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A look at conical inductors

There comes a time in RF and microwave work when one must introduce DC power into a signal carrying line without degrading the high frequency action of that line. A commonly used device is called a “bias-tee”. The following sketch illustrates two ways to use such a device.

Figure 1 Two common ways to use a bias-tee usage.

In the upper circuit, DC power is being delivered to the something-or-other that is receiving a signal from a something-else. In the lower circuit, DC power is being delivered to the something-else that is doing the delivering to the something-or-other. In either case, the inductor, an RF choke, should not exhibit self resonances at the operating signal frequencies and therein lies the challenge.

One approach to making the RF choke is to wind an inductor as a conical spiral as follows.

Figure 2 A diagram of a conical coil.

Arbitrarily denoted, the capacitances between turns are shown as a collection of “Cn” capacitors and the capacitances from each turn to RF ground are shown as a collection of “Cm” capacitances. With the conical arrangement of turns, the highest frequencies in the signal spectra are supported by the inductance near the conical tip while the lower frequencies are supported by the larger turns further away from the signal line itself. Both the Cm and Cn capacitances are kept small close to the signal line and allowed to be larger away from the signal line where their effects are less critical.

We can make a very crude circuit model for this to illustrate how self resonances are set up so that the lowest self resonant frequency of the RF choke is targeted at being higher than the highest operating frequency of the signal-input to signal-output path’s frequency span.

Figure 3 Circuit model of a conical choke.

Each turn of wire is represented by a parallel combination of Ln and Cn and a shunt capacitance of Cm. Although there will be cross coupling between the turns of wire, I have chosen here to assume zero cross coupling just to make the mathematics tractable versus my own personal analytic limitations. I must get my knees and beg your forgiveness for that, but a useful insight emerges anyway.

Taking our starting point to be on the right, we have an impedance that is assumed to be dominantly inductive, but which we shall choose to have an inductance value of zero so that what we see at the left end of this model will be an impedance that arises only from the conical coil itself.

For each section of this model, starting from a j X1, we may derive the equation for j X2.

Figure 4 Mathematical derivation of the impedance per turn of the conical coil.

The j X2 that is presented by each section can be used as a new j X1 for the next section to the left. This calculation gets repeated as many times as there are turns in the coil for each frequency that we choose to examine. That calculation process is thereby iterative.

We now look at part of a specification for an actual conical inductor to get a feel for the properties of a real-world component.

Figure 5 A real-world conical inductor.

Note the specified usable frequency range of this part as 100 kHz to 40 GHz. That is a span of 8.644 octaves or 2.602 decades which is quite frankly enormous. My attempted simulation result below isn’t nearly as good as that, but it still suggests enough bandwidth for satellite radio service.

Figure 6 Simulated inductive reactance of conical inductor.

Take note that the real-world parts are physically very tiny. Beginning with several admittedly blithe assumptions of an Ln of 50 nHy for the first turn, we choose to diminish the Ln value for each of the forty-four succeeding turns by a factor of 0.9452 which yields a final inductance value of 840 nHy which we test for at 1 MHz which is well below any resonance.

We also assume seemingly super-teeny-tiny capacitances of 0.001 pF each which we further diminish for the succeeding turns by a factor of 0.6 per turn. The simulation’s result is inductive behavior without resonance up to just a little below 10 GHz.

Are these results in the right ballpark? I can’t prove it, but I suspect they are. At least they seem to suggest the validity and merits of the conical structure of the purchasable item.

John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).

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