Analog Computer Prototyping for the Future
Analog Computer Prototyping for the Future
Carl Oskar Ahlqvist
Malmö University, [email protected]
Måns Ahlgren
Malmö University, [email protected]
Master thesis for a
degree in computer science
Faculty of Technology and Society
Malmö University
Sweden
2022-05-10
Supervisor: Sven Karlsson
Examiner: Dario Salvi
ACKNOWLEDGEMENTS
Firstly, we would like to thank our supervisor Sven Karlsson for believing in this somewhat unusual project. Your guidance
and limitless excitement for this project helped immensely when times were tough.
We would also like to express our sincerest gratitude for the excellent work of Bernd Ulmann and the team at Anabrid for
keeping analog technology both alive and relevant in a digital society. Without the information you have gathered over the
years, this project would not have been possible.
Lastly, we would like to thank friends and family for their input, especially Ellen Ek for her contribution of excellent
mathematical knowledge that elevated the paper to the next level.
The analog computer created during this project.
C ONTENTS
I
Introduction
1
II
Background
II-A
Relevancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
III
Purpose and Goal
2
IV
Research questions
2
V
Research Methodology
3
VI
Terminology
VI-A
Analog computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI-B
Boolean algebra and discrete mathematics in computer science . . . . . . . . . . . . . . . . . . . . . .
VI-C
Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI-C1
First order linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI-C2
Second order linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI-D
Simple oscillator circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI-E
Mathieu’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI-F
Spring mass dampening system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
4
4
4
4
4
5
VII
Related Work
VII-A The computational advantage of analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-B Analog computers in academia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-C Modern analog computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-D Modern applications of analog computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-E The different components of the analog computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-F Inverting and non-inverting operational amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-G Summer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-H Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-I
Potentiometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VII-J
Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
7
8
8
10
10
10
11
12
12
VIII Results
VIII-A Circuit design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-B Breadboard experimentation - Spring mass dampening . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-C Breadboard experimentation - Mathieu’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-D V1 of the analog computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-E Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-F Summer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-G Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-H Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIII-I Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
13
13
13
14
15
15
15
15
15
IX
15
15
16
16
16
16
16
16
Evaluation of the V1 Analog computer
IX-A
Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX-B
Inverter/Summer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX-C
Multiplayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX-D
Potentiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX-E
Design choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX-F
Usability and function testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX-G
Functionality evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X
Evaluation of the V2 Analog computer
X-A
Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X-B
Inverter/Summer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X-C
Multiplayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X-D
Potentiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X-E
Design choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X-F
Hybrid system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
17
17
17
18
18
18
XI
Evaluation
XI-A
Feasibility of creating a low cost modular single board analog computer . . . . . . . . . . . . . . . . .
XI-B
Capabilities of the modular analog computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XI-B1
Physical capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XI-B2
Capabilities for providing academic value . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XI-B3
The extended capability of the analog computer . . . . . . . . . . . . . . . . . . . . . . . . .
XI-B4
General applications of the analog computer in a digital society . . . . . . . . . . . . . . . .
19
19
20
20
20
21
21
XII
Conclusion & Future work
22
References
23
Appendix A: Integrator schematic
25
Appendix B: Power supply schematic
26
Appendix C: Summer schematic
27
Appendix D: Summer schematic
28
Appendix E: Hamilton Analog computer schematic
29
Appendix F: Integrator V2 Schematic
30
Appendix G: Summer V2 Schematic
31
Appendix H: Multiplier V2 Schematic
32
Appendix I: PSU and Potentiometer V2 Schematic
33
Appendix J: BOM-table of V2
34
after the release of Vannevar Bush’s analog computer). The
early digital computers relied on the user writing code to
solve a problem algorithmically, as the case is still to this
day. The early problem was that the digital machines often
were slow and expensive, thus hindering the user to solve
problems. The digital machines were and still are based on
an algorithmic sequence thus not being able to run many
calculations simultaneously natively.
Analog computers have a long history dating back to around
1000 years ago. The most famous example of this age is the
Antikythera mechanism dated to around 100 B.C [5].
