Cables that leave you stranded
Peter follows up on his recent series on the electrical aspects of battery-powered locos to dispel a few myths which clearly persist regarding electrical cables.
When I wrote my recent series of articles on the electrical aspects of battery-electric locos (Current Affairs, EIM Nov 2020 – Feb 2021), I did wonder if they might cause the editorial mailbag to experience an obesity crisis. The articles were actually quite difficult to write, as they needed to include sufficient detail to enable ‘experts’ to understand the basis for my assertions (and hopefully be satisfied of their validity) but not so much detail that the average (non-electrical) reader would get lost and give up.
I have received some excellent feedback (many thanks to those who wrote in), some of which shows that there is still a little confusion out there on a couple of points, at least in part due to me trying to steer clear of delving too deeply into advanced electrical theory. The purpose of this follow-up article is to try and clear up these points.
Again, I will simplify things as much as possible and hope that those with relevant expertise will forgive my over-simplifications. It would be very easy to tip over into writing a full scientific journal paper (I have written well over 100 of these over the years), however I hope to teeter on the edge of that particular precipice and try to avoid falling in... This is, after all, a popular model engineering magazine (with, no doubt, ambitions to remain popular) and not an academic journal.
A Walk Along the Strand
Cable rating is one area in which it is easy to get confused. In the articles, I discussed a cross-sectional area of 16mm2 for the main cables in a battery-electric loco, stating that they were adequately rated for a steadystate current of 62A. A quick consultation with the electrical expert, Dr Google, will reveal many entries advertising products that claim to be capable of handling well over 100A from this same cross-sectional area. So, who is right? The answer is, of course, that both are correct in their own circumstances.
The variables to consider are:
● For how long (continuously) will the cable be required to conduct its maximum current?
● Is the cable enclosed? How much cooling is available around the cable?
● What type of insulating material is used? If PVC, what type of PVC and at what temperature does it begin to lose strength?
There is a fallacy that car-battery cables (Photo 2), which typically have a large number of (small) strands for a given conductor cross-sectional area, can somehow defy physics and handle more current (all other things being equal) than can cables with the same conductor cross-sectional area and fewer strands. The reason normally given is the ‘skin effect’ (more on this below) which, whilst very real, is not relevant at DC.
The actual reason why car battery cables typically have large numbers of strands is that they need to be (relatively) flexible, to cope with both the bend-angles required in their vehicular application and also to cope with the inevitable (and constant) vibrations they will experience, plus the flexing due to occasional disconnection and re-connection for maintenance/replacement/charging. Cable failures due to metal fatigue would not go down too well with your typical motorist.
This is also a consideration in our locos and is a good reason to choose such multi-strand cabling, however increasing the number of strands will not magically increase their currentcarrying capacity (for a given total cross-sectional area of the conductors). Indeed, arguably, the current carrying capacity will decrease slightly relative to that of a single, solid, conductor of a comparable overall diameter (or piece of copper bar stock as we model engineers would more commonly refer to it) – see below for an explanation as to why this is true.
The products highlighted by Dr Google, with their 100A plus currentratings, will have one or all of the following characteristics:
● An insulating material rated to cope with a high temperature – this makes sense in a hot engine compartment anyway, even if the cable itself is not the source of most of the heat
● A rating which intentionally allows the conductor to run at a high temperature (e.g. 110A at 90 deg C – running a loco with cables at 90 deg C is probably not the wisest idea)
● A rating which is time-limited (for example the cable is capable of carrying 200A for up to two minutes)
In most automotive applications (let’s ignore battery-electric vehicles for the present), a high current is only
“Running
a loco with cables at 90 deg C is probably not the wisest idea...”
required to be passed for a short period of time, typically when the car is being started. The cable will therefore heat up, however its thermal inertia will ensure that it does not reach a temperature capable of melting its insulation before either i) the car starts successfully or ii) the battery is exhausted (or at least no longer capable of delivering greater than the cable’s rated current).
Getting Under the Skin
We can now turn to ‘skin depth’ and the ‘skin effect’, which I have heard a number of times as a reason for adopting cables constructed using very high numbers of strands.
The skin effect refers to a phenomenon by which the current carried by a cylindrical conductor
(such as a strand of normal wire) can, in certain circumstances, be concentrated close to the surface of the conductor, with a lower current density in the centre (Figure 1). It therefore follows (so the fallacious explanation goes) that more, thinner, conductors equals more outer surface available to conduct the current which equals a higher overall current rating for the multi-strand cable. A detailed explanation of the physics as to why the skin effect occurs is probably beyond the scope of a model engineering magazine, however those interested can, in the first instance, obtain a good explanation from the relevant Wikipedia entry.
