### Reverse-engineering an analog Bendix air data computer: part 4, the Mach section

In the 1950s, many fighter planes used the Bendix Central Air Data Computer (CADC) to compute airspeed, Mach number, and other "air data". The CADC is an analog computer, using tiny gears and specially-machined cams for its mathematics. In this article, part 4 of my series,1 I reverse engineer the Mach section of the CADC and explain its calculations. (In the photo below, the Mach section is the middle section of the CADC.)

The Bendix MG-1A Central Air Data Computer with the case removed, showing the compact gear mechanisms inside. Click this image (or any other) for a larger version.

Aircraft have determined airspeed from air pressure for over a century. A port in the side of the plane provides the static air pressure,2 the air pressure outside the aircraft. A pitot tube points forward and receives the "total" air pressure, a higher pressure due to the air forced into the tube by the speed of the airplane. The airspeed can be determined from the ratio of these two pressures, while the altitude can be determined from the static pressure.

But as you approach the speed of sound, the fluid dynamics of air change and the calculations become very complicated. With the development of supersonic fighter planes in the 1950s, simple mechanical instruments were no longer sufficient. Instead, an analog computer calculated the "air data" (airspeed, air density, Mach number, and so forth) from the pressure measurements. This computer then transmitted the air data electrically to the systems that needed it: instruments, weapons targeting, engine control, and so forth. Since the computer was centralized, the system was called a Central Air Data Computer or CADC, manufactured by Bendix and other companies.

A closeup of the numerous gears inside the CADC. Three differential gear mechanisms are visible.

Each value in the Bendix CADC is indicated by the rotational position of a shaft. Compact electric motors rotate the shafts, controlled by the pressure inputs. Gears, cams, and differentials perform computations, with the results indicated by more rotations. Devices called synchros converted the rotations to electrical outputs that are connected to other aircraft systems. The CADC is said to contain 46 synchros, 511 gears, 820 ball bearings, and a total of 2,781 major parts (but I haven't counted). These components are crammed into a compact cylinder: just 15 inches long and weighing 28.7 pounds.

The equations computed by the CADC are impressively complicated. For instance, one equation is:

$~~~\frac{P_t}{P_s} = \frac{166.9215M^7}{( 7M^2-1)^{2.5}}$

It seems incredible that these functions could be computed mechanically, but three techniques make this possible. The fundamental mechanism is the differential gear, which adds or subtracts values. Second, logarithms are used extensively, so multiplications and divisions are implemented by additions and subtractions performed by a differential, while square roots are calculated by gearing down by a factor of 2. Finally, specially-shaped cams implement functions: logarithm, exponential, and application-specific functions. By combining these mechanisms, complicated functions can be computed mechanically, as I will explain below.

### The differential

The differential gear assembly is the mathematical component of the CADC, as it performs addition or subtraction.3 The differential takes two input rotations and produces an output rotation that is the sum or difference of these rotations.4 Since most values in the CADC are expressed logarithmically, the differential computes multiplication and division when it adds or subtracts its inputs.

A closeup of a differential mechanism.

While the differential functions like the differential in a car, it is constructed differently, with a spur-gear design. This compact arrangement of gears is about 1 cm thick and 3 cm in diameter. The differential is mounted on a shaft along with three co-axial gears: two gears provide the inputs to the differential and the third provides the output. In the photo, the gears above and below the differential are the input gears. The entire differential body rotates with the sum, connected to the output gear at the top through a concentric shaft. (In practice, any of the three gears can be used as the output.) The two thick gears inside the differential body are part of the mechanism.

### The cams

The CADC uses cams to implement various functions. Most importantly, cams compute logarithms and exponentials. Cams also implement complicated functions of one variable such as ${M}/{\sqrt{1 + .2 M^2}}$. The function is encoded into the cam's shape during manufacturing, so a hard-to-compute nonlinear function isn't a problem for the CADC. The photo below shows a cam with the follower arm in front. As the cam rotates, the follower moves in and out according to the cam's radius.

A cam inside the CADC implements a function.

