A recent Twitter thread about a counterfeit analog multiplier chip attracted my attention since I'm interested in both counterfeit integrated circuits and how analog computers multiply. In the thread, John McMaster decapped a suspicious AD633 analog multiplier chip and found an entirely different Rockwell RC4200 die inside. Why would someone do this? Probably because the RC4200 (1978) currently sells for about 85 cents, while the more modern laser-trimmed1 AD633 (1989) sells for about $7.2

Die of the RC4200 analog multiplier with functional blocks labeled. Die photo courtesy of John McMaster.

## Analog multiplication

Analog multiplication has many uses such as mixers, modulators, and phase detectors, but analog computers are how I encountered analog multiplication. A typical analog computer uses voltages to represent values and is wired up through a plugboard to solve a particular equation. Adding or subtracting two values is easy with an op amp, as is multiplying by a constant. Integration seems like it would be difficult, but it's almost trivial with a capacitor; analog computers excelled at solving differential equations.

Multiplying two values, however, was surprisingly difficult; multiplication techniques were slow, inaccurate, noisy, or expensive. One accurate but slow multiplier used the Rube-Goldberg configuration of servo motors turning potentiometers.3 A 1969 multiplier circuit uses a light bulb and photocells. A fast and accurate approach was the "parabolic multiplier", built from numerous expensive high-precision resistors.4 The approach I'll discuss is to multiply by adding the logarithms and taking the exponential. Inconveniently, this approach magnifies even small differences between the transistors. It is also very sensitive to temperature. As a result, this approach was simple but inaccurate.

The Model 240 analog computer from Simulators, Inc. includes analog multipliers using the parabolic multiplier approach.

However, the development of analog integrated circuits created new opportunities for analog multiplication circuits. In particular, since the transistors in an integrated circuit were created together, they have nearly-identical properties. And the components on a tiny silicon die are all at nearly the same temperature.5

The first analog multiplier integrated circuit I could find is a television demodulator from 1967. The Gilbert cell technique was introduced by Barrie Gilbert in 1968 and is used in most analog multipliers today.6 The AD530 was introduced around 1970, and became an industry standard, but required external adjustments for accuracy. Laser-trimming the resistors inside the integrated circuit during manufacturing greatly improved the accuracy, an approach used in the AD633, the integrated circuit that was counterfeited.

Before explaining the circuitry of the RC4200 (the multiplier inside the counterfeit chip), I'll discuss the components that it is constructed from, and how they appear in an integrated circuit. This will help you recognize these structures in the die photo.

### Transistors

Transistors are the key components in a chip. The photo below shows an NPN transistor in the RC4200 as it appears on the chip. The different blue colors are regions of silicon that have been doped differently, forming N and P regions. The white lines are the metal layer of the chip on top of the silicon—these form the wires connecting to the emitter (E), base (B), and collector (C).

An NPN transistor on the RC4200 die. The emitter is embedded in the base, with the collector underneath.

You might expect PNP transistors to be similar to NPN transistors, just swapping the roles of N and P silicon. But for a variety of reasons, PNP transistors have an entirely different construction. They consist of a circular emitter (P), surrounded by a ring-shaped base (N), which is surrounded by the collector (P).[4] This forms a P-N-P sandwich horizontally (laterally), unlike the vertical structure of the NPN transistors. The diagram below shows one of the PNP transistors in the RC4200.

A PNP transistor has a circular structure.

The input and output transistors in the RC4200 are larger than the other transistors and have a different structure to support higher currents. The photo below shows one of the output transistors. Note the multiple interdigitated "fingers" of the emitter and base.

A larger output transistor with parallel emitters and bases.

### Capacitors

Capacitors are important in op amps to provide stability. A capacitor can be built in an integrated circuit as a large metal plate separated from the silicon by an insulating oxide layer. The main drawback of capacitors on ICs is they are physically very large. The 15pF capacitors in the RC4200 have a very small capacitance but take up a large fraction of the die area. In the photo below, the red arrows indicate the connection to the capacitor's metal layer and to the capacitor's underlying silicon layer.

