ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR SMALL SIGNAL HETERODYNE MIXERS WITH EXCESSIVE INJECTION AMPLITUDES Technical Report No. 62 Electronic Defense Group Department of Electrical Engineering By: J. F. Cline Approved by: J. A. Boyd Project 2262 TASK ORDER NO. EDG-6 CONTRACT NO. DA-36-039 sc-63203 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-042 arch, 1956

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv I. INTRODUCTION 1 II. METHOD OF ANALYSIS 2 III. CONVERSION TRANSCONDUCTANCE 7 IV. INJECTION FREQUENCY COMPONENT 11 V. INJECTION MODULATION COMPONENT 17 VI. EXPERIMENTAL RESULTS 22 DISTRIBUTION LIST 30 ii

LIST OF ILLUSTRATIONS Page Figure 1 Output Current (i or i') and Incremental Transfer Conductance (g or g') as Functions of the Instantaneous Total Injection Potential ei. 3 Figure 2 Relation Between Instantaneous Values of Injection Potential ei, Transfer Conductance g', and Output Current i' as Functions of Time When the PiecewiseLinear Approximations Are Used. Figure 3 Theoretical Values of Relative Conversion Transconductance Gcrel Obtained from Equations (14) and (15) Expressed in Decibels. 12 Figure 4 Graph of the Relative Amplitude Iirel of the Injection-Frequency Component in the Output Current, According to Equation (20), Expressed in Decibles 14 Figure 5 Graph of the Ratio of the Relative Conversion Transconductance Gcrel to the Relative Amplitude Iirel of the Injection-Frequency Component in the Output Current According to Equations (14), (15), and (20), Expressed in Decibels 16 Figure 6 Graph of the Relative Value Idc rel Of the D-C Component of the Output Current, According to Equation (25) 19 Figure 7 Theoretical Value of the Figure of Merit a, Defined by Equations (28), (29), (30), and (31), Expressed in Decibels 21 Figure 8 An Experimental Circuit Used to Illustrate the Application of the Piecewise-Linear Analysis 23 Figure 9 Measured Values of Instantaneous Output Potential eout and Incremental Transfer Conductance g in the Circuit of Fig. 8, as Functions of Instantaneous Total Injection Potential ei. 24 Figure 10 Measured Value of Relative Conversion Transconductance in the Circuit of Figure 8. 26 Figure 11 Experimental Measurements of Relative D-C Current in the Output of the Circuit of Figure 8 27 Figure 12 The Ratio of the Measured Values of Gcrel and Iirel for the Circuit of Figure 8, Expressed in Decibels 28 iii

ABSTRACT The theory of small-signal heterodyne mixers operating with very large injection potential swings is described. The mixer properties investigated are the conversion transconductance, the relative amplitudes of the difference-frequency and injection frequency components in the output, and the amplitude of the injection modulation-frequency component in the output in the case where the injection potential is amplitude-modulated to a small degree, as by power-supply ripple or noise, for example. The theoretical analysis is based upon two- and three-segment piecewise-linear approximations of the current and incremental transfer conductance curves, so that the mixer can be treated as a switch-type or commutator-type modulator. The method of constructing the approximations from the original curves is described. The result is a set of universal performance curves for all mixers of the particular class chosen to illustrate the method. The method can be extended to other classes of mixers. iv

