Download Mathematical Basis for Electronic Design

Mathematical Basis for
Electronic Design
Wentworth Institute
of Technology
Electronic Design I
Prof. Tim Johnson
Designs with a Purpose
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In order to design a component, you don’t
have to rely on inspiration or resort to the
Edison solution: methodical testing of all
possible variables until the key is found.
You can instead ask if the system can be
described mathematically?
Since a design is only a realization of a
solution to a problem, what you are really
asking is: can the problem or need be
described mathematically?
Is there a
formula that describes the operation?
Then implement the design solution using
electronics.
Designing by variable
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When working on the Bell project, you had the
benefit of knowledge of the formula describing an
electromagnet.
In the formula (N*I/L)*μ=B each letter represents
a variable or electrical parameter of the design.
The results of the calculation (B) was a value we
were trying to maximize by means of the inputs.
When the speaker worked, we could define B as a
parameter we had to reach.
We discovered its value by experimentation just
as Bell did in his design. Some of the inputs may
have limits.
Clarifying the purpose
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To find a formula in other designs, we
need to examine what it is that the system
or component does and how math plays a
part.
Assume a system takes a measurement of
something and reports the information
back to the user.
We’ll
look
first
at
some
various
measurement of a design then look at how
they are implemented as components.
Distance measurements
Distance can be measured directly.
 A
ruler can measure distance
between two objects (or marks)
directly.
 This is a counting of units.
 The units can be inches, centimeters,
volts, amperes, ohms, and etc.
 Counting is a summation process.
 Summation is addition.
Time measurements
Time can be measured.
 Time can be measured by an
electronic counter.
 Counters can display the sequence of
units.
 Stopping the counter freezes the
display giving the summation.
 You are measuring from the start
(t0=0) until the finish (tf=X).
Time, con’t.
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Summation for time is mathematically the
same as summation for distance…it’s
addition.
It’s also a difference between time zero
and time of finish: t0– tf =X
Alternatively, two different times can be
subtracted from each other giving the
difference.
Time can be measured by addition or
subtraction.
Indirect measurements
Determining size from a distance.
 This employs the trigonometry of similar
triangles.
 Knowing the height (H) and distance (D) of
one object (#2) and either the height or
distance of the other object (#1)
determines the unknown.
 Math used is multiplication and division.
 H1/D1=H2/D2 is the formula. Solving for the
height of unknown object: H1=(D1/D2)*H2
A simplification of math
used in electronics
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Multiplication is the summation of a
number, over and over for a specific
number of times.
Division is the subtraction of a number,
over and over for a specific number of
times.
Microprocessors are really stupid but fast.
If you are multiplying x by y, it merely
adds y to itself (x-1) times and keeps track
of the total in various registers.
Amplification
A telescope and a microscope are optical
amplifiers.
 The size of the object seen is a function of
the focal length of the lenses times the
angle the object actually occupies without
the lens.
 Tan =d/(2*f) or 2*tan *f=d
 Where  is the angle that subtends the
object, d is the image size we see in the
lens and f is the focal length (distance from
the lens that the object comes into focus).
 The
focal length is changed by the
curvature of the lens.
Amplification con’t.
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Mechanically, force can be multiplied using
a lever.
The mathematical formula is Fa*Da=Fb*Db
If we know Fb and Da&b then we can
calculate Fa easily by use of multiplication
and division: Fa=Fb*Db/Da
By rearrangement of terms: Fa=(Db/Da)*Fb
where (Db/Da) is the amplification factor of
Fb to get Fa.
Other means of amplification mechanically
is by using gears or hydraulics.
Transportation
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Transportation is the movement of
objects from x1 to x2.
This mathematically is the
multiplication of x1 by some value.
A*x1= x2
This is also known as translation or
projection.
Translation
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The value A can be a constant or
some function.
A is a constant (and linear) if for
example
you
are
rearranging
furniture in a room.
On the other hand if you are moving
to another apartment across town
then A is a function (and non-linear).
Systems
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Let’s expand our understanding of
systems beyond just measurements.
We’ll consider the system itself as a
mathematical entity.
There is an input, some work is
performed and there is an output.
Control
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If you can control the output,
there is a feedback loop.
System, Part II
Let’s consider a radio receiver
as a system.
 The input is normally a very
low powered electromagnetic
signal.
 The output is a audio wave at a
much higher power level.
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Transfer Function
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If we know the value of the input to
a system that we are measuring at
the output, we can establish a
relationship.
Value out/value in=amplification
Vout/Vin is a transfer function (the A
coefficient from a few slides back) for a
system that is measuring (for
example) voltage levels.
Communication
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Communication is the movement of
information with fidelity through a
medium from point x1 to point x2.
Transmission through a medium
causes a loss.
The loss is a known value and can
be expressed in units (the signal is
being measured in) per some
standard distance (mile, meter,
etc.).
Communication
Amplification
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Since the loss is a function of distance, if
the distance between a receiver and a
transmitter is known then I know what my
gain has to be.
V2=A*V1 where A is the gain which is
equal to loss per mile times the number of
miles. Since V2/V1=A; then A is also a
value for the system’s transfer function.
Gain is adjustable and accomplished by
amplifiers which is electronic components
made up of transistors.
The feedback
loops can vary or control the gain.
Our Design Tools
Resistor
 It’s math is Ohm’s Law: V=R*I
Capacitor
d
 It’s math is calculus based: ic  C
dt V c
Inductor
 It’s math is also calculus based:
d
V L  L dt iL
Differential Calculus
(in a nutshell)
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d/dt is an operator
Whenever you see it, it means it’s
measuring change.
d
Thus ic  C V c
dt
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Means the current (ic) is measured by the
change in voltage across the capacitor
times the value of C.
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What does
d
V L  L dt iL
mean?
Application of
Differential Calculus
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If we had a capacitor in a circuit across a
voltage that we wished to measure,
And an ammeter to measure current in
series with the capacitor, then
The value read for the current is actually
translatable into the value of the voltage.
You’ve all done differential calculus in
Circuit Theory 1 when you measured the
voltage rise across a capacitor.
Application of
Differential Calculus
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If we had an inductor in series with a
current that we wished to measure,
And a volt meter across the inductor, then
The value read for the voltage is directly
translatable into the value of the current
flowing if we knew the rate the voltage was
changing.
Since the time constant is R/L, this is
known. There is an additional “connection”
that can be made.
Application of
Differential Calculus
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Because of the time constant, changing
the resistor and/or the capacitor/inductor
will change the time for the device to
charge up, we have control over the
summation.
Switching resistors into a circuit that are
proportionately scaled to reflect the
multiplier allows different number to be
added.
Integral Calculus, part 1
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if we integrate both sides of the
formula for the Capacitor, we’d get:
ic  C
d
d
1


