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 system you don’t
have to rely on inspiration.
You can instead ask if the system can be
described mathematically?
Since a system only provides a means,
what you are really asking is: can the
problem or need can be solved
mathematically?
Then implement a design using
electronics to do the math.
Clarifying the purpose
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To answer the question, we need to
examine what it is that the system does
and how math plays a part.
Assume a system that takes a
measurement of some sort and report the
information back to the user.
We’ll look first at some various
measurement components of a design.
Component: Distance
Distance can be measured directly.
 A ruler can measure distance between
two object or marks directly.
 This is a counting of units.
 The units can be inches, centimeters,
volts, amperes, ohms, and etc.
 Counting is summation process.
 Summation is addition.
Component:Time
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 and either the height or
distance of the other object determines
the unknown.
 Math used is multiplication and division.
 H1/D1=H2/D2 ; solving for the height of
the first object: H1=(H2*D1)/D2
Mathematically basis of
indirect measurements
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.
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Amplification
A telescope and a microscope are optical
amplifiers.
 The size of the object viewed is a function
of the focal length of the lenses.
 Tan =d/(2*f)
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Where  is the angle that subtends the object, d
is the image size 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 gearing or hydraulics.
Transportation
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.
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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
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.
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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
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 for a
system that is measuring (for
example) voltage levels.
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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 are electronic
components made up of transistors.
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)
d/dt is an operator
 Whenever you see it, it means it’s
measuring change.
d
 Thus ic  C V c
dt
 Means the current (ic)measures the
change in voltage across the
capacitor (times the value of C).
d
 What does
mean?
L
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V
L
i
dt
L
Application of
Differential Calculus
If we had a capacitor 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.
 INSERT DIAGRAM
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Application of
Differential Calculus, con’t.
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.
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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
Is, if we want to sum the current going
into the capacitor, we only have to
measure the voltage across it.
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).
We now have a means to add.
Subtraction is a comparison of the differences.
Conclusions
Calculus only takes about two weeks
to explain…the rest is just practice,
familiarity, and some other tricks
they throw in.
 Since we can add, we can multiply.
 Since we can subtract, we can
divide.
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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.