Shielded Loop Aerials Rev 0

Trask, “Shielded Loop Aerials” 1 20 March 2010

Mastering the Art ofShielded Loop Aerials

by

Chris Trask / N7ZWYSonoran Radio Research

P.O. Box 25240Tempe, AZ 85285-5240

Senior Member IEEE

Email: [email protected]

20 March 2010


Figure 1 – Basic Loop Antenna Theory

Introduction

Shielded loop aerials have been a visiblepart of radio technology for almost a century now[1], yet their characteristics are at times poorlyunderstood. Their properties of shielding low-frequency electrostatic noise and good direc-tional characteristics have been long appreci-ated [2 - 9], and their application in directionfinding attests to their overall usefulness. Manyradio amateurs and shortwave broadcast(SWBC) listeners live in congested areas hav-ing substantial low-frequency electrostatic in-terference from a variety of sources such asfluorescent lighting and defective power mainstransformers. With it’s ability to provide a sub-stantial amount of immunity from such sourcesof interference, along with it’s relatively smallsize and directional characteristics, theshielded loop offers many desirable advan-tages over other types of aerials. This articlewill describe the theory and practical aspectsof the design of shielded loop aerials togetherwith a couple of significant variations, and willconclude with the design of a high-performancereceiving loop with remote tuning.

Basic Shielded Loop Aerial Theory

Loop antennas, shielded and otherwise,work on the basic priciple that a magnetic fieldpassing through a closed conductor results ina current along the conductor, which then re-

Figure 2 – Basic Theory ofShielded Loop Antennas

sults in a voltage across a load resistance atsome point in the loop, as illustrated in Fig. 1.The amount of current in the loop is dependenton the strength of the magnetic field togetherwith the surface area, number of turns, and ge-ometry of the loop. This, of course, is a bit over-simplified, but it is sufficient to convey the gen-eral concept of loop aerials.

Shielded loops are a bit more involved.Shown in Fig. 2, the shielded loop aerial con-sists of a pair of shelded arms that are groundedat the base end and open at the top, togetherwith an inner conductor. The shields act to re-duce noise from low frequency electric fields,which are the dominant form of electromagneticinterference (EMI) when in close proximity tothe ground. At higher frequencies, the outersurface of the shields are the actual antenna,which generate signal currents along the innersurface of the shield, which are then coupled tothe inner conductor by way of transverse elec-tromagnetic (TEM) coupling.


The frequency at which the shielded loopaerial goes from electric field shielding to mag-netic field reception is dependent upon thethickness and electrical characteristics of theshield material, while the overall effectivenessof thelow- frequency electric field immunity isdependent on the symmetry of the shield arms.These, as well as other aspects of the shieldedloop aerial will now be examined in detail.

General Frequency Characteristicsof Shielded Loop Aerials

The rejection of low-frequency E-fields isimproved by the split shielding of the outer con-ductor, which functions as a Faraday shield andwhich is not expected to carry significant cur-rents originating from either electric or magneticcoupling. With increasing frequency, however,the current flows increasingly along the exter-nal surface of the shield, until the shield be-comes the actual H-field coupling structure in-stead of the inner conductor. The outer con-ductor acts as a Faraday shield as long as itsthickness t is small with respect to the skindepth δ.also referred to as the 1/e depth of pen-etration [10], and is defined as [11, 12]:

(1)

where f is the frequency in Hz, µ0 is the perme-ability of free space (4π × 10-7 H/m), µr is therelative permeability of the material, and σ isthe conductivity of the material in S/m.

The skin effect dominates above a fre-quency limit fδ, which is dependent on the shieldthickness and conductivity, and for copper hav-ing t ≥ 4δ (13),

(2)

Thus, for practical thickness values theseparation of currents on the internal and ex-ternal surfaces of the shield begins at relativelylow radio frequencies (13). We will now dis-

cuss the high- and low-frequency characteris-tics of the shielded loop aerial in detail.

High-Frequency ShieldedLoop Aerial Characteristics

At all frequencies, the net current at theends of the shield at the gap is zero. By virtueof the fact that the currents along the inner andouter surfaces of the shield at high frequenciesare independent, the currents at the ends of theshield at the gap are equal and opposite (odd-môde), as shown in Fig. 3. In this circumstance,the outer sufaces of the shield are seen as aparallel-wire transmission line, while the innershield surface and the inner conductor are seenas coaxial transmission lines, the result beingthe transmission line model of the shielded looparial shown in Fig. 4.

By transposing the two conducting sur-faces of the inner conductor and the inner sur-face of the shield by way of a Marchand iden-tity [14], Libby [15] provides the simplified trans-mission line model of the shielded loop aerial

2-2107 tf ⋅=δ

σµµπ=

σµπ=δ

1

1

0rff

Figure 3 –High FrequencyOdd-Môde Gap Currents

Figure 4 – Shielded Loop AerialTransmission Line Model

GAP

ISi

SiI

antII

ISi

Si

ISe

SeI

ISe

SeI

00


shown in Fig. 5. This model eases the burdenof math in analytical approaches, but is not re-ally necessary when using computers. Still, itdoes help in the overall discussion.

The inner conductor is not necessarily asingle wire, but may actually be any number ofwires, which is common in designs that are in-tended for MF, LF, and VLF operation. Theactive loop antenna used with the HP100WWVB receiver is a good example of a multi-wire shielded loop. As the analysis of of theimpedance of such a shielded multi-wire bun-dle is overly complicated, the discussion herewill be limited to the analysis of a single-wireinner conductor, which is very common forshielded loop aerials intended for HF andhigher frequencies.

