*Last
changes 3 Jan 2009*

** **

**Very Weak Signal Reception with
Small Magnetic Loop Antenna**

*Chavdar Levkov LZ1AQ *

www.lz1aq.signacor.com lz1aq@abv.bg Sofia, Bulgaria

** **

** **

**Magnetic and
Ferrite rod antennas **

A loop with arbitrary shape with perimeter less than 0.1 of wavelength is called small magnetic loop. The ferrite rod antenna is a modification of magnetic loop and consists of a coil wounded on ferrite rod. Both antennas are very compact and are used extensively in long and medium wave AM receivers – a common name for these antennas is small magnetic loop (SML). It is interesting to use these antennas for receiving very weak signals – for example ham radio shortwave stations. Theory and construction of these antennas are widely available in the Net.

This paper is aimed to answer to the following questions:

1. What is the minimal possible size of a SML which still receives the atmospheric noise? That means that the antenna is limited by the electromagnetic environment not by the size.

2. How the antenna should be coupled to the hardware in order to reach maximal possible signal to noise ratio?

3. What is the maximal usable frequency range of a loop with fixed construction where a good signal to noise ratios can be reached.

The analysis and experiments are for the short wave band (1.5 – 30 MHz) but some of the results can be extrapolated to other frequencies. I will give here briefly the basic theory and equations for SML most of them well known in order to ease the reader.

**Equivalent
circuit**

The equivalent circuit of a small
magnetic loop (SML) is shown on **Fig.1**

** Å** is the electromotive force inducted by electromagnetic
filled;

*R _{A}*

*R _{L}*

** L** is SML inductance.

** Ñ** is parasitic capacitance of the loop.

** ****Fig.1**

If
the antenna is not made from very thick wire then *R _{A}*

The SML antenna has a Q-factor:

*Q =** **2ïfL / **R _{L}*

If SML is short circuited the current is equal:

*I = E / 2ïfL*** **

If
_{SML} is in resonance with capacitance ** Ñ **, the current is equal to:

*I = E / **R _{L}*

**SML
sensitivity. Effective height **

The
antenna is a transducer which transforms the electromagnetic wave energy into
voltage source. The effective height *h** *determines the sensitivity of the transducer*:*

*Å = h e***
**** ****
***(1)*

** Å** – is the voltage between antenna
terminals in

** å** – is the intensity of electromagnetic
wave in

** h ** has a dimension in
meters.

The basic equation for SML when the antenna is in optimal position to electromagnetic wave is given by:

* ** ** **(2)***
**

** λ** is the wavelength in meters

** w** - the number of ML turns;

** S** – is the area of the windings in m2;

*μ*_{R}** **is the effective
magnetic permeability of the ferrite rod SML.

The product:

** ***À*** ****=**** ***w **μ*_{R}* **S**
**(**3)*

is called effective area of the SML.

There is another equivalent equation where the magnetic component of the field is used:

*Å = (**2ï f w S**
**μ*_{R}* **μ*_{0}* **)**H*** ****
***(**4**)*** **

** H** is the intensity of the magnetic
field component in

*μ*_{0}* **= 4 ï 10 ^{-7}*

Both
equations (2 and 4) are equivalent taking into account that *e/H **= 377 [îì**]** *** . **In electromagnetic wave in free space (far zone) the ratio
between

For comparison the effective height of the half-wave dipole is given by:

*Å = (**λ
**/ ï)** e*** ****
***(**5**)*** **

All equations are referred to open circuit voltages (the antennas are not loaded) and the field is assumed to be homogeneous.

* *

**Voltage levels
that can be obtained from SML **

Let we have a SML with effective area ** À** = 0.1 m

If
the field density is 1
uV/m *(*this means that the
voltage between two points along the field gradient and distance 1m is 1 uV*)
*then we will obtain for inducted voltage Å* **= **0.0**08** uV**. *For comparison at the same frequency
a half wave dipole will give **Å **= 26 uV (open circuit voltages).

_{ }

_{ }

**Signal to
Thermal Nose Ratio **

Signal to noise ratio (SNR) is the ratio of signal voltage to noise voltage (effective values). We will use the well known Nyquist equation for resistor thermal noise.

