AD-ACAB 771 NAVAL RESEARCH LAB WASHINGTON DC F/S 20/3INTRINSIC SOURCES OF IM GENERATION U)JUL 80 B H STAUSS

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1. SUPPLEMENTARY NOTES

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Satellite communications Connector designIntermodulation interference Ferromagnetic materialsNonlinear conduction Nonlinear circuit analysisTunneling junctions Multiplex systems

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INTRINSIC SOURCES OF IM GENERATION.)-" ') Geoge U.Stauss

Magnetism BranchElectronics Technology Division -

In addition to avoidable sources of intermodulation signals introduced in ianufacture or assemblyof a multiplex system, there will be some sources inherent to the materials of which the system is con-structed. At some point these sources must provide a lower bound to possibe system sensitivity. Sucha limitation will only be significant if it is reached before thermal noise be/omes dominant. The thermmal noise power is PN = kTAY W for a frequency bandwidth-Ar. At 20 C and a bandwidth of 2500Hz, the noise power becomes -140 dBm. It is potentially possible to lower either the temperature orthe bandwidth by a factor of about 100, so an ultimate sensitivity limit of about -180 dBm is perhapssignificant. Above these levels, a number of intrinsic IM mechanisms can be identified. This chapterconstitutes a review of significant intrinsic IM sources.

1. RESISTIVE HEATING IN NON-MAGNETIC CONDUCTORS

This problem has been discussed in several places including Philco' and TRW2 studies. This sec-tion is aimed at unifying the discussions and eliminating apparent discrepancies in the conclusions. It isconcluded that this effect can be significant at high power levels at all frequencies.

The calculation which follows is done classically (no quantum effects; cf. Chapter 1) and to first

order. It is similar to that of the Philco study, but with fewer approximations. The TRW calculations,done differently and specifically for a waveguide configuration, reach substantially the same conclusions.Two primary frequencies oi1 and W2 ((01 > (02) are assumed present with electric fields of amplitudes Eland E2 = /3E I parallel to the metallic surface, and the normally strongest 3rd order IM signal at(2w, - ( 2) is sought. The surface is taken as part of an infinite plane and o 1 ,cu2 , (2wu, -W02) areassumed close enough in frequency that a single skin depth 80_ 1 ino temso smol/nh

appended list can be used for all.

To lowest order, the density of primary power being dissipated at a depth z below the surface isthen

piN(z)= L1 e 80 (1 + P2) + cos 2,1it - + p2 cos 20)2t - (1)2p0 08

+8 23Cos IWi 3) + 2p3 cos (W1 -W2

where P0 is the material resistivity at ambient temperature. [For copper P0 = 1.72 x I0 - 0 m at roomtemperature]. This power dissipation produces resistive heating and changes the local value of theresistivity. Heating can in principle also lead to local dimensional changes which could produce inter-modulation. However, with temperature changes expected to be <10-6 K even at the surface, a linear

65

~~n W c 4 L ~ . . -.. .- .. . - - . . . ./ "--

G. H. STAUSS

thermal coefficient of 1075 /K, weak dependence of the propagated signal on small dimensional changes,and considerable bulk thermal inertia, this effect is considered negligible. From (1), one finds that thetotal power absorbed at all depths per unit surface area, averaged in time, is

80E I (I + #I2)PIN = 1 4 (2)4P0

To find the change in local resistivity, one uses the fact that p is essentially proportional to,bsuime ,ermperature and then looks for the temperature distribution. This is obtained from thediffusion equation

- 2T(z) 8T(z)G -Z 2 + Ch -at -PIN(Z). (3)

Here G is the thermal conductivity and Ch the heat capacity. [For copper at room temperature, G =4.2 x 102 W/Km, Ch = 3.44 x 106J/Km3 ]. Since Eq. (3) is linear in T(z), the solution will be a3uperposition of solutions for the various terms in p(z) of Eq. (1). We neglect the steady state term;

for the others, boundary conditions are taken to be 6 T_ = 0 at z - 0 (no heat flow into air space)az

mu T(z).- - 0 (the thickness of ordinary components is sufficient to be considered infinite).

In solving for T(z), one will obtain contributions corresponding to all the different frequency-:endent power terms. When these are converted to changes in resistivity and combined with the

i. grals, they will yield currents at several frequencies. However, the desired (2w, - w2) currentiy arises either from modulation of the w input by an (W1 - 2) resistivity term, or from modulationthe w2 input by a 2w, term. Furthermore, evaluation of the current amplitudes shows that modula-

.. , by the (COI - W2 ) resistivity term is dominant. Hence, only this part will be discussed in what.,,, From the solution for T(z) we then obtain

T -el7 "oco 0(w cos) 4

Here W2 -- (c. '°2)C,, and tan4,---- W28°22G2

This resistivity will modulate the local currents produced by the primary input at ow1 . Writing thek'n effect in differential form, the current density at z, J(z), is found from

! dJ(z) (1+i) (1+) _ (1+1i) t - T(z) (5)

For an input at w1n one obtains

J'o) - -e 80 cos (,, -+ cos - -o + l (6)

HereT~o W2 (C 2) Ch and tan

IP0 W 0 + 0 -f-Q 0 4 T

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' " " . ... . .. " - ... " ... ' " :-: ' ".-./ - : - ; . -- .i£, .. ., -' i: (. .