However, the shape of the cam doesn't provide the function directly, as you might expect. The main problem with the straightforward approach is the discontinuity when the cam wraps around. For example, if the cam implemented an exponential directly, its radius would spiral exponentially and there would be a jump back to the starting value when it wraps around. Instead, the CADC uses a clever patented method: the cam encodes the difference between the desired function and a straight line. For example, an exponential curve is shown below (blue), with a line (red) between the endpoints. The height of the gray segment, the difference, specifies the radius of the cam (added to the cam's fixed minimum radius). The point is that this difference goes to 0 at the extremes, so the cam will no longer have a discontinuity when it wraps around. Moreover, this technique significantly reduces the size of the value (i.e. the height of the gray region is smaller than the height of the blue line), increasing the cam's accuracy.5

An exponential curve (blue), linear curve (red), and the difference (gray).

To make this work, the cam position must be added to the linear value to yield the result. This is implemented by combining each cam with a differential gear; watch for the paired cams and differentials below. As the diagram below shows, the input (23) drives the cam (30) and the differential (25, 37-41). The follower (32) tracks the cam and provides a second input (35) to the differential. The sum from the differential produces the desired function (26).

This diagram, from Patent 2969910, shows how the cam and follower are connected to a differential.

### The synchro outputs

A synchro is an interesting device that can transmit a rotational position electrically over three wires. In appearance, a synchro is similar to an electric motor, but its internal construction is different, as shown below. Before digital systems, synchros were very popular for transmitting signals electrically through an aircraft. For instance, a synchro could transmit an altitude reading to a cockpit display or a targeting system. Two synchros at different locations have their stator windings connected together, while the rotor windings are driven with AC. Rotating the shaft of one synchro causes the other to rotate to the same position.6

Cross-section diagram of a synchro showing the rotor and stators.

For the CADC, most of the outputs are synchro signals, using compact synchros that are about 3 cm in length. For improved resolution, many of the CADC outputs use two synchros: a coarse synchro and a fine synchro. The two synchros are typically geared in an 11:1 ratio, so the fine synchro rotates 11 times as fast as the coarse synchro. Over the output range, the coarse synchro may turn 180°, providing the approximate output unambiguously, while the fine synchro spins multiple times to provide more accuracy.

## Examining the Mach section of the CADC

The Bendix CADC is constructed from modular sections. In this blog post, I'm focusing on the middle section, called the "Mach section" and indicated by the arrow above. This section computes log static pressure, impact pressure, pressure ratio, and Mach number and provides these outputs electrically as synchro signals. It also provides the log pressure ratio and log static pressure to the rest of the CADC as shaft rotations. The left section of the CADC computes values related to airspeed, air density, and temperature.7 The right section has the pressure sensors (the black domes), along with the servo mechanisms that control them.

I had feared that any attempt at disassembly would result in tiny gears flying in every direction, but the CADC was designed to be taken apart for maintenance. Thus, I could remove the left section of the CADC for analysis. Unfortunately, we lost the gear alignment between the sections and don't have the calibration instructions, so the CADC no longer produces accurate results.

The diagram below shows the internal components of the Mach section after disassembly. The synchros are in pairs to generate coarse and fine outputs; the coarse synchros can be distinguished because they have spiral anti-backlash springs installed. These springs prevent wobble in the synchro and gear train as the gears change direction. The gears and differentials are not visible from this angle as they are underneath the metal plate. The Pressure Error Correction (PEC) subsystem has a motor to drive the shaft and a control transformer for feedback. The Mach section has two D-sub connectors. The one on the right links the Mach section and pressure section to the front section of the CADC. The Position Error Correction (PEC) servo amplifier board plugs into the left connector. The static pressure and total pressure input lines have fittings so the lines can be disconnected from the lines from the front of the CADC.8

The Mach section with components labeled.

The photo below shows the left section of the CADC. This section meshes with the Mach section shown above. The two sections have parts at various heights, so they join in a complicated way. Two gears receive the pressure signals $$log ~ P_t / P_s$$ and $$log ~ P_s$$ from the Mach section. The third gear sends the log total temperature to the rest of the CADC. The electrical connector (a standard 37-pin D-sub) supplies 120 V 400 Hz power to the Mach section and pressure transducers and passes synchro signals to the output connectors.

The left part of the CADC that meshes with the Mach section.

## The position error correction servo loop

The CADC receives two pressure inputs and two pressure transducers convert the pressures into rotational positions, providing the indicated static pressure $$P_{si}$$ and the total pressure $$P_t$$ as shaft rotations to the rest of the CADC. (I explained the pressure transducers in detail in the previous article.)

