4. Alternating Current Effects in Busbars
The apparent resistance of a conductor is always higher for a.c. than for d.c. The alternating magnetic flux created by an alternating current interacts with the conductor, generating a back e.m.f. which tends to reduce the current in the conductor. The centre portions of the conductor are affected by the greatest number of lines of force, the number of line linkages decreasing as the edges are approached. The electromotive force produced in this way by self-inductance varies both in magnitude and phase through the cross-section of the conductor, being larger in the centre and smaller towards the outside. The current therefore tends to crowd into those parts of the conductor in which the opposing e.m.f. is a minimum; that is, into the skin of a circular conductor or the edges of a flat strip, producing what is known as 'skin' or 'edge' effect. The resulting non-uniform current density has the effect of increasing the apparent resistance of the conductor and gives rise to increased losses.
The ratio of the apparent d.c. and a.c. resistances is known as the skin effect ratio:
where Rf = a.c. resistance of conductor
Ro = d.c. resistance of conductor
S = skin effect ratio
The magnitude and importance of the effect increases with the frequency, and the size, shape and thickness of conductor, but is independent of the magnitude of the current flowing.
It should be noted that as the conductor temperature increases the skin effect decreases giving rise to a lower than expected a.c. resistance at elevated temperatures. This effect is more marked for a copper conductor than an aluminium conductor of equal cross-sectional area because of its lower resistivity. The difference is particularly noticeable in large busbar sections.
Copper rods
The skin effect ratio of solid copper rods can be calculated from the formulae derived by Maxwell, Rayleigh and others (Bulletin of the Bureau of Standards, 1912):
where S = Skin effect ratio
d = diameter of rod, mm
f = frequency, Hz
r = resistivity, mW cm
m = permeability of copper (=1)
For HC copper at 20°C, r = 1.724 mW cm, hence
where A = cross-sectional area of the conductor, mm2
Figure 4 Skin effect in HC copper rods at 20°C. Relation between diameter and x, and between Rf/Ro and x where x = 1.207 x 10–2 Ö(Af)
(Note: For values of x less than 2. use inset scale for Rf/Ro)
Copper tubes
Skin effect in tubular copper conductors is a function of the thickness of the wall of the tube and the ratio of that thickness to the tube diameter, and for a given cross sectional area it can be reduced by increasing the tube diameter and reducing the wall thickness.
Figure 5, Figure 6, and Figure 7, which have been drawn from formulae derived by Dwight (1922) and Arnold (1936), can be used to find the value of skin effect for various conductor sections. In the case of tubes (Figure 5), it can be seen that to obtain low skin effect ratio values it is desirable to ensure, where possible, low values of t/d and Ö(f/r). For a given cross-sectional area the skin effect ratio for a thin copper tube is appreciably lower than that for any other form of conductor. Copper tubes, therefore, have a maximum efficiency as conductors of alternating currents, particularly those of high magnitude or high frequency.
The effect of wall thickness on skin effect for a 100 mm diameter tube carrying a 50Hz alternating current is clearly shown in Figure 5.
Figure 5 Resistance of HC copper tubes, 100 mm outside diameter, d.c. and 50 Hz a.c.
Figure 6 Skin effect for rods and tubes
Flat copper bars
The skin effect in flat copper bars is a function of its thickness and width. With the larger sizes of conductor, for a given cross-sectional area of copper, the skin effect in a thin bar or strip is usually less than in a circular copper rod but greater than in a thin tube. It is dependent on the ratio of the width to the thickness of the bar and increases as the thickness of the bar increases. A thin copper strip, therefore, is more efficient than a thick one as an alternating current conductor. Figure 7 can be used to find the skin effect value for flat bars.
Figure 7 Skin effect for rectangular conductors
Square copper tubes
The skin effect ratio for square copper tubes can be obtained from Figure 8.
Figure 8 Skin effect ratio for hollow square conductors
n the foregoing consideration of skin effect it has been assumed that the conductor is isolated and at such a distance from the return conductor that the effect of the current in it can be neglected. When conductors are close together, particularly in low voltage equipment, a further distortion of current density results from the interaction of the magnetic fields of other conductors.
In the same way as an e.m.f. may be induced in a conductor by its own magnetic flux, so may the magnetic flux of one conductor produce an e.m.f. in any other conductor sufficiently near for the effect to be significant.
If two such conductors carry currents in opposite directions, their electro-magnetic fields are opposed to one another and tend to force one another apart. This results in a decrease of flux linkages around the adjacent parts of the conductors and an increase in the more remote parts, which leads to a concentration of current in the adjacent parts where the opposing e.m.f. is a minimum. If the currents in the conductors are in the same direction the action is reversed and they tend to crowd into the more remote parts of the conductors.
This effect, known as the 'proximity effect' (or 'shape effect'), tends usually to increase the apparent a.c. resistance. In some cases, however, proximity effect may tend to neutralise the skin effect and produce a better distribution of current as in the case of strip conductors arranged with their flat sides towards one another.
If the conductors are arranged edgewise to one another the proximity effect increases. In most cases the proximity effect also tends to increase the stresses set up under short-circuit conditions and this may therefore have to be taken into account.
The currents in various parts of a conductor subjected to skin and proximity effects may vary considerably in phase, and the resulting circulating current give rise to additional losses which can be minimised only by the choice of suitable types of conductor and by their careful arrangement.
The magnitude of the proximity effect depends, amongst other things, on the frequency of the current and the spacing and arrangement of the conductors. The graphs in Figure 14 (Section 6) can be used to obtain values of proximity effect for various conductor configurations at 50 or 60 Hz. Methods of calculation for other frequencies are available (Dwight 1946). The unbalancing of current due to the proximity effect can be reduced by spacing the conductors of different phases as far apart as possible and sometimes by modifying their shape in accordance with the spacing adopted. In the case of laminated bars a reduction may be obtained by transposing the laminations at frequent intervals or by employing current balancers using inductances.
Proximity effect may be completely overcome by adopting a concentric arrangement of conductors with one inside the other as is used for isolated phase busbar systems.
The magnetic field round busbar conductors may be considerably modified and the current distortion increased by the presence of magnetic materials and only metals such as copper or copper alloys should be used for parts likely to come within the magnetic field of the bars.
Both skin and proximity effects are due to circulating or 'eddy' currents caused by the differences of inductance which exist between different 'elements' of current-carrying conductors. The necessary condition for avoidance of both these effects (and hence for minimum loss) is that the shapes of each of the conductors in a single-phase system approximates to 'equi-inductance lines'. Arnold (1937) has shown that for close spacing, rectangular section conductors most closely approach this ideal. Such an arrangement is also convenient where space is limited and where inductive voltage drop due to busbar reactance must be reduced to a minimum. In the case of heavy current single-phase busbars and where space is slightly less restricted, the single channel arrangement gives the closest approximation to the equi-inductance condition, the channels of 'go' and 'return' conductors being arranged back-to-back, while for wider spacing a circular section is preferable.
In the case of special conductor arrangements, or where high frequencies are employed, the alternating current resistance may be calculated using the earlier sections. It is often necessary to know the depth of penetration of the current into a conductor, that is the depth at which the current density has been reduced to 1/e, or 0.368 of its value at the conductor surface. This can be calculated using the following formula when its resistivity and the frequency are known.
depth of penetration
where d = depth of penetration, mm
r = resistivity of copper, mW cm
f = frequency, Hz