One of the more recent famous analog computers is the
Vannevar Bush’s differential analyzer released in 1931. The
analog computer Bush created was initially meant to model
power networks, but its value as a general-purpose analog
computer was quickly realized. The computer was a large
mechanical construction made of an assortment of gears
and shafts driven by electric motors. The disc wheel based
integrators could be connected to a number of rotating shafts.
The computer could solve sixth-order differential equations,
although this was complex to set up. The machine was used in
solving problems in physics, seismology, and ballistics [6] [7].
These early examples are all examples of mechanical analog
computers.
After the release of Bush’s analog computer, the electric
computer was on the horizon. The electric analog computer
was widely developed by many manufacturers during this
time including, Electronic Associates Inc., Applied Dynamics,
RCA, and Telefunken among others. The electric analog
computer was first used as an aid in missile and airplane design
as well as running aerodynamics calculations. This meant that
the space and aviation industries were the largest customers
during this time. The customer base later extended to nuclearreactor control [8] [9].
Analog computers are not high-accuracy devices. Therefore
their usage was not centered around tasks that required high
levels of accuracy. The analog computer was used often in
engineering and applied sciences as an analog for the problem
being studied. One of the most common and least complex
calculations is the linear ordinary differential equation (which
is described in the terminology section of this paper) [3].
Physics calculations are very well suited for analog computation. In the early days of space flight, analog computers
were used to calculate orbit trajectory and other orbit-related
equations such as satellite positioning [10]. Calculating particle trajectories was also a well-established field within physics
where analog computers were widely used [11] [3]. See fig
1 for an visual example of an commercial electro analog
computer.
Before the invention of the modern MOS (Metal Oxide
Semiconductor) the electric analog computer was based on
vacuum tube technology. The vacuum tubes acted as transistors
and were later replaced by modern solid-state transistors. The
programming of the analog computer was (and still is to
some extent) done by interconnecting different computational
modules through a patch panel. This way of programming
Abstract—This research paper focuses on analog computers
and creating a modular low-cost analog computer system in
a single board computer form factor. The single-board analog
computer will have the capacity to solve second-order differential equations. The capabilities and possibilities of the single
board Analog computer will be explored as well as analog
computing in general. The paper follows design science research
methodology (DSRM) with the goal of creating and evaluating a
working artifact. The artifacts’ functionality is evaluated based
on a demonstration of its ability to solve Mathieu’s differential
equation as well as simulate a spring-mass dampening system.
This paper proves that it is possible to create a low-cost analog
computer in a modern form factor. The artifact is also placed
in a larger contextual setting based on the empirical material
provided where its value of it in a digital society is presented.
For the world to continue its progression in computational power,
but still, limit the already high energy usage, a drastic change
is needed. This paper suggests adapting to analog/hybrid technology. To further the progression of analog/hybrid technology
it must be made accessible to a wider group of people compared
to today. The artifact in this paper offers a solution to this.
Index Terms—Analog computing, Hybrid systems, Differential
equation, Single board computer, DSRM, Sustainable IT
I. I NTRODUCTION
Digital computers are the norm today regarding computational power. Since the solid-state transistor became widely
available, the digital computer has been dominating computation. Today’s high energy consumption and potential stagnating progression of digital computers have led to a new
turning point in the progression of computational speed and
sustainable IT. In order to meet the global demand for greater
computing power, a drastic change has to be introduced to
the world of computation. This is a must if we wish to
continue our present computational progression and decrease
our ecological footprint. One of the possible solutions to this
problem is to change the fundamental view of computation
[1].
By switching to solving problems using analog computing
a positive impact could be achieved in today’s ecological and
computational scene compared to using strictly digital systems.
By creating hybrid systems that can utilize an analog coprocessor, that is optimal for solving differential equations,
low-energy analog computing can be utilized to a larger extent
compared to today. In order to achieve this, a new approach
to introducing the world to analog computing is needed [2]
[3] [1].
II. BACKGROUND
The three major computational paradigms are analog, digital
and quantum [4]. The Analog computer has seen a steep
reduction in prominence during the last 40-50 years due to
the rapid improvement in digital computing. This has led to
the fact that very little recent research has been conducted on
the analog computer and its potential role in today’s society
[3].