The large metal area on the upper left is a capacitor.

### Resistors

Resistors are a key component of analog chips. Unfortunately, resistors in ICs are very inaccurate; the resistances can vary by 50% from chip to chip. The photo below shows four resistors, formed using different techniques. The first resistor is the zig-zagging blue region on the left. It is formed from a strip of P silicon, with metal wiring (white) attached on the left and right. Its resistance is 3320 Ω. The resistor in the upper right is much shorter, so it is only 511Ω (long, narrow resistors have higher resistance than short, wide resistors). The remaining resistors are 20KΩ despite their small size because they are "pinch resistors". In the pinch resistor, the square layer of brownish N silicon on top makes the conductive region much thinner (i.e. pinches it). This allows a much higher resistance for a given size. (Otherwise, a 20 KΩ resistor would be 6 times as long as the first resistor, taking up excessive space.) The tradeoff is the pinch resistor is much less accurate.

Four resistors, one on the left and three on the right.

## Multiplying with logs and exponentials

This integrated circuit multiplies using the log-antilog technique.
The idea is that if you take the log of two numbers, add the logs together, and then take the antilog (i.e. exponential), you get the product of the two numbers.
Conveniently, transistors have a logarithmic / exponential characteristic:
the current through the transistor is an exponential of the voltage on the base.
Specifically, if V_{BE} is the voltage between the transistor's base and emitter, the current through the collector (I_{C})
is an exponential of that voltage, as shown in the graph below.
The analog multiplier takes advantage of this property.

Ic vs Vbe curve for a transistor, showing the exponential relationship. Generated by LTspice.

The main complication with this approach is that the curve above is very sensitive to the temperature and to the manufacturing characteristics of the transistor. Because the curve is exponential, even a small shift in the curve will radically change the current. This was a serious difficulty when building a multiplier from discrete transistors, since the properties varied from transistor to transistor. To stabilize the temperature, some multipliers used a temperature-controlled oven. However, using an integrated circuit mostly solved these problems. The transistors in an integrated circuit are well-matched since they were built from the same piece of silicon under the same conditions. And the transistors in an integrated circuit die will be at almost the same temperature. Thus, integrated circuits made transistor-log circuits much more practical.

The diagram below shows the structure of the RC4200 multiplier chip. The user provides three current inputs (I1, I2, and I4) and the chip computes the output current I3, where I3 = I1×I2÷I4. (The use of current inputs and outputs is a bit inconvenient compared to other multipliers, such as the AD633, that use voltages.)

_{OS1}and V

_{OS2}are offset adjustment pins to improve accuracy.

The four transistors in the middle of the diagram are the multiplier core, the key to the IC's operation.
The transistors are configured so their base-emitter voltages sum:
V_{BE3} = V_{BE1}+V_{BE2}-V_{BE4}.
Because the transistor current is related exponentially to the voltage,
the result is that I_{3} = I_{1}×I_{2}÷I_{4}.

In more detail,
first note that the voltages V_{BE1} through V_{BE4} control the collector currents I_{C1} through I_{C4} through the transistors (below).
The op amps adjust the base-emitter voltages so the input currents match the transistor currents, i.e. I_{1} = I_{C1} and so forth.
(This is accomplished by op amp feedback.)
Now, if you go through the loop of base-emitter voltages starting at the base of Q1 and ending at the base of Q4 (red arrows), you find that
V_{BE1}+V_{BE2}-V_{BE3}-V_{BE4} = 0. (The voltages must sum to zero since you start at ground and end at ground.7)
Now, because I_{C} is related to exp(V_{BE}), taking the exponential of the equation yields
I_{C1}×I_{C2}÷I_{C3}÷I_{C4} = 1.
(Details in footnote8.)

Traveling around the loop indicated by the arrows, the voltages must sum to 0.

