IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-13, NO. 2, MAY 1971

produced using the sequence generator method, and the63-bit code is shown in Fig. 12. Note that even though thecode is slightly distorted, it is still adequate for detectionpurposes since the autocorrelation is accomplished using adelayed form of the same signal.

REFERENCES[1][2][3]

[4]

M. I. Skolnik, Introduction to Radar Systems. New York:McGraw-Hill, 1962.R. S. Berkowitz, Modern Radar. New York: Wiley, 1965.B. Elspas, "The theory of autonomous linear sequential networks,"IRE Trans. Circuit Theory, vol. CT-6, Mar. 1959, pp. 45-60.

, "A radar system based on statistical estimation and resolu-tion considerations," Appl. Electron. Lab., Stanford Univ.,Stanford, Calif., Tech. Rep. 361-1, Aug. 1, 1955.

[5] W. McC. Siebert, "A radar detection philosophy," IRE Trans.Inform. Theory, vol. IT-2, Sept. 1956, pp. 204-221.

[6] S. W. Golomb, Shift Register Sequences. San Francisco, Calif.:Holden-Day, 1967.

[7] S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J. Stiffer,and A. J. Viterbi, Digital Communications with Space Applications.Englewood Cliffs, N.J.: Prentice-Hall, 1964.

[8] G. D. Forney, Jr., "Coding and its application in space com-munication," IEEE Spectrum, vol. 7, June 1970, pp. 47-58.

[9] E. F. Buckley, "Outline of evaluation procedures for microwaveanechoic chambers," Microwave J., Aug. 1963.

[10] , "Design evaluation and performance of modern microwaveanechoic chambers for antenna measurements," Electron. Com-ponents, Dec. 1965, pp. 119-1126.

[11] R. E. Hiatt, E. F. Knott, and T. Senior, "A study of VHF ab-sorbers and anechoic rooms," Univ. of Michigan, Ann Arbor,Rep. 5391-1-F, Feb. 1963.

[12] E. L. Ginzton, Microwave Measurements. New York: McGraw-Hill, 1957.

Prediction of Mixer Intermodulation Levels asFunction of Local Oscillator Power

ELI F. BEANE, MEMBER, IEEE

Abstract-During intermodulation testing with diode mixers an in-crease of intermodulation interference was observed due to an increaseof LO power incident to the mixer. This phenomenon conflicted with thetheory that increase of LO power reduces intermodulation output of thediode mixer. In these tests the intermodulation decreased as expectedwhen the LO power was further increased. Results of a theoretical andexperimental study of how the level of incident LO power affects theintermodulation output levels emanating from the mixer are presented.The predicted results lead to the following experimentally verifiedconclusions.

1) A drop in power at some intermodulation frequencies occurs foran increase of LO power, depending on LO operating point and orderof intermodulation.

2) Power at each intermodulation frequency will repeatedly increase,reach a maximum, and then decrease as power in LO signal increases,where the number of repetitions follows the orders of intermodulation.

3) The maximum intermodulation power at low-order intermodulationfrequencies occurs for higher LO power than higher order intermodula-tion frequencies.LO power operating point is shown to be a significant factor in mixer

intermodulation consideration. Application of these results to receiverintermodulation improvement is discussed.

Manuscript received May 15, 1970.The author was with the Electronic Warfare Laboratory, U. S.

Army Electronics Command. He is now with the Ground RadarTeam, Combat Surveillance Acquisition and System IntegrationLaboratory, U. S. Army Electronics Command, Fort Monmouth,N. J. 07703.

INTRODUCTION

T HE EFFECT of LO level on the spurious intermodula-tion product output levels of a nonlinear resistance

mixer was studied in [1]-[4]. These studies concluded thata 3-dB increase of LO power to a positive resistance pointcontact diode mixer will improve the intermodulation ratio(IMR) by 6 dB. IMR is defined as

IMR = PIMPIF

(1)

where PIF is the power output of the mixer at the desiredIF and PIM is the power output of the mixer at undesiredintermodulation frequency. Figs. l(a) and (b) show theinput and output spectrums of the mixer, respectively. Weare limiting our consideration at this point to the IF out-puts: SIFT at fIF1 and SIF2 at fIF2 (SIF1, SIF2, etc., are signalidentifiers and do not specify amplitudes). These are dueto the desired second-order mixing of input signals S,at frequency f1, S2 at frequency f2, SLO, the local oscillator,at fLo, and undesired fourth-order intermodulation, SIM atfIM and SIM' at fIM' in the following manner:

F, = fi - fLo

fTC2 = f2-fLO (second order) (2a)

56

BEANE: PREDICTION OF MIXER INTERMODULATION LEVELS

e (t)

MIXER DIODE

Ro

11i(t)

R)

i (t)

Fig. 2. Lumped-element low-frequency equivalent mixer circuit.