A typical cross-section of coaxial trans-mission line is shown in Fig. 6, where the innerconductor has a radius of r1 and the outer con-ductor has an inner radius of r2, an outer radiusof r3, and a thickness of t. The space betweenthe inner and outer conductors is filled with aninsulating dielectric material such as PTFE that

Figure 6 - Coaxial Transmission Line

Figure 5 – Libby’s SimplifiedTransmission Line Model

rr

t

ε

µr

has a relative permitivity (or dielectric constant)εr and a relative permeability µr. If the conduc-tors and dielectric are lossless, the character-istic impedance is determined by:

(3)

where C is the unit shunt capacitance, whichis determined by [16]:

(4)

and L is the unit series inductance, which isdetermined by [11]:

(5)

An additional quantity of interest is the ve-locity factor of the cable:

(6)

where c is the speed of light and εv is thepermitiviity of free space (8.854x10-12 F/m). Ingeneral the relative permeability of most, if notall insulating materials is close to unity, so Eq. 6can be comfortably approximated as:

(7)

As mentioned earlier, the outer surface ofthe shield is modelled as a parallel wire trans-mission line, illustrated in Fig. 7. The charac-teristic impedance of the shield, ZSHIELD, isdetermined in the same manner is for anunshielded loop aerial, which, for a cirular loopof circumference L operating below the first

L 2

r

r

0.2 r

r H / m

21

r2

1

=

=

=

µπ

µ µ

ln

ln

Z LCo ≈

VF c

1

p v v

v v

r v r v r r

= = =

= =

υ µ εµ ε

µ εµ µ ε ε µ ε

VF 1

r≈

ε

pF/m

rr

655

rr

2 C

1

2

r

1

2

=

=

=

ln

.

ln

ε

επ


parallel resonance, is easily approximated byway of: [15, 17, 18]

(8)

where A is the area enclosed by the antenna:

(9)

When operated sufficiently below the firstparallel resonance, the shield can be approxi-mated as an inductance LANT in series with aloss resistance RLOSS and a radiation resist-ance RANT. Applying this approximation toLibby’s simplified transmission line model ofFig. 5 results in the simplified high frequencycircuit model of Fig. 8. In this approximation,the inductance LANT is determined by way of:

(10)

where XL is the inductive reactance of the shieldwhich is determined by:.

(11)

where the wavenumber k0 is defined as:

(12)

For small shielded loop aerials, the lossresistance RLOSS is primarily a result of the skineffect of the material, and can be approximatedby [10]:

(13)

For small circular loops, the radiation re-sistance of the shield can be aprroximated as:

(14)

An alternative method for approximatingboth the radiation resistance and reactance ofthe shield up to the first antiresonance can bemade from the work published by Awadalla andSharshar [17] for single-turn loop aerials of spe-cific geometries, where the radiation resistanceis shown to be closely approximated as:

(15)

where the coefficients a and b are found inTable 1.

Power applied to the aerial is dissipatedas electromagnetic energy by the radiation re-sistance and as heat by the loss resistance.The ratio of radiated power to total power isreferred to as the aerial efficiency, and is read-ily defined as:

(16)

Figure 7 – Transmission LineEquivalent of Shield Outer Surface

Figure 8 – Simplified High FrequencyCircuit Model of Shielded Loop Aerial

24r A π=

=

r L

A 4 log 276 Z0

R L

d

Ld

0l = =

σ π δµ

π σf

R 320 A

Na =

π

λ4

2

22

k 0 0 0= ω µ ε

R a k Lb 0

a =

tan

2

=

2

L k Z X 0

0L tanj

ωjL

ANT

X L =

Eff R

R R=

+a

a l


Gap Parasitic Elements

Returning to the model of the shieldedloop aerial of Fig. 4, the high-frequency modelis completed by including the parasitic elemetsassociated with the gap as shown in Fig. 9,where the exposed centre conductor at the gapis modelled as a series resistance RGAP andinductance LGAP, and the ends of the shield atthe gap creates a small fringing capacitanceCGAP.

With the aid of Fig. 6 and Fig. 10, the in-ner conductor gap inductance LGAP is deter-mined by [19]:

(17)

where l1 is the length of the gap and r1 is theradius of the inner conductor. The inner con-ductor gap resistance RGAP is determined by[11]:

(18)

where δ.is the skin depth of the centre conduc-tor material, defined earlier in Eq. 1, and σ isthe conductivity of the material in S/m.

Calculating the gap capacitance CGAP isnot quite as convenient, being that no ready for-mula for the gap capacitance of an infinite cyl-inder with finite wall thickness over an infiniteplane is to be found. However, as shown inFig. 11 we can make a reasonable estimateby first calculating the gap capacitance of acoaxial resonator having a radius of r3 [20] andthen subtracting the capacitance of a circularplate having a radius of r2:

(19)

This is, or course, a rough approximation,however the gap capacitance is usually quitesmall and the overall model o the shielded loopwill suffer little, if at all, for frequencies belowthe first resonance of either the inner coaxialconductors or the shield outer surface.Figure 9 – Complete High Frequency

Circuit Model of Shielded Loop Antenna

nH 1r

2 ln 2

3GAP

−

= llL

Ωδσπ

= 2

3

GAP rR

l

Configuration L/λ ≤ 0.2 0.2 ≤ L/λ ≤ 0.5 a b a b

Circular 1.793 3.928 1.722 3.676Square (side driven) 1.126 3.950 1.073 3.271Square (corner driven) 1.140 3.958 1.065 3.452Triangular (side driven) 0.694 3.998 0.755 2.632Triangular (corner driven) 0.688 3.995 0.667 3.280Hexagonal 1.588 4.293 1.385 3.525

Table 1 - Coefficients a and b for Equation 15

pF r 8.85

r 2

102.5

r 2

104.5

r 2

0.09551 0.1977

r C

3

23

3

4

3

6-

2

4

3

4-

4

3

3GAP

l

ll

l

π−

−

×+

×−

−

+

=


Figure 10 – Model for ConductorExposed at Gap

r

Figure 11 – Model for Gap Capacitance

r

r

Low-Frequency ShieldedLoop Aerial Characteristics

Before we investigate the specifics of theshield at low frequencies, we should first un-derstand the characteristics of a bare loop aerialand evaluate the rejection of unwanted E-fieldsat low frequencies. We begin by examining thesimplified low frequency equivalent circuit of abalanced loop aerial, shown in Fig. 12 (13).Here, the various lumped elements are definedas (8):

