PROCEEDINGS OF THE I.R.E.- Waves and Electrons Section

Design Values for Loop-Antenna Input Circuits*JAY E. BROWDERt, MEMBER, I.R.E., AND VICTOR J. YOUNGt

Summary-Design formulas and charts for the choice of in-ductances, Q's, and coupling coefficients of loop-antenna couplingtransformers ae given on the basis of signal-to-noise characteristics.Akmethod of determining these values when a cable or other primarycapacitance is present is also given.

INTRODUCTIONTnHIS ANALYSIS is based upon the fact that the

power match of an antenna to a receiver is notnecessarily the proper way' to obtain best per-

formance. In the transmitting case it is always desiredthat a maximum' amount of power be transferred intothe antenna, but with a receiver additional gain canalways be supplied to make up for signal power. Thesignal-to-noise ratio that is obtained from the an-tenna cannot be improved in the receiver. The propercriterion for arranging receiver input circuits is that ofobtaining a maximum signal-to-noise ratio. In the caseof a loop receiving antenna, the power-matching criterionis never the proper one for determining the constantsof the input circuit.

J

Fig. 1-A typical loop antenna in schematic form.

In Fig. 1 is shown a schematic drawing of a typicalloop receiving-antenna input circuit. The shielded loopwith its air gap is connected to a coupling transformerin the receiver either directly or through a transmis-sion line. This coupling transformer, in turn, supplies thefirst vacuum tube and is tuned by a capacitor whichmust track with other tuning capacitors. It is thepurpose of this paper to present the criteria that shouldbe used in selecting the values of the various com-ponents for such an input circuit.

In specifying signal-to-noise ratios in terms of thecircuit parameters, it will be necessary to assume thatthe resistive impedance of an antenna generates noise,just as does any other resistance, and that the normalstrength of the noise generated in this way is obtained

* Decimal classification: R320.51 XR325.3. Original manuscriptreceived by the Institute, April 29, 1946; revised manuscript received,July 15, 1946.

t Sperry Gyroscope Company, Inc., Garden City, New York.

by considering an equivalent resistor at a normal am-bient temperature. It is well known that the noisepower available from such a resistor is given by

N = kTAf

where k is Boltzmann's constant, T is temperature in de-grees Kelvin, and Af is the bandwidth. In practice thismay not always be true because the radiation resistanceis concerned in this respect with external noise condi-tions. Such external noise fields may be thought of aschanging the temperature of space. Thus external noiseat a level other than the one used changes the quantita-tive results of the computation but does not affect theirqualitative correctness.

RELATIVE PERFORMANCE OF Lo6pRECEIVING ANTENNAS

In order to compare the performance of various loopreceiving antennas, it is necessary to assign a quantita-tive figure of merit to each antenna. As is common inmost communication problems, the figure that hasproven most versatile is a signal-to-noise ratio; it isthe ratio of the signal strength to the thermal noise asmeasured at the open-circuited antenna terminals, underthe assumption of a standard field strength and a stand-ard bandwidth of the measuring equipment. It is alsoassumed in practice that the signal field strength in theneighborhood of the receiving antenna is larger thanany noise fields which also exist in this region of space.This last assumption is, of course, not necessary as far astheoretical merit is concerned, but in practice if thenoise field is larger than the signal field, nothing can bedone to appreciably improve reception.

It is necessary to maintain both the standardizationof bandwidth and a standard signal field strength inmaking absolute measurements of antenna performance.However, it is usually more convenient to deal entirelywith ratios of performance between two or more an-tennas, whence it is necessary to specify the standardconditions to be maintained during any given set oftests. This means that the present discussion is notlimited to any particular bandwidth or field strength.The maximum open-circuit signal terminal voltage of

any antenna is given by Eh1ff, and the thermal noisevoltage of any resistor is given at room temperature by1.26 X 10-10/RV/2f. The condition for unity signal-to-noise ratio under the assumption of small noise fieldmay then be written

1.26 X 1o0-lOV/NR\/fhIeff (1)

:19P7 519

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

whereE is the received signal field strength measured in

volts per meter,R is the loop resistance, including radiation resist-

anceAf is the radio-frequency bandwidth

h6ff is the effective height of the loop antenna, meas-ured in meters.

