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.

4 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. 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.

5 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. 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.

6 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.