Circuit provides more accurate multiplication
Convert into and out of the sampled-data domain to get an accurate multiplication.
Yakov Velikson, Lexington, MA; Edited by Paul Rako and Fran Granville -- EDN, December 15, 2011
Common analog multiplying devices employ methods using transistor parameters. Precise versions of these devices use the logarithm method of multiplication. This method involves the addition of logarithms and an exponential conversion (Reference 1). Using these methods, you can achieve a minimal error of ±0.1%. This Design Idea reduces the error, employs readily available standard components, and maintains the correct voltage scale.
The structure squares the sum and
the difference of both components of
the desired multiplication. The difference
of these squared values yields the
result of the multiplication. You can
scale the desired multiplication of a
and b using the identity of 4ab=(a+b)2−
(a−b)2. In a conceptual diagram, blocks
1 and 2 represent the input part of the
device (Figure 1). They comprise identical
precise rectifiers. You implement
these rectifiers with amplifiers A1, A2,
A3, and A4 (Figure 2). They provide the
addition and the subtraction of input
voltages VA and VB. The rectifiers create
the output voltages k(VA+VB), k(VA−VB),
which have only positive polarity. You
connect these outputs to a two-channel
ADC, Block 3, and then to two identical
DACs: DAC1 (Block 4) and DAC2
(Block 5).
Click to enlarge
These codes establish the values on the
R−2R dividers of each DAC. The output
voltages of blocks 4 and 5, comprising
N1|k(VA+VB)| and N2|k(VA−VB)|,
pass through operational amplifier A7
in Block 6. You configure the op amp
with a differential input, which takes
the difference between the inputs and
creates the multiplication result on the
output. For example, if both voltages
VA and VB have a range of ±10V and
the input range of the ADC is 0 to 10V,
then coefficient k=R2/R1=0.5. The full
sum of each part should be ±10V. Table
1 provides the results for all four quadrants
of these conditions.
Editor's note: Figure 2 was updated on Feb 22, 2012.
Talkback
-
Is it possible to do a multiplication circuitry by just using an ADC to a DAC and connecting Va to the Input of the ADC and Vb to Vref of the DAC?
Henry Chan - 2011-20-12 00:00:15 PST -
the difference of squares approach is quite an old one, predating the Gilbert cell and WWII for that matter. This implementation is the most complicated one I've yet seen. It is clever in its own way, but analog squaring circuits as simple as a suitably biased MOSFET are satisfactory for many applications. Here, if you are going to perform an A/D conversion anyway, why not do the multiplication digitally, drive a single DAC if analog output is a must, and be done with it?
O. Laney - 2011-19-12 13:09:59 PST -
This is incomprehensible from the text alone. It is not until you look at the figure and see that he is using MULTIPLYING dacs that this makes any sense.
Al Cosand - 2011-19-12 10:39:31 PST -
since vref of dac's is full range,high dynamic,error is high.
jkl jkl - 2011-18-12 12:28:17 PST