NRL MEMORANDUM REPORT 4233

When the quantities T(0) and f,T(z)dz are evaluated and substituted into (6), the IM current densitywith frequency 2w, - W2 is founc[ to be

~~~lE? E,8J 1 4-/2-oJIM (Z) - P E (4 + X4V' 12 e x8 X

P0 Po 2GTo

- .L + --L cos [(2w I,- W2) - + ++ - +

sin (2W 1-(2)t- -+L + (7

cos (2, - s 2 - 0)t ++ -I + -

The dimensionless parameter x 80 W is of order of magnitude unity here, but is material dependent.

In treating an unperturbed input signal, one finds that PN, the power absorbed per unit surface

area, is related to the total current density in the same element, J -f 0 J(z)dz, by way of the surface

resistance R, T; i.e.,

- fR. (8)

We will apply this same expression to find the power lost by JIM fo~Jm(z)t in the same volume.

Interating Eq. (7), we obtainPE? I Eo IPo Poj cos 1(2,, - )t,0 + 1 + 2x x (9)

1 -2GTt(4 + -%4)2)4 + 4 0 +( 1+ x)

ji 14 113 00x)115) x

+ oil

Hence from 1) anti 190 1he totai I M power absorbed per unit area becomes

1'" Io 1 3 9X - x 5 (2o)1G2 T(4 + x4) 20 1Ox 2(x + 1)2 (

or, uiing Eq. (2).

Pl " (1 ;'82)3 G2T3 f(X),

where fx)= ! 1 13 (9x 2- x- 5)(2x + 1)(4 x4 ) 2 0 Ox 2(x + 1)2

67

G. H. STAUSS

Reasonable values for x in our appioximation that (O, - w2) << w, are in the range x - I to 5 forT(0) -rzT(z)azcopper. In order to satisfy the assumption of first order perturbation, - and in Eq. (6)Jo~ 0o n0q.(6

must be much less than 1. This requires that x > 0 which here roughly means x> 10- .

Hence, the infinity in f(x) at x. = 0 is meaningless. For reasonable x values, f(x) is controlled mainly

by the 4 factor. Explicit values are

x 1 2 3 4 5f(x) .085 .012 .0036 .0013 .0006

The reason for obtaining Eq. (11) is that the IM power received at the output load can beexpressed in terms of this quantity. We assume that we are dealing with a well designed circuit withmatched input and load. Then we can represent a unit IM source included in a linear system by theequivalent circuit illustrated.

RL RRS

From this one IM source the power received at the load lML I M R$ (2 RL + Rs) 2 . But the input

power lost in the unit IM source is + Rs)2 while that available to the load is -_; the ratio of2R4RsRL L 4 RL

these gives (2RL + RS) 2' Hence we can write, using (10) or (8) and (2)- PiNPM

PML m (12)I.

4 PN

for the delivered IM power from a unit IM source element. If the assumption is made that all IM sig-

nals arrive in phase at the load, the total PIML - (Area) 2PtML. This then gives the result that

4IML & (13)4IPIN

Using Eq. (1 1), we obtain finally

PIWL -I o 2 8 -2 (piN .', f(x)(1 + P2 2GTo I I (14)

As numerical examples, we consider two cases. (i) A high-Q UHF structure with high power den-sity in a very restricted area with P'I - 270 MHz, V'2 - 245 MHz, P - 1, ,N - 4.5 x 103 W/m2, I- 100 W, PIN o,, - 20 W, x - 3.2, f(x) - .0028. This leads to PIlL - .71 x 10- 7 W or -142 dBm.(ii) The x-band waveguide case considered in the Philco report, with ',1 - 8050 MHz, V,2 - 7900 MHz,

- 0.1, p,- 145 W/m 2 , PIN _ 1000 W, PINl t - 45 W, x 1.45, f(x) - .031. This givesPIML _ 1.1 X 10-22 W or -190 dBm. Correction of a numerical error brings the Philco value to -232dBm; the remaining difference is associated with the assumption in that report that modulation of the

68

NRL MEMORANDUM REPORT 4233

input at W2 by the thermal effects at 2(3 is dominart, whereas we find modulation of the input at w, by(W - w2) thermal terms produces an effect about 40 dB stronger. Both of these examples indicate thatthis mechanism is capable, under certain conditions, of producing significant IM signals.

A particularly important non-intrinsic IM source which has been identified is the pressure-closedjunction. Because of oxidation and surface roughness this junction contains insulating, semiconducting,and metallic regions in dimensions small enough to permit tunneling. However, consideration of therestricted area of metallic contact within the skin layer and across a gap of perhaps 100 A suggests thatthe junction would be an important IM source simply through resistive heating. For example, in thewaveguide discussed above, if a flange junction were introduced in which I W is dissipated orPN 2 x 10 W/m 2, then PIML would be increased 110 dB to a significant level of -122 dBm. It is tobe expected that increasing contact pressure would decrease this contribution to junction IM production.

To see the important dependences more clearly, we will write PIN - PIN A where A is theeffective conductor area; -L is generalized to ob in the expression p - po( + ob T);

PIN - QPINRS - QPINPO180, where Q is descriptive of the element in question and R, is the surfaceresistivity of the conductor; f(x) =-x - 3 quite well in the range I < x < 5 and[ -,2(.r I /X 1 G .• Thus Eq. (14) lead to

2GL ~ G 1326 PP QP (ML + 2)3 w) I _13" 2 (15 )4

P2 A22Q 4 112-cc(2 PNP080 1 - (15)

#2 " 2 p GI2 4p1315/

(1 +32)3 G/ 2/,2 (132wg1 - 2) / 2

Thus IM production due to resistive heating is enhanced by:

(a) Large power inputs,

(b) 13 values near I (equal power in both carriers),

(c) Large-area elements,9 -3

(d) High-Q elements,

(e) High carrier frequencies, .. .

(f) Small carrier separations,

• (g) Conductors with high resistivity, low heat capacity, low thermal conductivity, and high-thermal coefficient of resitivity qo.