*U _{n} = (4k T R*

** U_{n }**is the noise
voltage (effective value)

** k** is Boltzmann constant;

** T** is the Kelvin temperature;

*R*** ** is the resistance in ;

** B_{w}** is the bandwidth in Hz
where the measurements take place;

Let the loss resistance ** R_{L }**of the assumed loop is 1
ohm. If we substitute in Eq.6:

If Eq. 2 and 6 are assumed we will obtain the following equation for SNR:

* *

*
**(7)*

If
we substitute ** À** from Eq.(3), then the efficiency of
SML to receive weak signals is determined by the simple ratio:

*(8)*

Since I am not professional in this
field this result was interesting for me and I was not aware of it. Surfing in
the Net I understood that I have invented the wheel again. In one very
early paper *“Ferromagnetic Loop Aerials For Kilometric Waves,” **Wireless Engineer**, Feb. 1955, pages
41-46, J. S. Belrose** *àâòîðúò (VE2CV) has suggested an equation for SNR in small
ferrite rod loops. The Belrose equation is :

*Å/U _{n} =*

Unfortunately I could not find this
paper but I found a reference to it by *L**a**ngford http://www.kongsfjord.no/
*where he comments that this equation is assuming the thermal noise. This
equation is given in ARRL Antenna book ed. 2005 Chapter
5 but there is
not clear that this SNR is actually for thermal noise. The analysys of Eq.9
shows that it gives the same numerical results as Eq.8 so both equations are
equivalent. In this form equation (9) it is slightly misleading since there are
mutual related parameters in it. For example* **Å/** U _{n}*

The simple relation is that signal
to thermal noise ratio (denoted as **SNRt** further on to avoid confusions) depends
only from effective area and loss resistance as is given by Eq.8.

**What SNRt
is needed for the short wave bands ?**

It is well known that the limiting
communication factor in frequencies between 1 – 30 MHz is the atmospheric and
man made noise. Fig.2 shows the graphic of the noise versus frequency.
The value given for 7MHz (-9dB uV/m @ 9KHz)
can be
recalculated for 1KHz bandwidth - ** å_{n}** will be around 0.1 uV/m @ 1KHz.

**Ôèã.2**

Another similar chart is given on Fig. 3 calculated for bandwidth of 1 KHz.

**Ôèã.3**

*1** – daytime mean noise value; 2
– nighttime mean noise value; 3- local thunderstorm; 4 –
industrial noise in town;*

*5** – noise in rural area; 6 –
cosmic noise; ( from “Receiving equipment, Bobrov N.V. Ed.
Sovetskoe radio 1971; in Russian) *

* *

We will use a value of 0.2 uV/m (normalized to 1KHz bandwidth) as a mean value of atmospheric noise for all bands. This is quite low level of noise – for example field with such intensity will induct at the terminals of half wave dipole at 14 MHz 0.7 uV. A single mean value is chosen to ease the following analyisis.

**Signal
Pickup **

The SML antenna is a part of
parallel circuit (Fig.1). We can use the parallel circuit in resonance to
increase the very small signal voltage to reasonable value. The parallel
capacitance C is sum of self capacitance and external capacitor. The voltage ** U
**at resonance is:

*U= Q Å**
*** (8)**
where

The parallel capacitor will be
assumed with very low losses with Q>1000. This is true for air variable
capacitors and some other high quality dielectrics. The analysis is not changed
if the losses are higher since we always can add to the loss resistance ** R_{L}** the loss resistance of
the capacitor. See Appendix 2 about the influence of the capacitance Q-factor.

Example:

Let we assume that the example
SML has ** L** = 2 uH at F=3.8 MHz and

The important fact is that the SNRt (at fixed frequency) does not depend whether the circuit is in resonance or not. This SNRt depends only from the ration given by Eq.8. At resonance the signal voltage is increased but the noise is increased also by the same factor.

