The Analog Computer
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The analog computer is not dead,
but alive and well and living
in industrialPAUl CUTHBfRTSON
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he analog computer has been with us
in one fomi or another for some considerable time. Despite this, it could be
called the "forgotten computer." Today
the public imagination is swamped by
notions of word processing, high
resolution graphics and digital communications - all (rightly) the domain
of the digital computer.
However, there are many analog computers around They lack the glamour and
fascination of their digital counterparts
and can be found incorporated into industrial controllers, dedicated to keeping
steel at such a thickness and ketchup at
such a consistency. Otherwise, they are
mostly found languishing in dusty closets.
Analog computers deserve a better fate
than this. They are a valuable tool for the
scientist, engineer and mathematician, providing a direct means of modelling systems
as diverse as control mechanisms, vehicle
suspension units and animal populations.
The history of the analog computer is as
varied and interesting as that of the digital
computer. There were a few mechanical
versions around in the 19th century (the
slide rule is really a mechanical analog computer and you could argue certain ancient
navigational instruments are too) but the
first really successful mechanical design
arose about 1930 or so in such places as
MIT and Cambridge. Electronic versions
appeared in the 1940s. RCA built the first
accurate design in 1950, since then the advent of integrated circuits has ·made the
design of analog computers easier, in just
the same way as digital computers.
Many of the pre-war analog computers
had military purposes such as bomb or gun
aiming and were very successful. Connected directly to the airspeed, height and
heading instruments in the aircraft, even the
primitive versions of automatic bombsights
were vastly superior to eye alone.
Further improvements used a gyroscope to allow for the aircraft banking and
allowed the operator to input a drift rate
to compensate for the effects of the wind.
Anti-aircraft guns used a "computer
predictor" which computed a trajectory
for a shell, assuming that the target was
holding a steady course, or that any
change was at a constant rate.
Mechanical Matters
20
Mechanical analog computers use the
amount of rotation of a shaft or the length
of a piston as the variable. Multiplication
by a constant is achieved simply by meshing two gears of a certain ratio. Summation can be done by levers.
E&TT August 1988
Integration was performed in an intriguingly elegant manner by a "spinning disc integrator." A roller bears on the surface of a
disc which spins at constant speed This
roller is free to move along its axle towards
the periphery or the centre of the spinning
disc. The shaft of the roller will accumulate
a rotation depending on how near the roller
is to the periphery of the disc. If the roller is
at the centre of the disc, then no rotation
occurs. If the roller is moved right over the
centre and onto the other side, then the
direction of accumulation reverses. Hgure
1-4 show some examples of mechanical
computer functions.
One of my friends who works in a
fisheries research establishment tells me
that there used to be a mechanical model
of fish populations standing in one corner
of his lab. Nowadays electronics has taken
over and they use a big VAX computer
system for such things.
A digital computer deals with data in
the form of discrete numbers and processes these in turn according to a sequence
of instructions. The bit-length of the word
dictates the resolution. The electronic
analog computer represents quantities as
voltages. These voltages are analogs to the
quantities we wish to represent and vary in
a manner analogous to the manner in
which the quantities vary.
To make an example of the differences
in operation, suppose we fire a shell from
· an artillery piece and this shell will attain an
altitude of lOkm before its vertical motion
stops and it starts back to earth. In the digital computer we might calculate the altitude
of the shell at discrete intervals. If we calculate to the nearest metre, the number 2.(XX)
would represent 2krn, 100), 1krn, and so
forth. A binary word of 16-bits would easily
accommodate the maximum altitude of
10km.
However, in the analog computer the
altitude of the shell would be represented
by a continuously varying voltage - 1V
might represent 1km. This is a far more
direct method than the digital but each
has its own advantages and disadvantages:
• Noise and drift (due to temperature
and ageing) and tolerances in the circuitry all contnbute errors in the analog
computer. There are no such errors in
the digital computer, excepting gross
fault conditions which cause a bit to
change state.
• The digital computer suffers from
rounding errors. In fact a small number
added to a much larger one can vanish
entirely under certain conditions. The
resolution of the analog computer is inE&TT August 1988
finite (in any practical sense) and there
are no rounding errors. We can minimize rounding errors in a digital system
by increasing word length, but then we
suffer the cost of extra hardware or increased processing time.
Fig. 1A mechanical coefficient multiplier
using two gears at 2·1 ratio (rotary motion)
~ i-2x
FIXED
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w
•
xl
Fig. 2A mechanical coefficient multiplier
using levers (linear motion)
tx+y/2
•
Fig. 3 Mechanical summation.