(20)

(21)

(22)

(23)

where ζ0 is the intrinsic impedance of freespace:

(24)

The radiation resistances R0 and R1 aresmall with respect to the reactances L0 and C1,

and at low frequencies we can assume that(13):

(25)

The E-field generator is represented by(18):

(26)

and the H-field generator is represented by (18):

(27)

from which we can determine that the ratio of

Figure 12 – Simplified Low FrequencyEquivalent Circuit of Balanced Shielded

Loop Aerial (adapted from ref.13)

−

µ≈ 2

8

2

0

0 d

DLn

DL

4

000 2

6

ζπ

≈Dk

R

-1

01 2 8

−ε=

d

DLnDC

2

001 2

6

= Dk

Rζπ

Ω== 120 0

00 π

εµ

ζ

( ) 01 1 LC ωω ⟩⟩

( ) HDjV 20

2H πµπ−=

E DV 2E 2 π−=


GAP

ISe

SeI ISe

SeI

ISi

ISiI

ISi

Si

Iant

0 0

the H-field and E-field voltages delivered to theload resistance is (13):

(28)

For a plane wave in free space:

(29)

which reduces Eq. 28 to (13):

(30)

where c is the speed of light in a vacuum(3x108 m/sec) . From Eq. 30 we see that if wewere to desire an E-field rejection of 20dB, thelimiting frequency in MHz would be (13):

(31)

Common Môde Rejection

As frequency decreases, the manner bywhich the shielded loop aerial functions takeson a dramatic change. To begin with, the depthof penetration due to skin effect increases andlow frequency magnetic fields penetrate theshield, inducing currents on the inner conduc-tor directly. At the same time, the currents alongthe two shield surfaces become less independ-ent, and at very low frequencies the currentsalong the two surfaces become indentical. Theshield currents are no longer induced by mag-

E

H

fD

j

V

V

8

02

E

H

επ−=

Ω=== 120 0

00 π

εµ

ζH

E

fD

c

V

V2

377E

H

4

π=

Ω

Df

120

20dB π=−

netic fields, but are instead a result of electricfields inducing voltages on the two shieldhalves.

The noise immunity property of theshielded loop aerial is a result of the fact thatwhen close to the earths surface electromag-netic interference (EMI) is dominated by elec-tric fields, which are vertically polarized due tothe boundary condition (21 - 24), which dictatesthat electric fields are always perpendicular toa boundary and magnetic fields are always par-allel. As we shall see shortly, it is very importantthat the shielded loop aerial be mounted verti-cally.

EMI due to thunderstorms is propagatedover long distances by way of the earth-iono-sphere waveguide (25, 26), while man-madeEMI from a variety of sources such asflourescent lighting (27, 28), power line arcing(29), and faulty mains transformers and streetlighting fixtures (30) propagates by way ofground waves, also known as Zenneck waves,which is a boundary phenomenon (31, 32). Thesubject of low-frequency radio noise can be ex-hausting and is beyond the scope of this dis-cussion, and the reader is instead referred toan earlier publication by the author (ref)

Referring to Fig. 13, the signal currentsat the gap remain zero, and due to the longwavelegth of the signal and the close proximityof the two shield halves the shield currents areof the same phase, which results in little if anysignal current due to E-field EMI being coupledto the inner conductor by way of TEM.

The degree to which low-frequencyE-fields are rejected by the shield is depend-ent upon the centering of the gap, and tp betterappreciate that we begin by defining thelengths of the two arms as l2 and l3, repectively,as shown earlier in Fig. 2, and further define adisplacement paremeter ∆, where (34):Figure 13 – Low Frequency

Even-Môde Gap Currents


(32)

(33)

so that

(34)

The effective length of a receiving antennais defined as the voltage induced between openterminals of the antenna divided by the incidentelectric field intensity (Schelkunoff and Friis1952), and for electrically small loop antennaswhere A<0.01λ2 (35):

(35)

which allows us to define the effective lengthsof the two shield arms as:

(36)

(37)

We now define two terms for the voltages ofthe two arms

(38)

(39)

which now lets us define the common môderejection ratio (CMR) of low-frequency E-fields(34):

(40)

When the gap is perfectly centred and theaerial is vertical with the earth’s surface, theE-field CMR becomes infinite and the circuitmodel of the shielded loop aerial of Fig. 9 re-

Figure 14 – Simplified Low FrequencyCircuit Model of Shielded Loop Aerial

duces to that shown in Fig. 14.

Remote Tuning

As we noticed earlier, the impedance ofthe shielded loop aerial is characterized as hav-ing a very low resistance in series with an in-ductance. We also learned that it is best to op-erate the shielded loop aerial balanced in or-der to enjoy the full benefits of the shielding ofE-field EMI signals as well as maintain the ra-diation pattern null when used for direction-find-ing.

The low resistance of the shielded loopaerial, and for loop aerials in general, providesa unique opportunity to provide sigla filtering atthe earliest possibility in the receiver system.Many designers forgo this opportunity and sim-ply connect the aerial to a high input impedanceamplifier so as to achieve wideband operation.Although this is convenient, it deprives the userof the full benefits of loop aerials, and in the caseof shielded loop aerials it impacts the benefitsof the shield, often eliminating its effectivenessentirely.

For aerials having low resistance, suchas loops, tuning is best done in series to as toenjoy the best possible Q, or selectivity. At thesame time, the low resistance becomes a bitproblematic as tuning at the receiver is imprac-tical and the designer is faced with having to

( )∆−π= 1 41 rl

( )∆+π= 1 42 rl

12

12

llll

+−

=∆

( )∆+= 1 2

eff2eff

Ll

EV 1eff1 l=

−

=

2eff 2

A 1

A 2

λπ

λπ

L

( )∆−= 1 2

eff1eff

Ll

EV 2eff2 l=

2 log 20

2

log 20 log 20 CMR

21

21

AVG

DIFE

∆=

=+−==

VVVV

V

V


Figure 15 – Shielded Loop Remote Tuning Assembly Schematic

apply some form of remote tuning right at theaerial itself.