But the effective height of a loop antenna may be givenby

2rrAnheff =- (2)

whereA is the area of the loop in square metersn is the number of turnsX is the wavelength in meters.

Making this substitution yields

1.26 X 10-'0x/kVAfXf X327rAn

In this expression E is the minimum signal field strengthwhich can be received with at least unity signal-to-thermal-noise ratio.

I¢

c:

+150

+5 - - - - - - - - - - -

-F- - - - - - -

-io-- - - --

RELATIVE 05.0 10.0

Fig. 2-Effect of loop Q on antenna merit for a Ioop having a constantarea and form factor.

To compare one loop with another, it is only neces-sary to compare the corresponding minimum values ofE. Doing this in ratio form, and using subscripts a and bto distinguish between the constants of the two installa-tions,

Eb AanaV/Rb= _* ~~~~~~~(4)Ea AbnbVR(

This expression leads to the form

A bnbV\RaSuperiority of -20 log A aflVRb decibels. (5)loop b over loop a A ana\/Rb

This formula gives a decibel.comparison of the signal-to-noise merits of any two loops. When the expression

yields a positive nunfber, loop b is that many decibelsbetter than loop a. When the number computed is nega-tive, it is loop a which is the best.One interesting fact is immediately apparent from

this expression. When loops of a given area and formfactor (winding length divided by diameter) containseveral turns (at least more than one), their inductanceis proportional to the square of the number of turns.Thus, if under this condition the square roots of La andLb are substituted for na and nb, the formula can bemodified to include only a constant and square roots of'ratios of La to Ra and Lb to Rb. These ratios may, ofcourse, be written as Q's, and it can then be concludedthat the merit of two loop antennas is identical as longas their Q's, form factors, and areas are the same; ormore specifically, that adding or subtracting turns tosuch a multiple-turn antenna will not change its merit.

In Fig. 2 is shown the relative merit of a direct-con-nected loop antenna which has multiple turns and con-stant area and form factor. As is shown, the merit of theloop is then dependent only on Q.

INPUT-CIRCUIT NOISE FI GURE

It will be assumed in what follows that the receiversoperate at low radio frequencies. Under this conditionthe input conductance of the first vacuum tube is smallcompared to the circuit conductance and hence can betaken as a circuit characteristic. As a result, the noiseof the first vacuum tube, which is very important, maybe treated as a correction to the noise factor of the restof the input circuit. The method of making such a cor-rection will be discussed below, but first the operationof the input circuit without the vacuum tube connectedmust be considered.To describe further the operating characteristics of

a loop-antenna input circuit, it is necessary to developwhat is called the inherent thermal noise of a receivingantenna. Consider a loop or any other type of receivingantenna which is exposed to a radio-frequency field ofstrength E in volts per meter. Such an arrangementwill give rise to an open-circuit terminal voltage.

e = Ehkff volts (6)where h6ff is the effective height of the antenna. Thisvoltage e will be due to the received signal. In addition,there will also be a noise voltage present at the termin-als. The series-resistance term in the antenna imped-ance as measured at the open-circuited terminals is thefactor which determines this noise voltage. This is trueirrespective of what fraction of this resistance is at-tributable to radiation and what fraction depends uponthe ohmic resistance of the conductors. If this resistivecomponent of the antenna impedance is R, then atroom temperature the open-circuit terminal noise volt-age is given by

e. = 1.26 X 10-10 x/RVIA volts. (7)

May520

Browder and Young: Loop-Antenna Input-Circuit Design

In case the radiation resistance of the antenna pre-sents greater than the normal noise voltage, the excessnoise can be considered to be caused by an externalnoise field, and will not be considered in this analysis.