INPUT SHAFT
(LINEAR
MOTION)
----
CONSTANT
SPIN
OUTPUT
SHAFT
!ROTARY
MOTION)
Fig. 4. The principal ofthe spinning disc integrator.
• The digital computer is an essentially
serial device performing primitive
operations on fragments of numbers in
sequence. This makes for slow arithmetic. An analog computer is inherently parallel. A single summer could take
an unlimited number of inputs, multiply
each by a coefficient and add them all
in a few microseconds. There may be
tens or even hundreds of these "computing elements" working simultaneously.
• Results are available continuously from
an analog computer. In the digital computer the results will progress by discrete jumps an intervals. A number
which may be precise at the instant of
its calculation will usually be progressively less accurate until replaced by its
successor.
• There is a certain minimum hardware
requirement for a digital computer. We
have to have a processor, RAM, ROM
and 10 (even if these are all on the
same chip). A useful analog computer
which might be used to solve a second
order differential equation can be built
from a few op amps. The total cost of
the components for such would be less
than a few dollars. In fact an analog
computer model of a filter - a state
variable filter - needs three or four op
amps, a few resistors and two capacitors. The display for an analog computer can be a meter, an oscilloscope or a
DVM.
• The method of interconnection of the
analog computer elements is a very
direct way of numerically solving systems of equations, even those which
might defy analysis. Compared with
these methods the digital computer is
an abstraction, requiring massive underpinning of languages, operating system and such.
• A sensor such as a potentiometer can
be wired straight into the analog computer inputs. The outputs can drive an
audio amplifie:-, or servo amplifiers.
• The operator can interact directly with
the analog computer in an experimental
fashion - to try things out. This is less
easy on a digital computer.
• An analog computer cannot be used as
a word processor or the like as it has no
way of representing characters. The
digital computer is ideal for that task.
An analog computer is a purely
numeric machine.
• The parallel nature of the analog computer makes testing easy. Each ccmputing element can be tested independently and if needs be ignored until
a service is done.
• The digital machine can store information indefinitely. This is not possible on
an analog computer.
I would identify inability of the analog
computer to store information or to handle
text as the two major reasons for the ascendancy of the digital computer. Hybrid
21
The Analog Computer
machines do exist - where muneric comhas the slightly unconventional addition of
putation is performed by the analog com~
an op amp buffer after the potentiometer,
puter and the digital section is respollSlble
which does away with this problem. Some
1a 1
of our potentiometers are also double endfor generating functions, for storage of outed - neither end is taken to OV. This is ocput or for performing any long term inIN~
casionallyuseful and again unconventional.
tegration or summation where speed is not
101< . - . a ouT
The summer (Fig. 6) takes a number of
important. Connection between the two
parts is via DAC and ADC converters.
voltages as inputs and adds them together
Attempting to patch the analog cominverting in the process. The actual circuit
1b 1
puter connections from the digital comconsists of a single op amp and a number of
Fig. S1l1e electronic coefficient multiplier
resistors. In our version the input resistors
puter is a complex business. Interestingly
symbol and circuit
are trimmable through a limited range to
enough, the arrival of a new generation of
crosspoint switch chips on the scene a
eliminate initial tolerances. Any practical
~
circuit must also include a nulling potentioshort while ago may herald a more comv,
-fx+y+zl
pact and effective hybrid computer.
meter. The inputs on our version are each
If you were to see an analog computer
tied to OV via a 10k resistor. This means the
and one of the more usual desktop digital
input may easily be left open without dis1a 1
computers side by side, the superficial difturbing the impedance balance of the cirferences would be glaringly obvious. In fact
IN
tMo
tMo
cuit too much, thus minimizing offsets.
you might not recognize the analog comIMo
ouT
The input resisto::s are connected
direct to the op amp circuit, the general
puter as being a computer at all, as all the
more usual keyboard, video monitor, printtrend being to keep these separate. In a
conventional computer this gives access to
ers and disk drives are entirely absent. Inthe virtual earth point and allows the
stead we might have a large panel on which
is an array of sockets, a set of knobs, one or
operator to introduce feedback networks
two switches and an analog meter moveother than the one supplied. Figure 7
1b 1
ment(orpossiblyasimplescopeofDVM).
shows a typical analog computer summer
The array of sockets is known as the
Fig. 6. The srunmer symbol and ciratit.
element which illustrates this.
patch panel and the analog computer is ,----SJ-,;=======~---------------.