One such approach is to use a motor-driven variable capacitor, but this is rather ex-pensive and is not necessary for a receivingaerial. Instead, varactors may be used, whichrequires nothing more than a tuning voltagebeing sent up the feedline. Varactors, however,have a bulk resistance that can be pafrticularlylarge, especially for the hyperabrupt junction va-riety used for wideband tuning, and this bulk re-sistance can often be larger than the aerial re-sistance, which will result in signal losses andsubsequent increase in the antenna noise tem-perature, meaning an increase in the receiversystem noise figure (NF).

This problem can be easily overcome byusing an impedance transformer between theaerial and the tuning varactors, as shown in Fig.15. Here, transformer T1 is a Guanella 1:4 bal-anced-to-balanced (BalBal) transformer, whiletransformer T2 is a 1:9 BalBal, and finally trans-former T3 is a 1:1 balanced-to-unbalalnced(BalUn) transformer. Due to the low circuit im-pedance, the construction of T1 requires closeattention to the details of wideband transformer

design, which will be discussed later.

What’s All This Shield GapLoading Stuff, Anyhow?

An aspect of shielded loop aerials that israrely given any attention is that of capacitivelyloading the shield gap, as shown in Fig. 16. Theeffect of adding a capacitance across the shieldgap is that the signal voltage across the load isincreased and the tuning Q of the aerial is im-proved. For single-turn shielded loop aerialssuch as being discussed here, the signal volt-age can be almost doubled, representing analmost 6dB increase in power delivered to thereceiver.

A detailed study of the effects of shieldgap loading was conducted by R.E. Burgessof the National Physical Laboratory Radio De-partment in 1939 (2), and a portion was laterrepeated by Thourel (36). Burgess provides avery detailed theoretical analysis, which usedthe mid-band model shown in Fig. 17. Fig. 18shows curves for three cases from that study.The vertical axis is the ratio of loaded loop out-put voltage (V2) to unloaded loop output volt-age (V20) for a range of values for the gap load-


Figure 18 – Theoretical analysis of shieldgap loading for three cases (from ref x)

ing capacitance, where C10 is the capacitancerequired for series tuning the shield inductanceLSHIELD:

(41)

The quantites Q1 and Q2 seen in Fig. 18

are the unloaded Qs of the shield and loop, re-spectively, which are determined by:

(42)

(43)

Typical values for Q1 lie between 50 and100 while Q2 may vary from 100 to 200, owingto the larger amount of surface area and hencelower resistance of the shield (36). The peaksin the curves of Fig. 18 are the points at whichthe shield gap loading capacitance is optimal(COPT), and that value can be determined byway of:

(44)

where q is the ratio of the unloaded shield andloop Qs:

SHIELD2SHIELD10 L

1 C Cω

==

SHIELD

SHIELD

1

1SHIELD1 R

L

R

L Q Q

ωω ===

LOOP

LOOP

2

2LOOP2 R

L

R

L Q Q

ωω ===

Figure 17 – Mid-band Model Used by Bur-gess for Shield Gap Analysis (from ref x)

Figure 16 – Shielded Loop Aerialwith Shield Gap Loading

−+

+−=

2

2

2

SHIELDOPT 1

1 1

C Ck

qk

qkk


(45)

and the value of the coupling coefficient k is de-termined by way of:

(46)

where n is the number of turns of the loop, inour case being one. When the shield gap load-ing capacitance is adjusted for the optimal sig-nal output voltage, the effective inductance ofthe shield becomes:

(47)

Burgess (2) provided experimental evi-dence of the effects of loading the shield gap,the results shown in Fig. 19. Here, curve a ismeasured data with the signal injected into theantenna with a toroid around the shield, curveb is measured data where the signal is injectedby way of a separate antenna, and curve c isthe theoretical data, which shows that the ex-perimental data has a strong correlation to thetheoretical analysis. For these tests, the valueof k2 is 0.51, which was determined from meas-urements of the loop and shield inductances. Inthis series of experiments, the peak of the loopvoltage occurs approximately where COPT is

half the value of CSHIELD.

Adding remote shield gap tuning is fairlysimple, the schematic of which is shown in Fig.20. The free ends of the 1-turn windings areconnected to the ends of the shield, while thefree end of resistor R2 is connected to the cen-tre of the exposed inner conductor. The con-struction details for T4 will be discussed later.

In order for the gap and loop tuning beadjusted with a single tuning voltage passedup the feedline, the same varactors are usedfor both tuning units and the required scaling is

Figure 20 – Shield Remote Tuning Schematic

Figure 19 – Comparison of theoreticalanalysis and measured performance

of shield gap loading (from ref . 2)

LOOP

SHIELD

2

1

L

L

L

L nnk ==

( )qk 1 L L SHIELDEFF +≈

LOOP

SHIELD

2

1

Q

Q

Q

Q ==q


accomplished by way of the N winding in trans-former T4. To determine the value of N, we firstidentify the capacitance CLOOP, which reso-nates the loop to the same frequency as theshield:

(48)

By combining Eq. 41 with Eq. 48, we findthat the shield tunig capacitance CSHIELD andthe loop tuning capacitance CLOOP are relatedby:

(49)

and then continue, arriving at a relationship be-tween the loop tuning capacitance CLOOP andthe shield optimal capacitance COPT:

(50)

Comparing the schematics or the looptuning section of Fig. 15 and the gap tuningsection of Fig. 19, we equate the varactor di-ode pairs as being a tuning capacitance CTUNE,

and by way of transformers T1 and T4 we findthat:

(51)

(52)

Substituting Eq. 51 and Eq. 52 intoEq. 50, we arrive at:

(53)

from which we can solve for the turns ratio N ofthe gap tuning section transformer T4:

(54)