The upper limit of this resonance method is the self resonance frequency of the SML itself. The upper range can be increased by connecting parallel inductances to SML – in this way the resonance frequency is increased and the SML still can be used in resonance mode. If the parallel inductance is ideal (no losses) the added noise will be zero and this is a good solution. But adding real inductance the SNRt will be degraded . The Appendix 1 gives quantitative analysis of the problem. For example if a parallel coil is connected with the same inductance and Q-factor as SML at the same frequency the SNRt will be reduced two times. The benefit is that the working frequency range will be increased almost with 50 % .In most of the cases this compromise is acceptable since the same antenna mechanical construction can be used with extended range.

** **

**Excel
Spreadsheet **

A
spreadsheet in Excel is given:* *** Magn_loop_SN.xls **which calculates SNR
ratio and other SML parameters.

* *

**Increasing SML
sensitivity **

** **At fixed frequency we should
increase the number of turns** w** and area

In order to increase the SML sensitivity the following measures can be taken:

1. Conductor: The conductor must be with large cross-section and the material copper or aluminum.

2. Increase the area: At fixed conductor length the highest area is reached with 1 turn. (the area increases with the square of the linear dimensions)

3. The shape of the loop: At fixed conductor length the highest area is reached with circular shape. The circumference has the highest area compared to all other geometrical objects with the same perimeter.

4. Increasing the windings ** w** : This method does not increase SNRt proportionally. The loss
resistance

5. Use low loss high permeabillity ferrite material for the case of ferrite rod SML.

On **Fig.4** a very useful chart
is given. These are theoretical curves calculated from the spreadsheet.
The minimal effective area ** À **can be obtained for 1-turn circular loop with given
Q-factor. This loop will have 10 dB SNR for signal with field intensity
of 0.2 uV/m eff. The value of 0.2 uV/m is somewhat representative level of atmospheric noise in
rural environment. The sensitivity of SML with such

** Ôèã.4**

** **

** **

**Preamplifier **

On Fig. 5 a simple preamplifier with ferrite rod antenna is shown. Source follower is used to match the high resonance impedance of the parallel LC circuit with low 50 ohms impedance of the most receivers. The gain actually is performed in the resonant circuit. The voltage gain is Q times. The advantage of the FET follower is that there is no additional resistive load and the high Q-factor of SML is preserved. Since the signal source has high impedance the equivalent noise current of the transistor is important – and usually for the FET it is quite low and acceptable.

** **

**Ôèã.5 Ôèã.5à
**

** **

The gain of the preamplifier must be high enough and the total noise at the output must be larger with 6 to 10 dB from the level of the equivalent input noise of the main receiver. The equivalent input noise of the communication receivers is in order of 0.05 – 0.3 uV eff. at 1000 Hz bandwidth.

A popular low noise JFET 2N5486 is used. According to data sheets its noise figure is 1.2 dB for frequencies up to 100 MHz at 1 Kohm internal resistance of the signal source. In our case the internal resistance of the signal source is much higher between 5 – 100 Kohms. I do not know whether the extrapolation of this noise figure is correct for higher load resistances but in any case low noise JFET should be used.

The working bandwidth is from 1.8 up to 20 MHz. The ferrite antenna has 36 turns on 11mm x 200 mm ferrite core. The material was unknown but the measured Q-factor was very good up to upper frequency limit. There is a tap from the coil at ½ of windings. The output of the preamplifier is separated from the receiver with 1:1 wideband transformer.

Since the inductance is quite high
in order to use the same antenna for higher bands a parallel inductances ware
connected to the circuit. They must not have inductive coupling with the
ferrite rod and also must not have magnetic core. The other possibility is to
use switched taps from the coil. I can not say which method is better but
I prefer parallel coils method*. See the Appendix 1. *

The variable capacitor is a cheap plastic type for MW radios but connected in a butterfly mode - the leads are between two stators. The maximal capacitance is reduced 2 times but there are no sliding contacts and the capacitor Q-factor is higher.

The power supply is from two 1.2v
NiMh accumulators AA size. The supply current is around 8 mA. This JFET works
fine at this voltage and there is no need to use higher voltages and increase
the supply current and size. The dynamic range is good but I have not
measured it quantitatively. The input signal is with low amplitude and
there is also a very high Q input filter. Moreover , the FET follower has a
deep negative feedback which also benefits the dynamic range. Everything is
mounted in a small plastic box as shown on **Fig.6.**

** ****Ôèã.6**

** **

One precaution - this preamplifier tends to oscillate if both the Q-factor of the circuit (> 200) and the equivalent resonance resistance are very high. This is due to parasitic source-gate capacitance of the FET. Usually this is a problem on these bands where the capacitance becomes very low.

