Friday, March 13, 2020

Mediumwave Loop Efficiency For The DXer

Many DXers are aware that an external, passive air core loop antenna can be tuned and coupled inductively to a portable mediumwave radio. Signal enhancement is usually quite good.

DXers may be unaware that an external air core loop antenna can be wired directly to the current corral of DSP radios. It is spelled out right in the manufacturer's data sheet for the Silicon Labs radio chips. It replaces, and is soldered in place of the internal ferrite loop. The document suggests a loop of minimal turns connected to the circuit board through a 1:5 winding (the so-called 25x step-up ferrite core transformer) thus providing the correct coil inductance of 180-450 micro-Henries. It was apparent to me that using a full inductance loop was also possible, bypassing the need for the transformer. This would also result in greater signal gathering ability. Some time ago I did an article called A Hardwired Loop For DSP Radios on this blog.

Loop tuned by a capacitor

Using an air core loop and the signal measuring capabilities of many of these radios we can determine a number of interesting things not possible with other analog or digital superheterodyne radios off the shelf. Today I thought we'd take a look at the mathematics of these loops, both passive and directly-wired. Don't be scared off by the mathematics of it - the toughest thing you will have to wrestle with is multiplication and division, or possibly getting the logarithm of a number from a calculator.

We will answer some interesting questions:

  • How big of a signal can I expect from a passive loop antenna of a certain size?
  • How much better will it be if I increase its size?
  • How much signal voltage is generated at the loop output for a certain field strength?
  • What about the reverse of this - what is the field required to generate that voltage?
  • What is the gain of my loop over the internal ferrite loop or another loop?


The signal voltage induced in a loop is proportional - increases linearly - with the number of turns, the area of the loop, and the frequency being received. Bigger is better, within certain parameters. We can measure the output of the hard-wired loop in microvolts using the modern DSP receiver. This is indicated by the RSSI "dBµ" figure on the display. We can then use that figure in the same formula to calculate the apparent field strength of the signal.

Small loops, that is, loops where the total length of wire is less than 1/10 wavelength at the operating frequency, are called magnetic loops. In close proximity, within 1/10 wavelength, they respond to the magnetic component of the passing wave. The loop is a transducer which transforms the electromagnetic wave energy into a usable voltage source. The number of turns in the winding, the physical size or area, and the frequency determines the loop's efficiency at transducing the incoming wave.


First, let's review the following S-unit chart from the article: The Ultralight dBµ Mystery, S-Meters, And Field Strength. This will give us an idea of what dBµV values we might see on our radio's DSP display.

S-unit        µV  dBµV  dBm
S9+60dB  50000.0   94   -13
S9+50dB  15810.0   84   -23
S9+40dB   5000.0   74   -33
S9+30dB   1581.0   64   -43
S9+20dB    500.0   54   -53
S9+10dB    158.1   44   -63
S9          50.0   34   -73
S8          25.0   28   -79
S7          12.5   22   -85
S6           6.3   16   -91
S5+4.9dB     5.6   15   -92
S5           3.2   10   -97
S4           1.6    4  -103
S3+1.9dB     1.0    0  -107
S3           0.8   -2  -109
S2           0.4   -8  -115
S1           0.2  -14  -121

The modern DSP receivers like the Tecsun PL-380, 310, etc. which employ the Silicon Labs chips, measure and display dBµV as received at the tuned front end across a load. They call it the RSSI indicator. It is measuring the voltage output of the ferrite or air core loop at the radio's input.

dB = decibels of course, simply a way of expressing magnitudes of a value, like voltage, logarithmically.

µV = microvolts, or millionths of a volt.

dBµV is a voltage expressed in dB above (or below) one microvolt. This is measured across a specific load impedance, commonly 50 ohms.