(b)Fig. 1. (a) Mixer input spectrum. fIFi = fl - fLO; fIF2 = f2 fLO;Af = f2- f. (b) Mixer output spectrum. fAM = 2f- f2 - fLo;fAM = 2f2 fl fLo-

and

JIM = 2f I f2 fLO

IM' = 2f2 f- fLo (fourth order). (2b)

The orders of mixing are related to the order of the termin the power series expansion (given in (4)) of the nonlineardevice from which a particular mix results [4] or from thesum of the absolute values of the integer coefficients of thefrequencies involved in a particular "mix," as given in (2a)and (2b).

Laboratory observations in TV display intermodulationtests [4] of an X-band mixer that have been made with a

microwave mixer analyzer [4] verify the conclusion [I]-[4]that increased LO power does eliminate to a great extentthe effect of the fourth-order intermodulation product.However, this conclusion did not account for an increaseof intermodulation and IMR that occurred for an initialincrease of LO power incident to the mixer, although theintermodulation and IMR decreased when the LO power

was further increased. This phenomenon seemed to suggestthat intermodulation level is a function of LO power

operating point. For certain LO operating points the follow-ing may occur: 1) an increase in LO power will decreaseintermodulation and IMR; 2) a decrease in LO power

decreases intermodulation and IMR; 3) an increase or

decrease of LO power will decrease intermodulation andIMR (see Figs. 8-10).A theoretical explanation of this effect seemed to be

shown in an article by Rutz-Phillip [5] which deals withinterference phenomena related to corroded joints aboardships and other vehicles. She found by analysis and ex-

periment 1) that the power at intermodulation frequenciesgenerated in a nonlinear resistive element in an actual circuitcan become smaller although the power in some of theincident waves increases; 2) that the power at higher orderintermodulation frequencies generated in a nonlinear re-

sistive element in an actual circuit can become smalleralthough the power in some of the incident waves increases;and 3) that the power at higher order intermodulation

frequencies (e.g., fourth-order fJIM in (2b) and Fig. 1(b)) willreach its maximum at a lower level in the incident waves

than lower order intermodulation frequencies (e.g., second-order fIF in (2a) and Fig. 1(a)).

Rutz-Phillip derives power conversion relations for a

nonlinear resistive element in series with linear resistors.She applies the analysis based on a lumped element low-frequency circuit (Fig. 2) to a microwave circuit where thedc current-voltage characteristic [6] functionally repre-

sents the diode or nonlinear resistive element given by theequation

i = io(e" - 1) (3)where

v

io

Voltage across nonlinear element.Nonlinearity coefficient.Saturation current.

This current-voltage characteristic is expressed as a power

series of a time dependent voltage v = V(t):

i(t) = aO + alv(t) + a2v(t)2 + a3V(t)3.. (4)

where the coefficients ao,al,a2... are given in the form ofrecursive polynomials as derived by Mills [7]. These co-

efficients can thus be obtained numerically for any orderpower series no matter how high. In previous analyses [2],[3] of nonlinear resistive elements coefficients higher thanthe tenth order were not obtained. Rutz-Phillip indicatedthat the prediction of intermodulation rise and fall whichdepends on signal level incident to the nonlinear elementcould be arrived at only by evaluating a considerable num-

ber of terms of the power series representation of thecurrent-voltage characteristic. As indicated, this approachhas a strong connection to the study of mixer intermodula-tion, and hence its application to an X-band single-endedresistive diode mixer was undertaken.