On **Fig. 7** another design is
shown with the same preamplifier. This is universal box - here different
external ferrite or aerial loop antennas and parallel capacitors can be
connected to the terminals. There is internal variable air capacitor 2x
360 pF in parallel connection. A very lightweight loop (50cm diameter) made
from aluminum conductor ( Ô=3.4mm) is shown. This construction is very suitable
for experiments. On **Fig.7b** additional fixed silver mica capacitor is
connected in order to tune the loop on 3.5 MHz band.

**Ôèã.7**

If the main receiver has sensitivity MDS better than -135 dBm at 500Hz
the thermal noise of the antenna will be the limiting factor and there is no
sense of additional amplification. For conditions where the atmospheric
nose is very low (above 10 MHz) and the sensitivity of the main receiver is
insufficient additional amplification must be performed. In the
schematics on **Fig.8** there is additional wide band amplifier with bipolar
transistor with gain around 12 dB. The output again is decoupled with
wide band 2:1 transformer. The output impedance is somewhere between 50
-100 ohms. The transistor must be RF type with Ft>300MHz . The popular
PN2222 works well up to 18 MHz. The noise figure requirements for the second
transistor are not tight and NF bellow 6 is sufficient. The best result
was obtained with old BFY90 – the wide band gain was flat up to 30 MHz.
Diode limiter and resistor of 2 Mohms are connected at the input
terminals. The only purpose is to protect the gate of the FET since the box is
experimental and numerous times external components were mounted in “hot”
condition. In fixed construction these parts can be omitted.

Here is a file ** MA_table_engl.GIF **where the results of Excel spreadsheet
calculations are presented for several different magnetic loops which were
tested. Their physical dimensions are given and also the theoretical SNRt
figure that can be reached. On the air test show that in the evenings on
3.5 and 7 MHz for both loop antennas the limiting factor is the atmospheric
noise. The ferrite rod antenna is limited by internal thermal noise but is
still quite good.

**Ôèã.8**** **

**SML Balance **

The coupling between the preamplifier and the receiver is made with wideband transformer acting as a balun. Unbalancing at the output of the amplifier is much easier to be done than in the SML input. The coupling might be performed also with a classical current balun as shown on Fig. 5a. Coupling directly to the receiver is equal to connection of very massive conductors to the one side of the SML. The antenna direction diagram will be deformed and the direction sensitivity might be loosed. Large common mode currents can flow from antenna to the ground and it will become to act as a whip antenna. In any case the conditions become uncontrollable. In some very sensitive applications even the battery must be decoupled with chokes as shown on Fig.8 since it is relatively massive part and can be coupled capcatively to other massive bodies.

Shielding of the SML, as is made in direction finding receivers, is another way to balance the common mode currents. I do not recommend this method for our purposes – it is mechanically quite complicated. The easiest way to cope with common mode currents is to make the board physically small and symmetric. If for example a big air variable 2 section capacitor is used it can be used in symmetrical way by connection between the stators and thus the construction becomes very symmetrical.

The connection to the main receiver is made with coaxial cable with arbitrary length after unbalancing.

**Experiments**

Absolute measurements of the
sensitivity of SML are difficult to be performed and professional equipment is
needed. Relative measurements were performed comparing the SML with a full size
reference antenna. Two absolutely identical receivers must be used at the
same time, receiving the same signals. The idea of the experiment is to obtain
the signal-to-noise ratio in SML and reference antenna on the same signals.
If SNR is the same in two antennas this is an indication that the SML can
receive as well as the full size antenna and the SML sensitivity is limited by
the external factors (atmospheric noise) rather than the thermal noise of ** R_{L}** .

