Applications Engineer Approach to
Maxwell and Other Mathematically
Intense Problems
Or
Applications Engineers Don’t Do
Hairy Math
Marcus O Durham, PhD, PE
Fellow, IEEE
Theway Corp
Robert A Durham, PE
Member, IEEE
D2 Tech Solutions, Inc.
Karen D Durham, EI
Member, NSPE
NATCO
Objective
 Develop a structure for app engineers
to use when reading or working with
complex math concepts
Abstract
 EE taught w/ complex concepts and intense
math
 In practice, very little intricate science
 Problems solved with algebra
 Why is there a difference?
 Paper reduces all EE math to
two simple equations
Abstract too
 Single Unified Equation (SUE) for circuits
 Add distance to encompass Maxwell’s suite
 Math is vector algebra
 NO CALCULUS
y
X
  Y
y
Any Questions?
?
But First…
 Would you agree that apps engineers…
- Are results oriented?
- Solve problems w/out complex theory?
- Don’t even read articles w/ hairy math?
- Can’t remember Maxwell?
- Think a curl is part of the Winter Olympics?
 Then this article is for you.
Core Belief
 Kirchhoff is used to solve all problems
 Kirchhoff derived from Maxwell
 Ergo - Maxwell is at the core of all EE
 However, how many EEs can do derivation w/out
reference?
 And how many EE books can be used for
reference?
Core Belief
 We’re not messing with Kirchhoff
 We are cleaning up Maxwell
 We are eliminating Calculus
 And Diff-EQ and Partials
 And others that App Engineers don’t use
 Allows comprehension of intense articles w/out
following the hairy math
Are we together so far?
Okay then . . .
P’s and Q’s
 Three elements of matter
- Mass (m)
- Magnetic Pole (p or φ)
- Charge (q)
Equations use elemental
rather than derived
Time is on Our Side
 Time is always a denominator
 Three elements of time
- Fixed: t = 1
- Rate: 1/tt
~ Velocity (Current), Energy
- Acceleration: 1/(tt tr)
~ Potential, Power
To the Point
 Electrical and magnetic concepts can
be combined into one simple equation.
i.e.
 Electromagnetic energy is the change
in the product of charge and pole
strength over time

pq 
E
tt
E = [p q] / tt
 Equation is for point conditions (node)
 Concept so fundamental and inclusive
appears intuitively obvious
 However, NO previous references
What are Measurables?
 Can only measure three things
 Voltage :
V = [p]/t
 Current :
I = [q]/t
 Frequency: f = 1 /t
 All measurables derived from SUE
 That’s a strong statement!
Calculating…
Can only calculate three things
Measured components are
unique, so can’t add or subtract
Leaves multiplication and
division
Calculating…
 Product
S = V*I = [p/tr] * [q/tt] = [E]/tr
 Ratio
Z = V/I= [p/tr] / [q/tt]
 Delay or phase shift
td = tr – tt
 Anything else?
Are the Laws Legal?
 Concepts embedded in SUE are staggering
- KVL
- KCL
- Faraday
- Definitions of “Measurables”
 At a node, this is all encompassing
 No more complex than algebra
E = [p q] / tt
 This opens the understanding of
electromagnetic science to an entire new
level of application.
 The equation removes the constraints on
moving between electrics and magnetics
 “But what about Fields?”
Fields are a Gas
 E-mag fields considered “toughest” part
of EE
 Actually, no more complex than circuits
 As a circuit is analogous to liquid flow…
Fields are analogous to gas in a vessel!
Space, the Final Frontier
 Cartesian axes good for straight,
rectangular world
 Real world is curvilinear, spheroidal
space
 Fields live on the surface of a spheroid
 A coordinate system based on a sphere
is necessary
Spherical Coordinates
 Corresponds to navigation coordinates
t ~ latitude
s ~ longitude
y ~ altitude
y
x
t
s
z
Spherical Coordinates
dt
bs
bys
θ
sy
x
bys defines a point on the surface
relative to the origin
 dt defines the distance around
the sphere for a given “parallel”
st
y
ss
Moving and Mooning
 Consider the sphere to be a moon
orbiting around a “fixed” planet
 How does the moon move?
- Rotational (days)
- Orbital (months)
 The combination
creates sinusoidal motion
Or
 Consider the magnetic rotation of a
motor
 How does the motor work?
- Rotational (shaft)
- Orbital (coils)
 The combination
creates sinusoidal motion
Crank out the Volume
 Surface volume
- Calculated from longitude, latitude and altitude
- Uses vector algebra
- Vy = ss st sy
 Operational volume
- Region transcribed by motion of the sphere
(under sinusoid in 3D)
- Vy = bys dt sy
 Space vector (sy) describes
the orbital motion
If you build it…
So, what’s the deal with
spheres and volumes?
Here’s the pitch
The Simplified Unified Equation
Multiplied by the ratio of
Operational Volume to Surface
Volume
Yields electromagnetic field energy
[ pq] Voperational
E
*
t
Vsurface
Going, Going
What is the significance of this
simple product of flux, charge and
distances over time?
And it’s Outta Here
Every machines, transmission and
fields problem calculated from one
simple relationship
Complex, special problems solved
using simple program or
spreadsheet
[ pz  qy ] bys  dt  sy 
E

