Applications of AVR single chip controllers AT90S, ATtiny, ATmega and ATxmega DCF77 - A primer on resonance and the ribbon cable loop antenna |

Now, the answer to that is very simple: take what you like, but be aware which consequences that has.

Your magnetic loop now has an inductivity of 2 Micro-Henry (µH). Smaller loops have less, loops with more windings than one have higher inductivities (see below). The first thing we have to know is what resistance has this loop against the DCF77's magnetic waves? The formula to calculate this is:

This resistance is NOT Ohm's resistance, which could be much smaller than 1 Ω, if your pipe is made of a 2 mm material with a diameter of 20 mm. Or higher, if you make your loop out of 0.1 mm iron wire. No, Z instead of R is the resistance that your loop has against alternating currents with a frequency of 77.5 kHz. We will see later on that more windings have a very much higher Z than our one-winding loop.

The loop is in resonance as such at around 8 MHz frequency, because it has a parasitic capacitor of 186 pF. But 8 MHz is not the frequency you want to receive. To get the loop down to a resonance frequency of 77.5 kHz (103-fold smaller), you'll need a capacitor, so that your loop produces a resonance circuit at that frequency. How large has this capacitor to be?

Now, simple: its capacitive resistance, which is

has to have the same value than your inductive resistance. So, if we have a L of 2 µH, a Z

C = 1 / 2 / 3.141592654 / 77500 / 0.974 = 0.000,002,108 F or 2.108 µF

Our single loop now gets resonance to DCF77 if we add this capacitor to the loop's endings.

If you think that the resonance circuit has a combined coil and capacitor resistance of 0.974 / 2, just because the coil's inductivity resistance and the capacitor's capacitive resistance are parallel, you are not correct. The fact that your two components are in resonance increases their combined resistance. This increase factor is called Quality factor and is

with Z

This quality factor depends from lots of things, namely from the Ohm resistance of the coil. It can as well be as large as 100, for lower quality LC's it would be around 40. So, the LC's resistance can be up to 50 Ω instead of the 1 Ω of the L's and the C's resistance only.

The Q factor is also resüponsible for the bandwidth of the LC circuit. Within the bandwidth

the resonance is reduced by a factor of two. A Q of 100 at 77.5 kHz means that f

If you need to calculate the resonance frequency from a given L and C, you'll need the following formula:

2 * Π * f * L = 1 / 2 / Π / f / C, or

f

f = 1 / 2 / Π / √(L * C)

The frequency therefore drops by the radix of L and C, not linearly.

But as our inductivity rose by a factor of 4, the needed capacitor is four times smaller now to be resonant at 77.5 kHz. This also increases the inductance of the L and C to 3.9 Ω and increases the inductance of the LC already to 200 Ω, if the Quality factor would be 100.

We can add even more loops to that. A ten-loop coil is at 200 µH, and a 100-loop coil is already at 14.6 mH. To bring such a large 100-winding-loop coil to resonance with DCF77 you'll need a capacitor of 290 pF only, ideal for a Medium Wave variable capacitor of 365 or 500 pF.

You can calculate the inductivity from the windings for a 1-m-diameter loop by the formula

w = √ (L / 0.000,002)

So, for a 1 mH air coil with a diameter of 1 m we need √(0.001 / 0.000,002) = 22 windings. (Reference to below: Take a 26-wire ribbon cable and leave four wires unconnected.)

What does this increase in inductivity mean? Now, the Z

If in resonance, the Z

The 90 cm ribbon cable and the 5 cm distance between the IDC sockets yield a circular loop of a diameter of 302 mm. With the following parameters

- 26 windings,
- 0.09 mm
^{2}wire area, 0.339 mm wire diameter, - 1.27 mm wire distance, distance between outer diameter of wires = 0.931 mm,

To test this I used a 16.0 nF Styroflex capacitor that I had at hand, and fed a rectangle via a resistor of 10 kΩ to the LC circuit. I found resonance at 65.04 kHz, from which an inductivity of 374 µH results. The LC's resistance is obviously 3.33 kΩ, while the inductive and capacitive resistance is 153 Ω, from which a quality factor Q of 43.6 results. The drop in quality is resulting from the measured 5.7 Ω of Ohm's resistance, due to the more than 23 m long wire of the loop. A Q of 43.6 means that the LC has a bandwidth of +/-0.89 kHz, which is a rather narrow bandwidth.