So, what is wrong with this theory? After all, the skin effect is a real physical phenomenon and in many circumstances needs to be considered and designed-around (radio-frequency circuits, for example, where I have spent much of my career). Let’s look at the (simplified) form of the equation for skin depth (defined as: the depth from the conductor surface to that at which the current density falls to around 37 per cent of its value at the surface):
“The answer is infinity and calculators struggle
a little with this concept...”
Where:
P is the resistivity of the conductor (discussed in part 1 of my original series, EIM Nov 2020). For copper, this is 0.0178 Ohm.mm2/m. ƒ is the frequency of the current passing through the conductor (such as 50Hz for UK mains). μ is the permeability of the conductor. Permeability is a parameter related to the magnetic properties of a conductor. For copper the value is very close to 1 (0.999994) whereas for iron, for example, it is very large (~5000).
This is one reason why iron (or steel) is a very bad choice as a conductor for anything other than pure DC (direct current) – and even then, it’s not great – but why it is an excellent choice for the laminations in a transformer, for example.
The first thing to note about this equation is what happens if we set the frequency to 0Hz, representing the DC currents provided by a battery.
The bottom line of the equation becomes zero and our metaphorical pocket calculator then gives an error; this is because the answer is infinity and calculators struggle a little with this concept (don’t we all?).
In other words, at DC, the whole of the conductor will conduct the current, irrespective of the diameter of the conductors (strands), the number of conductors or even the material the conductors are made from.
Even if we set the frequency to 50Hz, to represent UK mains, the resulting skin depth is over 10mm, meaning that a solid conductor of 20mm diameter (i.e. >300mm2 cross-sectional area) will (notionally) use all of its cross-sectional area to conduct the current. Any cable we are likely to encounter in a domestic situation is likely to be well under this size and so we can effectively ignore skin-depth here, too. Since its effect is either negligible or non-existent in domestic or vehicular applications, its effect is not considered, by manufacturers, when specifying the ratings for any domestic or office/ light-industrial cables (other than communications cables, such as TV/ satellite coax or Ethernet twisted-pair).
In conclusion, from the perspective of the cable ratings published by DC and mains cable manufacturers, no account will be taken of skin effect in arriving at their maximum current ratings for any cable we are likely to encounter as model engineers or amateur electricians.
So job done?
Not so fast... As discussed in the original articles, our loco controllers are not sources of pure DC. They use a high-frequency AC (say between 50 and 250khz) pulse-width modulated (PWM) signal to control the motor speed. Figure 2 illustrates a PWM waveform switching between 0V and 24V and its relationship to loco speed (making assumptions about a level track, no head or tailwinds and such). This is a simplified diagram, for clarity, and the frequency of the changes between 0V and 24V will be much higher in a real situation.
Bearing this in mind and therefore entering a frequency of, say, 100khz into Equation 1, results in a skin depth of just under one quarter of a millimetre, so a conductor (strand) of greater than half a millimetre will begin to suffer a meaningful loss of current carrying capacity, assuming that little or no filtering takes place in the controller, prior to current entering the cable.
Filtering is used to remove the high-frequency component (100khz, say), leaving only the quasi-dc voltage which ultimately governs the loco speed (giving us the blue line in Figure 2). Such filtering is not just a ‘nice to have’ as its absence will turn your loco into a radio transmitter! Radio 4 longwave broadcasts on 196khz and your loco wiring will form a
(somewhat inefficient) antenna.
There are many aspects of the system which can contribute to this filtering (not just components built into the PWM controller), for example the cable’s inductance (notably if the cable runs adjacent to a steel loco frame) and the motor itself (its windings and magnetic components). In summary: it’s complicated – hence my desire to avoid discussing it in the original series!
At this point, taking a simplistic view, it would be possible to conclude that the use of highly multi-stranded cables begins to make sense (in other words that the skin effect has some relevance), however even if we take this view their use will only restore us to the situation which we arrived at with DC, namely that we will get back to a cable rating which equates to that published by the manufacturer for a given cable cross-sectional area; we will not magically increase this! Based upon this argument, a 60A rated cable (as specified by the manufacturer, at DC) will, at best, be a 60A rated cable when conducting a PWM signal.
Elephant in the Room
There is one more problem, however, which prevents us from making the above assumption, namely that of treating a multi-strand cable as a collection of individual strands, from the perspective of skin effect: the cable simply doesn’t act that way! The reason is obvious, when you think about it...