(it might also be noted that Eq. (14) is, to the level of approximations made, in functional and quanti-tative agreement with the calculations by TRW, after correction of a numerical error. The TRW result,obtained in the manner of Section VI (dielectrics), develops on the fact that for long signal paths andsufficient attenuation the value of PtuL will saturate and then decline for longer paths. For most appli-cations the critical length, where the primary signals are attenuated 4.8 dB, will be quite long.)

69

G. H. STAUSS

2. MAGNETORESISTIVE GENERATION IN NON-MAGNETIC CONDUCTORS

Magnetic fields applied to a conductor have the effect of altering its resistivity. Hence those fieldsassociated with currents in the material can potentially create IM signals. An approach similar to thatused in Part 1. can be taken. We will evaluate the variations produced in the local resistivity by the pri-mary currents and then find Jm. This effect is found to be small.

First consider a simple numerical result. With 80 _ 10-6m typical of copper, a dissipationP- 103 W/m 2 using (8) leads Jo J 240 A rmts as the current through a 1 m width of skin layer.

eBut in MKS units wedlwrite •- L01 where I is the current through the loop around which

is integrated. Taking the loop as a rectangle transverse to the current, I m wide and several 8 deep,with its upper side at the surface of the conductor, the contributions of the short ends to the integralcancel, and B on the lower long side is essentially zero. Hence B on the upper surface becomes roughlyB - 2 4

0g o Tesla, which is equivalent to about 3 Gauss. But for the transverse magnetoresistive effect(B perpendicular to current flow as it will be for self-produced fields) we know3 for copper that

P - P0 - 3.98 x 10- 17 H 2 in MKS units. Thus resistance changes of a part in 1012 are possible in ourP0

example.

Given p(z) = PO(1 + crH 2 (z)), (16)

which is quite general since a linear contribution does not produce a 3rd order IM of interest anyway,

we need to know H(z). Using the same path integral formulation used in the example above, if we

take H(z)z- = 0,

H (z) = fz JIN (z)z. (17)

For inputs at two frequencies, we can write the primary current, allowing for skin effect,

JIN(z) = R (ew"+ + +eb 2. o(18+ " (18)

1 P0 8oThis then leads to H(z) and to p(z) through (16),

p(z) = Po + + 2) + sinl2wi1 - -L + 2 CS1. (192)t

if Eq. (19) is combined with initial input signals, various frequency terms again result. The only parts

of p(z) which lead to IM products at (2 t - W'2) are

p(z) - E0 e 80 sinoit - L0 + 2/3 cos ((0 - W2) • (20)

Unlike the resistive heating case, here both terms in the brackets remain significant. When (20) is

combined with the input signals, we obtain a form similar to Eq. (6)

J(z) - ReI-(e I + (e")) - -P0 1 + o o p(z) - P0 - , (21)P0 P0 28 0 P0

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NRL MEMORANDUM REPORT 4233

The results analogous to Eqs. (7), (9), (10) and (1) then become

oE8je 7 z o_____)- 16 - + -e jCos- + Il+e-T° Sin- CO z

e sin 8 + i- e - cos sin (2(2w - 2)t (22)16o 11 2808

-513,E?80d= si cos(2J 1 - e I2)o + ssin (2w - 2)t], (23)

and thus 1_!38 I I# O1 1 2_3801~ ~ -1 25 2L... 'V8 ~~J*80 ?I FPN PM (24)16I PO I 4po T (I + #2)3 p 2J

Comparing this with Eq. (11) and (13) we find that for equal input power and input losses

PmL (magnutoresistive) PIm (magnetoresistive) 25 [E 1_!± G2 T0PIML (thermoresistive) - PIM (thermoresistive) --16 p2 f-x) (25)

Thus the same functional behavior exists as for Eq. (14), except that the result is not sensitive to rea-sonable changes in the sideband position, signified by the absence of the x dependence in (24). Nu-merically for copper the ratio (25) takes the value 1.34 x 10 7/f(x) and for the two cases considered inPart 1. with f(x) - .0028, or f(x) = .031, the magnetoresistive IM power is found to be down fromthe thermoresistive values by 43 dB and 54 dB, respectively. For non-magnetic materials considerationof reasonable values of f(x) indicates that the magnetoresistive effects are insignificant.

3. DIRECT VARIATION IN RESISTIVITY WITH CURRENT

If p(z)- po(l + fjV(z)2) where JIN(z) represents the primary current density. then theappropriate part of JN(z) 2 leading to the 3rd order intermodulation signal can be extracted from (1) as

JIN(z) - E 12 e 80 Cos 2wot - L + 2P cos (WI - W 2) (26)

Then writing

I dJ(z) (I + i) ( + i) (1 + i) (I - fJN(z)2/2). (27)J(z) dz 8o(p/po)" 2 80(l + fJIN(Z 1)/2 80

we obtain

J(z) -ReIJ(O) expI + 111+ 2o(i+) fojNZ)2 1 (28)

With J(O) - ... (e0" + Pe',2,) (j j-jN(O) 2) ' to lowest order one getsP0

71

G. H. STAUSS

Jl~i(z) - 1-__ 15 sin

2:) 1 2 , 8 8CS(W 0)

e il o cos L + le'72L sin 'sin .w- (29)118 8 o 2 80 80 W2)1 .

Integrated over z, then this leads to

JIM - EllE80 [3 cos (2W1 - (12)1 + sin (2Qw - W2)t]. (30)

Thus

Lo1 l j23= 01"--o =I~ l0) 0U S- _I 2 f 2 5 #2 1 1JR=80 IS 8

-__ 3 N3 (31)'-6o 0J 256 =4 802pj 2 4po 0 3)121PN(1

which can be compared with (1) or (24).

This effect is hard to evaluate numerically. Any experimental determination of the coefficient fwould have to be separated from the magnetoresistive effect dissussed in Part 2. Although theinfluence of skin depth is different in (23) and (31), one might anticipate IM contributions of similarmagnitudes in the two cases.

4. RESISTIVE HEATING AND MAGNETORESISTIVE IMIN FERROMAGNETIC COMPONENTS