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN SMALL SIGNAL HETERODYNE MIXERS WITH EXCESSIVE INJECTION AMPLITUDES I. INTRODUCTION This report describes the operation of small-signal heterodyne mixers in which the injection potential is a biased sinusoid which may be permitted to become much larger than in ordinary applications. Such excessive injection potentials are found either in applications where it is desired to obtain a decreasing conversion transconductance with an increasing injection potential or in applications where, for reasons independent of mixer performance considerations, it is either necessary or expedient to tolerate an excessive injection potential. The term "small-signal" is used here to indicate a condition where the signal amplitude is too small to have an appreciable effect upon the value of the conversion transconductance. This condition is found in most superheterodyne receiver converters, analog multipliers, spectrum analyzers, etc. The mixer properties which are investigated here are the conversion transconductance, the relative amplitudes of the difference-frequency and injection-frequency components in the output, and the amplitude of the injection modulation-frequency component in the output in cases where the injection potential is amplitude-modulated to a small degree, as, for example, by noise or by power-supply ripple.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN II. METHOD OF ANALYSIS The method of analysis described here is illustrated by its application to a class of mixers having characteristic curves of the particular shape shown as solid lines in Fig. 1. The results can easily be extended to other classes of mixers. The essential properties of the solid curves in Fig. 1 are that the output current characteristic i has two horizontal sections at different levels while the incremental transfer conductance characteristic g has two horizontal sections both at zero level. Both curves are plotted with the instantaneous injection potential ei as the abscissa. The details of the curved transitional sections which join the horizontal sections are unimportant. It will be convenient in the discussion which follows to use the term "transitional potential range" in referring to the range of instantaneous injection potential ei over which the transitional curved sections of i and g extend. In most mixer applications the swing of the injection potential is usuall not much larger than, if as large as, the transitional potential range. This is true in many superheterodyne radio receivers, for example. A discussion of the conversion transconductance of small-signal heterodyne mixers which are operating with injection potential swings of the magnitude often employed in superheterodyne receivers has been given by Heroldl, and can be found in popular textbooks.2'3 1. Herold, "The Operation of Frequency Converters and Mixers for Superheterodyne Reception," Proc. I.R.E., Vol. 30, February, 1942, p. 8. 2. Seely, "Electron Tube Circuits," McGraw-Hill, 1950, pp. 359-61. 3. Van Voorhis, "Microwave Receivers," McGraw-Hill, 1948, pp. 138-48.

Gg-98-01 wI 89-99-V z9zz i max. -2 -I +I -- ei -- g max /i ---D -- - -2 -I 0 +I FIG. I. OUTPUT CURRENT (i OR i') AND INCREMENTAL TRANSFER CONDUCTANCE (g OR g') AS FUNCTIONS OF THE INSTANTANEOUS TOTAL INJECTION POTENTIAL ei. THE DASHED CURVES i' AND g' ARE THE PIECEWISE-LINEAR APPROXIMATIONS.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN When the injection potential swing is not large in comparison with the transitional potential range, the details of the curve shape are quite important in the quantitative analysis. Herold's analysis, for example, uses several points on the transitional part of the curve in order to take adequate account of the details of the shape. On the other hand, when the injection potential swing is very large in comparison with the transitional potential range, the fine details of the curved section are found to be relatively unimportant in the quantitative analysis. This makes it possible to replace the solid curves in Fig. 1 with the piecewise-linear curves i' and g', which are drawn as dashed lines in the same figure. The mixer then becomes essentially a switch-type or commutator-type modulator, which has been discussed by Caruthers4 and by Peterson and Hussey.5 For any particular mixer, the curves corresponding to the solid lines of Fig. 1 are obtained experimentally and it is difficult in general to obtain suitable mathematical expressions for them. The piecewiselinear curves, on the other hand, are easily described in mathematical terms. The problem is to find a way of constructing the piecewise-linear curves so that when they are analyzed mathematically, the results will be, as nearly as possible, the same as the results obtained from the solid curves. Consider a mixer in which there is a resistive load RL and an applied signal potential of instantaneous value es. The incremental transfer conductance g is then defined as the partial derivative of the instantaneous load current i with respect to es. A distinction must sometimes be made between this incremental transfer conductance g of the circuit as a whole and the ordinary incremental 4. Caruthers, "Copper Oxide Modulators in Carrier Systems," B.S.T.J., Vol. 18, pp. 315-337, April, 1939. 5. Peterson and Hussey, "Equivalent Modulator Circuits," B.S.T.J., Vol. 18, Pp. 32- 37, Jan. 1939.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN transconductance gm of the mixer tube alone. The distinction is not necessary when RL is so small as to have no appreciable effect upon i, but when g and gm do differ appreciably, they are often approximately proportional to each other for different values of the instantaneous injection potential ei, so that when relative rather than absolute values are being considered, g and gm may often be considered the same. The above definition of g applies to both single-input and double-input mixers. In a single-input mixer, es and ei are applied in series to the same input terminal-pair, so that g is equal to the partial derivative of i with respect to either es or ei. In a double-input mixer, however, es and ei are applied to separate terminal-pairs and in general they have unequal effects upon i. Therefore g is not equal to the partial derivative of i with respect to ei in a double-input mixer. Upon casual inspection of Fig. 1, it might be concluded from the shapes of the solid curves of i and g vs. ei that a derivative relationship exists and that the solid curves of Fig. 1 therefore represent a single-input mixer. On the other hand, the piecewise-linear curves i' and g' certainly do not possess the derivative relationship. The analysis which follows is organized in such a way that a derivative relationship between the i' and g' curves is not required, and consequently the results may be applied to either single-input or double-input mixers. In constructing the piecewise-linear curves, the discontinuities in g', which define the dynamic range D, are placed so that the area under g' is equal to the area under g and also so that the centers of gravity of the two areas are located at the same value of ei. For convenience, the origin of the ei scale is placed at the right-hand limit of D and the ei scale units are chosen so that D equals unity. This serves to normalize the injection potential with respect to the dynamic range and thereby simplifies some of the mathematical expressions. 5