Consider a ‘multi strand’ cable, consisting of just two strands (Figure 3a). In this case, it can easily be calculated that the percentage of the overall cable cross-sectional area which is covered by the conductors is 50 per cent.
If we now increase this to four conductors (Figure 3b), then the percentage area covered by the conductors also increases, to a little under 69 per cent. With seven conductors (Figure 3c), we get a coverage of just under 78 per cent – you can see a trend emerging here, although it is not entirely linear. For example, with 19 conductors (Figure 3d) the percentage is just over 80 per cent, however with 20 conductors (Figure 3e), this reduces to just over 76 per cent.
In the limit, with a near-infinite number of conductors, the packing density tends to a little over 90 per cent, which is very close to our piece of bar-stock discussed earlier, but
“In our locos,
we do have some handylooking chunks of iron and, more commonly, steel with which to realise our conductors...”
FIGURE 3A-E: Conductor packing in a multi-strand cable – see text for details. almost 10 per cent smaller. In other words, even the best-packed multistrand conductor will not be as good as a single, solid, conductor, for a given overall cable diameter.
This doesn’t mean to say that I am advocating running solid copper bus-bars (as they are known) around our locos, however it does serve to illustrate that it is the total crosssectional area of the copper conductors which matters and not the diameter of the cable itself. To be fair, this copper cross-sectional area is what most cable manufacturers specify, but this might not be the case with private sellers on ebay, for example, so be sure to check what you are buying.
So, back to our problem... If we assume that the copper strands contained within our cables are formed (in the factory) without corrosion and remain that way during use, then our multiple copper strands in our multi-strand cable will all be in electrical contact with each other (or at least with each touching neighbour) throughout their length. This is hopefully obvious from the latter diagrams in Figure 3.
The cable will thus behave as a single ‘solid’ conductor insofar as the skin effect is concerned and not as a series of separate, insulated, standalone conductors. A multi-strand cable of this type, for example any form of ‘car battery’ cable, will have no advantage over any other cable of a similar conductor cross-sectional area, from the perspective of the skin effect, no matter how many strands there are in the cable, even supposing such an effect is a factor in the first place.
Is our ‘clean copper’ assumption reasonable? The short answer is yes. For evidence, find an old length of cable which has been lying around in your workshop for years and strip off some insulation, say 1 inch from the current (probably corroded) end of the cable. This cable will almost certainly appear shiny and bright, despite having received no care or attention in its entire history. Even cables which run underground, in nearpermanently damp conditions, will exhibit this property (although not those which have been filled with water, of course).
There do exist multi-strand cables which are deliberately constructed to enable the individual conductors to act independently of one another, hence enabling each to form its own ‘skin’ of current conduction. These are known as Litz wires and consist of a (typically large) number of strands, each of which is insulated from its neighbours by means of an enamel (or in the olden days, silk) coating, the conductors themselves often being silver plated. Remember, this has an even lower resistivity than copper and skin effect, where present, means that most of the current would then be conducted by this silver outer layer, so this extravagance makes sense.
I have never seen a high-current version of this type of wire, however, and even if such did exist, I dread to think what it would cost! Hi-fi enthusiasts favour this type of wire and pay hundreds or even thousands of pounds for it, yet requiring it to conduct only milliamps at most.
Irons are Flat
The above discussion has focused on copper conductors. In our locos, we do have some handy-looking chunks of iron and, more commonly, steel with which to realise our conductors, in the form of the frames. In the articles, I did mention their possible use for this purpose and it is tempting to do so (they’re large, handy and ‘free’), however the use of a PWM controller may give cause for us to re-evaluate this idea (as hinted above).
The permeability of mild steel is in the region of 2000, although this figure varies a lot depending upon its precise composition. If we substitute this figure into Equation 1, the resulting skin depth when conducting our 100khz PWM current is: 5µm (in other words, 5 thousandths of a mm)!
Needless to say, this is tiny and using the frames for PWM conduction purposes would not be very sensible.
So, in the same way that irons are flat, anything containing iron should only be used for ‘flat’ (i.e. DC) currents. Hopefully this will make it easy to remember...
What does this mean?
I probably lost some of you somewhere around ‘cylindrical conductor’ and ‘current density’. If that’s you, don’t worry, here are a few recommendations:
● Rate your cables based upon a conservative conductor area and a (relatively) small temperature rise. For example, 60-70A of continuous current through a 16mm2 cable is a reasonable maximum. This represents a current density of 4A/mm2. You may even want to be more conservative than this, using say 3A/mm2.