When components are composed of ferromagnetic materials, the kinds of components chosen inthe previous numerical cases become so lossy as to be impossible for consideration. For a more suit-able component we chose a 10 cm long coaxial line similar to RG-19 with Z0 - 50f0, ID - .435 cm,OD - 1 cm. For such a line, standard expressions give attenuation - 8.68c. dB/m, where a, - 1.05R, (cf. Eq. (8)). The frequencies will be chosen as P, - 270 MHz, v2 - 245 MHz. Thus with copperelements in this line there will be a loss of .00376 dB or 9 x 10 - 4 of input power. If nickel componentsinstead are used, the corresponding loss will change with skin depth depending on the proper value ofdBthe permeability gi. This will be taken as the average small signal value of - and will be presumed

dHconstant here (but see Part 5.). If u is assumed to be 100 I-o or 500 As0 the fractional loss of inputpower becomes .019 or .043, respectively. Further assuming 3 - 1 and a total input power of 60W, wecan then evaluate the IM products for copper or for nickel elements. In this part and in Part 5., thepermeabilities used are those suitable to low frequencies. They are still roughly appropriate in UHFsystems, but may decrease by an order of magnitude on going to microwave frequencies.4

The coaxial component will be treated in an average fashion, with equal power densities on bothconductors. One can alternatively distinguish the two, assuming equal total currents on both. In thislatter case, the current densities and the total powers absorbed vary inversely as the radii. This impliesthat the inner conductor is by far the more significant IM source. But since the ratios are known, onecan relate the separated conductor results to the average case with the same total absorption. We findfor q - 2- that Alm from the outer conductor alone is 1/q 2(l + q) 2 times the average calculated, while

that from the inner conductor is q4/(I + q) 2 times the average. With the coax dimensions chosen, thismeans that the resultant PmL from the inner conductor is 4 dB above that found from the average,while the outer conductor gives a result which is down 18 dB and negligible in comparison.

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NRL MEMORANDUM REPORT 4233

If we evaluate PML from Eq. (14), averaged as above, we find for copper, since PIN - 12 W/m 2

and PN lost = 0.054 W, that PmL - .62 x 10- 27W or only -242 dBm due to resistive heating. Magne-toresistive effects are still 43 dB below this, since the frequencies and thus f(x) were chosen as for thefirst example in Part 1.

But going now to nickel elements we must alter the material parameters: po(Ni) = 4 .5po(Cu),

G(Ni) - -G(Cu), C1,(Ni) - 1.15C,(Cu), and we will look at both j = 50OI0O and A - 100A0.6.5

We then obtain

A - 50(0o A - 1001A0

PIN jos = 2.58 W PINlot f= 1.14 WR, = 47.5 R,(Cu) R= 21 R,(Cu)1 180- - 80(Cu) 8o= -L 80(Cu)

x = 0.26x(Cu) x - 0.58x(Cu)f(x) - 0.136 f(x) = 0.0153PML - 0.59 x 10- 19 W or -162 dBm PUL = 0.25 x 10- 21 W or -186 dBm

The IM levels here are for resistive heating effects. Although they are still at low levels, they are con-siderably stronger than the corresponding copper value of -242 dBm.

To evaluate the magnetoresistive contribution, we need a value for the coefficient o in Eq. (16).Material is available in a discussion5 by Jan which shows that below about half of saturation magnetiza-

tion, - both transverse and longitudinal fields give at proportional to M2. Numerically, Jan'sti 2 P0

curves for nickel in transverse fields sho_ -4 x 1072 . Generalizing this somewhat, if aP0IA

strong transverse field H0 is applied, an expansion about H0 gives

Ap(H)-Ap(H0) = b 2M(HO) -L(H) (H - H0 ) + (32)P0 d

I [ (HOj + M(Ho) d (Ho) (H- Ho)2

= 4 x 10-2where b , . Only the quadratic term will contribute to the 3rd order IM. (For M near

Ap is experimentally quite linear in H, so saturation must drastically reduce IM production.)P0

Within the coefficient of the quadratic term, the first part is positive; the second will be positive andaugment the first at the lower end of the magnetization curve, but must eventually become nealive

and weaken the first. To simplify, we will take the first coefficient alone, giving r - M--M (H )11 2

b b I i. Since M -4.85 x 105A/m we obtain

JA - 50040 A = 100lo

or - -4.24 x 10- s (A/M) - 2 - -1.69 x 1079 (A/m) - 2

PIUL (magnetoresistive)____________i__ 1.3 x 109 or +91 dB - 1.8 x 107 or +72 dBPIAIL (thermoresistive)

Magnetoresistive PIML - -71 dBm - -114 dBr

73

G. H. STAUSS

Thus for magnetic materials, magnetoresistive effects are extremely strong sources of IM signals.Despite this, however, we find that an even more important source exists in the direct variation of thepermeability with current. This is discussed in the next part.

It should be emphasized that these results are only approximate. The value of 1 is a continuouslyvarying function of field strength and frequency and the skin depth is thus itself a function of depth.Also, especially with external fields applied, there are directional variations in effective permeabilitysince the ability of a small field to change the magnitude of magnetization is much greater for fieldsparallel rather than perpendicular to existing magnetization. For example, a solenoidal field parallel tothe coaxial component is parallel to the currents in the conductors and normal to the fields which theyproduce, and so should decrease the effective permeability and consequent IM production moreeffectively than a transverse applied field. Furthermore, shape effects on the demagnetizing field makethe solenoidal field much more effective in saturating the magnetic components, requiring only about10-2 T as opposed to several tenths Tesla for a transverse applied field.