Clearly, if one could use perfect input circuits, a sig-nal of strength equal to that of the noise could just bereceived. This occurs when e = e, or when

1.26 X 10-10vR\1fheff

(1)

With practical input circufits, E has to be larger thanthis in order that the signal-to-noise level of the sys-tem be equal to unity. Expressed in decibels, the amountthat E must be raised to yield the desired level with apractical input circuit is called the noise figure of the in-put circuit. Strictly, noise figure is always a negativequantity because it represents a reduction in systemperformance, as compared to a noiseless standard. Inthis paper it always appears as a negative number ofdecibels.

Although only loop receiving antennas are discussedherein, the concept of input-circuit noise figure canequally well be applied to any antenna connection. Thisquantity puts the calculation and measurement of theoperation of an antenna coupling circuit on an absolutebasis. Noise figure is most conveniently expressed indecibels. If it is zero decibels, then reception can occurat the minimum field strength as determined by h1ffand R of the antenna itself, and no further improvementin the coupling circuit can possibly be made. If, how-ever, the noise figure of a gi1en arrangement is minus 10decibels, then considerable improvement can be madeby redesign or improvement of the input circuit.

TRANSFORMER-COUPLED LooP INPUT CIRCUITS

The actual circuit of a transformer-coupled loop in-put circuit is shown in Fig. 3(a). L and R represent theinductance and resistance of the loop. The other L'sand R's specify the constants of the transformer asshown. For the purpose of analysis at the radio fre-quency, the circuit shown in Fig. 3(a) is replaced bythe equivalent form shown in Fig. 3(b). Fig. 3(c) isalso, in a sense, the same as that of Fig. 3(a) or 3(b).This follows from an application of Thevenin's theoremto the circuit of Fig. 3(b). Measurement of open-circuitinductance and resistance as seen at terminals AB willyield the values L2' and R2' and the complete circuitmay be represented as shown in Fig. 3(c).The circuits of Fig. 3 are used to determine the sig-

nal-to-noise ratio of the loop plus the input circuit.When a decibel comparison of this signal-to-noise ratiois made to the signal-to-noise ratio of the loop alone,the noise figure of the input circuit results.When the loop is exposed to a radio-frequency field

and properly oriented, a maximum signal voltage ofstrength e is induced in the antenna. In Fig. 3, this

voltage e may be thought of as being inserted' in serieswith the loop at point P. As long as terminals AB areopen, e will distribute itself only around the one avail-able closed circuit and (assuming the R's are small com-

R RI RIJ- A~~~~~~~

P (a) M

R Rl L1-M L2-M Rt A

L M

p (b)

Re_ ~~~A

OB(c)

Fig. 3-Actual and equivalent loop-antennatransformer-coupling circuits.

pared to the X's) the fraction of e which will appearacross the inductance M of Fig. 3(b) will be given by

MeeL=

l + Li (8)

Furthermore, as long as no load is placed on terminalsAB, this is the voltage which will be observed at thoseterminals. Replacing M with kVL1L2, where k is thecoupling coefficient of the transformer, the voltage atAB may also be written as

e. = 6e.L + L1 (9)

The impedance at terminals AB, called ZAB, may becomputed from Fig. 3(b). It is

ZAB=R2s+ j[w(L2-M) ]

(j+wM) [R + Ri + jw(L +Li - M)]R + R + jw(L + Li)

(10)

This expression may be simplified because the R's arenormally small compared to the (wL)'s. It is specificallyassumed that (R+R1)2 is very small in comparison to[co(L+Li)]2. This is equivalent to saying that the Q2'smust be large. In fact, for Q's as small as 10, the valueof Q2 is 100, and so the error of the approximation isequal to only 1 per'cent. With this approximationand with the substitution of kV\L1L2 for M, ZAB maybe rewritten as