Next, I'll explain how the V_{BE} voltages are generated. Each current input has an op amp associated with it that produces the "correct" V_{BE} voltage for the current using
a feedback loop9
For example, suppose I_{C} is too low so not all the input current flows through the transistor. The excess current will
raise the voltage on the op amp's negative input, causing it to reduce its output voltage and thus the transistor's emitter voltage.
This raises V_{BE} (since the base will now be higher compared to the emitter), causing more collector current to flow through the transistor.
Similarly, if too much current is flowing through the transistor, the op amp's input will be pulled lower, reducing V_{BE}.
Thus, the feedback loop causes the op amp to find the exact V_{BE} for the current input.10

### Correcting for emitter resistance

The above circuit works reasonably well, but there's a complication:
the transistors have a small emitter resistance R. The voltage drop across this resistance will increase V_{BE} by I_{C}R, disturbing the
nice exponential behavior.
This creates a nonlinearity that reduces the accuracy of the result. The datasheet says that "Raytheon has developed a unique and proprietary means of
inherently compensating for this undesired term." They don't explain this further, but by studying the die I have figured out how it works.

In the compensation circuit, each of the four multiplier transistors is paired with an identical "mirror" transistor with the corresponding emitters and corresponding
bases connected, as shown below.
These connections give the paired transistors the same base and emitter voltages, so they have the same collector currents.
In other words, they form a current mirror.
The mirrored currents are fed into special correction resistors that match the undesired emitter resistance, 0.1 Ω according to the datasheet.11
The voltage across the correction resistors will be the same as the excess voltage that needs to be compensated (since the resistance and current are the same).
The final step is the correction resistors are connected to the base of the multiplication transistors, replacing the connection to ground.
This will shrink V_{BE} by the amount it was erroneously increased, fixing the computation.

The main multiplier consists of four transistors. Each transistor has a mirror transistor generating the same current, used to correct for emitter resistance.

Why are there two correction resistors? Recall that the multiplier has two transistors adding and two transistors subtracting
(i.e. V_{BE1}+V_{BE2}-V_{BE3}-V_{BE4} = 0).
To handle this, the correction circuit is split in two. The left half sums I_{C1} and I_{C2} and applies this current to a correction resistor on the
Q3/Q4 side, while the right half sums I_{C3} and I_{C4} and applies this to a correction resistor on the Q1/Q2 side.
The addition and subtraction work out to yield the desired net correction.

## Schematic

The schematic below shows the complete circuitry of the RC4200; I've highlighted the main functional blocks. (Inconveniently, I didn't find this schematic until after I'd traced out the circuitry from the die photo.) The multiplier core and the correction resistors were discussed above The op amps circuits are fairly similar to the 741 op amp, which I've written about. They lack the output stage of typical op amps; the output transistor (Q112/Q212/Q412) corresponds to the intermediate gain state in a typical op amp. The bias circuit (orange, lower right) provides a fixed bias voltage for the op amps.12

## Conclusion

Before integrated circuits, analog multiplication was difficult to implement. However, integrated circuits made it easy to create matched transistors, leading to fast, inexpensive analog multiplication integrated circuits. Unfortunately, analog multiplier integrated circuits were introduced just as analog computers were dying out, killed by inexpensive digital microprocessors, so analog computing missed most of the benefit of these chips.

While most analog multipliers use a circuit called the Gilbert cell, the Raytheon RC4200 analog multiplier uses a different technique to multiply and divide values represented by currents. Although, it includes a special error compensation circuit to improve its accuracy, it is obsolete compared to accurate, laser-trimmed multipliers. Now, counterfeiters re-label RC4200 chips and sell them as the more-expensive AD633 multiplier.

Die photo of the RC4200, courtesy of John McMaster.

I announce my latest blog posts on Twitter, so follow me at kenshirriff for updates. I also have an RSS feed. Thank you to John McMaster for the die photos used in this blog post; the photos are here.