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The discontinuity in i' is placed so that the area under i' is equal to the area under i, which results in most cases in placing the discontinuity somewhere near e. = -1/2. It will develop subsequently that the three mixer properties with which this paper is concerned can be expressed in terms of the zero- and first-order coefficients of i and g when they are expanded in Fourier series as functions of time. The reason for drawing the piecewise-linear curves with the equal area and center of gravity properties described above is that the zero- and first-order coefficients are approximately the same as those of the solid curves when this is done. The reason for this can be explained in terms of the g and g' curves as an example. Consider the transitional range of injection potential for which g has an appreciable value. This will be an interval of ei somewhat larger than D, but much smaller than the total swing of the injection potential. If this swing extends quite far beyond both limits of the transitional range, time and potential will be quite linearly related within the range, and consequently the areas under g and g', which have been made equal to each other when these quantities are plotted against ei, will now be approximately equal to each other when g and g' are plotted against time. Now let g and g' be expanded in Fourier series as functions of time, with the zero reference chosen so that they are even functions. If t is time in seconds and if wi is the injection frequency in radians per second, the Fourier coefficient of order i is al = (2/n) f g" cos (wijt) d(wit), (1) 0 where g" represents either g or g'. Now if 01 and 92 are the angular limits of vijt corresponding to the interval of potential or time being considered, and if l this interval is quite small, and has been assumed, the variation of cos (wit) over this interval is small enough so that Wjit may be treated simply as a constant 6

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 8, where P is some angle between @1 and 92. Now (1) can be rewritten as an approximation 2 al 2 cos I 1g" d(cwit). (2) I @1 The integral in (2) is simply the area under g or g' when plotted against time, and since these areas have been made approximately equal, we conclude that this integral has approximately the same value for g' as for g. Its value can be obtained from the g' curve by choosing the angular limits @1 and @2 to be the exact limits of the nonzero segment of g', as indicated in Fig. 2. The value of P is the same for both g' and g because of the equal center of gravity condition. Thus al is the same for both g' and g. A similar analysis justifies the equal area conditions for i' and i. III. CONVERSION TRANSCONDUCTANCE It has been noted previously that when the load resistance R is not small a distinction may be made between the incremental transfer conductance g (or g') of the circuit as a whole, acting as a straight amplifier, and the incremental transconductance gm of the tube alone. For the same condition, a distinction may be made between what might be called the conversion transfer conductance of the whole circuit as a heterodyne mixer and the conversion transconductance of the mixer tube alone. However, for simplicity, this distinction will be avoided here, and the symbol Gc will be used to represent both quantities. Consider a mixer circuit whose characteristics are represented by the g' curve in Fig. 1. The development of the conversion transconductance Gc follows the familiar pattern, with a few exceptions. For the reader's convenience, we review it here. Suppose that a signal potential _ ~~~~~~~~~~~7

SS-9Z-O1 we OL-99-v z9zz AND3 2 r Oi t I a, 828 eimr AN -1/2.... 7 g max. --- t +, Imax I TIONS ARE USED.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN es = Es cos(ost), (3) where Es is very small, is applied to the appropriate mixer terminal-pair. Since the transfer conductance g' was defined previously as the partial derivative of the output current with respect to es, we have o f= g' des, (4) where io is the output current (not shown in Fig. 1) corresponding to g'. Since Es is very small, g' is essentially independent of es, so that io = g' I des = C + g'es (5) where C is the constant of integration. Now let the injection voltage be an even function ei = Edc + Ei cOS(Oit). (6) Then g' and ei are related in time as shown in Fig. 2, and g' can be written as a Fourier cosine series g' = b0/2 + bl cos(ait) + b2 cos(2uit) + etc. (7) If (3) and (7) are now substituted for g' in (5), the current becomes io = C + (bo/2)ES cos (wfst) + (bl/2)E, cos(0s - i)t + (bj/2)ES cos(c + oi)t + (b2/2)Es cos(cws - 2i)t + (b2/2)Es cos(ws + 2a)t + etc. (9) One of the terms in (9) has an angular frequency (wus - i) the ratio of the coefficient of this term to the amplitude Es of the signal potential is defined as the conversion transconductance Gc. Evidently, Gc = b1/2, (10) which is the well known relationship. The value of b1 for the piecewise-linear function g' is given by the formula for the Fourier coefficient of an even _ n~~