ANALYSIS

Consider the block diagram in Fig. 3 of a microwavemixer circuit for a two equal tone intermodulation test ofa single-ended 2-port microwave (X-band) mixer. Now,represent this microwave circuit by a lumped element low-

frequency equivalent circuit (idealized approximation) as

SLO

A-

-9 fl v

s, s2

A f

fI f2 vTLO

(a)

: :

57

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, MAY 1971

SIGNAL SOURCE

SIGNAL SOURCEVHF

( _~~~~~~~~~~MIE FIELD INTENSITY3 dB r3Sl+S2 +SLO .UNDER Q1: IF OUTPUT METER

HYBRID| U(COMBINED TEST . (INCLUDINGMIXER INPUT) M PRODUCTS) |50n INPUT Z

LOCAL OSCILLATOR

Fig. 3. Two equal tone microwave mixer intermodulation test circuit.

dc current and voltage are replaced by time-varying i(t)and e(t):

(6)

where e(t), the voltage across nonlinear element resistance rin Fig. 4, is given by

e(t) = v(t) - i(t)Rwhere

R = Ro + RL + rs.

Replacing the value of e(t) in (7) into (6), we find

i(t) = io{exp [oc(v(t) - i(t)R)] - 1}.

(7)

(8)

+ ~~~~>*c5

MIXER DIODE EQUIVALENT

Fig. 4. Equivalent circuit of nonlinear resistance mixer diode.

shown in Fig. 2, where

v(t) Instantaneous input voltage of signals S1, S2, andSLO.

e(t) Instantaneous voltage across diode.i(t) Instantaneous current induced in equivalent circuit.Ro Linear resistance equivalent of input impedance.RL Linear resistance equivalent of output load im-

pedance.

To complete the circuit in Fig. 2, the equivalent circuitfor the semiconductor nonlinear resistance mixer diode isshown in Fig. 4, where

rrs

Cs

Nonlinear resistance.Linear spreading or bulk resistance of semiconductor.Shunt capacitance.

In our analysis reactive effects attributed to c, will beignored; we are taking into consideration only the linearand nonlinear resistances in the circuit. This is similar tothe approach taken by Rutz-Phillip [5]. She justifies theuse of the low-frequency circuit model (Fig. 2) for a micro-wave circuit, disregarding frequency dependent impedancemismatches by having both input and output (intermodula-tion) frequencies all in the same spectral vicinity. We followa similar analysis and disregard frequency dependent im-pedance mismatches in order to facilitate measurementsand obtain results for comparison with theoretical calcula-tions which does not significantly alter the result.

This input voltage v(t) consists of three CW sinusoidalsignals, S1, S2, and SLO, which are two equal amplitudeinput signals and a local oscillator signal, respectively.Corresponding to these signals are their respective voltageamplitudes V1, V2, and VLO and angular frequencies w1l, w02,and WaLO such that

v(t) = V1 cos Co1t + V2 cos w)2t + VLO COS (OLOt. (5)

The nonlinear resistive diode in Figs. 2 and 4 is representedby the dc current-voltage characteristic in (3), where the

We express i(t) in (8) as a power series in v(t) (see (4)) aboutthe point v = 0. The derivatives

dnidv5 v=O

needed for a Taylor series, as obtained from the analysisof Mills [7], are expressed as polynomials in Z, H(Z),where

z di - iOadv v=O 1 + io0cR

and the power series expression for i(t) is given bym I

i(t) = E Hk(Z)vk(t).k=1 k!

The polynomials H(Z) in (10) are expressed as

zXk-1 2k-i

Hk(Z) = I ak,j(ZR)iR j=o

(9)

(10)

(1 1)

for k = 1,2,3- . The ak,j in (11) are computed by thefollowing recursion relation:

ak+1,J = jak,- 2(j - l)ak,j11 + (j - 2)ak,j-2

whereak,j=O, forj<Oorj>2k-1a1,1 = 1 and al,j = 0,

(12)

for] #= 1.

Note from (10) that the kth-order component of currentik(t) is given by

ik(t) = -Hk(Z)Vk(t). (13)

The general Fourier series expansion of the current i(t)as a function of the generator voltage v(t), where v(t) iscomposed of three sinusoids as in (5), will have the form

00 00 00

i(t) = E ixyzexp [j(xw1 + YW2 + ZOLO)t]x=-00 y=-00 z=-00

(14)where x, y, and z are integers and Ixyz is the current com-ponent at intermodulation frequency,

C)XYZ = XC)1 + Y()2 + ZCWLO (15)