● Use multi-strand cable with many strands (for flexibility/reliability), but base its current rating upon the above figures, not the seller’s claimed rating for automotive applications.
● Don’t use iron or steel conductors for PWM controller signals, in other words don’t use the frames of a loco as conductors for this type of controller!
There, that wasn’t so hard, was it?
“For the
small quantities we use in our locos the saving would not be worthwhile
– we’re not wiring up the National
Grid...”
Aluminium Foiled
Another topic of discussion which has emerged is the use of aluminium in conductors – this was prompted by my inclusion of a photo of a section of underground mains cable in the first article (repeated here as Photo 3), plus its (brief) mention of the possible use of aluminium loco frames as a conductor (does anyone make frames from aluminium? – I’ve never seen any, but perhaps someone out there has?).
Firstly, if Photo 3 left the impression that aluminium cables are a good idea in model locos, then this was not intentional! The photo was intended as a tongue-in-cheek reference, to raise a smile, and not a serious proposal (it is a hugelyoversized cable, with a bend-radius larger than most 71/4-inch gauge locos). Aluminium has many disadvantages when considering its application in a cable for use by ‘amateur’ electricians:
● Its resistivity is a fair bit higher than copper (see the first article in the series)
● Whilst it is cheaper than copper, for the small quantities we use in our locos the saving would not be worthwhile – we’re not wiring up the National Grid here (although it might feel like it at times...).
● It is actually pretty hard to get hold of wire of the size and (short) length we would need – your local motorist discount centre won’t stock it.
● Making (good) connections will be a challenge.
The last point is a key one and was raised by one correspondent in particular. When aluminium corrodes, to form aluminium oxide, the result is a very good insulating layer – the last thing we need in a high-current connection! Aluminium also oxidises extremely quickly when exposed to air, making it difficult to form a good connection before oxidation sets in. Making a copper crimp terminal-to-aluminium chassis/ frame connection is not quite so hard (a lot of radio-ham kit, high-end Hi-fi equipment and such uses aluminium chassis and copper wiring for earthing/grounding). It is important, however, to thoroughly clean/abrade the aluminium, tighten the connection quickly and then apply some grease or paint in order to keep it air-tight.
Having said the above, it is still not an approach I would necessarily advocate for our purposes, although it remains an option, if needed.
Chasing Ratings
Another point which has been raised (and, I must admit, one I half-expected when writing the articles) is that of using the published/claimed ratings for cables and maybe applying a ‘margin of safety’ to these ratings. This could be viewed as a simple (almost no maths needed) way of specifying the required cable. There are a couple of things to bear in mind here:
● Not all cable ratings are created equally. A cable sourced from your local motorist discount centre or an unknown online supplier may be rated at 200A (say), however the expectation in arriving at this rating may be for short-term use (such as running a starter-motor for a minute) and it will probably not specify the temperature to which the cable will rise during long-term use at this (or any other) current. Likewise, it won’t specify the temperature (or continuous current) at which its insulation will become weakened.
● Not all copper is pure – indeed, it could be argued that most ‘copper’ isn’t. Many years ago, I worked for a cable manufacturer (of very highpower radio coax cables) and they were being undercut by a rival
Chinese manufacturer. At the time, my employer was the largest buyer of copper on the planet (not just for cable use, but for any application), so they knew they were getting the lowest price for copper that was available. So how did the Chinese manufacturer undercut them?
A little was down to lower labour costs, but labour was a small part of the overall cost of the cable. The main answer was simple: impurities – the Chinese manufacturer added other (cheaper) metals to the copper, thereby reducing the overall cost of the raw material. This also (unsurprisingly) made it much more lossy. So, beware: ‘cheap’ copper cables, or those supposedly rated at X hundred amps (when others rate their cables lower), may not be all they seem.
Another way of looking at the first point is this: a 13A domestic mains fuse is rated for 13A of continuous current without failing, however I wouldn’t want to live in a house cabled with 13A fuse wire!
The upshot of this is: do the maths (it’s really very simple and compared to the amount of time you’ll spend diagnosing/fixing faults on your loco, it’s time wisely invested).
A Big Thank You
...to all who wrote in. It’s nice to know that the articles were studied in detail, with such thoughtful points being raised. Your letters helped me see the articles through different (readers’) eyes and hopefully this postscript has allowed some misunderstandings which may have arisen to be explained and corrected.
n For Peter’s previous articles you can download digital back issues or order printed copies at www.world-of-railways. co.uk/store/back-issues/engineering-inminiature or by calling 01778 392484.