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN function b = (2/i) g' cos(coit) d(it). (11) 0 From Fig. 1 or Fig. 2, it is seen that g' is equal to a constant maximum value g'1 over the dynamic interval and equal to zero elsewhere, so that if 91 and 62 are the angular limits of the g' pulse, as indicated in Fig. 2, bl= (2/'n) g cos (Cit) d(wit). (12) @1 By substituting (12) in place of bI in (10) and performing the indicated integration, we obtain the following expression for the conversion transconductance Gc: Gc = (g'max/)(sin @2 - sin 91) (13) The difference (sin @2 - sin 01) can vary from zero to unity, so that it can serve conveniently as a definition of the relative conversion transconductance, which will be denoted here by the symbol Gcrel. Since 01 and @2 are quantities not easily measured, it is more useful to have Gcrel expressed in terms of more easily measured quantities such as certain components of the injection potential. Three component values suggest themselves as possibilities: the d-c or bias component Edci the peak value Eim of the a-c component, and the peak instantaneous value eim of the total injection potential (see Fig. 2). If two of these are known, the third is determined, so that only two are needed in the expression for Grel. It happens that Gcrel is most sensitive to changes in eim, so that it seems logical to use eim as one of the values in the new expression. Eim and Ede are in general rather large quantities, while eim is the small difference between them, so that if Eim and Ede are taken as the measured values, a small fractional error in either results in a large fractional error in eim. Consequently it is desirable to use eim together with either one of the other two in the new expression for Gcrel. Edc has been chosen here. From trigonometry, we find that - 10

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN creli 2 (14) Gth t - Edc + eim - Edc ein tcrel - (15) Values of Gcrel, as given by (14) and (15) and expressed in decibels, are plotted in Fig. 3 for the more interesting ranges of values. For comparison, Fig. 10 shows some measured values of Gcrel taken from an experimental circuit which will be described subsequently. The comparison shows good agreement between theoretical and practical results when (-Edc) is large and eim is positive, as should be expected from the discussion in connection with (2). It is interesting to note that when (-Edc) is large in comparison with both eim and unity the relative conversion transconductance becomes approximately equal to,2/(c), which suggests that the circuit might be useful as an analog for an inverse square root function. In any practical circuit, where the transconductance is more accurately described by g than by g' (Fig. 1), it would be necessary to hold eim constant at zero or some small value in order to achieve this result. IV. INJECTION FREQUENCY COMPONENT Consider again the hypothetical mixer whose characteristics are represented by the curves in Fig. 1. Let the piecewise-linear approximation i' replace i in the analysis. (Note that i' is not the same as the io of the preceding section, which was obtained by integrating g' with respect to ed.) As shown in Fig. 2, i' is an even function of time, so that it can be represented