There's one complication though. The static pressure $$P_s$$ is the atmospheric pressure outside the aircraft. The problem is that the static pressure measurement is perturbed by the airflow around the aircraft, so the measured pressure (called the indicated static pressure $$P_{si}$$) doesn't match the real pressure. This is bad because a "static-pressure error manifests itself as errors in indicated airspeed, altitude, and Mach number to the pilot."9

The solution is a correction factor called the Position Error Correction. This factor gives the ratio between the real pressure $$P_s$$ and the measured pressure $$P_{si}$$. By applying this correction factor to the indicated (i.e. measured) pressure, the true pressure can be obtained. Since this correction factor depends on the shape of the aircraft, it is generated outside the CADC by a separate cylindrical unit called the Compensator, customized to the aircraft type. The position error computation depends on two parameters: the Mach number provided by the CADC and the angle of attack provided by an aircraft sensor. The compensator determines the correction factor by using a three-dimensional cam. The vintage photo below shows the components inside the compensator.

"Static Pressure and Angle of Attack Compensator Type X1254115-1 (Cover Removed)" from Air Data Computer Mechanization.

The correction factor is transmitted from the compensator to the CADC as a synchro signal over three wires. To use this value, the CADC must convert the synchro signal to a shaft rotation. The CADC uses a motorized servo loop that rotates the shaft until the shaft position matches the angle specified by the synchro input.

The servo loop ensures that the shaft position matches the input angle.

The key to the servo loop is a control transformer. This device looks like a synchro and has five wires like a synchro, but its function is different. Like the synchro motor, the control transformer has three stator wires that provide the angle input. Unlike the synchro, the control transformer also uses the shaft position as an input, while the rotor winding generates an output voltage indicating the error. This output voltage indicates the error between the control transformer's shaft position and the three-wire angle input. The control transformer provides its error signal as a 400 Hz sine wave, with a larger signal indicating more error.10

The amplifier board (below) drives the motor in the appropriate direction to cancel out the error. The power transformer in the upper left is the largest component, powering the amplifier board from the CADC's 115-volt, 400 Hertz aviation power. Below it are two transformer-like components; these are the magnetic amplifiers. The relay in the lower-right corner switches the amplifier into test mode. The rest of the circuitry consists of transistors, resistors, capacitors, and diodes. The construction is completely different from modern printed circuit boards. Instead, the amplifier uses point-to-point wiring between plastic-insulated metal pegs. Both sides of the board have components, with connections between the sides through the metal pegs.

The amplifier board for the position error correction.

The amplifier board is implemented with a transistor amplifier driving two magnetic amplifiers, which control the motor.11 (Magnetic amplifiers are an old technology that can amplify AC signals, allowing the relatively weak transistor output to control a larger AC output.12) The motor is a "Motor / Tachometer Generator" unit that also generates a voltage based on the motor's speed. This speed signal provides negative feedback, limiting the motor speed as the error becomes smaller and ensuring that the feedback loop doesn't overshoot. The photo below shows how the amplifier board is mounted in the middle of the CADC, behind the static pressure tubing.

## The equations

Although the CADC looks like an inscrutable conglomeration of tiny gears, it is possible to trace out the gearing and see exactly how it computes the air data functions. With considerable effort, I have reverse-engineered the mechanisms to create the diagram below, showing how each computation is broken down into mechanical steps. Each line indicates a particular value, specified by a shaft rotation. The ⊕ symbol indicates a differential gear, adding or subtracting its inputs to produce another value. The cam symbol indicates a cam coupled to a differential gear. Each cam computes either a specific function or an exponential, providing the value as a rotation. At the right, the outputs are either shaft rotations to the rest of the CADC or synchro outputs.

This diagram shows how the values are computed. The differential numbers are my own arbitrary numbers. Click for a larger version.

I'll go through each calculation briefly.