The popularisation of digital computers in today’s society
stems back to the early days of computing when both analog
and digital computers were common (the following decade
1
used 13.2 billion kWh of electricity to power data centers
and servers. This was an increase from 12.6 billion Kwh in
2016. The global consumption is estimated to be 350 billion
Kwh in 2017. Given the fact that there has been a steady
increase in energy consumption between 2010 and 2017, there
is evidence that points toward this trend continuing [15]. Most
of the high energy usage is due to today’s power-hungry
CPUs. A warehouse-scale data center’s total power usage is
approximately 33-60% only due to the modern CPU [16].
The digital computer running boolean algebra is a versatile
and well-developed system. One of its problems is the aforementioned high energy consumption of modern processors. A
hybrid system that utilizes an analog co-processor has been
proven to run faster and use less energy than a common
digital processor as proven by Köppel et al [2]. The digital
computer is also approaching the theorized Moore’s law where
the computer chips can not be made faster by physical means
[2].
Analog computing has been lost to history due to the digital
computer being more versatile and becoming easier to use.
The three major computational paradigms are digital, analog,
and quantum computing. Quantum computing is still too far
away to be implemented in today’s society. Therefore looking
to history and investigating analog computation could be a
way forward and continue the computational progression and
create more energy-efficient systems.
Fig. 1. GTE Analog Computer EA22 [12]
is very complex and required a skilled programmer to be
successful. This is also one of the reasons for their later
demise. Another contributing factor to the demise of the analog
computer was the introduction of the digital computer. The
more straightforward approach to programming, algorithmic
computational structure, the ability to easily store information,
high precision, and ability to handle any problem given the
increase in computational time was a clear advantage over the
analog computer. The advancements of MOS and chip technology made digital transistor-based chips faster and cheaper
thus quickly rendering analog computers obsolete [8] [5].
This paper will not focus on pre-electric analog computers
like the Antikythera mechanism and the Vannevar Bush’s
differential analyzer. The focus will lie on electric analog
computers, for example, the computer created by the research
team at Anabrid [4]. Anabrid has created an open-source
analog computer that aims to enable researchers and students
to explore the area of analog computing [13]. The device can
calculate and simulate several large computational challenges
like market economics, the spread of disease, and more [13].
Digital computers are approaching the end of continuous
improvement and Moore’s law (where the computer chips can
not be made faster by physical means) will take full effect.
To keep progress going, something has to be done in order
to further the improvements in computational power, since
quantum computing is still far off in the future as a viable
option. This could be done by creating a hybrid computer that
takes all the good aspects to form digital and analog computers
and combines them into a hybrid computer like Steiglitz [14]
proposes.
One of the most challenging parts of analog computing in
today’s society is the user experience, availability, and ease
of use. In order to have analog computing viable in today’s
society, a better way is needed to interact with the computer.
III. P URPOSE AND G OAL
An analog computer will be created that has the capability
to run different first and second order differential equations.
The actual component-based computation that is based on the
mathematical expressions for the differential equations will be
converted into circuits. These circuits will then be tested and
used as a tool to verify the functionality of the computer.
The purpose of this paper is to create an analog singleboard computer to make analog computing accessible for
academia and educational institutions. A low-cost, easy-to-use,
analog computer can be used as an educational tool to further
the innovation within analog technology. Given the fact that
most analog computers are either expensive or non-existent on
the open market, a low-cost modular analog computer could
have a similar impact as the Arduino single-board computer
platform had on academia. The created artifact will also be
placed in a larger contextual setting where its role in a larger
socio-material context will be investigated. The uses for the
artifact and general analog computing in a digital society will
also be investigated based on the empirical material provided
in this paper. A hybrid setup using an Arduino and the
analog computer as a co-processor will also be demonstrated.
Therefore this project has the potential to be an innovation for
change in a digital society for a more computation demanding
and greener future.
IV. R ESEARCH QUESTIONS
A. Relevancy
The global energy consumption by data centers, servers, and
other digital hardware is staggering. In 2017, Germany alone
Based on the information presented in this paper thus far,
the research questions for this project are as follows:
2
requirements of the analog computer such as components, size,
and complexity.
The functionality of the artifact will be tested with a demonstration of its ability to solve specific differential equations.
