Circuit to calculate square root of voltage?

Barry Gilbert himself freely admits in his publications that a certain Mr H.E. Jones has prior art on the "Gilbert Cell".

--
Rick

Barry is a quiet type of guy. He explained, improved, promolgated and popularized this circuit, so we insisted on naming it after him. He doesn't (didn't back then) use the name Gilbert Cell, hence his proposed name (IIRC), translinear. We liked Gilbert Cell better.

--
 Thanks,
    - Win

Come ON. (A "textbook" explanation, facile, unrelated to the established art that I referred to, and just what I was complaining about.)

I'll explain. Here are some things that emerge when you delve into translinear circuits, or even just their literature. This includes an explanation of them. (The following is supported in much more depth than I cite here.)

(a) The prototype translinear circuit, if the subject is viewed from a long focus, is the bipolar-transistor current mirror. (It's useful also in explaining.) It is both the simplest embodiment of the principle, and also is very obviously a current-input current-output function, underscoring the point that translinear circuits basically perform arithmetic operations on currents.

(b) In the current mirror, two junction devices (BJTs) share the same Vbe voltage. At the same time, these devices each have a nonlinear relation of Vbe to collector current (Ic) that is (to remarkable potential accuracy) exponential [1]. The connection of their Vbe voltages, and the exponential large-signal Vbe-Ic law, lead to a relation between the two collector currents, which prevails as long as the two transistors are happy (i.e., forward-active). In this very simple case, the two collector currents will tend to track, whatever else is happening, and this is useful. Notably, the two collector currents stay equal regardless of transistor construction (the notorious Js current-density parameter) or temperature (which shows up in kT/q, and even more so, implicitly, in Js). Of course, you must use matched (or deliberately area-ratioed) devices -- these designs are most often monolithic. Commercial monolithic dual or quad transistors or monolithic "kit parts" serve this need in a discrete design.

(c) Generalizing, if you put four Vbe voltages into a loop (two in series, equaling the voltage across another two in series), these voltages will sum algebraically to zero (KVL) and this again will impose a relation among collector currents. Depending on what you do with the collector currents, you can get explicit algebraic relationships among these currents, such as

I1 I2 = I3 I4

-- which, if I1and I2 are arranged to be identical and to equal the output current, and I3 is a fixed current and I4 an input current, will give a DC, large-signal, fundamentally accurate square-root relation between input and output currents. (Embed it with an op amp and resistors and you can get and give voltages.) Various simple, four-transistor square-root-law circuit cores pervade the "translinear" literature. Broader versions of the principle apply to general voltage loops of BJTs. Obvious further generalizations apply to other semiconductor devices (even with generalizations beyond voltage loops).

(d) The chief origin and exploration of these circuits in print is, as I already cited in the earlier posting, Barrie Gilbert's December-1968 JSSC "Amplifier" paper, not the companion "Multiplier" paper, which employed some of the results. I _strongly_ recommend reading them, they disabuse some notions cultivated elsewhere in print. The first is where the "legion" remark appeared. In 1975, BG coined the nickname "translinear" (because they rely on exponential I-V devices, which have TRANSconductance LINEAR in collector current) for this pervasive class of circuits [2]. Note well that --

(e) -- one of the basic four-junction translinear blocks is often called a "gain cell," focus of the 1968 "Amplifier" paper, and pretty clearly associated with Gilbert. I am using the language here carefully, read on. This cell contains two transistors at one DC current level, whose Vbe difference is then imposed easily onto a second pair at another DC current level. Current differences within the two pairs are input and output signals, and can have either polarity. Changing the relative DC currents, or "tail" currents, controls the _current_ gain in the signal path. If then you take two different output pairs and a single input pair, and drive the input pair with one differential current, and the "tail" currents of the two output pairs with a second differential current, you achieve an accurate four-quadrant (either polarity on either input) multiplication ("with subnanosecond response" even in 1967). Note carefully that the accurate relation is among currents (as always with classic translinear circuits). This is the true six-transistor multiplier core or "six-pack" as it was often called in the 1970s by IC designers. It and other variations are the upshots of the 1968 "Multiplier" paper that I cited.

(f) Some later authors, despite this black-and-white origin, took to calling a different set of six transistors (the two output pairs above, and the differential tail current source that drives them) a "Gilbert six-transistor multiplier" even though it is neither Gilbert's, nor a multiplier. The original "six-pack" is a true self-contained, current-mode four-quadrant analog multiplier and is the core of many monolithic bipolar-transistor multiplier products.