The conventional circuit also doubles
as an integrator if you should switch in
programmed by linking (patching) various
10 ~
of the computing element sockets
took
either of the capacitors, and another
together, rather in the manner of the old
ton--'\.,""M"'o' - - 4
element's input resistors could be hijacktime telephone exchange. The analog
ed if necessary. In our circuit the elements
are fixed and trimmed for accuracy,
computer software is easy to see - it is.
the wiring on the patch panel There is no
SJ2 o--;::::==~=::::::1
which does not allow this fleXibility.
confusion about where the software is on
The integrator element integrates the
SJ
a11 analog computer.
~>-::-::---r-.A./\1\/'----oOP'
sum of the input voltages with respect to
Let's examine the individual computing
'--"..1\/\."---1---oC\ 1c
time. If we suppose that the input x is a
constant, the output voltage will change
elements before discussing how they
by xV in 1 second Note that there is an
might be interlinked. The three most
commonly used are the coefficient multiinherent sign reversal as in the summer.
plier, the summer and the integrator. UseThe Aberdeen unit is unconventional
ful work can be done on systems of linear
in that the initial conditions input is not
sign reversed The relays are to do with
equations with no more than these three Fig. 7 'D1e circuit for a basic conventional computt';pes of elements. We built our own ingelement
setting the element to its initial conditions
analog computer at Aberdeen University
~
or holding the computation at any point.
recently. It incorporates all these three. Our
We chose IC analog switches instead, principally because they do not bounce. Figure
approach has been slightly unconventional
and where there are differences between
8 shows the symbol for an integrator.
the Aberdeen unit and the usual case, I'll
-j ~ dt • c
Figure 9 shows the elements of the Abermention them.
deen unit as they might appear on a
The coefficient multiplier multiplies an
Fig. 817te integrator symbol.
problem flow chart. The triangle is an iJ?.incoming voltage by a constant. The coefverter. Normally one would press a sumficient must be bet\veen zero and one.
mer into service as an inverter because of
Physically the multiplier is usually a poten- ~
the way our circuit is built, there is a spare
tiometer, wiili one end connected to OV
·
:0
inverter with each summer, which is
(F"~g. 5).
brought out to the front panel The numbers refer to gams" - 10 is a xlO input.
When the output of this arrangement is
Ia I
IcI
patched to the input of the next element,
Use of stackable hermaphrodite connec1b 1
tors remove the need for the usual multithe set coefficient will tend to droop, due
to the next element's non-infinite input Fig. 9 Symbols for the Aberdeen unit computing ele- ple outputs on elements.
impedance. Our own analog computer ments (a) Szunmer (b) hwerter (c) httegrotor
There are numerous other circuits
-<]---
22
E&TT August 1988
which can be used on analog computers.
compressed and the force it exerts is upAmong the most important we could
ward hence the negative sign in front of the
mention are four quadrant multipliers and
~
spring's force. Similarly when the motion of
.
the mass is downward (negative) then the
the various diode circuits for modelling
nonlinearity, discontinuities and
damper exerts an upward force.
hysteresis. In fact, any circuit which beThese forces make the mass accelerate.
haves in a fashion analogs to a physical
Newton (bless him) said that force is mass
times acceleration, so:
system can be pressed into service. None
of these non-linear elements are incorFig, 10 The mass-spring-damperproblem.
mx = -dx-kx
porated on the Aberdeen unit ... yet.
All right then, that's our model of how
So how do we patch those together to
the system behaves. How to get it into the
~
computer? Let's indulge in some algebra
produce something useful? We can appreciate what is happening better if we
~
and get m (the mass) out of the way to leave
devise a model of a system and set out to
x on its own
solve it. I have chosen the classic mass
x = -dx/m-lov'm
Fig. 11 First steps
spring damper model of a car suspension,
I mentioned earlier that integrating is the
beloved of generations of long suffering
opposite of differentiating so if we fix up an
fifth formers ever since Newton. It is not too
'"" ~-.,.__
integrator as in F~g.11, it's a good start. We
complex to imagine what is happening in
_ ;._( )- x, m
i '~
get-xout(rememberthesigninversion).
the mind's eye but at the same time it is not
-..1-If I integrate a constant times xI will get
a trivial example. Figure 10 shows the arthe same constant time -x. So, if I put in a
rangement.
