The Mathematics of Mixers: Basic Principles

By Gary Breed Editorial Director

ixers are classic This month’s tutorial is RF/microwave a first introduction to the circuits that f1 + f2 M f 1 mathematical principles make it possible to trans- f1 – f2 that describe the operation late RF signals from one of frequency mixers frequency to another. Ideally, they implement this frequency change with no effect on the amplitude and frequency components of the f2 signal’s modulation. Figure 1 · The frequency translation scheme Frequency Translation that is the goal for a frequency mixer. Mixers are nonlinear circuits; they rely on near-perfect nonlinearity. This sounds like a contradiction, but it means that perfect tude, with the actual rate of decrease versus switching—discontinuity being the ultimate order determined by the quality of the mixing nonlinearity—will result in ideal mixer behav- circuit. In all cases, the second order respons- ior. We will describe how this switching takes es will have the highest amplitudes: place in a circuit later on, but first let’s review

the overall behavior of the mixing process. f1 + f2 Nonlinear response creates new signals f1 – f2 (actually: |f1 – f2|) where none previously existed. In the case of

two unmodulated signals applied to the input 2f1 and 2f2 are also second-order outputs, of a nonlinear device, there will be a series of but nearly all practical mixers use a balanced output signals that contain multiples of the design to suppress these outputs, as well as all input signals (harmonics), plus sums and dif- other even-order harmonics. ferences of ALL signals, fundamental and har- Figure 1 shows the frequency translation monic, as described by [1]: scheme we want to obtain from an ideal mixer. If there are no other outputs, if components

fout = |nf1 ± mf2| are ideal (lossless), then the circuit performs the function of multiplication [1], represented

where fout represents all output signals, f1 and as the trigonometric identity: f2 are the two input signals, n and m are the ω ω order of the harmonics, from zero (fundamen- cos( 1)cos( 2) = ω ω ω ω tal) to infinity. [cos( 1 + 2)]/2 + [cos( 1 – 2)]/2 Mathematically, this is an infinite Fourier ω ω type of series, where the amplitude of each where cos( 1) and cos( 2) are the time-domain discrete output frequency dependent on the representations of f1 and f2. The 1/2 factors order. Higher order results are lower in ampli- simply show that the input amplitude is divid-

34 High Frequency Electronics High Frequency Design MIXER THEORY ed between the two output terms. In practice, this represents a 6 dB conversion loss. Usually, we want only one of the mixer’s outputs, so the unwanted signal must be removed, either by filtering, or by implementing an image-reject mixer topology that is actually two mixers with phase shift circuitry that results in a single sum or difference output. Filters have finite stopbands, and image-reject mixers have finite rejection of the unwanted signal. In a sensitive receiver, these imperfect responses may allow strong signals outside the desired passband to be detectable. To minimize this pos- sibility, the relationship of input and output signals must be considered. f1 + f2 should be chosen so higher-order responses do not fall within the passband of the interme- Figure 2 · An ideal mixer has devices (diodes or tran- diate frequency (IF) filter. Rather that repeat the equa- sistors) that act as perfect switches. tions and charts for this type of analysis, References [2, 3] should be consulted. ⎡π2 ⎤2 Real Circuit Performance ⎢ ()RR+ ++ RR⎥ ⎣ 4 gSWLSW⎦ An ideal mixer requires perfect switches, as illustrat- L = 10 log conv π2 ed in Figure 2. In this double-balanced circuit, switches A- RRLg D, and B-C are alternately activated at the local oscillator frequency, which is the unmodulated signal that deter- Using the above equation, an ideal switching mixer mines the amount of frequency difference between input would have a conversion efficiency (in dB) of: and output signals. In this ideal mixer, the local oscillator signal is not a sine wave, but an ideal square wave with = 4 Lconv 10 log normal and inverted polarity providing the “push-pull” or π2 balanced LO control to the switches. However, practical circuits do not have zero loss resis- which is –3.92 dB. Thus, all mixers will have greater than tance or instantaneous transition times, so an analysis of 3.92 dB conversion loss. performance must include these terms. Oxner [4] pro- Finally, Oxner provides the following expression that vides the following description: describes the switching function relative to the rise/fall An ideal square wave drive will result in switching time of the LO switch driver signal (for FET switches): action according to the Fourier series: 2 ⎡ V ⎤ ⎢t ω s ⎥ 1 2 ∞ sin[]21nt− ω ⎣ rLOV ⎦ Fx( ) =+∑ 20 log c π − 2 n=1 []21n 8

The switching function is derived from this equation where Vc is the peak oscillator voltage, Vs is peak signal as a power function by squaring the first term. Thus the voltage, and tr is the rise/fall time of Vc. output power deliverable to the output (IF) is: References V 1. K. McClaning, T. Vito, Radio Receiver Design, Noble P = 0 or, out Publishing, 2000, Ch. 3, “Mixers” (now distributed by RL SciTech Publishing).

2 2. L. Besser, R. Gilmore, Practical RF Circuit Design VR = in L Pout for Modern Wireless Systems, Vol. 1, Artech House, 2003, ⎡π2 ⎤2 + ++ Ch. 3, Section 3.2.6.3 “Spurious responses.” ⎢ ()RRgSWLSW RR⎥ ⎣ 4 ⎦ 3. R. Carson, Radio Communications Concepts: Analog, John Wiley & Sons, 1990, Ch. 9 “Spurious where RL is the load impedance, Rg is the internal loss Responses.” and RSW is device loss (diode junction, or FET RDS). 4. E. Oxner, “A Commutation Double-Balanced Mixer Conversion efficiency is obtained by the ratio of Pavg of High Dynamic Range,” Proceedings, RF Expo East, and Pout: 1986, p. 73.

36 High Frequency Electronics