Figure 21 – Remote Shielded Loop Dual Tuning Schematic

LOOP2LOOP L

1 Cω

=

=

SHIELD

LOOPLOOPSHIELD L

L C C

=

SHIELD

OPT

SHIELD

LOOPLOOPOPT C

C

L

L C C

TUNELOOP C 4 C =

TUNE

2

OPT C N

2 N C

+=

=

=

+=

+

=

SHIELD

OPT

SHIELD

LOOP

2

TUNE

TUNE

2

LOOP

OPT

C

C

L

L

N 2

2 N

C 4

C N

2 N

C

C

1 C

C

L

L 2

2 N

SHIELD

OPT

SHIELD

LOOP −

=


The loop tuning schematic of Fig. 15 re-quires some modification to enable passing thetuning voltage to the shield tuning section byway of the inner conductor. Shown in Fig. 21,transformer T3 of Fig. 15 is replaced with trans-former T5 and the direction of varactor diodesD1 and D2 is rever\sed. Because of the lowcircuit impedance at the input, capacitor C2should be a good quality device having a verylow equivalent series resistance (ESR), suchas an ATC series 100 porcelain device.

And Now for SomethingCompletely Different

An interesting development in the designof shielded loop aerials is that of Carl Baum,who, while assigned to the Air Force WeaponsLaboratory (AFWL) in the 1960s, devised asensor for ionizing radiation fields (37). Shownin Fig. 22, the sensor is essentially a shieldedloop aerial in which the inner conductor is bro-ken at the gap, then the inner conductor endsare attached to the opposite shield arms. Baumnamed this configuration as the Moebius StripLoop, referring to the German mathematicianAugust F. Mobius who is generally recognizedas first conceiving a three-dimensional objecthaving only one side, which was later immor-talized by the Dutch artist Maurits C. Escher inhis work entitled “Mobius Strip II” and is pres-ently used in the form of the unversal recyclingsymbol.

Baum’s work was first recorded in 1964and remained classified until being cleared forpublic release in 1994. The mathematical de-velopment is somewhat limited, but was laterembelished in 1973 by Paul Duncan of theMcDonnell Douglas Corporation (38, 39), andwhich also remained classified until cleared forpublic release in 1997. Baum later describedMoebius Strip loop aerials having multiple turns,in which he referred to the single-turn configu-ration of Fig. 22 as being first-orderg (40).

The mathematical development by bothBaum and Duncan is focused on the perform-ance of the sensor to measure magnetic fieldsat frequencies where the currents of the innerand outer sufaces of the shield are independ-ent, so there is no investigation into the low-frequenciy noise performance of the sensor.

Baum initiates his development by firstexamining the basic shielded loop, which herefers to as a “split shield loop”. In Baum’slumped element model, shown in in Fig. 23, thesensor is seen as a signal voltage source V,equal to the product of the sensor area and themagnetic field strength, in series with an induct-ance L. The current sources ICL and ICR repre-sent the common-mode currents of the left andright arms of the shield, respectively. The valueZ is the differential-mode load resistance, whilethe value Z’ is the common-mode load resist-ance

Using this notation, Baum determines the

Figure 22 – Basic Moebius StripShielded Loop Aerial


common-mode voltage VCOM and differential-mode voltage VDIF of the shielded loop sensoras being:

(55)

(56)

Note here that the differential-mode voltageVDIF is derived for the differential output volt-age due to the common-mode currents ICL andICR while the signal cource voltage V is zero.

To later appreciate the characteristics ofthe Moebius Strip loop aerial, we return to Fig.2 and examine the signal output voltage of theshielded loop aerial. We begin by establish-ing a signal voltage VR at the right side of theshield gap having a value +V, for which there isa corresponding signal voltage VL at the leftside of the shield gap having an equal and op-posite value -V. To satisfy the boundary condi-tion, the signal voltage at the centre of the ex-posed inner conductor is zero, or rather a vir-tual ground.

We now view the two arms of the shieldedloop aerial as being a pair of coaxial transmis-sion line transformers (TLT). By way of TEMcoupling, the voltage across the both ends ofthe coax are equal in both magnitude andphase. Since the shield arms are grounded atthe base of the aerial and the signal voltage ofthe inner conductor at the centre of the gap iszero, the output voltage becomes:

(57)

In a similar analysis of the Moebius Sriploop, Baum used the model shown in Fig. 24,which he used to determine the common-modevoltage VCOM and differential-mode voltageVDIF of the shielded loop sensor as being:

(58)

(59)

The common-mode performance of theMoebis Strip loop will be of considerable inter-est when we later examine the low-frequencynoise immunity performance.

Figure 23 – Shielded Loop Aerial Model Used by Baum

( )Z I I V CLCRCOM ′+=

0 V

CLCRDIF Z

L Z

L

2

I I V

=

+

−=

ωωj

j

V 2 V V V LROUT =−=

0 V V V LRCOM =+=

0 V

CLCRDIF Z

L 4 Z

L 4

2

I I V

=

+

−=

ωωj

j


Figure 24 – Moebius Strip Shielded Loop Aerial Model Used by Baum

We now examine the signal output volt-age of the Moebius Strip loop in a similar man-ner to the earlier examination of the shieldedloop. Here, we retain the earlier designationsof VR and VL for the signal voltages at the endsof the shield at the gap, as well as their respec-tive values of +V and -V. With the inner con-ductor now broken at the gap and connectedto the opposite shield ends, the TLT voltagemapping shows that the output voltage is now:

(60)

which confirms that the output signal voltage ofthe Moebius Strip Loop aerial is twice that ofthe shielded loop aerial, making it the equiva-lent of a shielded loop aerial having an innerconductor with two turns but without the me-chanical complexities.

It is interesting to note at this point thatthe transmission line model for the MoebiusStrip loop aerial is exactly that of Libby’s sim-plified transmission line model for the shieldedloop aerial that was shown earlier in Fig. 5.

