A model of current distribution in copper clad steel conductors at RF – capturing conductor curvature

A model of current distribution in copper clad steel conductors at RF laid out a model for current distribution, though ignoring curvature of the conductor in calculating current density vs depth.

A model for current distribution in a conductor is that for a homogenous conducting half space with surface current parallel to the interface. Current density at depth d is given by the expression \(J_r=J_R\frac{J_0(kr)}{J_0(kR)}\) where δ is the skin depth \(δ=(ω \cdot µ \cdot σ)^{0.5}\) and \(k=\frac{1-j}{\delta}\), σ is the conductivity). This takes into account curvature of the conductor surface, albeit with slower compute time.

Let’s compare the two algorithms on a test case at 1.8MHz being copper cladding of 67µm copper over a steel core for an overall diameter of 1.024mm (#18).

Above is a stacked image, the simpler algorithm is the feint plot.

There is a quite small difference in this case. When the expected loss of 400Ω line using the conductor is calculated, the result with the simpler algorithm is 1.3% less than the later one using the Bessel distribution.

Distance to fault in submarine telegraph cables ca 1871 – the leap expanded

Distance to fault in submarine telegraph cables ca 1871 gave a mathematical explanation of the location of fault…

Now it is in terms of the three known values u,v,w and unknown x.

\(w(v-2x+u)=(v-2x+u)x+(v-x)(u-x)\\\)

\(x^2-2wx+vw+uw-uv=0\) from which you can find the roots.

\(x=w – \sqrt{(w-v)(w-u)}\\\)

I have been asked to expand the last ‘leap’.

So we have \(x^2-2wx+vw+uw-uv=0\) which is a quadratic, a polynomial of order 2.

The solution or roots of a quadratic \(ax^2+bx+c=0\) are given by \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

So, for our quadratic \(a=1, b=-2w,c=vw+uw-uv\), so \(x=\frac{2w \pm \sqrt{(2w)^2-4(vw+uw-uv)}}{2}\).

Dividing the top and bottom by 2 we get \(x=w \pm \sqrt{w^2-(vw+uw-uv)}\) which can be rewritten as \(x=w \pm \sqrt{(w-v)(w-u)}\).

We want the lesser square root \(x_-=w-\sqrt{(w-v)(w-u)}\) because x must be less than w, a constraint of the physical problem.

So when measurements gave \(v=1040 \Omega\) and \(w=970 \Omega\) we can calculate that the distance to fault is the lesser root, 210.3km from Newbiggen-by-the-sea. (The greater root would imply a -ve value for x or y which is not physically possible.)

Distance to fault in submarine telegraph cables ca 1871

In the early days of submarine telegraph cables, the cable technology was a single core steel wire wrapped in gutta-percha worked against ground. Now the gutta-percha was not a uniform or durable insulation and leaks to ground (sea) were inevitable, and when the leakage became sufficient the cable could not longer be used and had to be repaired.

The earliest method of locating a cable fault was a binary chop… which would mean deploying a cable ship, grapnelling for the cable, hauling it to the surface with a special dividing cut and hold grapnel that severed the cable when tension was too great, buoying off one end and steaming back to the other to haul it on board, clean it up and test to the far cable station. New cable was spliced and the cable ship steamed back to find the buoy and pull that end on board, clean and test to the other end. This was done to localise the fault, and eventually replace a fault section of cable. During this longish period, the cable was out of service. Continue reading Distance to fault in submarine telegraph cables ca 1871

nanoVNA – tuning stubs using TDR mode

From time to time I have discussions with correspondents who are having difficulties using an antenna analyser or a VNA to find / adjust tuned lengths of transmission lines. I will treat analyser as synonymous with VNA for this discussion.

The single most common factor in their cases is an attempt to use TDR mode of the VNA.

Does it matter?

Well, hams do fuss over the accuracy of quarter wave sections used in matching systems when they are not all that critical… but if you are measuring the tuned line lengths that connect the stages of a repeater duplexer, the lengths are quite critical if you want to achieve the best notch depths.

That said, only the naive think that a nanoVNA is suited to the repeater duplexer application where you would typically want to measure notches well over 90dB.

Is it really a TDR?

The VNA is not a ‘true’ TDR, but an FDR (Frequency Domain Reflectometer) where a range of frequencies are swept and an equivalent time domain response is constructed using an Inverse Fast Fourier Transform (IFFT).

In the case of a FDR, the maximum cable distance and the resolution are influenced by the frequency range swept and the number of points in the sweep.

\(d_{max}=\frac{c_0 vf (points-1)}{2(F_2-F_1)}\\resolution=\frac{c_0 vf}{2(F_2-F_1)}\\\) where c0 is the speed of light, 299792458m/s.

Let’s consider the hand held nanoVNA which has its best performance below 300MHz and sweeps 101 points. If we sweep from 1 to 299MHz (to avoid the inherent glitch at 300MHz), we have a maximum distance of 33.2m and resolution of 0.332m. Continue reading nanoVNA – tuning stubs using TDR mode

Finding the electrical length of the branches of an N type T – #4

Finding the electrical length of the branches of an N type T – #1 posed a problem, this article looks at one solution.

For convenience, here is the problem.

An interesting problem arises in some applications in trying to measure the electrical length of each branch of a N type T piece.