C
 ic  dt V c 
dt V c
C
i V
c
c
• Plus some initial current flow (assume zero)
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The integral sign means: the sum of
what it’s applied to. In this case the
current going into the capacitor.
Integral Calculus, part II
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As a result of the information on the
previous slide…
The meaning of:
1
  ic  V c
C
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Is: if we sum the value of the current into a
capacitor, we determine the voltage. If a
meaning is assigned to the current then the
voltage is an answer.
Application in a design
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Laser light travels at the speed of light.
Most microprocessors operate at too slow
a speed to allow accurate measurement.
How can you tell the distance without
parsing the time delay of the return pulse.
In a distance measuring device, if a light
pulse is sent out and a photo-voltaic cell
operates on the reflected levels received,
more current flows because the pulses
return quicker.
Thus a higher voltage
across the capacitor indicates a shorter
distance being measured.
You merely
calibrate Vc to the distance.
Integral Calculus, part III
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Integrate both sides of the inductor
equation to get:
1
 iL
V

L
L
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Which means that to know the sum of the
voltages applied to the inductor, we only
have to measure the current going into it
(x’s the inverse of the value of L).
Integration can be implemented
electronically to add.
Subtraction is an addition of a negative
value.
Conclusions
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Calculus only takes about two weeks to
explain…the
rest
is
just
practice,
familiarity, and some other tricks they
throw in.
It’s summation and becoming familiar with
strange symbols.
Since we can add, we can multiply.
Since we can subtract, we can divide.
Basic arithmetic can be implemented
electronically.
Your Task
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Write a memo that explains the
basis for the math that we used
to control the temperature of the
soldering iron project.