The common-mode properties of theMoebius Strip loop noted earlier in Eq. 58 nowneed to be examined with respect to low-fre-quency noise. To do this, we return to the work

of Carobbi and Millanta, following their proc-ess and making the necessary adaptations.We begin by recognizing that Baum’s VL andVR are now V1 of Eq. 38 and V2 of Eq. 39, re-spectively.

Using this notation and applying it to Fig.22, we use the same method of TLT voltagemapping as before and find that the signal out-put voltages of the Moebius Strip loop are:

(61)

(62)

which results in a low-frequency output differ-ential-mode voltage VDIF:

(63)

Substituting Eq. 63 into Eq. 40, we arrive at theCMRE of the Moebius Strip loop aerial:

(64)

( ) ( )

( ) ( ) V 4 V 2 V 2

V V V V V RLLROUT

=−−=

=−−−=12LR OUT V V V V V −=−=+

21RL OUT V V V V V −=−=−

12

- OUT OUT DIF

V 2 V 2

V V V

−=

=−= +

4 20

4 20

20 CMR

12

12

AVG

DIFE

∆=

=+−=

==

log

VV

VVlog

V

Vlog


neither of these equations take the electricallength of the Moebius Strip Loop aerial intoaccount, and as we shall see later the first reso-nant frequency of the Moebius Strip Loop aerialis approximately half that of a shielded loopaerial of the same diameter.

Still, the quadrupling of the terminal im-pedance at frequencies sufficiently below thefirst resonance allows us to make a significantmodification to the shielded loop remote tunerassembly described earlier in Fig. 15. Shownin Fig. 26, a comparable remote tuner for theMoebius Strip loop aerial does not require the4:1 impedance transformer T1 of the shieldedloop remote tuner. This component is a criticalelement of the shielded loop aerial remote tuner,as will be shown shortly, and eliminating it isboth a savings in material cost and a furtherimprovement in performance.

The doubling of the output signal voltagetogether with the same S/N performance makesa Moebius Strip loop aerial comparable to add-ing a low-noise amplifier (LNA) having a gainof 6dB and a NF of 0dB (not attainable) to asingle-turn shielded loop aerial having the samediameter, and this is readily attainable withoutthe burden and cost of any additional material.At the same time, one may use a Moebius Striploop aerial that is 70.7% the diameter of ashielded loop aerial and still have the sameamount of signal power delivered to the load.

Comparing Eq. 64 with Eq. 40 shows thatthe Moebius Strip loop aerial has twice the low-frequency output noise as does the shieldedloop aerial. Additionally, Eq. 57 and Eq. 60show that the Moebius Strip loop aerial hastwice the output signal voltage as does theshielded loop aerial, thus the Moebius Strip loopaerial has the same signal-to-noise ratio (S/N)as the shielded loop aerial, so there is no in-crease in the antenna noise temperature andthe receiver system dynamic range remainsunchanged.

We now examine the output current of theMoebius Strip Loop aerial, using the gap cur-rent illustration of Fig. 25. Since there is aphysical connection between the shield and theloop, the signal current at the gap ends are nolonger necessarily zero. Instead, we can seethat:

(65)

By virtue of TEM coupling between theinner conductor and the inner surface of theshield, we know that:

(66)

therefore, we conclude that:

(67)

Normalizing Eq. 67, we can readily see that theoutput current for the Moebius Strip Loop aerialis half that of the shield current, which again issimilar to that of a shielded loop having two turnsbut without the mechanical complexity.

Since a Moebius Strip Loop aerial and ashielded loop aerial having the same diameterwill have a similar, if not identical shield current,we might conclude that, by virtue of Eq. 60 andEq. 67, the terminal impedance of the MoebiusStrip Loop aerial will be approximately fourtimes that of a shielded loop aerial. However,

Figure 25 –Moebius Strip Loop AerialOdd-Môde Gap Currents

GAP

ISe

ISe

ISi

ISi

Iant

0 0

antIantI

Iant

SiANTSe III −=

ANTSi II −=

ANTSe 2 II =


These factors make the Moebius Striploop aerial a higly suitable option when overallperformance, physical size, and ease of con-struction are desirable design goals.

Low Impedance WidebandTransformers

The impedance of loop aerials, bothshielded and non-shielded, is characterized ashaving a very low resistance in series with aninductance. Therefore, in order to not degradethe efficiency of the aerial, the transformers inthe loop and gap tuning sections need to bedesigned in such a way as to not offer any ad-ditional losses (41, 42, 43).

In the design of wideband transformers,there are a number of details that need to beconsidered (44, 45, 46). To begin, the ferriteor powdered iron material in the core must beappropriate for the operating frequency range.Choosing the wrong material can lead to eitherinsufficient coupling at low frequencies, or ex-cessive core losses at high frequencies. For awideband transformer being used at low imped-ances, either of these can prove fatal. Gener-ally, for HF frequencies the choices can be co-balt-nickel-zinc ferrites such as Ferronics mix

K, or a high permeability carbonyl powderediron material such as Micrometals mix 8. Forlow HF frequencies, nickel-zinc ferrites such asFerronics mix J or Fair-Rite mix 61 are goodchoices, while for frequencies below HF, man-ganese-zinc ferrite materials such as Fair-Ritemix 77 will give good performance. It shouldbe kept in mind that powdered iron materialsare lossier than ferrites (47), but they are pref-erable for wideband transformers at high HFfrequencies and beyond.

The shape of the core is also an impor-tant factor. Wideband transformers wound ontoroidal cores are going to have the highest de-gree of leakage inductance, which will lower themaximum usable frequency of the transformerand which will lower the high end of the aerial’stuning range. Balun (or binocular) cores aregenerally the best shape for all practical pur-poses. For low frequencies, pot cores giveeven better performance as the leakage induct-ance is lowest and they deliver inductors andtransformers of much higher Q. Not all materi-als are available for all shapes, so this will fur-ther decrease the number of options available.The materials mentioned earlier are all avail-able in the form of balun cores, while additionalmaterials are available in the form of toroids

Figure 26 – Moebius Strip Loop Remote Tuning Assembly Schematic


and pot cores.

