Analog multipliers and mixers, what do people actually use usually

[Infraviolet]

I've been trying to find a fit more about performing basic mathematical operations on analog signals, summing, integrating and differentiating are all pretty easy, but what about multiplying one signal by another.

Multiplier chips seem remarkably expensive, are there therefore other methods people often use instead, or is it just that multiplying analog signals is only done very rarely in modern designs?

There doesn't seem all that much discussion of multiplier circuits, Horowitz and Hill seems to have nothing about them except one or two mentions in the context of things to be added to other circuits. Are there any good resources about different methods of performing analog multiplication, and how they compare against each other in performance?

Thanks

[Kleinstein]

Many mixers in RF application don't muliply with with a sine function, but closer to a square wave. So the muliplication is more like switching the gain between +1 and -1. This way the exact amplitude of the LO signal has little effect. I am not an RF expert. There are different types of mixers used. The ones I know of are:   
1) nonlinear amplifiers and mixing by shifting the operation point (more 0 and +1 multiplication). Especially in older low cost applications
2) Diode ring mixers with a ring of 4 (Schottky) diodes and 2 transformers.  Good, but needs a relativel high LO level.
3) Mixers with active controlles switches (e.g. MOSFETs / JFETs) may be 2 or 4.
4) Gilbert cells (e.g. MC1496 or NE614 and similar). A kind of simplified multiplier with not so temperature stable gain for the LO.

Besides mixers there is also some muliplication in a variable gain amplifier. Here usually a rather slow control signal effecting a fast signal and usually no change of sign. This can be speciall chips, or amplifiers with a variable bias.  Another way are attenuators based on PIN diodes.

[iMo]

You may try with logarithmic amplifiers.
An opamp with a diode or a transitor(s) in the feedback.
ln(a*b) = ln(a) + ln(b)

I played once with the cheapest AD633, and indeed, the chips are rather expensive.

It depends on your application. The RF multipliers (mixers) are usually not linear as Kleinstein indicated, those "precision and linear" ones which you perhaps target are low freq and expensive (special trimmed chips with special on-chip references).

PS: the RF mixers' task is to create a big mess in the resulting spectra where you then filter out the spectral components you actually wants. The amplitude of the inputs and results does not matter much.

The precision ones do "real math" within some precision, like 1-2% error max, within a +/- 10V range, for example..

[rstofer]

Analog Devices AD534K factory trimmed to 0.25% is used in one of my analog computers.  +-15V rails, +-10V workable signal range.


https://www.analogmuseum.org/english/homebrew/vogel/

Good thing I only needed 1, they're quite expenive.  Digikey lists some and actually has stock on a few  variants.

[robowaffe]

If you can make a scalar and gate,  that is multiplying a number from 0 - 1 with another 0 - 1.

[David Hess]

The Gilbert cell and logarithmic multipliers are the most common.

Another one not mentioned yet uses variable amplitude pulse width modulation.  One input controls the amplitude and the other controls the duty cycle of a pulse output.  The area under each pulse is then the product of the inputs and can be recovered with low frequency filtering or integration.  This can be very accurate but has lower bandwidth.

A multiplier can also be made with an ADC and DAC.  One input is converted by the ADC, and then used to control the DAC.  The other input is applied to the DAC as the analog reference.

Another way, that I have only seen Tektronix and myself use, relies on a pair of matched FETs with one being used as a variable resistance element and the other compensating for the non-linearity of the first.  So a linear voltage versus resistance curve is produced and then the variable resistance can be used as part of a divider.  With some cleverness, the resistance can be floating instead of single ended.  Tektronix used this method for some variable gain controls up to 100 MHz.  I know that in the past this kind of multiplier/divider was also made from a pair of CdS photocells.

Operational transconductance amplifiers are inherently multipliers, but can also be used to synthesize a voltage variable resistance which can then be used for multiplication.  Two operational transconductance amplifier can be used to synthesize a floating voltage variable resistances (or inductance, or capacitance) instead of a single ended voltage variable resistance.