The power output Px,_ at frequency f,Y_ (= cowY,/27r) isshown, following Rutz-Phillip [5], for simplified equivalent

58

i(t) = io(eae(t) 1 )

59BEANE: PREDICTION OF MIXER INTERMODULATION LEVELS

I/ -u 1- /vn

(P ult) lJ

Fig. 5. Simplified equivalent microwave mixer circuit.

microwave circuit in Fig. 5, where ZO is the characteristicimpedance of the circuit and ZL = ZO is the matchedterminating impedance to be

Pxz= 2I 2ZL (16)Identify ZO and ZL = ZO with RO and RL of Fig. 4, ZL = wherZO = RL = RO; hence IxyzI

Pxz= 2IX2 Ro. (17) tion

Each vk(t) in the expression for ik(t) in (13) is given by

vk(t) = (V1 COS (Olt + V2 COS w02t + VLO COS )LOt) (18)which upon expansion yields an explicit expression for the shouamplitude of each frequency component generated by k =

vk(t). The amplitude or the Fourier coefficient of the TIfrequency shou

X£ o 1 + Y-2 + Z()LO (24),xyz =2 of t:

indicdue to vk(t), (Vxyz)k (following the work of Wass [11]) is enccgiven by poly

a =23.1V-1io= 2x10-6AIs= 15QR -I 15 Q

0.7V in VOLTS

Fig. 6. Plot of diode mixer current-voltage characteristic.

re Hk(Z) is given by (11). Thus the Fourier coefficientof the current associated with the output intermodula-frequency,

XWt)1 + Yct2 + ZO)LOJ xyz - 27

ild be computable by the formula in (24) for any orderm of the power series in (4).he power output at any intermodulation frequencyild now be obtained by substituting Iyz, computed frominto (17), the only limiting factor being the capacity

he computer used to evaluate the triple summationcated in (24). In fact serious computer limitations were)untered in the numerical evaluation of the Hk(Z)'nomials, given in (11). The computer evaluation of

k-i k-b ( Vla2VLO k!

bO c=O 2k1[(a + x)/2]! [(a- x)/2]! [(b + y)/2]! [(b-y)/2]! [(c + z)/2]! [(c-

where a,b,c are positive integers such that

a + b + c=kor

a = k-b-c. (20)

Using (VXy),)k from (19), we may write vk(t) as a Fourierseries

cc 00 GO

vk(t) = E E (VXyz)kx=-oo y--cc z=-oo

x exp [j(xwo + Y(02 + ZWDLO)tl. (21)

Substituting this expression for vk(t) into (10), we obtainm 1 cc °° cc

i(t) = , Hk(Z) S E Y (vXyz)k. (22)k=1 k. x=-00 Y=-co Z=-

Comparing (22) with (14), we find

Ixyz = E (VXyz)k HJ) (23)

Replacing (Vxyz)k by its equivalent from (19),

Hk(Z) involves terms of such immense magnitude, for allbut the smallest of the k (k - 8) that it is numericallyunfeasible to use the series in (10) to represent the current-voltage characteristic of the mixer circuit under investiga-tion. Thus the theoretical analysis which leads to theRutz-Phillip conclusions regarding intermodulation in a

resistive diode circuit based upon evaluation of higher orderterms (k > 8) of the current-voltage relationship cannot beregarded as valid.A simpler analytic examination of behavior of the cur-

rent-voltage relationship (8) representing the diode mixercircuit leads to a basic understanding of the correlation oflocal oscillator power with the intermodulation output of aresistive diode mixer without a complicated and numericallycumbersome power series. Fig. 6 shows a plot of the timeindependent (i.e., dc) form of the current-voltage charac-teristic in (8):

i = io{exp [oc(v -iR)] - 1}. (25)First, examine the behavior of the current i as the voltage

v becomes very large. In (25), separating terms involving i

m k-1 k-b VlaV2bVOHk(Z)xyz k=l b= O C kl '[(a + x)/2]! [(a -x)/2]! [(b + y)/2]! [(b- y)/2]! [(c + z)/2]! [(c -z)/2]!

(19)

(24)

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, MAY 1971

and v to opposite sides of the equation gives

i + io . (aiR 2e

and from (32):

di--. 1-(constant),dv R

Let v -+ oo. We find that e"v -- oo. Then by virtue of the

equality in (26)

+ °. eR -x. (27)io

This cannot be, unless i -+ oo. Hence

i -oo, as v (28)

Solving (25) which is implicit in i explicitly for v we obtain

V=1 l i + io + iR. (29)

oc io

Differentiate (29) with respect to i:

dv I+ R. (30)

di c(i + io)

Take the reciprocal of both sides of (30):