These differential equations are the spring-mass dampening
system and Mathieu’s differential equation. A simple oscillator
will also be used to evaluate the computational elements.
The result of which will be compared to circuitry specifically
developed to solve these two equations. The larger role of the
artifact and analog computing in a digital society will also be
investigated based on the empirical material provided in this
paper. The steps this paper will take in order to develop the
artifact is:
• Investigate the different computational elements of an
analog computer.
• Translate the computational elements to circuitry.
• Develop the PCB that will make out the analog computer.
• Evaluate the functionality by demonstrating the predetermined differential equations and compare the result of the
analog computer with the custom circuitry.
• Present the results found in the evaluation in the context of an ever growing digital society with a focus
on education and practical application of analog/hybrid
technology.
"RQ1 - What is the feasibility of creating a low cost
modular single board analog computer that can solve second
order differential equations?"
"RQ2 - What are the capabilities and usages of a low cost
modular single board analog computer?"
V. R ESEARCH M ETHODOLOGY
The methodology used in this paper is heavily influenced
by design science research (DSR) and its methodology as
described by Peffers et al. [17]. The focus of the research
conducted is to create an artifact that has the potential to
further the development of analog computers, therefore being
actually useful in its domain. The methodology does not focus
on rigid design processes and formal theory.
The design science research methodology (DSRM) presented by Peffers et al. [17] is influenced by the DSR forerunners March and Smith [18], Nunamaker [19], and Walls [20].
These authors were focusing on building actual artifacts that
would later be used in case studies to examine their impact
on a specific institution.
A DSRM research project is well suited to start with a
loosely specified problem such as a stakeholder request or
further development on an existing product. Important to note
is that this method of conducting research is more of a
suggestion on "how to do it" rather than stating that it is
the best way to go about that particular research according
to Peffers et al. [21].
The role of theory for DSRM is to create a technical base
for the artifact being created to rest upon. Very little contextual
theory is included since it is assumed that the artifact contains
the context needed in order to justify its creation. This means
that more focus can be allocated to the technical base for
the artifact and limit contextual inclusions to a DSR paper.
Although the artifact created holds context by its very creation,
some context will be provided given the lack of prevalence of
analog computing in today’s society.
To evaluate the artifact, DRSM uses demonstration-based
evaluation in order to prove the functionality of the artifact.
If the artifact can be demonstrated using a predetermined
functionality test, the functionality of the artifact can be proven
instantly. If it can be demonstrated over a range of contexts,
it can be evaluated [17].
In order to create the artifact, that is the low-cost analog
computer, the different computational elements that make up
an analog computer need to be investigated. This includes the
analog computer programming schematics that is prevalent in
a lot of textbooks from the time of the analog computers’
height of popularity. These computational elements will then
be translated to circuitry that can be manufactured on a PCB
(Printed Circuit Board). This includes choosing ICs (Integrated
Circuits), components, and component values based on the
technical literature and experimentation. This will form the
VI. T ERMINOLOGY
A. Analog computer
Analog is derived from the Greek words ana-logon. The
translation is interpreted as "according to a ratio". This could
be explained as a connection between different relationships
in regards to material and constructs. A sail catching the wind,
an airplane wing creating lifting force, and the submarine deep
in the Atlantic ocean are all different constructs but analogous
in the way they deal with physical forces.
The electric analog computer’s output from operational
amplifiers is a response to its input signals. The output varies
depending on the input. Utilizing this, a system that can
analogously respond to different conditions can be modeled
to simulate different real-world phenomena [22].
Analog computers were and still are based on variable
voltages to calculate and represent the solution to a given
problem. Analog computers handle computation by utilizing
logic built by different operational amplifier-based circuits for
different mathematical calculations based on variable voltages
in the circuit [5].
The analog computer is highly suited for calculating second
order differential equations. The problem the user wants to
solve (by the equation) is first translated into an equation. The
equation is then rearranged, in this example using the Kelvin
feedback technique [5], where the highest derivative is isolated
on the left side of the equation. By then feeding this derivative
into a specific chain of computational elements (integrators,
summers, etc.) the right side of the equation will be obtained.
