The Mathematics of Mixers: Basic Principles

1 34 High frequency ElectronicsHigh frequency DesignMIXER THEORYThe Mathematics of mixers : Basic PrinciplesBy Gary BreedEditorial DirectorMixers are classicRF/microwavecircuits thatmake it possible to trans-late RF signals from onefrequency to , they implementthis frequency change with no effect on theamplitude and frequency components of thesignal s TranslationMixers are nonlinear circuits; they rely onnear-perfect nonlinearity. This sounds like acontradiction, but it means that perfectswitching discontinuity being the ultimatenonlinearity will result in ideal mixer behav-ior. We will describe how this switching takesplace in a circuit later on, but first let s reviewthe overall behavior of the mixing response creates new signalswhere none previously existed. In the case oftwo unmodulated signals applied to the inputof a nonlinear device, there will be a series ofoutput signals that contain multiples of theinput signals (harmonics), plus sums and dif-ferences of ALL signals, fundamental and har-monic, as described by [1]:fout= |nf1 mf2|where foutrepresents all output signals,f1andf2are the two input signals,nand mare theorder of the harmonics, from zero (fundamen-tal) to , this is an infinite Fouriertype of series, where the amplitude of eachdiscrete output frequency dependent on theorder.

2 Higher order results are lower in ampli-tude, with the actual rate of decrease versusorder determined by the quality of the mixingcircuit. In all cases, the second order respons-es will have the highest amplitudes:f1+ f2f1 f2 (actually: |f1 f2|)2f1and 2f2 are also second-order outputs,but nearly all practical mixers use a balanceddesign to suppress these outputs, as well as allother even-order 1 shows the frequency translationscheme we want to obtain from an ideal there are no other outputs, if componentsare ideal (lossless), then the circuit performsthe function of multiplication [1], representedas the trigonometric identity:cos( 1)cos( 2) = [cos( 1 + 2)]/2 + [cos( 1 2)]/2where cos( 1) and cos( 2) are the time-domainrepresentations of f1and f2. The 1/2 factorssimply show that the input amplitude is divid-This month s tutorial is a first introduction to themathematical principlesthat describe the operationof frequency mixersf1f2f1 + f2f1 f2 Figure 1 The frequency translation schemethat is the goal for a frequency mixer.

3 From January 2011 High frequency ElectronicsCopyright 2011 Summit Technical Media, LLC36 High frequency ElectronicsHigh frequency DesignMIXER THEORYed between the two output terms. In practice, this repre-sents a 6 dB conversion , we want only one of the mixer s outputs, sothe unwanted signal must be removed, either by filtering,or by implementing an image-rejectmixer topology that isactually two mixers with phase shift circuitry that resultsin a single sum or difference output. Filters have finitestopbands, and image-reject mixers have finite rejectionof the unwanted signal. In a sensitive receiver, theseimperfect responses may allow strong signals outside thedesired passband to be detectable. To minimize this pos-sibility, the relationship of input and output signals mustbe + f2 should be chosen so higher-orderresponses do not fall within the passband of the interme-diate frequency (IF) filter. Rather that repeat the equa-tions and charts for this type of analysis, References [2, 3]should be Circuit PerformanceAn ideal mixer requires perfect switches, as illustrat-ed in Figure 2.

4 In this double-balanced circuit, switches A-D, and B-C are alternately activated at thelocal oscilla-torfrequency, which is the unmodulated signal that deter-mines the amount of frequency difference between inputand output signals. In this ideal mixer, the local oscillatorsignal is not a sine wave, but an ideal square wave withnormal and inverted polarity providing the push-pull orbalanced LO control to the , practical circuits do not have zero loss resis-tance or instantaneous transition times, so an analysis ofperformance must include these terms. Oxner [4] pro-vides the following description:An ideal square wave drive will result in switchingaction according to the Fourier series:The switching function is derived from this equationas a power function by squaring the first term. Thus theoutput power deliverable to the output (IF) is:or,where RLis the load impedance,Rgis the internal lossand RSWis device loss (diode junction, or FET RDS).Conversion efficiency is obtained by the ratio of Pavgand Pout:Using the above equation, an ideal switching mixerwould have a conversion efficiency (in dB) of:which is dB.

5 Thus, all mixers will have greater dB conversion , Oxner provides the following expression thatdescribes the switching function relative to the rise/falltime of the LO switch driver signal (for FET switches):where Vcis the peak oscillator voltage,Vsis peak signalvoltage, and tris the rise/fall time of K. McClaning, T. Vito,Radio Receiver Design, NoblePublishing, 2000, Ch. 3, mixers (now distributed bySciTech Publishing).2. L. Besser, R. Gilmore,Practical RF Circuit Designfor Modern Wireless Systems, Vol. 1, Artech House, 2003,Ch. 3, Section Spurious responses. 3. R. Carson,Radio Communications Concepts:Analog, John Wiley & Sons, 1990, Ch. 9 SpuriousResponses. 4. E. Oxner, A Commutation Double-Balanced Mixerof High Dynamic Range, Proceedings, RF Expo East,1986, p. Lconv=1042log LRR RRRR convgSW LSWLg=+()++ 104222log PVRRR RRoutinLgSW LSW=+()++ 2224 PVRoutL=0 Fxntnn()=+ [] []= 12221211 sinFigure 2 An ideal mixer has devices (diodes or tran-sistors) that act as perfect switches.

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