(g) Though the translinear idea was first explored in depth in Gilbert's

1968 "Amplifier" paper, analog circuits embodying the principle can be found earlier, even beyond the trivial example of a current mirror and its variations. Grasselli and Stefanelli's 1966 BJT current-gain cell is "almost" translinear. Certain JFET circuits falling outside Gilbert's original definition embodied useful generalizations of the idea as early as 1965, I've cited them in print in the past.

(h) Some of the oldest cases of what are called generalizations of translinearity (early to middle 1960s) employ the "square-law" relationships found in FET devices. An aspect of translinearity that carries over to its generalizations is that unpredictable device parameters can be cancelled out inherently. Temperature- and manufacturing-independent square-root functions have been designed using FETs cleverly, though the FET square laws have not the fundamental accuracy of the BJT exponential.

(i) Please note finally that translinear circuits are _not_ "log-antilog" analog circuits as found in classic analog-computer texts of Jackson or Korn and Korn or in the App Notes of makers of temperature-compensated log amplifiers (or classic discrete log-amp app circuits with Tel-Labs Q81 thermistors). They are compact, specialized circuits that directly implement large-signal, temperature-independent linear or nonlinear input-output relations, such as square roots.

I did not mean to claim that one of the simple four-transistor dedicated BJT square-rooting circuits, or the various FET square-rooting circuits, was always the best or simplest solution, but rather that they should be suggested, and considered, for this task.

Your servant -- MH

[1] Gibbons and Horn popularized this point in an ISSCC 1963 paper, still cited, demonstrating ten decades of current range, with attention to second-order effects. [2] _Electronics Letters_ vol. 11 no. 1 pp. 14-16, January 1975.

In article , Winfield Hill wrote: [snip]

For powers and roots I used to be a fan of a circuit taken from an Analog Devices publication, called the Multifunction Circuit. Two pairs of transistors and a few opamps.

That was in the days when packaged multiplier/dividers were a fearsome price........

--
Tony Williams.

In me, the Tony Williams:

Do you have the schematic of this circuit? I wish to see it.

[]s

-- Chaos Master®, posting from Canoas, Brazil - 29.55° S / 51.11° W

"Now: the 2-bit processor, with instructions: 1. NOP - does nothing, increase PC. 2. HLT - does nothing, doesn't increase PC 3. MMX - enter Pentium(r) emulation mode; increase PC 4. LCK - before MMX: NOP ; after MMX: executes F0 0F C7 C8 "

Did you cancel that? Could you repost it? It didn't appear.

--
 Thanks,
    - Win

Analog Devices Model 433. It was essentially a log-antilog multiplier/divider with a 4-element resistor bridge between the log-antilog sections. It would take powers/roots over an exponent range of 5/1 to 1/5. I used the circuit extensively after realising the curves of many of the exotic thermocouples could be simply described just by raising to a specific power.

Copied below, all opamps were inverters with their +ve inputs connected to 0v.

+---||--+ +----[R3]--+-Vout | | | | | | Vx--[R1]-+--|->--+---[R?]----+ +--|->-----+ | |/ | | |/ | | | +----+ | | | +---+---+ | +----|---+ | B | | +----+ | | | [Ra] [Rc] | | | | |/ | | | |/ | | 0v---|--| Q1a/b |----+A C+----| Q2a/b |--|---0v | |e e/| | | |e e/| | | | | [Rb] [Rb] | | | | +-+-+ | | +-+-+ | | | +---+---+ | | | [Rt1] | [Rt2] | | | -+-0v | | +---||-+ +-||---+ | | | | /| | Vz--[R2]-+--|->-+ +-1), B-C would be shorted and m = (Rb + Ra)/Rb.

For roots, B-A would be shorted and m = Rb/(Rb +Rc).

For a straightforward XY/Z multiplier/divider, short ABC.

The Model 433 had an internal 9v Vref and in manufacture the ratio R4/R3 was adjusted to equal the value of Vref/10.

--
Tony Williams.

About 100mS after upload you see the typo, ambiguity, whatever.

Rt1/2 are not temperature senstive resistors. They are simply Rtail-1 and Rtail-2 (of the long-tailed pairs). AFAIR I ran I-tail at about 125uA full scale using the RCA 5-transistor array.

--
Tony Williams.

No I didn't cancel it and it did appear back here.

Text of the original article reposted below........