Fig. 12Accountingfor mass
coefficient multiplier set to 1/m as in Fig.
The deviation of the spring from its
~ we can see the result. Then we can add
natural (unstretched) length I have called
in a coefficient multiplier for d (Fig. 13)
x. This is a distance of course. The rate of
and then another integrator and coeffichange of distance with time is called
cient multiplier fork (Fig. 14). Finally we
velocity. The rate of change of velocity
can add these two in a summer (Fig. 15).
with respect to time is acceleration. I have
It's fairly easy to see how the patching is
called the velocity x (x-dot') and the acbuilt up. Figure 15 shows the "open loop"
celeration x (''x-double-dot") which is
flow diagram for the problem.
mathematicians' parlance for the derivaFig. 13 Damping
But there's still one last thing. Where do
tive and the double derivative of x. Now,
we get x from in the first place? Lo and
you needn't worry about all this calculus.
behold, we have what seems to be the
The only important point to remember
right thing coming out of the surinner. We
for this purpose is that integration is the
can make the left hand and right hand
opposite of differentiation.
sides of the equation equal if we connect
As the spring is stretched or comthe input and the summer output together
pressed, it will exert a force equal to the
as shown by the loop in Fig. 16. This is the
stiffness times the distance we have
closed loop flow diagram and is the patch
stretched it. If we call the stiffness k, the
that we need to solve the problem.
Provided we've got the plusses and
force is kx. So far so good There is also a Fig. 14 Adding the spring.
force exerted by the damper. .--- - - - - - - - - - - - - - - - - - -- - - , minuses right, the solution is a
The damper only exerts force
·-• tm
decaying sine wave. Mathe[
matically it's possible to have a
when we try to move it. If we
call the damping factor d, then
"draft damper'' which assists
the force exerted by the
motion instead of retarding it.
damper will be d times the
That gives an increasing sine
velocity which is dx.
wave. It's also possible to have a
- Xd/ m - xk /m
l(k / rn
These are the only forces on
------<
"silly spring'' which pushes in
the mass, so we can add them
the wrong direction as we
together to get the total force:
Fig. 15 The complete patch
stretch it. The solution in this
F = -dx-kx.
case would probably be an exThere are two important
ponential (depending on the
points to note here. We have
- x
ratioofkandd).
---o•----.--i
This problem is quite easy to
ignored such complications as
[ ------air resistance and mass of
solve analytically. The analog
spnng· (and a good thing too, I
computer really comes into its
kim
hear someone saying). We also
own where we encounter sets
-xd/m - xk lm
of differential equations which
have to decide which direction
is positive and I have decided
xk tm
are difficult to analyse. These
are no more difficult in printhat up is positive. When distance is negative, the spring is Fig. 16An alternative patch withfewercoefficientmultipliers
ciple to solve on an analog
1
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E&TT August 1B
23
The Analog Computer
computer. For example air resistance,
double acting dampers, spring masses and
the like can all be built in. All we have to do
is derive a set of equations which describe
the system. We can build several separate
models and interconnect to feed the results
of one into the next.
The model we have just used does not
account for gravity or a "'bumpy road". We
can add in any acceleration we like at the
summer, including that of gravity. We can
connect an oscillator to the same place, to
inject ''bumps." (This oscillation is known
as a forcing function). This illustrates the
direct nature of working with an analog
computer.
So far we have not attempted to quantify
the settings of the pots. To get a useful
quantitative result we must scale the
problem. Ideally the model will use full
dynamic range of the machine ( + lOV in
our case) without going appreciably outside
those limits (which may cause clipping and
invalid computation). It's a similar problem
to that encountered by anyone confined to
integer arithmetic or the user of a slide rule.
The slide rule user has to keep track of all
the zeroes or he will end up a factor oftento-the-something out. Similarly, the integer
user may run out of bits.
I don't propose to go into scaling in any
detail, except to say that there are well
defined procedures for doing it which con- sist basically of writing out an equation for
each computing element, estimating the
maximum value a variable can be expected
to take and dividing through, calculating the
pot settings as we go. Some operators get by
using try-it-and-see methods.
Anyone who is particularly interested in
the rigorous scaling of problems is recommended to read "Systematic Analog Computer Programing' by Charlesworth and
Fletcher which gives a detailed treatment of
this and other facets of analog computing.
My own interests in analog computers
started when I was asked to look at one
which appeared faulty. Unfortunately it was
an extremely poor design and was sent
packing. Subsequently we decided to develop our own system. The photographs
shows views inside and outside the
machine.
