# Minimum frequency transmitted by PCB?

## What is the minimum frequency that can be transmitted by a PCB?

Of course this question is size and shape dependent. So I read somewhere that the antenna lenght must be about λ/4 to transmit those frequencies.

So I was not sure if you take the longest or the shortest side, I believe you take the longest one. So in the case of a rectangular PCB, say 10cm x 5 cm, I took 10 cm for example.

# Near Field

Now we need to define the near field of the antenna, so one precise way, described by ednmag.com, is:

r = (2* l^2 ) / λ , where l is the longest size of the antenna

another general definition is (by Ott & White):

r = λ / (2 * π )

In fact the Ott & White definition, can actually be applied to PCB's too.

So going with the first formula, in case of our example (10x5 cm PCB),

# Calculation

At 500 meters, using the first formula for the fraunhofer boundary definition, using the path loss equations (Friis , Schantz) :

The first formula tells us that the minimum frequency that can be transmitted by this PCB is roughly 47713 Hz, since the path loss is positive before this, so this is our minimum reference distance I believe.

Also using the general formula λ / (2 * π ), the min frequency is 72273 Hz Electric and 76303 Hz Magnetic field, roughly.

So there is some discrepancy between the two formulas, but it's roughly in the same range.

However the first formula already puts the limit in the far field where we have planewaves with 377 Ω impedance, while the second one is still in the range where the magnetic and electric field has discrepancies between their impedance.

So I assume the first formula is more accurate in our case since I'd expect the impedance to be 377Ω.

The question is as, stated in the beginning, whether my thoughtprocess is correct, and given that the first formula is more accurate, is the minimum frequency in this case roughly 47713 Hz?

I have read that you need a λ/4 length antenna to transmit those waves. So in our case that would be 12km long antenna, which is obviously larger than 10cm.

• Are you asking about intentional emissions from a PCB antenna? Or about unintentional emission from a PCB designed to do something other than transmit radio waves? – The Photon May 22 '17 at 21:23
• @ThePhoton The minimum frequency that can be used to transmit data, where the circuit of the PCB is acting like the antenna. So the minimum frequency that can be used for communication. So that doesn't include the noise inside the PCB nor the background noise. – David K. May 23 '17 at 0:02
• @ThePhoton if there is such great difference between the two, then you can tell me about both cases. – David K. May 23 '17 at 0:02
• With a big enough PCB, I would imagine you could make a quarter wave whip for just about any frequency. Just depends on your fab house; it's probably hard to get a PCB that's a kilometer or so on a side. – uint128_t May 23 '17 at 0:07

Well a printed transformer can (very inefficiently) work at only a few KHz, and there is no issue with placing such things on a board that size, so that implies that your math must be wrong....

I could easily build a magnetic loop antenna on that board that would sort of work at 25KHz or so, Q would be rather low and I would worry about dielectric losses and maybe breakdown if running any power into the thing but it would sort of work.

An odd multiple of a quarter wave is desirable because it makes the feed point appear both resistive and reasonably low impedance, but it is hardly necessary (I can send on 173Khz using a vertical in the garden very much shorter then a quarter wave it is just very inefficient (Think very much less then 1%), and extremely narrowband given the matching network required.

• My formula is a function of distance from the transmitter. So you may have a 25KHz antenna, but would it work if you'd place the receiver a couple meters from it? In my theory it wouldn't. Of course in a transformer I think you measure the distance how long are the wires that are wrapped up. So a 5 cm long transformer might have a 20m wire wrapped up there, so then the antenna is actually 20m long. ..... And we haven't even considered the GAIN, the gain itself in a whip antenna is around -65 dBi, so that is already pretty improbable. – David K. May 23 '17 at 20:02
• Nope, in general coiling an antenna does not retain its effectiveness if the coil is a very small fraction of a wavelength, I cannot build an effective 1/4 wave antenna for the 80M band by coiling up 20m of wire on a 5cm bobbin, be nice if I could (Google the Hatfield antenna technology crossed field antenna for one notorious failed attempt).... My printed mag loop would be horribly inefficient, but stuff enough power into it and I will be able to hear it a few meters away. There is some interesting work with metamaterials on small LF aerials that may be worth looking at. – Dan Mills May 23 '17 at 21:02
• I see it needs the space to be able to collect all the energy from all directions, I misunderstood your answer, I thought you were referring to the shape of the antenna, whereas you were talking about power. I am studying a paper now, It looks like the gain can be as low as -180 dBi (Gtx*Grx) at low frequencies for a good loop antenna (even lower for our improvized one), so I don't know what kind of power you can push into that, to overcome that attenuation. – David K. May 23 '17 at 21:27
• Longwave radio works just fine with a loop antenna wound on a hunk of ferrite a few inches long, granted one does not usually transmit that way..... Also loops are anything but omni directional, they have a deep null, which is actually a useful feature. – Dan Mills May 23 '17 at 21:39

The Motorola pagers (smaller than a cigarette pack), transmitted in the hundreds of Megahertz using "short antennas", basically heavy rectangular loops of copper strapping for low losses. The "short antennas" are less than quarter wave, perhaps much much less.

• Yes it depends what components it has, as I told above, if it has a transformer or some other component where a long circuit is wrapped up in a small compact space, then the total length is actually the length of the circuit. In my example I was only considering a simple PCB with a SOC, a few resistors and capacitors, so the printed wires are actually the source of emanation, and the lenght of them which is aproximately as long as the board itself, unless the path is a "maze like", however I've also read that 90 degre or sharper turns inside the path, also lower the efficiency of the signal. – David K. May 23 '17 at 20:07