If we exchange the sides of the IDC plus by exchanging the two sides of the ribbon cable when pressing it into the IDC female plug, we'll get this connection scheme for the ribbon-cable-loop-antenna. It is a little bit simpler to solder, but works in the same way like the other option.

In general: increasing wire lengthes increases Ohm's resistance linearly, increasing wire diameters increases Ohm's resistance by the square of the diameter. The same applies roughly for the reduction of Q.

Nevertheless, I plan a 1 m diameter loop with 50-wire ribbon cable as well, but still need some time. I'll report on the results later on.

When measuring coils with a higher inductivity, it can be recommendable to use a smaller capacitor than 16 nF, e. g. 1 nF, so that the oscillation does not stop.

With the capacitor of 16 nF the 26-windings-loop oscillated at a frequency of 661 kHz. With the capacitor removed, the loop oscillated at 1,075 kHz.

The first approximation yields an impedance of 362.341 µH for the LC circuit. Calculating the capacitor for the frequency without the fixed capacitor arrives at a parasitic capacitor of 60.5 pF. This is not very large and in an acceptable range.

Repeating the calculation of the inductivity with the parasitic capacitor leads to a slightly different inductivity of 360.976 µH. Additional approximations do not change the resulting inductivity any more, so the parasitic capacitor of this loop is at 60.7 pF.

This amplifier can also be used as a remote driver for the loop antenna circuit, for operation in some distance to the receiver. The DC operating voltage is supplied via the inner wire of a shielded cable and is decoupled by the 33 mH coil and the two capacitors of 10 µF and 100 nF.

I adjusted the gain to roughly 100 and this is how the DCF77 signal looks like (scope on 20 mV per division). So, the sensitivity of the loop is rather limited, as compared to a ferrite antenna: roughly 0.3 mV HF. But: at least the resistance is rather low, well below 10 kΩ, so the antenna circuit can directly drive a silicon transistor amplifier and does not need an FET driver stage.

This is such an NPN amplifier with a gain of roughly 200.

This amplifier can, like the one with the OpAmp above, also be used as a remote driver for the loop antenna circuit, for operation in some distance to the receiver. The DC operating voltage is supplied via the inner wire of a shielded cable and is decoupled by the 33 mH coil and the two capacitors of 10 µF and 100 nF.

The design, here photographed in the evening sun, is indeed impressive. Something that an electronic and radio freak loves to have in his living room. If he declares to outsiders that this receives an atomic clock in every direction, guests may be impressed, too.

That meant to add two additional male 26-pin IDC plugs and to connect the additional two sides by soldering those with enamed copper wire. This time I decided that this second loop is changing the sides, so that the ribbon cable on pin 1 on the left arrives at pin 26 on the right.

This is the base plate that holds the wooden double-cross with the four IDC connectors.

These are the results of the measurements with the CMOS oscillator described above. The two loops are nearly identical to each other, as regarding their inductivities as well as their frequencies without a fixed C. It is interesting that combining the two loops by connecting them serially leads approximately to the sum of their inductivities. It is like both loops were not at all interacting, the increase of inductivity is only 1.8%. This is completely the opposite to the cross antenna with a ferrite core: there the inductivity of the cross rises much higher, the two windings are coupled much closer than here.

Also interesting: the parasitic capacitor of the combination is not combined but decreases by a factor of almost two, if compared to a single loop. This is obviously an advantage of the crossed loop and makes it a better receiver than the single-loop version.

Now: this cross antenna, equipped with two 2n2 and one 1n5 capacitors, brings already a visible result on my oscilloscope, even without a pre-amp. Here it is:

Relevant differences are only for the serially connected cross antennas: the inductivity is roughly the sum of the two single loops, their inductivity does not increase by the square of the windings. Both antenna loops in crosses are not interacting.

Again: practice is the master: build it and you'll be surprised.

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