The 'dB' or decibel measurement is a logarithmic ratio as you may know. In terms of voltage, an increase of 6 dB is a doubling of voltage. So, if our little DSP radio receives a signal at 28 dBµ and it increases to 34 dBµ, the received voltage has doubled. Coincidentally, this is also an increase of one S-unit!

Use the following formula to convert dBµV to microvolts, or millionths of a volt:

                    µV = 10 ^ (dBµV/ 20)

To convert microvolts back to its decibel representation:

                    dBµV = 20 * Log(µV)

(Log is the common logarithm, or base 10).


One more concept we must address which is rarely mentioned in technical articles: A tuned loop produces higher voltage output levels than an untuned loop. In fact, much higher. When connected to our radio the loop is effectively tuned by the DSP receiver, and the voltage output "at the tuned frequency" is greatly increased from that of the untuned, unterminated loop sitting out in the open. Think of the DSP receiver as a variable capacitor which tunes the loop's inductance. This is true for the ferrite loop as well. Combined with the loop inductance, it forms a tuned circuit that literally concentrates the signal's field causing greater current flow at the tuned frequency.


Let's talk about loop efficiency. The efficiency of the loop determines the sensitivity of the loop. How is one loop better or worse than another? We can calculate a loop's efficiency if we know its area, number of turns, and the wavelength we wish to receive on it. Once that is known we can make comparisons to other loops.

An 18 inch untuned loop

Our efficiency factor here is often called the "effective height" H, in meters.

Effective height of an 18 inch loop of 12 turns:

                    H = (2 * pi * N * A) / wavelength

                         pi = 3.14159
                         N = number of turns
                         A = area of the loop in square meters
                         wavelength = wavelength of the frequency, in meters.

Effective height, in meters, is what the popular articles call it, sometimes referred to as He. That's a bit of a misnomer. It is actually a ratio or percentage of the wavelength, 0 - 1, since its value is derived by division by wavelength of the received signal.

Referencing the formula above, wavelength is easily calculated. It is the speed of light in meters/second (299,792,458) divided by the frequency in Hertz (Hz).

Wavelength example for 640 KHz:

     299792458 / 640000 = 468.425 meters.

Next, area of the loop in square meters. In the US we commonly use imperial measure, feet. Area conversion to meters is easily done. One square meter = 10.7639 sq ft. Length of a meter is 3.28084 ft. We square this, as such:

     1 sq meter = 10.7639 sq ft. =  3.28084 ft. * 3.28084 ft.

Area, for practical example:

     Area of 48 inch loop = (4ft * 4ft) / 10.7639 = 1.486 sq. meters
     Area of 18 inch loop = (1.5ft * 1.5ft) / 10.7639 = 0.209 sq. meters
     Area of 12 inch loop = (1ft * 1ft) / 10.7639 = 0.093 sq. meters
     Area of 9 inch loop = (.75ft * .75ft) / 10.7639 = 0.052 sq. meters

So, plugging in the values for our 18 inch loop, assuming a frequency of 640 KHz:

     Effective height H = (2 * 3.14159 * 12turns * 0.209area) / 468.425wavelength = 0.03364

Our loop's effective height H is 0.03364.

If we double the number of turns to 24 the effective height is doubled to 0.06728.

If we double the area to 0.418 sq. meters the effective height is doubled to 0.06728.

If we double the received frequency to 1280 KHz the effective height is doubled to 0.06728. Aha! More signal output as we go higher in the band!

The loop's voltage output will be directly related to its efficiency, or effective height. As you can see, efficiency increases linearly with the number of turns, the area of the loop, and the frequency. Voltage output will track right along with that too.


Let's measure the voltage output of our loop. We find the loop's voltage output by converting the RSSI value right off the display of our DSP receiver, marked "dBµ" (which is actually dBµV).

63 dBµV from a station on 1040 KHz

Here's an example. In western Arizona, at mid-day we'll tune to Los Angeles station KFI-640, a 50 KW outlet. At 240 miles, it's a fairly weak signal (about 17 dBµV) using the radio's ferrite loop. However by removing the ferrite and hard-wiring an 18 inch square loop in its place it generates a commendable 42 dBµV at the receiver.