[Infraviolet]

David Hess, can you link to a schematic of that matched FET method. I'd be interested to see how it operates.

CdS photocells would, I assume, be only suitable for pretty slow signals, slow enough that even really slow ADC-->MCU-->DAC setups could outpace them.

iMo, where can I find more about the logarithmic method? Does that need matched diodes, I don't think such things are manufactured, I searched once when hoping to get some for a totally different application.

[David Hess]

David Hess, can you link to a schematic of that matched FET method. I'd be interested to see how it operates.

Tektronix did it in their 7D20 digitizer as shown below.

I have done with with JFETs and 2N7000 MOSFETs.  A constant current into the drain of one FET produces a voltage at the drain which is controlled by an operational amplifier driving the gate.  The same voltage is then applied to the gate of the matched FET yielding a constant channel resistance.  The channel resistance does vary slightly with drain voltage, but this can be compensated for by altering the gate voltage slightly with drain voltage.

[daisizhou]

AD834 looks more suitable and the price is very cheap

[macboy]

...

iMo, where can I find more about the logarithmic method? Does that need matched diodes, I don't think such things are manufactured, I searched once when hoping to get some for a totally different application.

You don't need well matched diodes necessarily, you can calibrate out a lot of matching error. However, since temperature is a huge factor in the V-I curve of a junction, you absolutely need thermally tracked diodes. Any temperature mismatch will result in errors. Ironically, a diode-connected transistor behaves more like an ideal diode (tracking the logarithmic VI curve) than a diode does. This is fortunate, as monolithic transistor arrays are available and can work well. Beware that non monolithic arrays are also common enough and must be avoided. They use individual dice which do not thermally track well.

[Terry Bites]

By far the biggest application for analog multipliers is as RF mixers and demodulators. The product of two sinusoids and such. DC precision doesn’t matter too much for that. AD534 and the like are more for computational applications or where you want AC precision at a relatively low bandwidth, say in lock in amplifier. Transconductance amplifiers and multiplier cells suffer with the heat. The multiplier cells rely on the very highly temperature dependent logarithmic I V characteristics of the PN junction, getting all that to cancel out in the cell is very difficult- hence laser trimming and associated production costs. OTA's are slow and work in only two quadrants only. See the LM13700 datasheet for some truly horrifying specs!
I'd put it firmly in the box marked novelties.  I have a friend who suffered a nevous breakdown working in a lab that was trying to create a "precsion OTA". (That's not a joke BTW).

Log/Antilog multipliers are limited to one quadrant. They are very fussy about transistor or diode matching and often need unobtainium temperature compensation resistors. Hmm, anyone ever ovenised one? Must get back to that synth controller.

Here's funky idea:
www.semanticscholar.org/paper/Analog-multiplier-using-operational-amplifiers-Petchmaneelumka-Songsataya/b3d0fac5fc510455dae543ea41b910658706759e

[David Hess]

Log/Antilog multipliers are limited to one quadrant. They are very fussy about transistor or diode matching and often need unobtainium temperature compensation resistors. Hmm, anyone ever ovenised one? Must get back to that synth controller.

Jim Williams designed several ovenized log converters when he worked at National Semiconductor and Linear Technology.  He used transistor arrays with one transistor acting as a temperature sensor and a second transistor operating as the heating element, leaving the remaining transistors free and temperature stabilized.

The earlier design he did used the LM389 which was a combined audio amplifier with 3 uncommitted transistors.  Unfortunately it has been out of production for a long time.

[edavid]

The Nonlinear Circuits Handbook is a classic reference on traditional analog multipliers, and is available online:

https://www.analog.com/en/education/education-library/nonlinear-circuits-handbook.html

These days it's very common to use digital multiplication instead.