di cx(i + io) (31)dv 1 + a(i + iO)R

Let i -s o, which is equivalent to v -s o

find(by (28)). We

di >I- (constant). (32)dv R

Next examine (25) as v 0. From (26) we see that e'v -*

as v 0 from which it follows that

*io

as i 0. Hence

i-Oasv -> 0. (33)

Differentiate both sides of (31) with respect to v; then

d2i di 1 _ (i + iO)R 1 (34)

dv2 dv [1 + (i + io)R [1 + R(i + io)]2JIn (34) let i -+ 0, which is equivalent to v -+ 0 by (33):

d2i a didv2 (1 + cioR)2 dv

Substituting

k = (constant)(1 + i0R)2

in (35) we find:

d2i di2-k = 0.

dv2 dv

Therefore

i'ekv as v -*0

(35)

The results in (37) and (38) indicate that the current-voltagecharacteristic of the resistive diode mixer circuit behaveslike an exponential for small v(v 0) and like a straight

line of slope I/R for large v(v >> 0). (See Fig. 6.)In terms of a power series representation of the current-

voltage characteristic, higher order coefficients would berelatively large for low voltage. As v is increased, the higherorder coefficients of a power series representation of thefunction become smaller relative to the lower order co-efficients, and the current-voltage characteristic tends tobecome a linear function. This functional behavior relatesto intermodulation of a resistive diode mixer and leads tothe previous conclusions, namely, power at intermodulationfrequencies, especially of higher orders, can become smalleralthough power in some of the incident waves (i.e., LO)increases, and power at higher order intermodulation fre-quencies (e.g., fourth, sixth, etc.) will reach its maximum atlower levels in the incident waves than lower order inter-modulation (e.g., second order). This is so since increasingincident LO power to the resistive diode mixer correspondsto the effect of increasing the voltage v, i.e., the bias in thedc current-voltage relationship. Hence higher LO powercauses the incident waves to intercept a portion of the diodecharacteristic less nonlinear than that intercepted by lowerLO powers.

Further understanding of the intermodulation behaviorof the point contact diode mixer may be obtained by succes-sive differentiation of (29) yielding the derivatives of v withrespect to i. From these derivatives algebraic expressionsfor the coefficients, ak of the kth-order terms of the Taylorseries expansion of the diode mixer current-voltage relation-ship expanded about a voltage v = v0 (i.e., bias) can beobtained by reversion of a power series as shown in [10,appendices I and II]. The algebraic expression for the firstfour Taylor series expansion coefficients is shown to be

a, = (i + i0)oc

a2 =

(X + 1)3 2!

a3 = [-2x + 11 (i + io) 3

(x + 1)5 3!

a 6x2 - 8x + 1 (i + io)4(X + 1)7 4!

(39a)

(39b)

(39c)

(39d)

where x = c(i + io)R. These coefficients may now be com-

puted as a function of i which in turn may be correlated tothe bias voltage vo by means of (29).

(36) In Fig. 7 the magnitudes of the coefficients ak, for k =

1,2,3,4, are plotted (semilogarithmically) as a function ofthe bias voltage v0 computed from (39) and (29). Observehow the magnitudes of the coefficients oscillate (circled

(37) nodes indicate zero crossings) about the v0 axis with the

as v -+ 0o. (38)

60

(26)

BEANE: PREDICTION OF MIXER INTERMODULATION LEVELS-20_

-30_

Em

ESL

Vo IN VOLTS

Fig. 7. Diode mixer coefficient magnitudes as functions of biasvoltage (semilogarithmic plot).

number of oscillations increasing with increasing k. Thebehavior of these coefficients relates directly to the mixerintermodulation output product levels. It can be shown [10]that the most significant contribution to a kth-order inter-modulation product of small input signals is due to thecoefficient of the kth-order term of the Taylor series expan-sion. Hence variation (i.e., peaks and valleys) in powerlevels of kth-order mixer intermodulation products follow-ing the oscillatory behavior of the kth-order coefficient kare to be expected as the value vo is varied across the diode.Although the variation of the kth-order coefficient ak is