On the right is a panel (the control unit)
which contains a large analog meter
movement, three rotary switches, three
push buttons and a variety of lamps and
4mm sockets. On the left are four narrower panels.
The control unit is the nerve centre of
the computer. As well as controlling the
hold and reset functions, it allows for
24
monitoring of the progress of a computation and it also provides access for BBC
micro to gain control and monitor and
store the results. Thus the Aberdeen
analog computer is "hybridisable." Two
D-type connectors on the rear can be
fitted with cables which plug into the user
port and the analog port of the BBC.
The meter is used to set up the potentiometers and to monitor the progress of
computations. There are four yellow sockets which are used to input signals to the
meter. A meter select switch routes the signals, as well as selecting reference or supply
voltages to be monitored. A hold and a
reset button toggle the hold and reset states
on and off - an lED shows which state is
selected. An unusual feature is the
bandwidth control which switches
capacitors in all the integrators, to allow
faster operation.
Of the four smaller panels, one contains
five integrators, one has five summers and
five inverters and the two remaining
panels each contain six coefficient multipliers, along with sockets to provide
+ lOV and OV to the programmer.
These four panels can be plugged into
the frame in any of the seven possible
positions as the bus structure is not position sensitive. The three spare slots allow
the introduction of similar or other panels
as they become available. The control unit
must however be in position at the far
right.
All the sockets are colour coded. Blue
sockets are outputs. Yellow sockets are x1
inputs. White are xlO inputs. The initial
conditions sockets on the integrators are
brown. The red, black and green are for
+ 10, -10 and OV respectively.
This makes it easy to find your way
about. There are no legends, hieroglyphics
or diagrams on the panels but there are
group markings encircling sets of sockets
which are associated with the same computing· element. The five indicators on the
summing and integrating panels are overload indicators. They latch on in the event
an output exceeds about 115V. Resetting is
by a common pushbutton marked OVV on
the control unit.
The control unit houses a motherboard
and several daughter boards. There is a
logic board which controls the hold and
reset functions and a meter amplifier
board which is controlled by the meter
range switch.
There are two each of the others generalized optical interface boards used
by the BBC interface and generalized
analog conditioning used to switch signals
or to attenuate and shift the normal
+ lOV range of the analog computer to
suit the BBC ADC inputs.
The power supply board is on the rear
panel of the frame, along with the transformer, filter, rectifier and reservoir
capacitors which are all off board. This
power supply performs well. No voltage
deviation registers 4-1(2 digit D MM when
full load (500mA) is applied. I couldn't
believe it at first. No current limit is neCessary as the supplies are not available externally. The supplies are + 15V for the
analog circuitry and + 7V for the digital
circuitry, which is all CMOS. The OV line
is not a supply, and does not carry supply
currents. It is purely a reference. This also
helps lessen noise. An interesting feature
of the supply is its sequencing. The 15V
rails cannot come right up until the 7V
rails are established. This prevents
damage to the CMOS analog switches.
The master references are on this
board too. These are trimmed to within 1
millivolt. We can claim lO.OOV in fact, or
0.01% accuracy. The supplies are
trimmed to within a few millivolts too. The
primary reference is a band gap diode and
the setup is remarkably stable in the long
term. Each reference socket is individually
buffered to prevent loading of the master
reference. Incidentally the integrators and
summers on the Aberdeen unit are
trimmed to within 0.01% as well. It's quite
possible to set up the zero point, with the
aid of a decent DVM, to within a few tins
of microvolts. However, having said that,
the time constant on the integrators is the
very devil to set up accurately.
This, like most analog computes, has
proven to be a valuable tool Even in these
days of fast digital arithmetic, the analog
computer should not be despised or cast
aside. It offers direct, easily interpreted
evaluation of problems.
One of my little projects for the near future will be to make up a dedicated analog
computer which multiplies six variables by
a coefficient matrix, giving six outputs.
One of my colleagues will use it to investigate the motion of buildings during
earthquakes. It has to perform 36 additions and 36 multiplications. It could be
made to work at up to 100kHz (although it
won't need to in this instance). An
equivalent digital system would need to do
these 72 operations every five
microseconds to keep pace - a good few
transputers worth may be, or a very fast
DSP chip, plus converters, etc, etc.
The hardware fm using? A few op
amps and a few dozen resistors. •
E&TT August 1988