eim -I ~-0.5 0 + 0.5 +1 + 1.5 0~~~~~~~~~~~~~~~~~~~~~~~~~ Edc=- -5 -l0 w 1 5,cj 00 w-5 02 w 0~~~~0 rb.)~~~~~~~~~~L 30 FIG. 3. THEORETICAL VALUES OF RELATIVE CONVERSION TRANSCONDUCTANCE Gcrel OBTAINED FROM EQUATIONS (14) AND (15) EXPRESSED IN DECIBELS. THE SYMBOL Edc REPRESENTS THE d-c INJECTION BIAS POTENTIAL AND eim IS THE INSTANTANEOUS MAXIMUM TOTAL INJECTION POTENTIAL.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN by a Fourier cosine series: i' = Bo/2 + B1 cos(owit) + B2 cos (alit) + etc. (16) The coefficient B1 is evidently the amplitude of the injection-frequency component of the output current, and will hereafter be denoted as Ii. From the formula for the Fourier coefficient of an even function, we have I = (2/n) f i' cos (cit) d(wit) (17) Let 93 denote the value of alit for which ei = -1/2. It is evident from Fig. 2 that i' has a constant maximum value i'mx between 0 and 03 and a value of zero between @3 and i so that (17) may be rewritten as 33 Ii = (2/g) it I cos (aotit) d(wit) = (2/i) i'max sin @3 (18) As a function of 93, Ii evidently has a maximum value equal to (2/t)i'max when 93 = t/2 and a relative value Iirel sin 9.3 (19) From trigonometry, Iirel can be evaluated in terms of Edc and eim as E -dc 1/2 2 irel (20) -Edc + elm Values of (20), expressed in decibels, are plotted in Fig. 4 as functions of ei, with Edc as a parameter. By dividing either (14) or (15) by (20), depending upon the value of eim, we obtain a number which is proportional to the ratio of the amplitude of the desired difference-frequency component to the amplitude of the undesired injection-frequency component in the output current. Since this number is only proportional to, not equal to, the ratio of the amplitudes, its actual value is of less interest than the way in which its value changes with eim for different 13

eim -0.5 0 0.5 1.0 0.5 0 Ee~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~( -5 I c'.) W 4 m '~J -I1 F~ -201,/ _C'5~2~0 1 - 25 FIG. 4. GRAPH OF THE RELATIVE AMPLITUDE Ii re OF THE INJECTION FREQUENCY COMPONENT IN THE OUTPUT CURRENT, ACCORDING TO EQUATION (20), EXPRESSED IN DECIBELS.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN values of Edc. This is shown in Fig. 5, where ordinates are expressed in decibels. The curves of Fig. 5 may be obtained simply by subtracting the curves of Fig. 4 from those of Fig. 3, for the corresponding values of Edc. In the limiting case where Edc approaches minus infinity, Gcrel sin 2 sin 2 s /(eim + 1) 2eim (21) Iirel sin 3 2/ 2eim + 1 2eim + 1 for the case where eim is positive and Gcrel sin 2 - sin 1 / 2(eim + 1) (22) Iirel sin @3 2e im+l1 for the case where eim lies between -1/2 and zero. These are relative values. The actual value of the ratio of the amplitude of the difference-frequency component of the output current to the amplitude of the injection-frequency component of the same current can be obtained, in decibels, by adding the term 20 log10 (Esg'ma/2it' x) to the curves of Fig. 5. Here, Es is the peak amplitude of the signal potential, as indicated by (3), and g'max and i'max are the values used in (12) and(18) and pictured graphically in Fig. 2. This ratio is important in cases where the injection frequency and difference frequency are not widely separated, since the filter which usually follows the mixer must be designed to pass one and rejct the other. The curves of Fig. 5 exhibit sharp corners at eim = 0 and steep slopes as eim approaches -1/2 because of the discontinuities in the i' and g' curves at these points. In curves representing measurements in actual circuits, these features are missing, as would be expected. Figure 12, for example, shows curves of measured values in a particular circuit to be described subsequently. 15

10 05 - 0. I- - - bLI _m -5 -10 10 0dc -J20 -25 -0.5 00. 5.0 1.5 eim ---,FIG. 5. GRAPH OF THE RATIO OF THE RELATIVE CONVERSION TRANSCONDUCTANCE Gcrel TO THE RELATIVE AMPLITUDE Iirel. OF THE INJECTION-FREQUENCY COMPONENT IN THE OUTPUT CURRENT, ACCORDING TO EQUATIONS (14), (15), AND (20), EXPRESSED IN DECIBELS THE LIMITING CASE. WHERE Er.=-0o, IS GIVEN BY EQUATIONS (21)