being considered as a function of vo (the dc bias), similarresults are to be expected if the LO voltage incident to thediode were varied. This is reasonable since VLO creates aneffective dc bias voltage across (or dc bias current through)the diode due to rectification. The results of measurementsshown in Figs. 8-10 indicate that this conclusion corre-sponds very well with experiment. Note that the oscillatorybehavior of the series coefficients comes about only whenthe diode mixer's linear circuit resistance (R = Ro +RL + rj) is taken into account. Were these resistancesneglected in the analysis, the resulting coefficients wouldbe independent of the bias voltage vo or the effective LObias level.

RESULTS

Intermodulation measurements of a microwave X-bandsingle-ended mixer, utilizing a IN23WE crystal diode weremade with circuit represented (power monitoring andcalibration instrumentation not shown for simplicity) inFig. 3, where

S1 is at f1 = 9055 MHz

S2 is atf2 = 9065 MHz

-40

-50

-60

-70

-s8

-sC

-100

-80

-70

-60

p =psI s

a = 23.

jO= 2.1

rS = 15_

F--50

m -50

a -40

-30

-20 _

-10 _

0 -50

=-15 dBmt2 p

.5 V1 -*PI F

IX 10-' A,J

- -\ \ Pim4

Pim6- ~Iti1

-40 -30 - 20 -10 0 +10

PLO in dBm

(a)

PS = Ps2=-5dBma = 23.5 V-

io 2. 1x 10-6ArS= 15IxOX

1\1IMR6

zo/

-40 -30 -20 -10 0 +10

PLO in dBm

(b)Fig. 8. (a) PIF at fIF = fl - fLo as function of PLO at fLO (second-

order intermodulation). PIM4 at fiM4 = 2fi- f2 - fLo as functionof PLO at fLo (fourth-order intermodulation). Plm6 at fiM6 = 3fi-2f2 - fLo as function of PLO at fLo (sixth-order intermodulation).(b) IMR4 at fiM4 as function of PLO at fLo (fourth-order inter-modulation). IMR6 at fiM6 as function of PLO at fLo (sixth-orderintermodulation).

In all measurements the power PLO of SLO was variedfrom -40 to +10 dBm (+ 10 dBm was not exceeded dueto saturation effects). In each test the power levels of S1and S2 remained fixed, i.e., they served as parameters.Power outputs of the mixer were measured at frequencieswhich are in the vicinity of the desired IF which resultfrom mixing signals (S and S2) within the mixer inputbandwidth. These intermodulation products turn out to beall of even order, as follows:

fIF = fi- fLo

= 9055 MHz - 9000 MHz

= 55 MHz second-order IM, desired IF

JIM4 = 2f f2 - fLO

= 2(9055 MHz) - 9065 MHz - 9000 MHz

= 45 MHz fourth-order IM, undesired

JIM6 = 3f- 2f2 - fLO

= 3(9055 MHz) - 2(9065 MHz) - 9000 MHz

(41a)

(41b)

= 35 MHz sixth-order IM, undesired. (41c)

-110

61

SLO is atfLo = 9000 MHz. (40)

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, MAY 1971

Fig. 8(a) shows the results of equal-tone (i.e., PS, = Ps2)intermodulation measurements for PIF, PIM4, and PIM6 asfunctions of LO power PLO with Ps1 = Ps2 = - 15 dBm.For PIF the rise and fall predicted by the analysis with theincrease of PLO did not occur within the range of PLO(-40to +IOdBm).The theory is not necessarily wrong since second-order

intermodulation might be reduced if the diode were ableto withstand PLO greater than + 10 dBm on the IN23WEdiode. However, the data presented for PIM4 and PIM6 inFig. 8(a) does show the strong dependence of intermodula-tion on LO operating point. Both PIM4 and PIM6 fall sharplyfor PLO = + 6 dBm and rise for an increase or decrease ofPLO about this point. Furthermore, PIM6 rises, reaches amaximum, and falls for a lower PLO than PIM4, although itrepeats the rise and fall pattern as PLO is further increased.

Fig. 8(b) shows how the IMR for fIM4 and fIM6O

IMR -PIM4PIF

(dB)

IMR - PIM6 (dB)IF

(42a)

(42b)

respectively, vary with LO power. Note how both IMR4and IMR6 reach a sharp minimum for PLO = +6 dBm,and how in addition IMR6 reaches a sharp minimum forPLO = -11 dBm. The repeated rising and falling of PIMwas not predicted at all by Rutz-Phillip, but is quiteapparent from the simple analytical approach which weemploy in the preceding section.Note that the number of maxima and minima are related