Then there is the matter of the wire. Forboth good balance and coupling, the wiresshould all be twisted together. This implies thateach strand will be of the same length. Also,the number of wires should be chosen in orderthat the interwire coupling be uniform (48).Fig. 27 illustrates the reason for this last state-ment. In the first case there are only two wires,and obviously the interwire capacitance, hencethe coupling, is uniform. In the second casethere are three wires, and the interwirecapacitances are equal between all threewires, again making the coupling uniform be-tween all the wires. In the third case wherethere are four wires, the capacitances are notequal. For the immediately adjacent wires, theinterwire capacitance is of value C, but for thediagonally opposite wires the capacitance isreduced to 0.707C, which means that the cou-pling will not be equal between all four wires.

In general, twisted wires will provide abetter coupling coefficient for small gauge wiressuch as are used here and are very convenientwhen constructing wideband transformers forsmall-signal applications (44, 45, 48, 49). Par-allel wires could be used in the construction ofsmall signal transformers, however there is lit-tle difference in the performance between thetwo methods (49). Parallel wires are a far bet-ter option for applications where winding ratiosare not convenient integral multiples or wherelarger gauge wire is called for in higher powerapplications (49).

Transformer T1 is very problematic dueto its being located at the lowest impedancepoint of the shielded loop and dual remote tun-ing assemblies of Fig. 15 and Fig. 21, respec-tively. Even minor losses in this component willhave a noticeable impact on the receiver sys-tem NF, so the choice of core material and con-struction technique requires close scrutiny. Itwas found earlier that using ferrite cores for thiscomponent resulted in significant losses regard-less of the core shape, and a considerableamount of trial-and-error experimentation wasneeded so as to arrive at a suitable design thatwas easily reproducible. The final configura-tion, hown in Fig. 28, is constructed as twowindings of twelve turns each of #30 AWG bifilarwire on a Micrometals T44-6 powdered ironcore, and a T44-2 core may also be used forHF frequencies, as may others depending onthe frequency range of interest.

Transformers T2 and T3 are less de-manding, and for HF frequencies both may beconstructed with four turns of #30 AWG trifilarwire through the holes of a Fair-Rite2843002402 binocular core. Fig. 29 illustrateshow the wires are grouped together for the vari-ous connections.Fig. 27 - Twisted Wire

Interwire Capacitances

Fig. 28 - Construction Detailsfor Transformer T1


Transformer T5, which is used in the re-mote dual tuner of Fig. 21, is also constructedon a Fair-Rite 2843002402 binocular core, thistime winding four turnseach of #30 AWG bifilarwire through each hole and along the outside,as shown in Fig. 30 which also illustrates howthe wires are grouped together for the variousconnections.

The construction of transfomer T4, usedin the shield gap tuning section of Fig. 19, is abit more demanding as its general nature pre-cludes the use of twisted wires, while at thesame time it has the same concerns of low im-pedances as does transformer T1, which wasdiscussed earlier. In this transformer, the twosymmetrical windings of unity ratio are closelywound as a parallel pair on a portion of aMicrometals T44-6 powdered iron core, sameas used for transformer T1, The N ratio wind-ing is then added as a monofilar windingin theremaining space of the core, thereby keepingall windings as a single layer so as to minimizeexcessive leakage inductance and ensure

good coupling.

Shielded Loop AerialMechanical Construction

The mechanical construction of aweather-proof shielded loop aerial is muchmore demanding than for a non-shielded loopaerial, the difference being the gap. For a non-shielded loop, the aerial element is a single con-tinuous piece of metal which requires nothingmore than a simple vertical support. For ashielded loop, the gap in the shield weakensthe mechanical integrity. In addition, the posi-tion of the gap needs to be carefully centred,and some means needs to be provided forpossibly including the dual tuning feature dis-cussed earlier.

Using a method suggested by RobertoCraighero (21), the familiar SO-239 UHF con-nector is used to attach a pair of shielded aerialelements to a metal base assembly and a PVCmasthead assembly. The base and mast as-semblies make use of commonly available elec-trical conduit fittings, which helps minimize thecost while making the design easily reproduc-ible.

The mechanical design described hereis divided into three distinct portions, being themast head, the mast base, and the shieldedaerial elements.

Mast Base Construction Details

The construction of the mast base is cen-tred on a 3/4” cast aluminum electrical conduitfitting that has threaded nipples at each end,often referred to as a conduit C body. Thesefitttings usually come with weatherproof seal-ing gaskets for the cover, and are well-suitedfor outdoor antenna projects such as this. Theremovable cover provides easy access to thematching circuitry, which is a nice feature if onewishes to experiment with different forms ofmatching circuitry and perhaps an amplifier.

Fig. 29 - Construction Details for Transform-ers T2 (top) and T3 (bottom)

Fig. 30 - Construction Detailsfor Transformers T5


Shown in Fig. 32, a pair of SO-239 con-nectors are added to the sides of the conduitfitting for later attaching the shielded aerial ele-ments. A small plate made from aluminum an-gle stock is added internally for mounting a BNCconnector to which the feed line will be at-tached. The BNC connector is of the type thatmounts to a PC board, which simplifies themounting of the small PC board remote tuningassembly.

Mast Head Construction Details

The construction of the mast head is simi-lar to that of the mast base, this time using a1/2” PVC electrical fitting that has a nipple atone end and a second one on the back side,often referred to as an LB fitting. These PVCfittings also come with weatherproof sealinggaskets as do the metal fittings used for themast base.

Shown in Fig. 33, a pair of SO-239 con-nectors are added to the sides of the conduitfitting for later attaching the shielded aerial ele-ments. A short piece of #10 solid copper wireis attached between the connector centre con-ductor pins. Constructed as such, the gap ca-pacitance CGAP is measured as 2.0pF, whilethe gap inductance LGAP and resistance RGAP

are too small to measure accurately and areinstead calculated to be 0.00uH and 0.00 ohms,respectively.