Let’s make some assumptions that the device is of quality, that the connection from each connector to the internal junction is a uniform almost lossless transmission line of Zo=50Ω. Don’t assume that the left and right branches above are of the same length (though they often are) and we should not assume that the nearest branch is of the same length as the others (and they are often not).

Before we start, we will calibrate the VNA entering the offsets appropriate to the OPEN and SHORT cal parts. In the case of my nanoVNA, measurement above 900MHz is very noisy, so the scan will be 100-890MHz (to avoid a glitch at 900MHz due to harmonic mode switching).

So, the problem is all the uncertain things that connect to the internal T junction. Lets connect a calibration quality 50Ω termination to the left hand port. We now know that the path from the male port to the remaining female port comprises lengths l2 and l1 of low loss 50Ω transmission line with a 50Ω resistor shunting at the junction of l1 and l2. Continue reading Finding the electrical length of the branches of an N type T – #4

Finding the electrical length of the branches of an N type T – #3

Finding the electrical length of the branches of an N type T – #1 posed a problem, this article looks at one solution.

For convenience, here is the problem.

An interesting problem arises in some applications in trying to measure the electrical length of each branch of a N type T piece.

Let’s make some assumptions that the device is of quality, that the connection from each connector to the internal junction is a uniform almost lossless transmission line of Zo=50Ω. Don’t assume that the left and right branches above are of the same length (though they often are) and we should not assume that the nearest branch is of the same length as the others (and they are often not).

Before we start, we will calibrate the VNA entering the offsets appropriate to the OPEN and SHORT cal parts.

So, the problem is all the uncertain things that connect to the internal T junction. Lets connect a calibration quality 50Ω termination to the left hand port. We now know that the path from the male port to the remaining female port comprises lengths l2 and l1 of low loss 50Ω transmission line with a 50Ω resistor shunting at the junction of l1 and l2. Continue reading Finding the electrical length of the branches of an N type T – #3

Finding the electrical length of the branches of an N type T – #2

Finding the electrical length of the branches of an N type T – #1 posed a problem, this article looks at one simple solution.

For convenience, here is the problem.

An interesting problem arises in some applications in trying to measure the electrical length of each branch of a N type T piece.

Let’s make some assumptions that the device is of quality, that the connection from each connector to the internal junction is a uniform almost lossless transmission line of Zo=50Ω. Don’t assume that the left and right branches above are of the same length (though they often are) and we should not assume that the nearest branch is of the same length as the others (and they are often not).

So, the problem is all the uncertain things that connect to the internal T junction. Lets connect a calibration quality 50Ω termination to the left hand port. We now know that the path from the male port to the remaining female port comprises lengths l2 and l1 of low loss 50Ω transmission line with a 50Ω resistor shunting at the junction of l1 and l2. Continue reading Finding the electrical length of the branches of an N type T – #2

Finding the electrical length of the branches of an N type T – #1

An interesting problem arises in some applications in trying to measure the electrical length of each branch of a N type T piece.

Let’s make some assumptions that the device is of quality, that the connection from each connector to the internal junction is a uniform almost lossless transmission line of Zo=50Ω. Don’t that the left and right branches above are of the same length (though they often are) and we should not assume that the nearest branch is of the same length as the others (and they are often not).

Something to keep in mind is that the reference plane for the female connectors is about 9mm inside the T, and you can see the reference plane on the male connector, the nearest end of the shield connection.

With a ruler, the physical length of the left and right female branches looks to be about 13mm, and around 28mm for the male branch… but electrical length will be longer due to an unknown (as yet) deployment of dielectric of unknown type inside the T.

So, put your thinking caps on.

A solution to follow…

nanoVNA – measuring cable velocity factor – demonstration – coax

The article nanoVNA – measuring cable velocity factor discussed ways of measuring the velocity factor of common coax cable. This article is a demonstration of one of the methods, 2: measure velocity factor with your nanoVNA then cut the cable.

Two lengths of the same cable were selected to measure with the nanoVNA and calculate using Velocity factor solver. The cables are actually patch cables of nominally 1m and 2.5m length. Importantly they are identical in EVERY respect except the length, same cable off the same roll, same connectors, same temperature etc.

Above is the test setup. The nanoVNA is OSL calibrated at the external side of the SMA saver (the gold coloured thing on the SMA port), then an SMA(M)-N(F) adapter and the test cable. The other end of the test cable is left open (which is fine for N type male connectors). Continue reading nanoVNA – measuring cable velocity factor – demonstration – coax

nanoVNA – measuring cable velocity factor

With the popularity of the nanoVNA, one of the applications that is coming up regularly in online discussion is the use to measure velocity factor of cable and / or tuning of phasing sections in antenna feeds.

‘Tuning’ electrical lengths of transmission line sections

Online experts offer a range of advice including:

  1. use the datasheet velocity factor;
  2. measure velocity factor with your nanoVNA then cut the cable;
  3. measure the ‘tuned’ length observing input impedance of the section with the nanoVNA; and
  4. measure the ‘tuned’ length using the nanoVNA TDR facility.

All of these have advantages and pitfalls in some ways, some are better suited to some applications, others may be quite unsuitable.

Let’s make the point that these sections are often not highly critical in length, especially considering that in actual use, the loads are not perfect. One application where they are quite critical is the tuned interconnections in a typical repeater duplexer where the best response depends on quite exact tuning of lengths. Continue reading nanoVNA – measuring cable velocity factor