Two channel direct conversion
receiver was used for this purpose. This receiver was used for SDR
(Software defined radio) experiments. It is a two channel I/Q (quadrature)
receiver. The input mixer was slightly modified as shown on **Fig. 9** in
order to use separate antennas. Thus two identical receivers were
obtained with common heterodyne oscillator. The 90 degrees phase shift between
receivers is unimportant and not used in this experiment. The outputs of the
receivers are fed into two channel sound card. Two channel commercial software
spectrum analyzer was used to process these two independent signals.

Remark:

*This setup can be used to compare any
two antennas in real time in-band tests with great precision. Unfortunately as
far as I am aware, the existing ham radio SDR software have no independent two
channel mode to make such a processing possible. I suppose that it
will be relatively easy to make this modifications in the existing
software. This will be invaluable tool to compare antennas and many other
applications can be mentioned also.*

**Ôèã. 9**

*Experimental setup:*

*Location:** Yard in a house in
rural environment with relatively quiet electromagnetic conditions. There
were no industrial or house sources of noise in immediate proximity. The
sun activity was close to the absolute minimum (June 2008). The same day
the sun spots number was 0. *

*Reference antenna**: Asymmetrical fed dipole
with length 80 m. 15 m mean height above the ground. Antenna tuner
and 1:1 current balun was used for matching the antenna to the receiver input.*

*Small Magnetic Loop:** 1 turn Circular
loop with 0.5 m diameter. Effective area is 0.2 m ^{2}. Aluminum wire
with 3.2mm diameter was used. The measured inductance of the loop was 1.7
uH. In Table 1 measured (Q-factor) and computed (Rloss, Req, BW and
SNRt) parameters are given. SNRt is theoretically predicted ratio for 0.2
uV/m field intensity. *

* *

* ***Table 1**

* *

*The SML with preamplifier according
to Fig.5 was placed on a table approximately 1 m above the ground outside
the house on the loan. 8 m long coaxial cable connects the preamplifier with
the receiver. Both devices were powered with batteries to limit the possible
ground loops. *

*Receiver:** **Two channel direct
conversion receiver with sensitivity -128 dBm @ 500Hz. The measured
crosstalk between two channels including the sound card was < -32 dB. *

*Software and computer:** Notebook **Dell ** Inspiron **1501. **Sound card **Creative Live USB 24bit. **Software Spectrum
analyzer Spectralab v.**4.32.13** from **http:/www.soundtechnology.com/**. **The computer was also
powered by the internal battery during the experiments. *

All these spectrums were taken in the amateur radio bands. On each figure there are two pictures - the spectrum from SML and that of a reference anrenna. These pictures are the spectrum images of a live band with different stations with different modulations. The CW signals are with narrow peaks and SSB signals are wider – 2 to 3 KHz. The two spectral curves are absolutely synchronized. The amplitude scale is logarithmic in dB and frequency scale is linear.

Sampling rate was 96 KHz and the receive band was also 96KHz since the image frequencies are not cancelled. That means that these spectrograms are with superimposed baseband and image signals + - 48 KHz around the heterodyne oscillator. The spectrograms are averaged for time between 30 and 100 sec to reduce the influence of the random noise and fading. The noise limit is clearly defined – this is the base line. Power density is measured normalized for 1 Hz bandwidth.The other FFT parameters are visible on the pictures.

The gain scale of the two channels is not calibrated and that is not needed. What we want to measure is the SNR ratio between signals and the noise base line in each channel. The electromagnetic gain is quite different for two antennas. The reference antenna is giving much stronger signal than SML but what we are interested is the SNR of each antenna compared to its own noise level – that is important in receiving. Then for each antenna the SNR ratio can be measured and compared to the SNR ratio of the other antenna for the same spectral band and signal.

**Ôèã. 10 ***3.5 MHz, low level of atmospheric
noise during daytime. SNR of the reference antenna is 6 dB better than SML.*

* The effective area and Q-factor
of SML is insufficient in these conditions. On this band Q=160 with C=975 pF. *

** **

**Ôèã.
11 ***3.5 MHz band at night time. The band is open and the atmospheric
noise is much higher. *

*There is no significant difference in
SNR of two antennas. *

**Ôèã.
12 **** ***7 MHz**, low level of atmospheric
noise during daytime. SNR of reference antenna is 10 dB better than SML.*

*The effective area and Q-factor of
SML is insufficient in these conditions. On this band Q=215 with C=300 pF. *