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The theoretical and measured curves show reasonably close agreement, however, for values of eim well removed from -0.5 and 0. V. INJECTION MODULATION COMPONENT The injection source always has a slight modulation in its amplitude because of noise, power-frequency ripple, microphonics, etc. When the injection potential in a heterodyne mixer becomes very large, these variations become important. The d-c component Idc of the output current of the mixer is a function of the amplitude of the injection potential. Consequently, when the injection potential varies in amplitude in accordance with noise, ripple, or microphonics, the d-c output current varies in a corresponding manner. The variation in injection potential amplitude is equivalent to an equal variation in the peak value eim while Edc remains constant. Consequently, the relative amplitude of the injection modulation-frequency component in the output current can be expressed in terms of the slope of a curve of the d-c current plotted against eim, for various fixed values of Edc. In Eq (16), the d-c component of i' is Idc = B0/2. The formula for this Fourier coefficient is Idc = Bo/2 = (1/n) If i' d(aoit) (23) From inspection of Fig. 2, it is evident that i' is equal to a constant value i'max between the limits oit = O and wit = @3, and equal to zero from ait = 33 to Wit = n, so that (23) reduces to Idc = B/2 = (l/) i'max d((wit) = (93/) i' (24) In general, Edc is a large negative quantity. For the present, it is assumed that 17 -

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Edc is at least as negative as -1/2. For this limiting case, @3 = r/2, so that the corresponding limiting value of BO/2 is (Idc)max = (1/2)i'max The relative value (Idc)rel of the d-c current component Idc is therefore equal to (Idc)rel = = Idc (25) (Idc)max (1/2)i'max 7/2 A graph of this quantity is given in Fig. 6 as a function of eim, with Edc as a parameter. It is evident in Fig. 6 that all curves approach zero very rapidly as eim approaches -1/2, where Fig. 1 shows a discontinuity in i'. This would lead to the conclusion that the injection modulation component, which is proportional to the slope of the curves, as mentioned above, should have a very large value as eim approaches -1/2. However, in any practical circuit, a sharp discontinuity in i' such as that shown by the dashed line in Fig. 1 does not occur; the actual behavior is more like that represented by the solid line in the same figure. Consequently, steep slopes in the d-c curves are absent in practice, and the curves of Fig. 6 are not particularly useful for negative values of eim. For example, Fig. 11 shows some relative d-c current values measured in a mixer circuit which will be described subsequently. It can be seen that the agreement between these measured characteristics and the curves of Fig. 6 is quite good for positive values of eim. Many mixers are operated with eim in this positive region, so that for many practical situations equation (25) and Fig. 6 are quite satisfactory. From elementary trigonometry, we find that the angle @3 is (25) is @3 = arcsec Edc + eim (26) -Edc - 1/2 18

IC) N ~~~~~~~~~1.0 ~ ~ ~ ~ ~ 0.8 0.6 0.44 ~D~~~~~~~~~~~~~~~~~~~~~~~\ 0.2 0 - 1.0 -0.5 0 0.5 1.0 i.5 2.0 eirn FIG. 6. GRAPH OF THE RELATIVE VALUE Zdc rel OF THE D-C COMPONENT OF THE OUTPUT CURRENT, ACCORDING TO EQUATION (25). ALL CURVES APPROACH UNITY AS elm BECOMES VERY LARGE.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The derivative of (26) with respect to eim is d (-Ed - 1/2) de 3 2 im (-Edc + eim) (-Edc + eim) - (-Edc - 1/2)2 which is a measure of the relative amplitude of the injection modulation-frequency component. The ratio of the relative conversion transconductance as given by (14) and (15) to the relative amplitude of the injection modulation-frequency component given by (27) is a figure of merit for the mixer which is worth some attention when the injection modulation component falls within the passband of the difference-frequency filter following the mixer. The symbol a will be used here to represent this ratio. When eim lies between -1/2 and 0, a is found by dividing (15) by (27), with the following result: eim -2Edceim-2Edc-l eimdc2Edceim-Edc-l/4 (28) (-Edc - 1/2) When eim is positive, a is found by dividing (14) by (27), with the result [ i 2 d 2 eimd-2Ecl-2de -1/ _elm d Ecem2dc-1 -Ee /eim dcEdei im Edc-/4 (-Edc - 1/2) (29) The value of a, expressed in decibels, is plotted in Fig. 7 as a function of eim, with Edc as a parameter. The value evidently changes very little as a function of Edc, so that only three curves, drawn for Edc = -1, -2, and - co, are sufficient to describe it. In the limiting case where Edc = - a, (28) and (29) reduce, respectively, to a = /2(eim + 1) (2eim + 1) (30) and a = /2(eim + 1) - 2eim 2eim +1 (31) 20