By creating an analog system, a physical system is configured and translated to mathematical abstraction. By introducing an initial condition that corresponds with the problem’s
3
initial value the analog computer will simulate the problem
over time starting at T0 to Tf (meaning the start and stop of
the simulation) and present the results continuously [8] [5].
Digital computers rely on relays representing ON (1) and
OFF (0) to calculate and display the solution to a problem.
This is called boolean algebra and is the norm for digital
computers today.
dN
= λdt
N
2) Second order linear differential equations: A second
order differential equation is defined by the equation below:
B. Boolean algebra and discrete mathematics in computer
science
This is essentially the same as a first order differential
equation, except that y second derivative ÿ also is included.
Boolean algebra is the logical language where you are
able to express any statement through a binary form [23]. In
computer science this often refers to machine language, ones
and zeros or on and off pulses. Due to the vast versatility
of the different logic gates that can be created by combining
transistors boolean algebra and discrete mathematics quickly
rose as the leading language of computer science [24].
There are many advantages with discrete mathematics and
boolean algebra when it comes to computer science, mainly
giving in this case the programmer the ability to be very
precise when running calculations. It takes a lot for a one
to become a zero in comparison with a 1,00001 becoming a
1,00002 in the field of continuous mathematics [25].
Despite being the computer language of today, discrete
mathematics and boolean algebra has a big bottleneck when it
comes to its use in computer science. This disadvantage stems
from the lack of variability in the throughput which forces
digital computers to work sequentially. A digital computer can
only process either a one or a zero at a time which limits
the computational speed to the rate in which you can switch
between a one and a zero [25].
D. Simple oscillator circuit
•
Radioactive decay:
−
F (t, y, ẏ, ÿ) = 0
The oscillator circuit is described in Ullmann’s book “Analog and Hybrid Computer Programming” [5] the equation for
the oscillator goes as follows:
ÿ = −ω 2 y
where ω 2 is some arbitrary weight. The logic schematic is:
Fig. 2. Simple oscillator [5]
The oscillator can be used as an effective tool to test the
functionality of analog computers (integrators, invertors and
summers). It was used in this project as the first functionality
test and the output (solution) should be displayed as a steady
waveform.
C. Differential equations
As Braun and Golubitsky state in there book "Differential
equations and their applications" differential equations occur
in many areas of science [26]. In this section we will briefly
explain what a first and second-order linear differential equation is and give some examples of what they are used for.
Important to understand is that a derivative can be interpreted
as a rate of change seen over time. The first derivative is the
rate of change and the second derivative is the acceleration of
change.
1) First order linear differential equations: A first order
differential equation is defined by the equation below:
Fig. 3. Simple oscillator output [5]
F (t, y, ẏ) = 0
E. Mathieu’s differential equation
In other words, an equation consisting of the unknown
function y and its first derivative ẏ with respect to time t.
First order differential equations occur in a lot of different
fields of science, a few examples of those are:
• Newton’s law of cooling:
Mathieu’s differential equation is a well known, linear
homogeneous, second order differential equation. The equation
has many different applications ranging from the vibrations of
an elliptic drum to balancing an inverted pendulum [27].
As described by Ulmann [27], the general expression of
Mathieu’s differential equation is:
ẏ = k(M − y)
•
ÿ + (a − 2qcos(2t))y = 0
Eikonal equation:
H(x, ∇u(x)) = 0
With the initial conditions:
4
y0 = 1
ẏ0 = 0
a and q are parameters that has been assigned a value based
on the given problem. To simplify the calculations one can set
a = 2q. By doing this one gets the calculation:
ÿ + (a − acos(2t))y = 0
One can then factor out a out of the parentheses which
results in:
ÿ + a(1 − cos(2t))y = 0
Fig. 5. X for Mathieu’s equation [27]
The resulting equation is now dependent on y (the sought
after value) and t (the solution to the equation when taking
time into account a is still a parameter one can assign any
value to. The parts that is dependent on t can be seen as an
"input signal" that can be modeled. Therefore one can call
it x(t) or just x. To ease the calculations we introduce the
x variable to the aforementioned equation by the following
statements:
x(t) = 1 − cos(2t)
ÿ + axy = 0
Based on the aforementioned explanation, these statements
holds true.