* *

**Ôèã.
13 ***7 MHz band at night time. The band is open and the atmospheric
noise is much higher.*

* There is no significant
difference in SNR of two antennas. *

* *

**Ôèã.
14 ***14 MHz, very low level of atmospheric noise. SNR is almost 20 dB
better in reference antenna.*

* In this case there is also
limitation from insufficient sensitivity of the main receiver. *

* Additional gain is needed as
shown in Fig.8 that will improve the SNR. On this band Q=250, C=75 pF.*

* *

* *

* *

**Ôèã.
14 À ***14 MHz, very low level of atmospheric noise. Additional preamplifier
with 12 dB gain is used for SML as shown on Fig.8*

* The SNR of SML is only 2-3 dB
less then the reference antenna. *

There are limitations in these experiments :

- The absolute level of the atmospheric noise is not known.

- Different directivity of SML and reference antenna

- The reference antenna has much grater height than SML and its gain is grater for low angle reception. This automatically leads to increased SNR ratio for signals located in long distance.

Irrespective of these limitations this experiment permits to measure whether the SML sensitivity is limited by its physical parameters. One more time I want to emphasize that the goal of these experiments is to measure whether the SML sensitivity is limited by external noise or by its physical dimensions. If the limiting factor is the external noise there is no need to refine the SML parameters any more. I do not declare that SML is equivalent in reception to a full sized antenna. The results of these experiments are in agreement to the theoretical assumptions.

**SML in local
noise**

The conclusions from previous experiments are that the SML is not an extraorinary quiet antenna for the ionospheric signals (there are such a myths in ham radio community). Is this antenna more immune to local noise?

Most of the local noise is produced by the power lines, cable TV lines an Ethernet lines. The name of the enemy is called pulsed power supply since in all these lines a power voltage (10 to 90 volts) is carried to switches and amplifiers or energy saving lamps. Usually the power supply unit for a Ethernet switch is of pulsed type with bad filtering or not filtering at all. These pulsed supplies emit wideband noise which propagates along the cables. These noise sources do not have proper matched antenna so they are radiating mostly as near zone (read about it in ARRL antenna book) radiators where the near (also called reactive) field might have very high intensity. In the near zone the intensity of the field decreases with the square or even to the third power from the distance to the source. In near zone any meter of additional distance from the noise source is very important.

My experiments show that there
is no such a special immunity of SML compared to full-sized antenna. On the
contrary – a full sized antenna placed highly above the house is in better
situation for near zone radiations. A common mode balanced SML is
very sensitive to near field magnetic component sources for example a TV set.
If the SML is not balanced it will be sensitive also to near field
electric component noise sources. But I think that
the answer for
the good reputation of SML has a firm ground. The SML is a *directional
to near filed* and *very compact* antenna. It might be possible
to find some very quiet place on the yard or balcony or somewhere else where
for some instances the noise is much lower or cancelled.

** **

**DX reception
with SML**

It
is interesting to know what SML must be designed for DX reception. The analysis
shows (see **Fig.4**) that the size of SML can be very modest. I will
suggest the following lower edge sizes of loops.

1.8 and 3.5
MHz 1 turn, D= 0.88
m, À= 0.6 m^{2} , Q>150, L = 3 uH, wire diam. d = 4 mm, conductor copper or aluminum. The preamplifier
should be as in **Fig. 5**.

Universal:
1 turn, D= 0.5 m, À= 0.2 ì^{2} , Q>200, L = 1.7 uH, wire diam. d = 4 mm, conductor copper or aluminum. If the main receiver
has sensitivity < 130 dBm @500Hz ,the preamplifier should be as in **Fig.
8** (additional 12 dB gain).

Do not expect any miracles - the SML is not a low angle narrow beam antenna as Beverage but is very very compact.

For every day use my personal choice is the Universal SML with aluminum conductor. This SML has high Q-factor, it is very simple and lightweight and can work with very large capacitance values to very low frequencies.