ZAB R2+ (R±Ri)kL1L2] +jw L2- (11(L+L1)2 J L L+Lj

1947 521

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

We may calculate the thermal noise level at terminalsAB from the resistive component of this impedance inthe usual way. Denoting the noise voltage as en, thenthe signal-to-noise ratio at terminals AB is e8/e,. Writ-ten out completly, the expression is

Signal-to-noise ratio at terminals AB

eL+Ll

1.26 X 10-1A/R2 + (R +RV)k2L1LjfV ~~(L +L1)2

This last equation gives the variation and values ofthe transformer-coupling noise figure with any values orchanges of the parameters of the input circuit. In Fig. 4,

5E-j

(12) cl

From this expression and from the signal-to-noiseratio of the loop terminals as obtained before, the noisefigure of the coupling circuit is obtained by

Noise figure of coupling circuit

Signal-to-noise at AB= 20 log decibels.

Signal-to-noise in loop

Making the appropriate substitution yields

Noise figure of transformer coupling

= 20 logk

R1 R2(L + L1)2R RL,L2

(13)

(14)

Fig. 4-Noise-figure variation with Lo/L.(1) Qa=Ql Q=Q k=0.6(2) Q2=Q1 Qi=Q k=1.0(3) Q2=2Q0 Q1=2Q k=0.6(4) Q2=2Q, Q =2Q k=1.0(5) Q2=2Q1 Q1=3Q k=1.0(6) Q2= 2QQi = lOQ k=1.0

Since it is more convenient to work with Q's than withR's, the appropriate coL/Q is next substituted for eachR. The expression then becomes

Noise figure of transformer coupling

=2O0lkog2Q k2Q Li Q L L1=2Olo~O~~ Q2 Ql L QAL Li)

Fig. 5-Noise-figure variation with Ql.(1) Li/L=0.75 Q2=Q k=0.6(2) L1/L=0.75 Q2=Q k=1.0(3) Lu/L=0.75 Q2=2Q k=1.0(4) Ll/L=0.75 Q2=1OQ k=1.0

Q2/QFig. 6-Noise-figure variation with Q2.

(1) L1/L=O.'75 QI=Q k=0.6(2) Lt/L=0.75 QI=Q k=1.0(3) L1/L=O. 75 Qi=2Q k=1.0(4) Lu/L = 0. 75 Qi= lOQ k=1.0

K

Fig. 7-Noise-figure variation with k.(1) L1/L=O.75 Q2aQi Q1=Q(2) -L1/L=0.75 Qa2=2 Q =2Q(3) L1/L=0.75 Q2=2 Q = OQ

522 MIay

Browder and Young: Loop-Antenna Input-Circuit Design

this equation is plotted so as to show the variation ofnoise figure with L1/L for various values of the Q

ratios. Since all of these curves show optimum operationfor L1/L equal to about 0.75, this value of L1/L ischosen, and in Figs. 5, 6, and 7 it is used in showingvariation with the various Q ratios.

EQUIVALENT NOISE OF INPUT TUBE

In the actual loop-coupling arrangement, two ele-ments are connected across the terminals AB of Fig. 3.One is a capacitor which is tuned to resonate the loop,and the second is the grid of the first vacuum tube. Thecapacitor does not affect the noise figure which has beencomputed, and the tube adds a noise correction whichmay be evaluated.To show that the capacitor does not affect the noise

figure of the loop, refer to Fig. 3(c) and imagine that a

source e is connected at point P' and a resonating ca-

pacitor connected across terminals AB. The resonantcurrent may be computed so that the drop across thereactance, which is the terminal voltage at AB, becomes

CwL2'e. = R, e. (16)

To-obtain the signal-to-noise ratio under this condition,this quantity must be divided by 1.26X 101-R4',where RTeg is the resistance component of the resonantimpedance as measured at AB. It can easily be shownthat Rre-= (L2'cL)2/R2'. Thus the signal-to-noise- ratiocan be written as

Signal-to-noise ratio

to be to the left of the terminals AB. The tube noise isrepresented by the equivalent resistant Req connectedin series with grid. Aside from the noise it generates,this resistor has no circuit significance as long as thegrid is assumed to draw little or no current.The effect of R,,q expressed in the same terms as were

used for the input circuit proper, yields

Noise figure correction due to tubeReq

= -lO0logI 1+-IRree(18)

Values of this quantity may be obtained from Fig. 9.This measure of noise increase due to the first vacuumtube must be added to the noise figure of the input cir-cuit to obtain the complete noise figure for the over-allperformance of the system.