Our conversion formula again is:

                    µV = (10 ^ (dBµV/ 20))

Substituting values:

                    125.89(µV) = 10 ^ (42/ 20)

We have a loop output of 125.89 microvolts.

Now that we know the loop's effective height H and the voltage output of our loop we can calculate the received signal's field strength. Be aware, there is a slight hitch here. The calculated E field is the "apparent" E field, not the actual one. Follow along and I'll explain further.

Let's calculate the apparent electric field E required to produce that loop output. The formula becomes simple at this point:

                    Erms(V/m) = Vrms / H

Vrms is the loop output V in Volts or mV or µV
Erms is the electric field E in V/m or mV/m or µV/m (volts, millivolts, or microvolts per meter)

Be sure to use the same factors in the formula: microvolts to microvolts, millivolts to millivolts, and volts to volts. If we use microvolts in the equation we will have the answer in microvolts, as such:

Substituting values:

                    E(µV/m) = 125.89 / .03364
                    3742.27(µV/m) = 125.89 / .03364

Our E field is 3742.27 µV/m (microvolts per meter). This is equivalent to 3.74227 mV/m (in millivolts per meter).

But wait, there's more. 3.742 mV/m seems awfully much for KFI Los Angeles at 240 miles. Its groundwave field strength on computed charts is only 0.209 mV/m. What's going on here?

Recall above I said a tuned loop produces higher voltage output levels than an untuned loop. Remember that the tuned loop literally concentrates the signal's field, the same as a ferrite loop does. This concentration results in an apparent increase in the E field, or a "gain" if you will. The Erms field we just calculated includes the literal "gain" of the loop as well. The gain of the tuned 18 inch loop makes the apparent field equivalent to 3.742 mV/m! This is a ratio increase of 3.742 / .209 or 17.904.

This is a hard concept to wrap your head around. The "efficiency", or effective height formula, does not tell the whole story of how we get from actual field strength - the E field passing our loop - to signal strength - the loop's output. There is also gain involved, plus a little thing called Antenna Factor.

In the previous article Decoding Antenna Factor In Ferrite Loops on this blog, we dived into antenna factor. Antenna factor applies to all kinds of antennas, not just ferrite loops. Since we're here let's calculate the antenna factor of our 18 inch loop.

From a chart, KFI-640 will produce an actual E field here of .209 mV/m, or 46.4 dBµV/m.

You might recall from the The dBµ vs. dBu Mystery: Signal Strength vs. Field Strength? article on this blog the conversion formula to get from millivolts per meter to dBµV/m, also known as dBu, or engineer's dBu.

                    dBµV/m = 20 * Log(mV/m * 1000)        ...a.k.a. dBu (lowercase 'u')

Substituting values:

                    46.4(dBµV/m) = 20 * Log(.209 * 1000)

We are receiving KFI at 42 dBµV on the receiver's RSSI display for our 18 inch loop. Since we are dealing with decibels on both sides of the equation, we can use simple subtraction to arrive at our antenna factor. Antenna factor of our 18 inch loop is then 4.4 dB (46.4 - 42).

9 inch Helper Loop

I have a little 9 inch loop I built which I call my "Helper Loop". The side length is exactly half of the 18 inch loop, so the area is one-fourth that of the 18 inch. Thus, we should see about one-fourth the signal output. Let's compare it to the 18 inch. First we calculate the efficiency, or effective height again.

Plugging in the values for our 9 inch loop, assuming a frequency of 640 KHz again:

     Effective height H = (2 * 3.14159 * 24turns * 0.052area) / 468.425wavelength = 0.01674

Our 9 inch loop's effective height H is 0.01674.

Directly-wired to the DSP radio, we tune to KFI-640 again at mid-day and see an RSSI dBµV of 30.