[mawyatt]

We have a couple texts, "Research Perspectives on Dynamic Translinear and Log-Domain" (Kluwer Academic) by Serdijn and Mulder 2002 which is a collection related topics. Another is "Design and Analysis of Integrator-Based Log Domain Filter Circuits" by Roberts and Leung 2000 (Kluwer) which is a good reference for Log Domain Filters. Dr Robert Fox at University of Florida was also instrumental in development of Log Domain Filter synthesis & design back around 2000.

We developed a high order tunable lowpass filter chip in IBM 8HP SiGe BiCMOS (utilizing the superb SiGe bipolar transistors as log and exponential functions) back around 2000. The goal was to improve on-chip DR for active filters by utilizing the log compression of the input, then filtering by selective scaled integration, then subsequent expansion with the exponential function. The math behind filter type translation into the Log Domain is quite involved, and only trivial for simple 1st and 2nd order filter functions. What's intriguing is the filter function gets scaled in frequency linearly by the active device integrator bias current (collector current), and with good bipolar transistors this can scale many decades.

Recall this was an 8th or 9th order Inverse Chebyshev LPF, current tunable over 5 decades to ~600MHz but the overall filer noise figure was compromised.

The input signal and its' noise get compressed by the log function, and during the filter analog computations, active device noise as well as other internal noises contribute to the log domain signal path noise, similar to how active and passive devices contribute noise to a conventional active filter. However, upon output signal expansion by the exponential function back into the Linear Analog Domain, the signal noise which was compressed along with the signal itself at the Log input gets expanded, but the internal log function noise sources also get the same exponential expansion (they were not compressed by the input Log function, and thus higher relative to the input signal)) and subsequently corrupt the overall filter output SNR. Believe this is the fundamental reason Log Domain Filters never really caught on, they had poor overall NF.

BTW back then the IC simulators didn't handle noise properties very well in the time domain, especially with log compression and exponential expansion, one of the many reasons we and others began looking into Time Domain Noise Analysis simulations.

Anyway, this was an interesting adventure back then when we were trying to improve on-chip analog signal processing, and worthy of looking into for folks interesting in Log Domain circuits and functions.

Best,

[EPAIII]

In today's world multiplying two analog signals would be done by two A-D converters, a processor, and a D-A converter.

Back in the day, I designed an audio mixer where I did not want to pass the audio through a variable resistor because they always became noisy, sooner or later. So I searched and found a chip made by RCA: CA3060. It is was called a four quadrant, transconductance amplifier. To my great surprise, DigiKey has, IN STOCK, a version made by Harris at $301 per 100. They also list them as out of stock in smaller quantities but perhaps a phone call might shake some loose at a price you can afford. I doubt they are a stellar seller so they may be happy to sell as many as they can.

The Harris data sheet is not much help in designing with them.

https://rocelec.widen.net/view/pdf/nhy2au7dgv/HRISD034-3-35.pdf?t.download=true&u=5oefqw

But, as luck would have it, I just happen to have a paperback published by RCA ("Linear Integrated Circuits", 1970, RCA) with about three whole pages on their version of the chip, showing a number of suggested circuits. If you decide you like this chip, send me a PM with your e-mail address and I will forward those pages to you. Unfortunately I no longer have the circuit that I designed with them.

Edited: To remove incorrect statement about difference in RCA and Harris versions of the chip. They are apparently identical.

[EPAIII]

I seem to recall that there are other analog ICs that were designed to implement electronic gain controls in analog circuits. You might look into analog video and audio chips, probably proprietary designs by OEMs of professional and home equipment. However, I do not know if they are strictly multiplication of the two signals as the CD3060 does. They could be very non-linear or should I say inaccurate in that respect.

[edavid]

The CA3060 is not a dedicated multiplier, it is a triple OTA.  OTAs are usually used to implement VGAs (2 quadrant multipliers) as in your mixer application.  It's not in production - the parts you see at DigiKey are actually the Rochester Electronics surplus stock.

There's nothing special about the CA3060, so there's no reason to buy it except for repair stock.  The similar LM13700 dual is still in production
(but who knows for how long).