to the order of intermodulation as our analysis predicts.Thus for PjIM4 (fourth order) we have two peaks at PLO =0 dBm and PLO = + 8 dBm and two sharp falls at PLO =+6 dBm and PLO 2 + 10 dBm. For PIM6 (sixth order) wehave three peaks: PLO = -23 dBm, PLO = + 2 dBm, andPLO = + 8 dBm; we also have three minima: PLO =

-11 dBm, PLO = +6 dBm, and PLO > +10 dBm. Wecan thus relate the total number of maxima and minima tothe order of intermodulation product, e.g., fourth-orderintermodulation would have four maxima and minima,while sixth order has six maxima and minima. A similarphenomenon has been observed in measurements of spuriousresponse level as a function of dc bias voltage by Donaldsonand Moss [12] and as referred to by Herishen in hisstudy [10] of mixer response versus bias current of a non-linear diode mixer. They [12] indicate from their data thatspurious responses of the same order display similar proper-ties as bias is changed and that the number of minimarelative to the absolute maximum is equal to the order ofthe mix product. They also noted the influence of localoscillator power in increasing and decreasing spuriousresponse level.Both the effects of LO power and bias on spurious

response levels or intermodulation are related in that dif-ferent bias values and LO input levels subtend differentportions of the mixer nonlinear characteristic (3) andthereby subject input signal voltages to different mixer

Eco

Sai-

-20

-30

-40

-50

-60

-70

-80

-90

-100

-80

-70_

-60_

-50

-40-

2 -30

-20

-10-

PS Ps = -2OdBma= 23.5 V-1io= 2.1 x 10-6Ar,a= 15S

--O-o- //

" ICo

- IF

-0_

a0 -a a- a0\ I \\

,\

N \ \aA m,/ ° ! i' 4

IM4

°PIM6

-50 -40 -30 -20 -l0 0 +10

PLO in dBm

(a)sp s2 =-2OdBm

a = 235 V Iio = 2-1 x 10- A

rs = 15Q IMR8I

A /I/P\ /?\ i

/I

,6 \\ /1 V/

-a--Al/I~IDO. /,)d'

o-50 -40 - 30 - 20

PLO In dBm-10 0 +10

(b)

Fig. 9. (a) PIF at fIF = fl - fLo as function of PLO at fLo (second-order intermodulation). PIM4 at fIM4 = 2f, - fLo as function ofPLO at fLo (fourth-order intermodulation). PIM6 at fiM6 = 3f, -2f2 - fLo as function of PLO at fLO (sixth-order intermodulation).(b) IMR4 at fIM4 as function of PLO at fLo (fourth-order inter-modulation). IMR6 at fiM6 as function of PLO at fLo (sixth-orderintermodulation).

nonlinearities. In our study we considered only the effectof LO power levels without any external bias.

Figs. 9 and 10 show results of equal tone intermodulationand IMR measurements for Ps, = PS2 = -20 dBm andPSi = PS2 = -30 dBm, respectively. Data for PIM6 andIMR6 do not appear in Fig. 10 since the level of PIM6 wastoo low for accurate measurements. Observe how for lowerlevels ofinput signals S1 and S2 the intermodulation maximaand IMR minima are shifted to lower levels (i.e., the inter-modulation maxima and IMR minima occur for lower LOlevels). Although the IMR for lower level input signals,-20 and -30 dBm at PLO = +10 dBm, are less thanthose at the preceding minima (i.e., at +4 dBm for Ps, =PS2 = -20 dBm and -6 dBm for Ps, = PS2 = -30 dBm),the likelihood of "burnout" at PLO = +10 dBm precludesoperation of the single-diode mixer at such a high LOlevel. (This problem pertains only to the single pointcontact diode mixer configuration used in the precedingtests; multiple diode mixers or mixers using Schottky barrierdiodes may overcome this power limitation.)

-----L--------L--------L.[IV -110

62

C.-

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BEANE: PREDICTION OF MIXER INTERMODULATION LEVELS

P3 =P --3OdBma =23.5xV-1iO= 2.1 xI0-6 A

rS = 15Q _ o- o pIP

50 -40 -30 -20

PLO in dBm

'b PIM4-10 0 +10

(a)PS, T-PSi-30dBma = 23.5V-'ie 2.1X10 A

rS= 15aIR

so 'IMR4

O/

0- --- -0*---O-

-lot-

-50 -40 -30 -20 -10 0 +10

PLo in dBm

(b)