Substituting values again:

                    31.62(µV) = 10 ^ (30/ 20)

We have a loop output of 31.62 microvolts.

Now we'll calculate the apparent electric field E again:

                    Erms(V/m) = Vrms / H

Substituting values:

                    E(µV/m) = 31.62 / .01674
                    1888.88(µV/m) = 31.62 / .01674

Our apparent E field is 1888.88 µV/m (microvolts per meter). This is equivalent to 1.88888 mV/m (in millivolts per meter).

Back to our measured RSSI outputs again. Our 9 inch loop's output is 31.62 µV. Our 18 inch loop's output was 125.89 µV. That's virtually four times the output of our 9 inch loop which is one-fourth its area. In dB (voltage), exactly +12 dB greater signal output is generated by the 18 inch loop which of course is 4 times the output as well. Remember, 6 dB is a doubling of the voltage, and another 6 dB doubles it again.

The gain of the tuned 9 inch loop makes the apparent field equivalent to 1.888 mV/m. This is a ratio increase of 1.888 / .209 or 9.033.

Wrapping up, using simple subtraction again to arrive at our antenna factor for the 9 inch loop, (46.4 - 30) = 16.4 dB Antenna Factor. This, again, is a 12 dB difference.

Here's a slightly different way to express our original  formula:

The induced voltage V of an untuned loop (the loop's output) is:

                    V(µV) = ((2 * pi * N * A) / wavelength) * E(µV/m) * Cos(theta)

                    Remember, our effective height, H, is this part:  ((2 * pi * N * A) / wavelength)

                    V = H * E * cos(theta)
                    E * cos(theta) = V / H

                         V is in µV (loop output)
                         H is the loop effective height
                         E is the field strength of the passing wave in µV/m
                         Cos(theta) is the cosine of the angle between the antenna and the transmitter

For untuned loops, the calculated E field is the actual passing field. Tuned, the calculated E field is the apparent passing field. Think of it this way: tuning a loop does not change the loop's core efficiency H which is determined by turns, size, and impressed wavelength, but it will indeed change the loop's output V.

A note on angle theta: Theta is the angle that the plane of the loop makes to the station's passing field. Our desired angle is almost always zero, pointed directly at the station, for max signal pickup. Since the cosine of 0 = 1, we can leave that factor out of the equation. Note that if you rotate the loop 30 degrees away from the station you have reduced the signal pickup by Cos(30), or 0.866. 60 degrees, Cos(60), or 0.5, half!

A 42 inch tuned loop

Above, a 42 inch loop tuned with a variable capacitor. Remove the capacitor and directly-wire this loop to a DSP radio after removing the radio's ferrite. You will see results! Watch out for overload!


Some interesting facts about loops, both passive and directly-wired:

1. A 48 inch loop gathers 16 times more signal than the 12 inch loop because it has 16 times the area of the 12 inch loop. A 9 foot loop gathers 81 times more signal than the 12 inch loop! Loop area is the determining factor here.

2. Halving the received frequency (let's say from 1200 KHz to 600 KHz) results in half the induced voltage given the field strengths at 1200 KHz and 600 KHz are equal at the reception point.

3. More turns are better. Pack as many turns as you can into your loop. Additionally for passive tuned loops: weigh turns over capacitance when calculating tuning parameters. For SiLabs DSP type radios, try to keep your loop inductance at the upper end of the range, 450 micro-Henries.

4. Be sure the plane of your loop is aligned at 0 degrees to the station. Rotating the loop 30 degrees off the station reduces the maximum induced voltage to 86.6%, because cosine(30) = 0.866. Rotating 60 degrees off the station reduces it to 50%, because cosine(60) = 0.5. Rotating 90 degrees to the station reduces it to 0%. The total null is a theoretical value of course, and not attainable in actual practice as there is no perfectly nulling loop.

5. Remember that tuned loops generate lots more signal than untuned loops.