### log static pressure

The static pressure is calculated by dividing the indicated static pressure by the pressure error correction factor. Since these values are all represented logarithmically, the division turns into a subtraction, performed by a differential gear. The output goes to two synchros, geared to provide coarse and fine outputs.13

$log ~ P_s = log ~ P_{si} - log ~ P_{si} / P_s$

### Impact pressure

The impact pressure is the pressure due to the aircraft's speed, the difference between the total pressure and the static pressure. To compute the impact pressure, the log pressure values are first converted to linear values by exponentiation, performed by cams. The linear pressure values are then subtracted by a differential gear. Finally, the impact pressure is output through two synchros, coarse and fine in an 11:1 ratio.

$P_t - P_s = exp(log ~ P_t) - exp(log ~ P_s)$

### log pressure ratio

The log pressure ratio $$P_t/P_s$$ is the ratio of total pressure to static pressure. This value is important because it is used to compute the Mach number, true airspeed, and log free air temperature. The Mach number is computed in the Mach section as described below. The true airspeed and log free air temperature are computed in the left section. The left section receives the log pressure ratio as a rotation. Since the left section and Mach section can be separated for maintenance, a direct shaft connection is not used. Instead, each section has a gear and the gears mesh when the sections are joined.

Computing the log pressure ratio is straightforward. Since the log total pressure and log static pressure are both available, subtracting the logs with a differential yields the desired value. That is,

$log ~ P_t/P_s = log ~ P_t - log ~ P_s$

### Mach number

The Mach number is defined in terms of $$P_t/P_s$$, with separate cases for subsonic and supersonic:14

$M<1:$ $~~~\frac{P_t}{P_s} = ( 1+.2M^2)^{3.5}$

$M > 1:$

$~~~\frac{P_t}{P_s} = \frac{166.9215M^7}{( 7M^2-1)^{2.5}}$

Although these equations are very complicated, the solution is a function of one variable $$P_t/P_s$$ so M can be computed with a single cam. In other words, the mathematics needed to be done when the CADC was manufactured, but once the cam exists, computing M is easy, using the log pressure ratio computed earlier:

$M = f(log ~ P_t / P_s)$

## Conclusions

The CADC performs nonlinear calculations that seem way too complicated to solve with mechanical gearing. But reverse-engineering the mechanism shows how the equations are broken down into steps that can be performed with cams and differentials, using logarithms for multiplication and division. The diagram below shows the complex gearing in the Mach section. Each differential below corresponds to a differential in the earlier equation diagram.

A closeup of the gears and cams in the Mach section. The differential for the pressure ratio is hidden in the middle.

Follow me on Twitter @kenshirriff or RSS for more reverse engineering. I'm also on Mastodon as @oldbytes.space@kenshirriff. Thanks to Joe for providing the CADC. Thanks to Nancy Chen for obtaining a hard-to-find document for me.15 Marc Verdiell and Eric Schlaepfer are working on the CADC with me. CuriousMarc's video shows the CADC in action:

## Notes and references

1. My articles on the CADC are:

There is a lot of overlap between the articles, so skip over parts that seem repetitive :-)

2. The static air pressure can also be provided by holes in the side of the pitot tube; this is the typical approach in fighter planes.

3. Multiplying a rotation by a constant factor doesn't require a differential; it can be done simply with the ratio between two gears. (If a large gear rotates a small gear, the small gear rotates faster according to the size ratio.) Adding a constant to a rotation is even easier, just a matter of defining what shaft position indicates 0. For this reason, I will ignore constants in the equations.

4. Strictly speaking, the output of the differential is the sum of the inputs divided by two. I'm ignoring the factor of 2 because the gear ratios can easily cancel it out. It's also arbitrary whether you think of the differential as adding or subtracting, since it depends on which rotation direction is defined as positive.

5. The diagram below shows a typical cam function in more detail. The input is $$log~ dP/P_s$$ and the output is $$log~M / \sqrt{1+.2KM^2}$$. The small humped curve at the bottom is the cam correction. Although the input and output functions cover a wide range, the difference that is encoded in the cam is much smaller and drops to zero at both ends.

This diagram, from Patent 2969910, shows how a cam implements a complicated function.

6. Internally, a synchro has a moving rotor winding and three fixed stator windings. When AC is applied to the rotor, voltages are developed on the stator windings depending on the position of the rotor. These voltages produce a torque that rotates the synchros to the same position. In other words, the rotor receives power (26 V, 400 Hz in this case), while the three stator wires transmit the position. The diagram below shows how a synchro is represented schematically, with rotor and stator coils.