Fig. 10. (a) PIF at fiF = fl - fLo as function of PLO at fLo (second-order intermodulation). PIF at fIM4 = 2fi- f2 - fLo as functionof PLO at fLo (fourth-order intermodulation). (b) IMR4 at fIM4 asfunction of PLO at fLo (fourth-order intermodulation).

A study of the PIF in Figs. 8(a) and 9(a) in the vicinityof PLO = +10 dBm shows a drop in PIF with increasedPLO. (This may be related to the down shifting in the PLOof the maxima and minima.) This lends credence to theassertion that a drop of PIF would occur for higher (+ 10dBm) PLO, as shown in Fig. 8(a), where PSI = PS2 =

-15 dBm. Also, as to be expected, from Taylor seriesexpansion of the diode characteristic, our data shows thathigher orders of intermodulation generally have lowerpower levels.

CONCLUSIONS

The critical nature of the LO power level operating pointof a point contact diode mixer in reducing intermodulationhas been shown. Depending on signal strength of inter-modulating signals, there exists a critical LO operatingpoint at which even ordered undesired intermodulation

products and IMRs are at a minimum and any excursionof LO power about this point results in a sharp rise inintermodulation. This optimum operating point rangedfrom -12 to + 6 dBm for mixer input interfering signalswith levels from -30 to - 15 dBm.Although even smaller IMRs are indicated for higher LO

power levels, they are not presently feasible for a singlepoint contact diode due to "burnout" considerations. Thesimilarity between dc bias and LO drive with regard tointermodulation rejection is shown both analytically andexperimentally.

Further studies under consideration are the following:1) develop an intermodulation prediction computer pro-gram based on the present indicated simple analyticapproach; 2) study combined effects of bias and LOpower operating point on intermodulation and IMR;3) study intermodulation and IMR of a Schottky barrierdiode mixer with regard to LO and dc bias operatingpoints; 4) investigate multiple diode mixer configurations;and 5) study effect of mixer output impedance on inter-modulation and IMR.

ACKNOWLEDGMENT

The author wishes to acknowledge the work of M.Morris, L. Doyle, R. Poulos, and S. Ippolito, whoseassistance was employed in the preparation of this paper.

REFERENCES[1] B. B. Bossard et al., "Interference reduction techniques for

receivers," U. S. Army Electronics Lab., Fort Monmouth, N. J.,Contract DA36-039 AMC-02345(E), Tech. Rep., June 1964-July1965.

[2] B. B. Bossard, P. Torrione, and S. Yuan, "Theory and improve-ment of intermodulation distortion in mixers," in Proc. 10thTri-Service Conf. Electromagnetic Compatibility, Nov. 1964.

[3] L. Becker and R. L. Ernst, "Nonlinear admittance mixers,"RCA Rev., Dec. 1964.

[4] R. L. Ernst, P. Torrione, W. Y. Pan, and M. M. Morris, "Design-ing microwave mixers for increased dynamic range," IEEE Trans.Electromagn. Compat., vol. EMC-11, Nov. 1969, pp. 130-138.

[5] E. M. Rutz-Phillip, "Power conversion in nonlinear resistiveelements related to interference phenomena," IBM J., Sept. 1967.

[6] H. C. Torrey and C. A. Whitmer, Crystal Rectifiers (M. I. T.Radiation Lab. Series), vol. 16. New York: McGraw-Hill, 1948.

[7] H. D. Mills, "On the equation, i = io[exp oc(v - Ri) - 1],"IBM J., Sept. 1967.

[8] L. M. Orloff, "Intermodulation analysis of crystal mixer," Proc.IEEE, vol. 52, Feb. 1964, pp. 173-179.

[9] L. D. Neidleman, "An application of FORMAC," Commun. Ass.Comput. Mach., vol. 10, 1967, pp. 167-168.

[10] J. T. Herishen, "Diode mixer coefficients for spurious responseprediction," IEEE Trans. Electromagn. Compat., vol. EMC-1O,Dec. 1968, pp. 355-363.

[11] C. A. A. Wass, "A table of intermodulation products," J. Inst.Elec. Eng. (London), pt. III, Jan. 1948, pp. 31-39.

[12] E. E. Donaldson, Jr., and R. W. Moss, "Study of receiver mixercharacteristics," Eng. Experiment Station, Georgia Inst. Technol.Atlanta, Final Rep., ECOM-01426-F, Sept. 1966.

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