5. IM PRODUCTION DUE TO VARIATIONS IN PERMEABILITYIN FERROMAGNETIC COMPONENTS

To illustrate the significance of this effect, ignoring other contributions including imaginary/j", we

allow the permeability to vary in the form

IAz) -,, (1 + DH(z) 2) (33)

since linear variations will not lead to 3rd order IM's. We are interested here in fields produced by thecurrents in the conductors. From the primary currents, the parts of H 2(z) which contribute to the 3rdorder IM become (cf Eqs. (16) and (20))

4p e sin [2wt -0 + 2#3 cOs (W t -W12)• (34)

This is to be incorporated into an expression analogous to Eq. (5)

I dJ(z) (1 + i) (1+) 1 + DH2(z) (35)J W) dz 8o( A / (z))'/' 2o

for which the solution, similar to (21), becomes

A~z) = Re - e + - (1+ i)D 2(Z) de so (36)J~)= IP0 280 f

From this we obtain

JIMD(Z) 8=- e - -810 - 1 cos-E? + 2 sin -z cos (2OI - 2) (37)

+ (e- - 1) (sin- -° - 2 cos-I 2)

The integrated form of (37) gives

JM 32p--- [cos (2w, - W2)1 + sin O- 2)11, (38)

74

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NRL MEMORANDUM REPORT 4233

in close correspondence with Eq. (23), as we would expect from the similarities of (16) and (33). (Onepoint of difference arises from the fact that at the surface (z - 0) the present model introduces no IMcurrent whereas magnetoresistivity does.) From Eqs. (37) and (23) we can thus write

Jim (variable A) D PIML (variable A) I D 2 (39)Jim (magnetoresistive) 5a- PIML (magnetoresistive) 5-J

In this formulation, Eq. (33) shows that D = d where A,, represents the average value of

,A"dH2

permeability in the average static field present. Typically we expect A to be an increasing function of Hfor small fields and to be decreasing in fields above a few hundredths Tesla. This suggests that asapplied fields increase 1A will pass through regions of constant slope, which should produce minima inIM power levels. The ratio in (39), using (32), becomes

M (45a dH2 / dHJ d1 2 (0

Using M = - - I H, (40) can be written as

D _ I d2, !.. 1 12 2 1 2 j.' (1OA + 41 1t- _L _ 1 + - - 2 + IJA_ I 1AH2 (41)

5 5bu dH2 Io 1 0 A Mo Ao dH

If we are concerned mainly with small external applied fields, curves for the magnetization of iron 6 canbe used to estimate these values. (Nickel and iron values are similar enough to give comparableresults.) As in Part 4., we use low frequency permeability data. At microwave frequencies we can anti-cipate an order of magnitude decrease ;4. 4 We also assume that the b value of - 04/M 2 is approxi-

mately correct. For iron, at H = 80 A/m (10 e), MAu = 3000 p.0, -2H '1 o anddH dH2

M, = 1.6 x 106A/m. Hence - becomes about 106 which then implies a power ratio of about 40 dB.5o-

This factor is difficult to establish accurately, and is very subject to material and environmentalinfluences. Nonetheless, this contribution to IM signals is obviously so extremely significant that itreinforces the argument that ferromagnetic materials must be entirely excluded from high sensitivitymultiplex circuits.

6. INTERMODULATION DUE TO NON-LINEAR DIELECTRICS

If dielectrics are present in a system, any variation in dielectric properties with applied field willserve to modulate incoming signal voltages in a manner analogous to the modulation of currents in Part1. This effect has been addressed in both the Philco' and TRW 2 studies cited, and again here the aim isto rectify some apparent discrepancies. The approach below is similar in general outline to that of theTRW study. A solution has also been obtained in the manner of Part 1.; it yields results substantiallyequivalent to thosc below when the largely reactive nature of the IM source impedance is taken intoaccount.

Variations in dielectric properties may occur in direct response to electric fields, through heating,or by electrostriction. The general form of the solution is applicable to a variety of dielectric media,such as molecular absorbers or ionizing gases. In each case, of course, the nature of the response toapplied fields must be determined.

We start by writing equations for a generalized transmission line in the formclV 1 L1 i V-y - - - Ri and -y - - Coe-- (42)

8Y at

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G. H. STAUSS

where L, R, and Co (vacuum value) are the inductance, resistance, and capacity per unit length of line.The conductance of the dielectric will be included in a complex relative dielectric constant c. We nowtake

f=C-1(1 +tV 2) - if2(1 +)V 2) (43)

both because quadratic terms are needed for IM production and because the dielectric response isexpected to be insensitive to the sign of the applied field. The relationship between the factors a and ,and corresponding factors suitable for multiplying E2 will be discussed later. Substitution of (43) intoEq. (42) leads to

02 V L 0 02 V V (44)0 2 --- - 0"- = LC --0 -2 'Ed + LCO(aE - i1lE - V2 (