The schematic symbol for a synchro.

A control transformer has a similar structure, but the rotor winding provides an output, instead of being powered.

7. Specifically, the left part of the CADC computes true airspeed, air density, total temperature, log true free air temperature, and air density × speed of sound. I discussed the left section in detail here

8. From the outside, the CADC is a boring black cylinder, with no hint of the complex gearing inside. The CADC is wired to the rest of the aircraft through round military connectors. The front panel interfaces these connectors to the D-sub connectors used internally. The two pressure inputs are the black cylinders at the bottom of the photo.

The exterior of the CADC. It is packaged in a rugged metal cylinder. It is sealed by a soldered metal band, so we needed a blowtorch to open it.

9. The concepts of position error correction are described here

10. The phase of the signal is 0° or 180°, depending on the direction of the error. In other words, the error signal is proportional to the driving AC signal in one direction and flipped when the error is in the other direction. This is important since it indicates which direction the motor should turn. When the error is eliminated, the signal is zero.

11. I reverse-engineered the circuit board to create the schematic below for the amplifier. The idea is that one magnetic amplifier or the other is selected, depending on the phase of the error signal, causing the motor to turn counterclockwise or clockwise as needed. To implement this, the magnetic amplifier control windings are connected to opposite phases of the 400 Hz power. The transistor is connected to both magnetic amplifiers through diodes, so current will flow only if the transistor pulls the winding low during the half-cycle that the winding is powered high. Thus, depending on the phase of the transistor output, one winding or the other will be powered, allowing that magnetic amplifier to pass AC to the motor.

This reverse-engineered schematic probably has a few errors. Click the schematic for a larger version.

The CADC has four servo amplifiers: this one for pressure error correction, one for temperature, and two for pressure. The amplifiers have different types of inputs: the temperature input is the probe resistance, the pressure error correction uses an error voltage from the control transformer, and the pressure inputs are voltages from the inductive pickups in the sensor. The circuitry is roughly the same for each amplifier—a transistor amplifier driving two magnetic amplifiers—but the details are different. The largest difference is that each pressure transducer amplifier drives two motors (coarse and fine) so each has two transistor stages and four magnetic amplifiers.

12. The basic idea of a magnetic amplifier is a controllable inductor. Normally, the inductor blocks alternating current. But applying a relatively small DC signal to a control winding causes the inductor to saturate, permitting the flow of AC. Since the magnetic amplifier uses a small signal to control a much larger signal, it provides amplification.

In the early 1900s, magnetic amplifiers were used in applications such as dimming lights. Germany improved the technology in World War II, using magnetic amplifiers in ships, rockets, and trains. The magnetic amplifier had a resurgence in the 1950s; the Univac Solid State computer used magnetic amplifiers (rather than vacuum tubes or transistors) as its logic elements. However, improvements in transistors made the magnetic amplifier obsolete except for specialized applications. (See my IEEE Spectrum article on magnetic amplifiers for more history of magnetic amplifiers.)

13. The CADC specification defines how the parameter values correspond to rotation angles of the synchros. For instance, for the log static pressure synchros, the CADC supports the parameter range 0.8099 to 31.0185 inches of mercury. The spec defines the corresponding synchro outputs as 16,320° rotation of the fine synchro and 175.48° rotation of the coarse synchro over this range. The synchro null point corresponds to 29.92 inches of mercury (i.e. zero altitude). The fine synchro is geared to rotate 93 times as fast as the coarse synchro, so it rotates over 45 times during this range, providing higher resolution than a single synchro would provide. The other synchro pairs use a much smaller 11:1 ratio; presumably high accuracy of the static pressure was important.

14. Although the CADC's equations may seem ad hoc, they can be derived from fluid dynamics principles. These equations were standardized in the 1950s by various government organizations including the National Bureau of Standards and NACA (the precursor of NASA).

15. It was very difficult to find information about the CADC. The official military specification is MIL-C-25653C(USAF). After searching everywhere, I was finally able to get a copy from the Technical Reports & Standards unit of the Library of Congress. The other useful document was in an obscure conference proceedings from 1958: "Air Data Computer Mechanization" (Hazen), Symposium on the USAF Flight Control Data Integration Program, Wright Air Dev Center US Air Force, Feb 3-4, 1958, pp 171-194.