** **

**Conclusions:**

1. The thermal noise of loss resistance is the limiting factor of the SML sensitivity.

2. An equation for the signal to thermal noise ratio is given as a function of the effective area and loss resistance.

3. Maximal sensitivity is reached with a low noise source follower JFET preamplifier.

4. The working band of the SML can be increased in higher frequencies with parallel inductances.

5. The SML can be unbalanced with wideband balun transformer at the preamplifier output.

6. The minimal effective area of the SML is estimated for a given SNRt ratio. A chart is given where the Q-factor and area of the loop are involved.

7. Experimentally is proven that SML sensitivity can be limited by atmospheric noise as is the case of a full sized antenna.

8. Single turn non shielded circular SML is mechanically simple and works satisfactory with large parallel capacitor on lower frequencies.

9. An Excel spreadsheet is given to estimate SNRt and to compute and optimize the basic SML parameters.

**Appendix 1. Signal to noise ratio degradation
when SML is tuned with additional parallel inductances**

In order to ease the analysis, admittances
and current sources are used since all elements are in parallel
connection. The equivalent circuit on **Fig. 16a** can be transformed
into that on **Fig.16b**. The parallel capacitances are neglected
since they do not influence the analysis. The SNRt ratio of circuit on **Fig
16b** must be compared to that on **Fig.16c** where there is
additional parallel inductance.

**Ôèã.16**

If *X _{L}*

*I*_{A}* = E/ **X _{L0}
*

The signal power at terminals 1 and 2 is:

*P _{S}*

The noise power of admittance *G _{0}*

*
**P _{N} =*

The signal to noise ratio is:

*(9)*

If a parallel inductance is added as in Fig. 16c this ratio will be:

*(10)*

If (9) is divided by (10) a new quantity *K***_{D }**will be obtained which is
the impairment of signal to noise ratio.

*
**(11)*

Going back to impedances which are
more familiar and taking into account that *G = 1 / (QX _{L}) : *

*
**(12)*

Here Q is Q-factor of
corresponding inductance. ( Note that *K***_{D }**is a

Equation (12) gives quantitative answer to some intuitively obvious relations – that the additional inductance must have high Q-factor and must be not with very different value from the SML inductance. For example if we connect parallel coil L1 with the same inductance and Q-factor as SML the SNRt will be reduced with 3 dB (2 times power). From the other hand the bandwidth of the SML will increase almost with 50%. In many cases this compromise is acceptable since the mechanical construction of the SML must not change.

** **

**Appendix 2. Signal to noise ratio degradation
of SML from parallel capacitances**

The same equation (12) is valid for connection of arbitrary reactive component in parallel – a capacitor for example. Then for parallel real capacitor the equation is:

*(13)*

Let
** Q_{L}** = 200 and

** **

**Links and
Papers**

*1. **ARRL
Antenna **book ed. 2005 **Chapter 5*

*2. **Small Antenna Design**, ** Douglas B. Miron,* *2006, Elsevier Inc*.* ISBN-13: 978-0-7506-7861-2*

*3. **Ïðèåìíüå
ôåðèòîâûå àíòåííû , Ãîñýíåðãîèçäàò 1963, Â.È.Õîìè÷*, *http://publ.lib.ru/ARCHIVES/M/''Massovaya_radiobiblioteka''/_''Massovaya_radiobiblioteka''_0400-0499_.html#0485*

*4. N4YWK
http://www.vlf.it/octoloop/rlt-n4ywk.htm
*

*5. Nikolova N.
http://www.ece.mcmaster.ca/faculty/nikolova/antenna_dload/current_lectures/L12_Loop.pdf*

*http://www.antentop.org/004/files/tr004.pdf*

*6. Butler VK5BR **http://users.tpg.com.au/ldbutler/VLF-LFLoopAerial.htm*

*http://users.tpg.com.au/ldbutler/Loop18MHz.htm*

*7. Grechihin A.
**http://rf.atnn.ru/s4/an-b92.html*

*8. Ferrite Rod Antennas for HF? R.H.M.
Poole, BBC WHP091 http://www.bbc.co.uk/rd/pubs/whp/index.shtml*

*9.
Forum: **http://www.radioscanner.ru/forum/index.php?action=vthread&forum=5&topic=22993&page=52*

*Chavdar,**
LZ1AQ*
*April –
June ** 2008*