E I I= -I 0of,9 IEdc= -I z ts 0 -2 0I -2 -3....... -0.5 0 0.5 1.0 1.5 2.0 eim - FIG. 7. THEORETICAL VALUE OF THE FIGURE OF MERIT cr, DEFINED BY EQUATIONS (28), (29), (30), AND (31), EXPRESSED IN DECIBELS.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The sharp points in the curves at eim = 0 are due to the discontinuity in g' at this point, while the large attenuation as eim approaches -1/2 is due to the discontinuity in i', as mentioned previously. In a practical circuit, the sharp points are rounded off and the attenuation is less severe. All of the curves in Fig. 7 approach zero as eim becomes more positive. Evidently the conversion transconductance and the injection modulation component both decrease at the same rate as eim becomes large, so that nothing is to be gained or lost as far as a is concerned by changing from one large positive value of eim to another. VI. EXPERIMENTAL RESULTS As an illustration of the piecewise-linear method of mixer analysis, the experimental circuit shown in Fig. 8 is used. The signal and injection frequencies are 1000 and 60 cycles per second, respectively. The output current through the 24,000 ohm resistor R3, is proportional to the output potential, the components of which are measured by means of a high-resistance d-c voltmeter and a high-impedance audio-frequency wave-analyzer. The measured values of the d-c output voltage and the relative transconductance are plotted as solid lines in Fig. 9, as functions of the measured d-c component of the input potential. The dashed lines in Fig. 9 are piecewise-linear approximations determined in the manner described in connection with Fig. 1, with which Fig. 9 may be compared. For convenience in relating the measured values to the theoretical values, the 1.5 volt cell E1 and the potentiometer R1 are included in the input circuit and the potentiometer is adjusted to make the right-hand step in g' fall at exactly zero volt on the ei scale. The cell E2 and the potentiometer R4 in the output circuit are used to adjust the d-c level of the output potential to the desired position. 22

E, I.5 VOLTS E2 1.5 VOLTS I 1 RI 400 SI R44004 lEdc l' l6H6 Edc eOUT Ei( (TO D-C VOLTMETER 60cps eiN R3 AND 24K&2 WAVE-ANALYZER) R2 Es, IOKQ I000 cps ' FIG. 8. AN EXPERIMENTAL CIRCUIT USED TO ILLUSTRATE THE APPLICATION OF THE PIECEWISE-LINEAR ANALYSIS. 23

0.2 e OUT 0.1 -1.0 -0.5 -0.31 0 0.5 -- elN -- - 1.0 0.8 g rel 0.6 e - I DUCTANCE g IN THE CIRCUIT OF FIG. 8, AS FUNCTIONS OF INSTANTANEOUS TOTAL INJECTION POTENTIAL ei. 24

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN By applying a graphical integration method to the g curve, the proper dynamic range of the g' curve is found to be 0.8 volt. This value then serves as the basis of normalization for the input voltage components Edc and eim where they appear in Figs. 10, ll,and 12, so that these graphs of the measured values can be compared directly with the theoretical curves in Figs. 3, 5, and 6. In Fig. 9, the abscissa represents the actual measured input potential,not the normalized value. When R1 is adjusted so that the right-hand step in g' falls exactly at zero, the left-hand step then falls at -0.8 volt (normalized value -1). Another graphical integration, this time applied to the i curve, then establishes the correct location for the step in i' at -0.31 volt (normalized value about -0.39). Relative values of conversion transconductance for the circuit of Fig. 8 are measured directly with the audio-frequency wave-analyzer. The results are plotted in Fig. 10. There is good agreement between these curves and the theoretical curves of Fig. 3, especially for large values of -Edc and +eim. The sharp corners in Fig. 3 are rounded off in Fig. 10, and the rapid decrease as eim approaches -1 is less pronounced in the actual circuit, as is to be expected. Relative values of measured d-c output potential (or current) are plotted in Fig. 11. These curves compare quite well with the theoretical curves of Fig. 6 for positive values of eim, but, as expected, not well for negative values. As mentioned previously, however, many mixers operate with positive values of eim. Figure 14 shows the ratio of the measured values of Gcrel and Iirel for the circuit of Fig. 8, expressed in decibels. These curves compare favorably with the theoretical values plotted in Fig. 5, except for values of eim near 25