ay a a a t+RCo( - 2V RCo(a - i*)E2) V2 aV

We assume that we can write V = Vo + V1+ ... in descending order of magnitude based on the small-ness of a and -q. Equations for V0 and V1 are then given by

02 Vo L~( 02V 0V°

LC0 (e - if2)- - - RCo(EI - iE 2) = 0 (45)

and

a 2 Vv 2 V1 V 02 1- LCo(eI - if2) 2 - RC(eI - if 2)

| L LCO(a f - iVE 2 ) 1V"ay2a 2 - R8o It at at~a

+ RCo(ae, - i?, 2)V 1 ?( 0°. (46)

The lowest order solution for one signal propagating to positive y can be written in the form

V0 = U Re eIc'*e- y = U Re e' "-"')')e-'v . (47)

Here W2 = wCo(WL - iR)(E1 - if 2) (K - Y)2.

We now assume that there are two primary input signals, so thatVo = U Re [e - e + #eI 22e-2"] (48)

Then Eq. (46) will contain on the right side terms at several different frequencies. We are interested inthe IM at (oiM = 2w, - (02. After evaluating Vd - we find that we can rewrite Eq. (46) as

_V - V1 (49)y2-LCO(e 1 - iE2)--T-- RC0(E, iE2)- 7

S1U3 (LCow 2 - iRCoo)(ae1 - i7jf2)e((t-,Ky)e-1/v

where we want to keep the real part of V1. Here (0 =IM 2c' - 2, K 2K)- K2, and

y = 2y, + Y2. Note that K and y are not in general the propagation constants for a signal at cuIM,although to lowest order K will be. These quantities will be discussed again later.

The solution of Eq. (49) which satisfies the conditions that V, - 0 both at y = 0 and as y - oo

is, to first order in a and ,e-

V" U3(0C4 Re eWIv (WL - IR)(a- -iR)(-(e 0 I 2 ) e-- (50)

4 ~V+ iK) 2 + ojC0&L - AR)(e I - if 2))

76

IMM~~ 'T, -TT M:

NRL MEMORANDUM REPORT 4233fIf the expression for T- (Eq. 47) when wJ = w/M is used to define new quantities KiM and Y/M we canthen write

Vim -" -- R e e" " ({ TIM( af I -i- 12) ) e -iux MY e- 1(2KI-2)y e- (2(

)+.y2)

17r2 - (2K I-K2- 2i3 I- iY21)2) (,EI_-/e 2))

Evaluated at the full length Y of the line, this then lead toPIML - VI2M(Y)IZo • (52)

The characteristic impedance Zo will also be frequency dependent, so that ideal matching cannot occurR R E2

at all frequencies if is significant. It is to be expected, however, that both and - will be

small for real cases of interest in which case Z0 can be taken to sufficient accuracy as L 112

Two cases of special interest are the general case when OIM == -w 2, and the case where -2

and -- are both small, but frequencies are general.oiL

When the IM frequency is very close to both carriers, we can approximate 0 1iM = Wl - (02,

KtM KI = K2, and IM -' =-Y2. From Eq. (51) we then find

VIM- = Re ei '(w -K ''Y) ((KI - iyl)2(aEl - intA2 )} (e-yY - e-h)Y) (3)16 1(23'2 + iK 1y)( 1 - iE2))

The y dependence of the amplitude, all contained in the last parentheses, leads to a saturation effect as

described by TRW, with a maximum at e- 2-fy _ 1 i.e., where PIN is down 4.8 dB. This will generally3'

involve transmission lines significantly longer than 3 m so that the effect would usually not beobserved. A similar saturation effect will occur also in other IM processes.

E2 R

In the case where - 2 and - are both small, the quantities K and y can be writtenEl

K-o w(LCoEI)2 and - [- 2 '+- . (54)

Hence we can take KIM - 2K I - K 2 and yVIM - 2-yI - 'Y2. Taken to lowest order, Eq. (51) leads to

IMKIM asin((0It-KMY) + !2 (a-7) COS(OIM t-KIMY)• (55)

Note that to this approximation, the exponential factors have vanished from the result. If, in fact,E2 - R - 0, so that only the real part of E is changing, Eq. (46) can be solved directly to obtain thefirst term here. It can be seen that generally a will dominate 71 as an IM contributor. Only if a isessentially zero, perhaps as in an absorbing gas, will the'n contribution be significant.

For the two special cases above, we can readily write the IM power at the load, taking

L 7Zo- . From Eq. (53) we obtain

PIML 02U6 (K? + /1 (22i? + I -2v1 ) 2 (56)512Zo y?( + ) (e2 +*2)

77

, ' ,- , -7-:772 '--- 'i_ _ _ _ _ _ _ _ __-_ _ __T...A

G. H. STAUSS

From Eq. (55), the result for small E2 and R becomes2_2_y2

PIML = 02U6 Kla •(57)128Z 0

In the absence of a, a2 in Eq. (57) or (58) below is to be replaced by y12 I-__ In terms of the input

p2w adP the quantity IU3) = 8Z 3 P2 p 2, while the propagation constantpoes Zl'0 adP=2Z 0 0I!

KIM in Eq. (57) can be taken as 21r/AkM. Hence Eq. (57) can also be written

PIML = Vi1.2Z pp 2 a2 . (58)

Here AIM is the IM wavelength in the transmission line. If Y2 is large enough it should be replaced inthe next order of approximation by e-2y' Y (-e- 2y,) 2/4y 2. Then the maximum PIML, whenC2yY- I , is given by Eq. (58) if Y2 is replaced by 1/27y2. On the other hand, a discrete dielectricelement can also be treated using Eq. (58) if Y is taken to be the length of that element.

In order to connect the parameter a with the more significant coefficient a' which expressesdependence on electric field, we can treat the transmission line structure as a capacitor with * depen-dent on E 2 and then find to lowest order the corresponding connection between capacity and V2. Forexample, if the volume occupied by the dielectric does not change, a structure with plane parallel platesaresults simply in a = --- , where d is the spacing of the plates. In the case of coaxial cylindrical plates,

th rlaioshp ecme a 1, (f_2 r 2 )3

th r b a I - r2 1 -. The same relationships, if needed, will con-

nect q and 71'.