The general expression for y can be exemplified by analog
computer schematics, see fig 4.
Fig. 6. Typical solutions for Mathieu’s differential equation [27]
F. Spring mass dampening system
The spring mass dampening system is a good example of an
oscillator with dampening. This simulation is an analog of a
weight suspended on a spring with the mass m. The oscillation
is the weight bouncing up and down with the position y, thus
the vertical position. According to ´ Ulmann [5], to simplify
the calculations, the gravitational forces acting upon the weight
will be neglected. The force due to the moving mass (weight)
can be expressed as:
Fm = ma = mÿ
The spring will introduce a force depending on the strain
applied to it (pulling the string) and the force of the linear
velocity dampening can be expressed as:
Fig. 4. Y for Mathieu’s equation [27]
The general expression for x in the Mathieu’s differential
equation (5) can be expressed as:
Fs = sy
ẍ + 4x = 4
Analog computers are ideally suited for solving differential
equations. To specify what differential equation to use, the
Mathieu’s differential equation was chosen because of its
prevalence in academic resources.
The analog computers schematic for solving Mathieu’s
differential equation can bee seen in fig 4 and 5. Typical
solutions for Mathieu’s differential equation can be seen in
fig 6
Fd = dv = dẏ
The spring mass dampening analog is a closed physical
system. This means that all the forces in the system needs
to add up to 0. The second order differential equation based
on this information can be expressed as:
mÿ + dẏ + sy = 0
5
The m is the mass, d is the dampening constant and s is the
spring constant. Important to note, with out Fd no dampening
will occur.
The actual equation the analog computer can solve, and has
the possibility to be easily converted to a schematic, is:
dÿ + sy
m
The schematic that will be used in this paper to create a
circuit is a simplified version in order to save one summer.
The disadvantage of this approach is that constants s and d
can not be set independently. The simplified approach has s
and d as a fraction of m. The simplified version introduces two
new parameters called v and µ. The mathematical expression
for these two are:
s
v=
m
ÿ = −
Fig. 8. Typical solutions for spring mass dampening. Top graph v = 0.6
µ = 0.8 Bottom graph v = 0.8 µ = 0.6 [5]
The comparison between analog and digital is something
academia has covered in regards to time, the complexity of
problem, and energy usage. A paper written by Köppel et al
[2] uses a common differential equation. Their experiment
showed that the analog computer did not take longer to
compute given the complexity of the problem but the digital
computer did. The energy consumption did also increase for
the digital computer (which was significantly higher initially)
when the complexity increased, whereas the analog computer
remained at low levels of energy consumption throughout the
experiment.
This is due to the fact that the analog computer does not
run on a sequential boolean algorithm. Instead, the Analog
computer runs continuously and directly using the op-amp
(operational amplifier) based computational elements. This is
the main advantage of using an analog computer according to
the writer.
The common digital computer is a so-called a "stored
program computer" and has a fixed internal structure. This
fixed internal structure is then controlled by an algorithm
which is stored as a program in memory inside the machine.
This structure has the tendency to create data dependency
and memory bottlenecks that impact the overall performance
negatively. The analog computer is in comparison not reliant
on memory or having a fixed inner structure. The different
computational components are arranged in a way to create an
analog of the mathematical problem that is being simulated.
This speeds up the process of computing and simulation of
a given task due to all calculations running in parallelism to
each other [28] [2].
In a paper written by Holzer and Ulmann [28] they discussed the integration of analog computing when working
with machine learning (ML). They created a hybrid system
where the analog computer was closely connected to a digital
machine. The analog computer was continuously simulating
the balancing of the inverse pendulum and the ML algorithm
used this information as data for reinforced learning.
The analog computer was utilized as a powerful coprocessing unit to solve the differential equation-based prob-
d
m.
The two potentiometers −ẏ0 and y0 are used to set the
initial condition of this analog. y0 sets the initial deflection
and −ẏ0 sets the initial velocity of the mass which is set to
1. Different values for s and d will result in different results
as the spring and dampening will be seen in the oscillation of
the mass. The analog computer schematic can be seen in fig
7 and typical solutions can be seen in fig 8.