Analog Devices Model 433. It was essentially a log-antilog multiplier/divider with a 4-element resistor bridge between the log-antilog sections. It would take powers/roots over an exponent range of 5/1 to 1/5. I used the circuit extensively after realising the curves of many of the exotic thermocouples could be simply described just by raising to a specific power.

Copied below, all opamps were inverters with their +ve inputs connected to 0v.

+---||--+ +----[R3]--+-Vout | | | | | | Vx--[R1]-+--|->--+---[R?]----+ +--|->-----+ | |/ | | |/ | | | +----+ | | | +---+---+ | +----|---+ | B | | +----+ | | | [Ra] [Rc] | | | | |/ | | | |/ | | 0v---|--| Q1a/b |----+A C+----| Q2a/b |--|---0v | |e e/| | | |e e/| | | | | [Rb] [Rb] | | | | +-+-+ | | +-+-+ | | | +---+---+ | | | [Rt1] | [Rt2] | | | -+-0v | | +---||-+ +-||---+ | | | | /| | Vz--[R2]-+--|->-+ +-1), B-C would be shorted and m = (Rb + Ra)/Rb.

For roots, B-A would be shorted and m = Rb/(Rb +Rc).

For a straightforward XY/Z multiplier/divider, short ABC.

The Model 433 had an internal 9v Vref and in manufacture the ratio R4/R3 was adjusted to equal the value of Vref/10.

--
Tony Williams.

This circuit is described at some length (about 9 pages as I recall) in the Nonlinear Circuits Handbook, published by Analog Devices and edited by Dan Sheingold. There's a lot of old-time circuits wisdom contained therein. I don't know if it's still in print, or available on the web - I keep a copy on my desk. Tony's quick description appears to cover most of the important bits.

Steve

A totally valuable book to analogue designers. I lost mine for about 15 years but recovered it earlier this year (from a customer's bookshelf of all places). The spine is broken and there appears to be a chunk missing, but what is left is still worth keeping for reference. Those 9 pages of the Multifunction Converter are still there and the posted sketch was copied directly from them.

Resistor [R?] was not in the A-D circuit. It is neccessary to avoid oscillations in the log-ratio circuit and to limit the voltage excursions on the Q1,Q2 bases. I arbitrarily sized [R?] so that, at full opamp output swing, the voltage at point B did not exceed about 1V.

--
Tony Williams.

Good! Thanks, I will check this circuit and maybe simulate it.

[]s

--
Chaos Master®, posting from Canoas, Brazil - 29.55° S / 51.11° W

"Now: the 2-bit processor, with instructions:
 1. NOP - does nothing, increase PC. 
 2. HLT - does nothing, doesn't increase PC
 3. MMX - enter Pentium(r) emulation mode; increase PC
 4. LCK - before MMX: NOP ; after MMX: executes F0 0F C7 C8 "

Speaking of square root circuits and design methodology (another thread), here's an interesting paper:

Best regards, Spehro Pefhany

--
"it's the network..."                          "The Journey is the reward"
speff@interlog.com             Info for manufacturers: http://www.trexon.com
Embedded software/hardware/analog  Info for designers:  http://www.speff.com

When m=1 the 433 circuit does Vout= X*Y/Z. If Z is connected back to Vout then X*Y = Vout-squared. If X is the input and Y is a fixed Vref then it functions as a square rooter.

However, (AFAIR) Win's suggested AD734 (and say the B-B MPY600) can function as a direct square-rooter, without the need for an extra Vref.

This is because this type of multiplier has a high gain internal opamp, which has X*Y on one input and Z on the other. Connections are always made so the the negative feedback forces (X*Y - Z) = 0.

If Z is made the input and Vout connected back to both X and Y then circuit is forced to do Vout= sq-rt(Z). No Vref needed.

--
Tony Williams.

What weed were those guys smoking ?:-)

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
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I love to cook with wine.      Sometimes I even put it in the food.

Nice. Also QPC19, Q77, Q184 and Q258.

Mike Monett

Page 6, Figure 8. Q123 looks a little suspicious.

--
Tony Williams.

[...]

Perhaps they never had to fix an autorouted board:)

Although it should be possible add more checking to the software to prevent those errors. For example, don't connect the collector to the emitter:)

Mike Monett

Works well? It has 0.1% accuracy over three input decades, with 10% supply-voltage and 50C temperature effects < 0.1%? Works over the usual production variations in BJT parms, and doesn't require a certain beta value or leakage current, etc.?

--
 Thanks,
    - Win