-J

of

mx -5 __1- I _ _ I

8 -10I-l-o-

W2_1 _ __ _ __- _

0

0

Li.-25

LiJ(I)

Q n

.a1 .05 01 0.5 1.0 5 10 50REQ/RRES

Fig. 9-Noise-figure correction variation with ReQ/RrU..

100

1 e= 20 log - decibel.

A/21.26 X 10-10 /\If(17)

Likewise, with no capacitor connected at terminals AB,

e= e

dividing noise io10-\7RV2,gives the same value of the signal-to-noise ratio. Theseresults also show that resonating a transformer coup-

ling does not affect the noise figure of that coupling.

D

Fig. 8-Equivalent noise resistance of the input tube.

To evaluate the noise correction due to the presence

of the tube, refer to Fig. 8. In this figure is shown theantenna connection to the first vacuum tube, and alsothe capacitor used for resonating the loop. The loop, or

the loop and the coupling transformer, are considered

EFFECT OF LooP CABLE

When the length of the cable connecting the loop tothe transformer is at all significant, its effect on theperformance of the system must be taken into ac-

count. If L' is defined as the inductance of the loop andconnecting cable as seen at the primary terminals ofthe transformer, and R' is the resistance as measuredat this same point, then in the formula which has beengiven for noise figure, L' can be substituted for L, andR' can be substituted for R.Two-cases are of interest: (1) when the connecting

cable has appreciable electrical length; and (2) when thefrequency and the length of the cable which is used is

such that the cable length is very short in comparison

to a quarter-wavelength.For case (1), the most important parameter of the

connecting cable is its characteristic impedance Zo. Inthis case the cable is comparatively long and is beingused as a transmission line. The values of L' and R'can be calculated from the equation

Zo sinh I + (R + jwL) cosh I1=0Zo coshI+ (R+jwL) sinhl

FS1]i -i-'s ii [lll

5231947

Lc

L

ILi

c

IccLc

z -au

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

where I is the electrical length of the transmission linemeasured as an angle.

Case (2), in which the cable is relatively short incomparison to a quarter wave, covers most of the casesgenerally encountered. Now the cable can be con-sidered to represent lumped-constant elements in thecircuit. Along with the capacitance of the cable, which isusually the most important factor, may well be in-cluded the distributed capacitance occurring at theloop and transformer ends of the cable.The values of L' and R' for this lumped-constant case

are obtained from the expression

R' + jwoL' = C1

R + jwL-j-IC

(20)

where C is the capacitance just referred to. If C is smallenough so that

-1 )2t- WL

is very large compared to RI, this expression may besimplified to yield

RRI'=(1 - wLwC)2 (21)

and

taken not to use this approximation where extreme ac-curacy is expected.

10

.l .2 .3 .4 .5 .6i'.1.8.910XLXc

Fig. 10-Change of loop parameters with shunt capacitance.(1) Change of reactance with cable capacitance.(2) Change of resistance with cable capacitance.

SYSTEM PERFORMANCE CHANGE WITH LooP Q

If a loop antenna is coupled to a receiver, and ifthe Q of the loop is varied while all other circuit param-eters are held constant, two factors will change. As isshown in Fig. 2, both the merit of the loop itself andthe noise figure of the coupling circuit will be changed.