0 -5 PI -0'4I cf)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ O 15 0 i. -o o..... I r.. 0 I ~ I -- o I_ 5.'x. _ _ - _o______ I1,,'/ _/~ ~.x. I u w / I,,.0 /qIIl~Z P', I e I x~.. j ~~ ~~ -20 P -25 -30 -35 - I0/ 0.5 0 0.5 1.0 1.5 N "0 N:::3 0 / ~K N__ ---ei --- — K / FIG. 10. MEASURED VALUE OF RELATIVE CONVERSION TRANSCONDUGTNT I THE CIRCUIT OF FIGURE 8. THIS MAY BE COMPARED WIT H -THEORETICAL VALUES OF FIGURE 3. - o'~~~~,,.,~~~~~~~~~~~~~~~ -20~ — __ _ _ _ _ //N -25 I N-I "N_ I~~~~~~~~~~~~~~~~~~~~~~~~~~ ' -30 o........ -1.0 -0.5 0 0.5 i.0 1.5 2.0 2.5 FIG. 10. MEASURED VALUE OF RELAiTIVE CONVERSION TRAiNSCONDUCTANCE IN THE CIRCUIT OF FIGURE 8. THIS MAY BE COMPARED WITH THE THEORETICAL VALUES OF FIGURE 3.

1.0 Edc~i 00 0.8 0.6 -- E /O /00.2 ~ ~ ~ ~ ~ - I/ 1 -'O.PP~~~~~~~~~~~~~~~~~~~~~cc 00 -~~~~~~~~~~~~~~ 0 ro 0 1~~~~~~~~~~~~~~ I100.~ 0.2 ___ 00 I~~~~~~ 0 10 0 _ ___ __ __ _ _ _ - 1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 eim - FIG. I I. EXPERIMENTAL MEASUREMENTS OF RELATIVE D-C CURRENT IN THE OUTPUT OF THE CIRCUIT OF FIGURE 8. THESE CURVES MAY BE COMPARED WITH THE THEORETICAL CURVES IN FIGURE 6.

10~~~~~~~~~~~~~~~~~~~~~~~~~~~ 5i 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 m0o UL) (0 ~~~~~~~___J Li -5 ~~~~o m~~~~~~~~~~~~~~~~d a -20~ ~ ~ ~~~~~~2 525 z-10 - 0. 5 0 0.5 1.0.5 () — ~~~~~~~~~~~~~~~~~ '-4~~~~~~~~~~l (9 ~ -15 -20c -25 - 3 0 1_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - 0.5 0 0.5 1.0 1.5 2.0 2.5 eim --- FIG. 12. THE RATIO OF THE MEASURED VALUES OF Gcrel AND 1irel FOR THE CIRCUIT OF FIGURE 8, EXPRESSED IN DECIBELS. THESE CURVES MAY BE COMPARED WITH THE THEORETICAL CURVES OF FIGURE 5,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 0 or -0.5, as mentioned previously in connection with the discussion of Equations (20), (21), and (22) and Figure 5........... 29

DISTRIBUTION LIST 1 Copy Director, Electronic Research Laboratory Stanford University Stanford, California Attn: Dean Fred Terman 1 Copy Commanding General Army Electronic Proving Ground Fort Huachuca, Arizona Attn: Director, Electronic Warfare Department 1 Copy Chief, Research and Development Division Office of the Chief Signal Officer Department of the Army Washington 25, D. C. Attn: SIGEB 1 Copy Chief, Plans and Operations Division Office of the Chief Signal Officer Washington 25, D. C. Attn: SIGEW 1 Copy Countermeasures Laboratory Gilfillan Brothers, Inc. 1815 Venice Blvd. Los Angeles 6, California 1 Copy Commanding Officer White Sands Signal Corps Agency White Sands Proving Ground Las Cruces, New Mexico Attn: SIGWS-CM 1 Copy Commanding Officer Signal Corps Electronics Research Unit 9560th TSU Mountain View, California 60 Copies Transportation Officer, SCEL Evans Signal Iaboratory Building No. 42, Belmar, New Jersey FOR - SCEL Accountable Officer Inspect at Destination File No. 22824-PH-54-91(1701 ) 3o

1 Copy H. W. Welch, Jr. Engineering Research Institu University of Michigan Ann Arbor, Michigan 1 Copy J. A. Boyd Engineering Research Institute University of Michigan Ann Arbor, Michigan 1 Copy Document Room Willow Run Research Center University of Michigan Willow Run, Michigan 11 Copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan i Copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan 31

UNIVERSITY OF MICHIGAN 11111 1 02827II II 2527 1 i III[[1111[ 111 3 9015 02827 2527