As one example, we now apply Eq. (58) to a case considered in the Philco study: moleculalabsorption by 1% water vapor in air, all contained in an x-band waveguide. We use the same conditionsgiven in the Philco study, P = 6 kW, P2 - 60 W, Z0 - 448 fl, Y - 10 m, XlM - 0.0445 m, d - .01m, el - 1, a : 0, dielectric attenuation 2.9x 10-6 dB/m, 10E2 -4 2 x 10- for E = 1.4 x 105 Vim.Treating the waveguide as a parallel plate capacitor leads to -qf _ 10- 14, 62-4.7 x 10- , and toPIML - 1.2 x 10- 7 W or -139 dBm. This value is below the -129 dBm estimated in the Philco study,but is not negligible. With gi'eater power on either input, higher humidity, a longer waveguide, or the15 to 20 dB increase estimated by Philco for more refined analysis of the molecular absorption process,water vapor becomes a significant source of IM signals. (In comparison, the same waveguide, astreated in Part 1, with 33 dB added for increased power and length contributes only -157 dBm throughresistive heating.) This water vapor effect is a result of the molecular rotational absorptions at 22.2GHz and will become more pronounced as that frequency is approached. Below x-band frequencies,this is the only absorptions process of interest in normal air, but at higher frequencies both water vaporand molecular oxygen absorption (at 60 GHz) will result in strong IM generation as well as significantsignal propagation losses.

When a solid dielectric is considered, the mass rather than volume of dielectric is conserved.

Allowance for the change dielectric volume modifies the relationships between a and a' above. These

will then be given by a - at'--2 or a - l Irl 2r- r 2 - in--I lnL2 , respectively, ond 2[ a2 j r? r~l I ril

the assumption that A - Am (X, susceptibility; m, density), as in electrostriction.

Electrostriction is expected to be the principal source of non-linearity in good non-polar dielec-trics.7 The TRW study considers the effect of electrostriction on teflon in a coaxial line. Using the

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• 2i

NRL MEMORANDUM REPORT 4233

conditions of the TRW example for a 300 MHz IM signal, with a'= (0.8 to 25) x10-21 (V/m)- 2, r2 = 2.4 x 1073 m = 3r1 , XIM = 0.7 m, Z0 = 47f0, P, - P2 = 30 W, 4E - 2, the aboverelationship for a combined with Eq. (58) leads to PlL = (.01 to 10) x 10-21 Y2 W. A line I m longwill thus yield PML of -170 to -200 dBm, probably below detectability.

The value of a' is obtained from the expression

Ae = j 2 - 1) 2 (59)Kwhere K is the bulk modulus of compressibility. Equation (59) is given by B6ttcher 7 based on thermo-dynamic arguments. The spread in values reflects the range of K values quoted by the TRW report.

Since K is about as low as possible for teflon, the effect of electrostriction is unlikely to besignificant with other well chosen dielectrics. If materials with resonance absorption losses, polar pro-perties (permament dipole moments) or anisotropic polarizabilities were used, however, the IM pro-ducts could be significantly increased, so such materials should be avoided.

Functionally, Eq. (58) and the relationships between a and a' show that PML will increase qua-dratically with the number of wavelengths in the transmission line length, and except as Zo is altered bydimensional changes will vary inversely with the fourth power of the transverse structural dimensions.Since structure size and power levels will generally be correlated, the conclusion is that this kind ofdielectric IM production will be insignificant in all but extreme cases. (For instance, if in the exampleabove, P and P2 are increased to 1000 W while r, and r2 are trebled, PML will be increased 27 dB.)The one configuration in which IM production might be significant with good dielectrics is a high-Qresonant structure where the power densities are correspondingly enhanced. Thus it is desirable toexclude all dielectric materials from such elements.

7. SUMMARY

The intermodulation signals contributed by the intrinsic non-linear properties of materials (otherthan semiconductors) used in multiplex circuits have been calculated. Functional dependences, interms of parameters accessible externally, have been obtained for the normally dominant 3rd-order IMsignals.

Non-magnetic conductors are considered in which resistivity changes either by resistive heating,by associated mangetic fields, or by current density directly. Resistive heating is the most important ofthese effects.

Ferromagnetic metallic components are treated separately. These materials are entirely unsuitedto small signal applications. In addition to causing very high losses, they can generate IM signals 100dB above thermal noise in realistic circumstances. The danger in the use of such components is illus-trated in Chapters II-IV.

Dielectric elements, distributed or not, in which the dielectric properties are functions of electricfield, are treated in a generalized fashion. It is found that water vapor in air produces non-negligibleIM signals, particularly at microwave frequencies. Dielectric media with no polar properties or reso-nance losses are generally insignificant as IM sources, although in resonant structures they might leadto observable IM signals.

The results of this chapter lead to the following recommendations concerning materials in multi-plex systems:

