded at least in one point to solve the problems of the electric field (E) inside the cable insulation, and this contributes to the formation of screen contours. Due to the magnetic field (H) of the line, these screen contours can have induced voltages and currents, as well as associated power losses. An important condition for the occurrence of screen currents and losses is that the screens of phases A,B,C are isolated from each other. This is typical in the case of single-core cables and even for some of three-core cables.

✅ Suppose that three single-core cables A,B,C are arranged in a row as shown in the figure on the left. Let’s assume that the core currents A,B,C have a conditional direction towards us, which is shown by a bold dot in the centre of the core (the tip of a flying arrow). Then three magnetic fields (H) are formed around the three cores, the direction of rotation of which is indicated by an arrow (the right-hand rule).

Consider the contour formed by screens A and B (contour sAB). It can be seen that the magnetic fields of cores A,B,C penetrate this contour in different directions. As a result, their mutual compensation does not occur, and voltage is induced in the circuit (UsAB). If the contour is closed at the ends, then there will be current in the contour (IsAB).

✅ Suppose that the three cables A,B,C are arranged in trefoil, as shown in the figure on the right. It can be seen that the magnetic field C penetrates the contour of sAB twice, with different signs. As a result, it is generally eliminated and cannot affect the contour. Therefore, fields A and B mutual compensation do not occur, and voltage is induced in the circuit (UsAB). If the contour is closed at the ends, then there will be current in the contour (IsAB).

As it was shown, AC 50/60 Hz currents arise in screen contours (closed frames formed by pairs of screens). Contour currents lead to the currents in each of three screens. However, the sum of the induced currents in the three screens is zero. This important circumstance means that no current occurs in the grounding circuit – and it does not matter whether the screens are well grounded (0.5Ω) or poorly (10Ω) or not grounded at all. Grounding does not affect the circulation currents in any way. A similar situation is known with the neutral of the star of a power transformer – in normal operation, currents in the phases A,B,C pass regardless of how the transformer’s neutral is grounded.

The current in the screen (Is) is defined as voltage (Us) divided by impedance (Zs). There are 1️⃣ and 2️⃣ which follow from here.

1️⃣ The currents in the screens do not depend on the length of the line (Us and Zs are proportional to the length, but when they are divided Is=Us/Zs, the length goes away). Similarly, with power transformers, the transformation coefficient does not depend on the length of the windings. However, unlike the currents, power losses and their cost are proportional to the length.

2️⃣ There are two ways to achieve the same result Is=Us/Zs=0 where we do not have currents/losses in the screens:
👉 make Zs=0, which is achieved by disconnecting the contours (called single-point bonding, although the effect is associated with the transfer of closed contours to idle contours, and the ground itself has nothing to do with it);
👉 make Us=0, and this is done by cross-bonding the screens.

Single-point bonding or cross-bonding provide the same result – no circulating currents and no power losses associated with them. However single-point bonding is more common for relatively short cable lines (due to induced voltage Us), and cross-bonding – for other cases.

All of this is shown in the book “High Voltage Cable Lines“. Also you will find there easy-to-understand calculation methods to select the optimal bonding/grounding type. These methods are more convenient than provided by CIGRE and IEC documents.