L 1 \ 2

C -WiC , cL' = (

1(2- WL

WcC

(22)

Frequently R and C are small enough so that

R2 L<<

,C W, cC )

and in this case the expression for L' may be furthersimplified to

L. L =

1 - wcoC(23)

These formulas are of particular importance in cal-culating the apparent inductance as seen at the primaryterminals of the coupling transformer. Graphical dataconcerning their solution are given in Fig. 10.

In those cases where the cable inductance is of im-portance in comparison to the loop inductance, it isusually adequate to add these two inductances and as-

sume that the capacitances are in shunt with the totalinductance. This is an approximation, and care must be

Q/QIFig. 11-Relative change of system performance with change of Q.

(1) Li/L=0.75 Q2=Q1 k=1.0(2) Iu/L=0.75 Q2=2Q1 k=1.0(3) L1/LL=0.75 Q2=10QQ k=1.0

From the computations which have been made, thesetwo decibel measurements of operation can be addedso as to yield a measure of over-all system performanceas a function of Q. This has been done, and under therestrictive conditions already mentioned, Fig. 11 showsthe result in graphical form.The curve of Fig. 11 increases steadily with increased

Q. The practical limitation on Q depends upon

524 May

Browder and Young: Loop-Antenna Input-Circuit Design

bandwidth. Just how this dependence occurs is dis-cussed below.

APPARENT Q OF THE TRANSFORMER SECONDARY

The apparent Q, the ratio of the reactance to theresistance, of the loop and coupling arrangement asshown at terminals AB of Fig. 3 is the quantity thatdetermines the antenna bandwidth. We shall call thisquantity Q2'. Q2' also determines the manufacturingtolerances which can be allowed for the loop induct-ance, transformer inductances, and tuning-capacitorcapacitance.

Here Q2' is actually given as a ratio to Q2. This has beendone only for convenience in the graphical representa-tion of Fig. 12.

APPARENT INDUCTANCE OF THE TRANSFORMERSECONDARY

In order to design t,he loop-coupling transformer sothat it will track with the other coils in the receiver, itis necessary to know the inductance of the secondarywith the loop connected to the primary of the trans-former. An expression for L2' was obtained as theimaginary part of (11). For convenience of graphicalrepresentation, this expression may be rewritten as

Fig. 12-Apparent Q of the(1) L1/L O . 75(2) L1/L=O. 75(3) LI/L 0.75

Q2/QI

secondary of the coupling transformer.Q=1OQ k=1.0Q@ = 2Q k=1.OQI=Q k=1.0

L2' Li-=1-k2L2 LL+L1

Fig. 13 shows graphical values of this expression.

I.A1 _-=.6 - r - -

It I [1 11 7N', II I I lIt I I

'_)1 .05 01

2L 31 11 x 1I --IFig.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~12ApaetQoh

C-L A -- .- - - -

(26)

Q5 1.0 5O 10 50 DO

To calculate Q2' we must evaluate wL2' and R2'.From Fig. 3(b), ZAB has already been found, and sinceonly inductances are involved, R2' and wL2' can be im-mediately taken from the real and imaginary parts ofthat expression. Thus

M2L2-I

RH- (R + R1)M2 (24)(L +L)2

Substituting Q's for R's, this may be rewritten as

1-k2 ( LAQ2' \

_ 3- * ~~~~~~~~~(25)Q2 Q2 L L( Q2 L )2

Q (L+Ll) (L+L1) Q1 L+L1

Fig. 13-Apparent inductance of the secondary of the coupling trans-former.

(1) k=1(2) k=0.9(3) k=0.8

CONCLUSION

The concept of the noise figure of an input circuit atcomparatively low radio frequencies is a powerful aidin the analysis and improvement of input circuits.Formulas and charts which have been given cover theloop-coupling-transformer input circuit. Further designdevelopment of these transformers will proceed morerapidly if effort is expended on those parameters whichtheoretically will give the greatest benefits with theleast change of the circuit elements. Unquestionably,there is much to be gained over those designs whichhave been so generally used in the past. The formulasand charts which have been given can be used directlyfor that purpose.

525

L'..

LtI