i.) Ferromagnetic materials (and also semiconductors) should be totally excluded from any partof a system in which multiple signals exist. Leakage effects may require shielding of such materials.

79

G. If. STAUSS

ii.) Water vapor should be excluded from all transmission lines and cavities, particularly formicrowave frequencies. Attenuation in the atmosphere is probably the main deterrent to operatingabove x-band frequencies, but if higher frequencies are used, oxygen also should be removed from thes) stem.

iii.) High power densities should be avoided; since these are inevitable for cavity filters, materialsused there (even more than in other parts of the system) must have low resistivity, high heat capacity,high thermal conductivity and low thermal coefficients of resistivity. Surfaces must be clean and free ofcai iosion products.

iv.) Dielectric materials should be excluded from resonant structures. If used elsewhere theythould be non-polar and with minimum loss characteristics.

v.) Paths in which multiple signals pass should be kept to minimum lengths, preferably a meteror less. Mechanical junctions and closures should be kept to a minimum, placed to minimize currentsacross them, and made to best insure metallic contact.

If the above conditions are satisfied, IM production will probably be dominated either by the junc-on effects or by resistive heating in resonant elements. In the latter case the example given in Part 1

suggests that for an input power level of I kW, 3rd order IM power can be held below about -140 dBmfor UHF signals. Because of the functional dependences illustrated in Eq. (15), this level will increase:is input power rises or system temperature falls, and will increase as the separation of primary signalirequencies shrinks or as the primary frequencies rise.

S YMBOLS USED

Quantities used only where defined not listed. MKS units throughout. The subscript IM on aq:4ULty means the same quantity as without, but restricted to the IM signal (3rd order, 2w, -w2only).

A effective conductor surface areaR magnetic inductionb defined by Eq. (32)C capacitance per unit length of transmission line with vacuum dielectric

heat capacity of conductorI) coefficient of quadratic magnetic field dependence of permeabilityd spacing of parallel plate capacitor!K electric field strengthf(x) defined by Eq. (11)G thermal conductivity of conductorH magnetic field strength; H(z) instantaneous local valueI instantaneous ctrrent in transmission lineJ (z) instantaneous local current density; JN (z) lowest order

value due to input signalsinstantaneous total current at all depths in conductor perunit width of surface

./ rms value of Jk Boltzmann constantL transmission line inductance per unit lengthM magnetization; M saturation magnetizationPIN (Z) instantaneous local density of power dissipated, lowest

order due to input signalsPN rms power dissipated at all depths per unit surface area,

lowest order due to input signals

so

NRL MEMORANDUM REPORT 4233

PIML rms IM power dissipated at load due to unit area of IM sourcePIN total rms input power available to loadPIN I, total rms input power lost in transmission to loadPtuL total rms IM power delivered to loadQ circuit Q factorR resistance of transmission line per unit lengthR, surface resistivity. cf. Eq. (8)r1 ,r 2 inner, outer radii of coaxial transmission lineT temperature; To ambient; T(z) ku.al instantaneous value above ambientU amplitude of rf voltage across transmission lineV instantaneous voltage across transmission line; V0, Vlowest, next lowest

order values for changes in dielectric constants.W defined by Eq. (4)x defined by Eq. (7)Y length of transmission line

y distance along direction of propagationZ0 characteristic impedance or matching load impedancez depth below surface of conductor

ratio of signal amplitudes for two inputs, signal 2/signal 1.8o skin depth for small signal levels; 8 (z) instantaneous local valuef = (E I- if2) relative dielectric constant for small signal levels

coefficient of quadratic current density dependent change in resistivityAwavelength in transmission line

magnetic permeability; IAs, permeability for very weak

signals; A (z) instantaneous local valuev circular frequency

radian frequency

Po resistivity for small signals; p (z) instantaneous local valuea coefficient of quadratic magnetic field dependent change in resistivity'defined by Eq. (4)

generalization of T6J, coefficient of linear thermaldependent change in resistivity

', vj' coefficients of quadratic electric field dependent changesin real and imaginary dielectric constants

a, qa', v' modified by transmission line geometry to give changesin dielectric constants with voltage across transmission line

T = (K - iy) complex propagation constant

REFERENCES

1. Philco Study Report WDL-TR 5242 (1973).

2. TRW FLTSATCOM Document 24200-540-006-01 (1974). J.Z. Wilcox and P. Molmud, IEEE

COM-24, 238 (1976).

3. J. DeLaunay, R.L. Dolecek, and R.T. Webber, J. Phys. Chem. Solids 11, 37 (1959).

4. R.M. Bozorth, Ferromagnetism, (Van Nostrand, 1951) p 798ff.

5. J.-P.Jan in Solid State Physics, F. Seitz and D. Turnbull, eds. (Academic Press, 1957), Vol. 5, pp70-71.

81

*. 'i. Y

G. H. STAUSS

6. AlP Handbook (McGraw Hill, 1957), p 5-2 14.

7. C.J.F. BN3tcher, Theory of Electric Polarization (Elsevier, 1973), 2nd Ed., Ch. VII.

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