µ=
Fig. 7. Simplified spring mass dampening [5]
VII. R ELATED W ORK
A. The computational advantage of analog
The analog computer does not run on a sequential basis
like its digital counterpart. The added flexibility of the digital
computer is due to the ability to run it on a joined clock system
(enabling time sync between two digital computers enabling
precise simultaneous operation) and being able to solve a
variety of tasks due to the aforementioned sequential algorithm
structure. The main problem with the current development of
the digital computer is that the digital computer is rapidly
approaching Moore’s law, thus hindering further progress [2]
[5].
6
lem of balancing an inverted pendulum. The reinforcement
learning was done episodically where one episode is from
the start to the pendulum tipping over or the cart moving
outside the designated area. The digital computer asked the
analog for the real-time simulated values, the communication
went through a hybrid controller (by serial communication)
that converted the information between the two systems. The
digital computer, running the ML algorithm, then decided if
the episode was to be terminated and reset or continued. If
the pendulum was mounted on a fixed cart the equation would
have been:
g
φ̈ − sin(φ) = 0
l
Since the cart was not fixed, the mathematical expression
was greatly more complex than if the cart would have been
stationary. The authors goes great lengths proving and simplifying the differential equation actually being calculated by
the analog computer. The differential equation that was being
solved was:
The usage of electronic analog computers can also be
applied to academic endeavors in other ways than simulating
physics or engineering problems, such as a pedagogical instrument (see fig 9). Students can be enrolled in mathematics
classes without ever visualizing a differential equation. Students are thought to instead create an expression that satisfies
the equation through a systematical (perhaps even mechanical)
approach that gives little deep understanding of the process.
The implementation of analog computers in this context is
both cost-effective and shows a positive impact on the learning
outcomes due to an increase in intuitive calculations by
the students. The analog computer allowed the students to
work on nonlinear and linear systems while emphasizing the
relationship between the mathematical and physical models.
This meant that there was a knowledge combining element
between the math and physics-focused classes [30] [31] [32].
φ̈ = ẍcos(φ) + gsin(φ)
And the acceleration of the cart can me modeled as:
M ẍ = F
The length of the pendulum was assumed to have the length
1 (l = 1) and is therefore removed in the above equation. The
φ is the angle between the the pendulum and the vertical axis
(y). g is the gravitational acceleration, x is the position along
the horizontal axis (x), M is the mass of the cart and F is the
force applied to the cart to stabilise the pendulum.
The results of the study by Holzer and Ulmann [28] showed
that a hybrid system is possible when dealing with ML and that
having an analog computer simulation the problem was more
stable and more energy efficient than a purely digital system.
The paper also suggest that great computational improvements
could be achieved if there was an integrated analog coprocessor embedded in modern digital systems, thus increasing
computational and simulation capabilities.
Fig. 9. A demonstration of the solution to a differential equation based
problem using an analog computer [30]
A low-cost, desk-sized, analog computer for use in academia
has been of interest to technical faculties in the past. Hamilton
[33] proposes a design for a machine (see appendix E for
schematic) in a paper called "An Analog Computer for Educational Laboratories". The machine is a general-purpose analog
computer that features several computational elements such as
integrators, summers, dividers, and coefficient potentiometers.
The design also features an overload detection, meaning when
an overload of the operational amplifiers occurs.
An example of a desk-sized analog computer that saw a long
life in academia is the Comdyna GP-6 released in 1966 [34].
The GP-6 longevity in academic institutions can be explained
by its use in control unit education. The example given in
Spiess [35] paper describes the students acquiring more knowledge of electronics and the connection to mathematics while
using an analog computer. The computer used in the paper is
a GP-6 and is a staple in the control systems laboratory at
the University of Illinois. Programming an analog computer
requires an understanding of the fundamental building block
B. Analog computers in academia
Since the invention of the general-purpose electronic analog
computer the usage in engineering and physics laboratories has
been prevalent. Given the nature of the analog computer and
its ability to solve differential equations efficiently, the analytical solution to problems was not the default approach. The
analog computer made the numerical approach possible as the
approximation needed was easily introduced in programming.
In the early days of analog computing, the experimentation
process was greatly sped up by having the possibility to easily
program (compared to its contemporaries) a problem and see
the results almost immediately. An example