RCA did have a multiplier chip, the CA3091, but I don't think it was a success.

[TimFox]

A trick used in analog computers (back in the day) is the "quarter-square multiplier", that required two analog circuits that approximated the square function  U = V²/V₀.
The exact equation implemented with these approximate squarers is

A x B = (1/4) x [(A+B)² - (A-B)²] = (1/4) x [4AB]

[Nominal Animal]

That reminds me of the Karatsuba multiplication algorithm:

Basically, if you have two values \$x = x_0 + x_1 B\$ and \$y = y_0 + y_1 B\$, where \$B\$ is your base, then
$$xy = z = z_0 + z_1 B + z_2 B^2$$
where ordinarily
$$\begin{aligned}
z_0 &= x_0 y_0 \\
z_1 &= x_1 y_0 + x_0 y_1 \\
z_2 &= x_1 y_1 \\
\end{aligned}$$
except that Karatsuba found and showed that you can calculate the middle one last via
$$z_1 = (x_1 + x_0)(y_1 + y_0) - z_2 - z_0$$
essentially trading one multiplication for one addition and two subtractions.


Another square-related detail I stumble onto all the time is \$x^2 - 1 = (x + 1)(x - 1)\$.

The "weirdest" case is when linearly interpolating between RGB color values –– it's not the best colorspace to do this in, but it works acceptably well for many use cases –– using integer math, with enough zero bits between components so that the color triplet can be used as a single integer.  For example, with 24-bit RGB color, you need a 48-bit intermediate word.  The "weirdness" is that although \$2^8 = 256\$, the blending range is actually \$0 \dots 257\$, because each color component is restricted to \$0 \dots 255\$; and the multipliers for the source and target colors are \$257-x\$ and \$x\$ for \$0 \le x \le 257\$.

(When only the eight most significant bits of each color component is used, both \$x = 256\$ and \$x = 257\$ will produce the exact same results.  But, if all 16 bits per color component are used, this does matter.)

[macboy]

...
Log/Antilog multipliers are limited to one quadrant. They are very fussy about transistor or diode matching and often need unobtainium temperature compensation resistors. Hmm, anyone ever ovenised one? Must get back to that synth controller.
...

It is true that a log amp or an anti-log amp requires very careful temperature compensation. However, when you are always following your log amp with an anti-log amp (like in a multiplier), then it is sufficient to merely have temperature tracking between the PN junctions used for the two stages. Dual transistors or monolithic arrays achieve this well enough. Then, the temperature-affected term kT/q naturally disappears in the math and in practice. The major error sources are non-ideality of the PN junction, input offset/bias voltage/current of the opamps, and temperature stability of the resistors and opamps.

The Analog Devices Nonlinear Circuits Handbook referenced earlier by edavid provides a more thorough treatment of the log-antilog multiplier than any other source I've seen. It states:
"The accuracy and temperature-stability of log-antilog multipliers is excellent, approaching the performance of the more-complex pulse-modulation multipliers. Errors of less than 0.25% of fullscale, with drifts of 0.01%/C are readily achieved."

If single-quadrant operation is sufficient, then the log-antilog multiplier is one of the simplest and best performing options.

[TimFox]

The -hp- 7563A from 1969 put the semiconductors for logging in a conventional oven.
See p. 10 of  http://hparchive.com/Catalogs/HP_New_Electronics_for_Measurement_Analysis_Computation_Autum_1969.pdf

There is a clever method of cascading long-tailed pairs to get reasonable "log compliance" over the wide dynamic range required in spectrum analyzers and receiver IF amps at high frequencies.
This gives a "video" output corresponding to the AC input level over a > 100 dB range.
An example is the TI TL441 IC:  https://www.ti.com/lit/ds/symlink/tl441a.pdf?ts=1692805243587&ref_url=https%253A%252F%252Fwww.ti.com%252Famplifier-circuit%252Fspecial-function%252Flogarithmic%252Fproducts.html