Solder lugs may be added to two of themounting screws for attaching a small PC boardfor the gap tuning assembly if one is to be used.In addition, a 1/2” PVC pluming cap is ce-mented to the unused nipple on the rear of thebody, using a short piece of 1/2” PVC pipe asa flange.

The mast head for use with the MoebiusStrip loop requires a slightly different configu-ration. Shown in Fig. 28, the centre conductorpins of the SO-239 connectors are cross-cou-pled to a solder lug or wire terminal beneath anut of the opposite connector. A good degreeof care must be applied when making these twoconnections so as to ensure that the connec-tions are symmetrical so as to ensure goodlow-frequency noise immunity.

The mast head is attached to the mastbase by way of a suitably long section of 1/2”PVC electrical conduit. The top end of the con-duit is cemented to the mast head body whilethe bottom end is cemented to a 3/4” PVCthreaded adapter and a 3/4” to 1/2” PVC re-ducer, which is then threaded into the mast base

Fig. 32 - Mast Base Assembly

Fig. 33 - Shielded Loop AerialMast Head Assembly


body, to which a small rubber O-ring should beadded so as to ensure a good weatherproofseal.

Shielded Aerial ElementConstruction Details

The PL-259 reducer, which is made forthe purpose of using the smaller diameterRG-58 and RG-59 cable with the PL-259 con-nector, is a very fortunate item for the design ofloop aerials. First, the inside diameter is slightlymore than 1/4”, which allows for easily sweatsoldering them to 1/4” copper tubing. And theoutside diameter of the boss at the one end issuch that it will fit very snugly inside 1/2” coppertubing, though some slight amount of effort maybe required and some small holes should bedrilled at the end of the tubing to allow for moresecure soldering. After the reducers are at-tached to the copper tubing, the PL-259 bodyis simply screwed on to complete the assem-bly. Fig. 3 shows the reducer as attached toboth 1/4” and 1/2” copper tubing.

The aerial element shields are made fromeither 1/4” or 1/2” flexible copper tubing, thelarger size being preferred so as to reduce theshield resistance and increase the shield Q.

The aerial inner conductor is commonlymade using coaxial cable from which the outerinsulating jacket and shield braid have beenremoved, using RG-8 or RG-11 cable with 1/2”tubing and RG-58 or RG-59 cable with 1/4” tub-ing. This is both convenient and economical,though some manufacturers such as Beldenadd a layer of metalized mylar beneath thebraided shield which must be removed in or-der that the shield inner surface currents arenot divided between mutiple metal surfaces,otherwise the coupling to the inner conductorwill be impaired..

An interesting and worthwhile alternativefor the inner conductor is to use a copper con-ductor that is slightly smaller than the inside di-ameter of the shield. Such an inner conductorwill have far less resistance than would onemade with coaxial cable, thus providing a sub-stantial improvement in the loop Q. In addition,the characteristic impedance willbe consider-ably lower, which is an aspect of shielded loopaerial design that is rarely if ever given any at-tention.

For a shield made with 1/2” flexible cop-per tubing, the inner conductor can be madewith 5/16”” flexible copper tubing. A pin for con-necting to the PL-259 body centre pin is added

Fig. 35 - PL-259 Reducers Attached to 1/4”(left) and 1/2” (right) Copper Tubing

Fig. 34 - Moebius Strip Loop AerialMast Head Assembly Detail


at each end, fashioned from a short piece of#10 or #12 copper wire which is soldered tothe inside of the tubing. After the pins are at-tached, two layers of PVC heat shring tubingare applied to the tubing, allowing it to fold overthe ends, providing a means of preventingshorting the shield and the innder conductorwhen the PL-259 reducer is soldered in place.If available, a small fibreglass washer may beadded so as to further reduce the chances ofshorting the inner conductor and shield.

Both the shield and the inner conductorassembly are kept straight so as to ease thepassage of the inner conductor through theshield..Afterwards, the PL-259 reducers aresoldered in place using a minimum amount ofheating so as to not damage the PVC insula-tion on the inner tubing. The assembly is thebent to shape around a suitable form madefrom plywood or other convenient material. Af-terwards, an isulator made from nylon rod isadded to the exposed centre conductor, thenthe PL-259 body is screwed on and the centrepin is soldered to the inner conductor, cthusompleting the aerial element assembly.

Tuning Control Assembly

The matching network tuning voltage issent up the coaxial cable, and the source neednot be more complicated than a variable powersupply and a simple bias tee, such as the onedepicted in the schematic of Fig. 36. Here, thetransformer T1 is made with 6 turns #26 AWGbifilar wire on a Fair-Rite 2843000102 or simi-lar balun core. As shwon in Fig. 37, this biastee fits comfortably inside a 3.25x2.13x1.13”aluminum enclosure and is very convenient forother applications such as powering mast-mounted preamplifiers.

A slightly more complicated but self-con-tained tuning controller is shown in Fig. 38.Here, the transformer T1 is the same as usedin the bias tee of Fig. 16, and the potentiometer

R3 is of the 10-turn variety, or it can be a sin-gle-turn unit with a reduction drive. Power issupplied by way of a 24VDC wall transformeror any other suitable means, and the controllercan be built within a 4.25x2.25x1.5” aluminumenclosure. If a single-turn potentiometer is usedfor R3 in conjunction with a 2-speed reductiondrive, then a tuning scale can be added to theenclosure, which is very convenient.

Synopsis

Despite the fact that shielded loop aeri-als have been widely used in radio communi-cations for almost a century, they remain widelymisunderstood. The theoretical aspectspresentled here should put most of the miscon-ceptions to rest. Much remains to be done in

Fig. 37 -Bias Tee Assembly

Fig. 36 - Biasing Tee Schematic (see terxt)


Fig. 38 - Control Unit Schematic (see text)

the way of constructing prototypes and provid-ing measured impedance data to this body ofwork, and updates will be provided as the workcontinues.


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Documents

Shielded Loop Aerials Rev 0