tt
ss  st  sy
Density
 Current not point but dispersed
 Skin Effect
 Circumference determines cross-sectional
area (At)
 Current Density (Jt) = current over area
 Charge Density (ρ) = charge over volume
Intensity of the Density
 Field Intensity
 1 / (time * length)
 Field Density
 1 / Area
 Energy is the product of intensity, density
and volume
 All four foundational relationships can be
derived from the fields equation
[ pz  qy  bys  dt  sy ]
E
tt  ss  st  sy
z
Electric Intensity
 pz 
E

 tr  st  t
Magnetic Intensity
 qy 
H

 tt  ss  s
dt
bys
bz
θ
by
bs
y
ss
[ pz  qy  bys  dt  sy ]
E
tt  ss  st  sy
z
Electric Density
 qy
D

 Ay



y
Magnetic Density
 pz 
B 

 Az  z
dt
bys
bz
θ
by
bs
y
ss
The Bottom Line
 All four relationships, which are the
basis of all field analysis, can be
extracted from the single e-m field
relationship
D
H
E-M
Equation
B
E
The suite of equations developed by Maxwell contains four relationships.
For the Details
 Correspondence to Maxwell
is straightforward,
if ever needed
XE =
- dB/dt
Volt/m2
 X H=
J+ dD /dt
Amp/m2
D =

Cb/m3
B =
0
Using the common internal, radial vector ‘1/sy’, rather than the del, the
suite of four equations can be calculated from the single unified
electric-magnetic energy field relationship.
E=
[pz qy bys dt sy]
tt Vy
First the intensity or density relationship will be shown as previously
defined. Next, to obtain volumetric terms, both sides of the equation will
be multiplied by the inverse of the vector along the y-axis, ‘1/sy’. The
subsequent equations manipulate the vector algebra. The result is a
relationship that is equivalent to one of the del equations.
This simple process uses a unified electromagnetic equation with a
vector along an axis. This eliminates the complex calculus of Maxwell
in exchange for a simple algebra operation.
Intensity: The distances we have used in the dynamic or intensity
relationships are relative to the external reference axes ‘st, ss, sy’.
These inherently contain the cross product of the del ‘’. The vector in
the radial direction ‘sy’ multiplied by a vector on the surface yields an
area in the other surface direction.
Equation of electric intensity
Et
=
[pz / tt st]t
Volt/m
(1/sy)Et
=
[pz / tt sy st ] Volt/m2
=
[pz] / tt A-s
=
[B / tt]-s
= -[B / tt]s
=
xE
Equation of magnetic intensity
Hs
=
[qy / tt ss]s
Amp/m
(1/sy)Hs
=
[qy / tt sy ss ] Amp/m2
=
i / At
=
Jt
=
xH
Equation of charge density
Dy
=
[qy/Ay]
[Dy / tt]
=
[qy / tt Ay]
=
i / Ay
=
J
=
xH
 Check Appendix for details
Cb/m2
Amp/m2
Density: the distances in the static or density relationships are relative
to the internal, reference axes ‘sx, sy, sz’. These inherently contain the
dot product of the del ‘’.
The vector in the radial direction ‘sy’ multiplied by the plane area in the
direction of the displacement yields a volume. In the magnetic equation,
the radial and the plane area are in different directions. Hence, the
result of a dot product in two different directions does not exist.
Equation of electric density
Dy
=
[qy/Ay]
(1/sy)Dy
=
qy/Ay sy
=
qy / Vy
=
y
=
D
Equation of magnetic density
Bz
=
[pz/Az]
Bz / s y
=
pz/Azsy
=
0
=
B
Cb/m2
Cb/m3
Wb/m2
Wb/m3
It is fascinating that all the action is on the radius axis ‘sy’. However, it
is the understanding of physical relationships that make the unified
electric-magnetic equations possible.
Conclusions 1/3
 Electro-magnetics is made up of
electrical charges and magnetic
poles moving in some time frame
 E = [p q] / tt
Conclusions 2/3
The circuit, or rotational
motion, makes a sphere
By maintaining directional
orientation, all fields, one
equation
[ pz  qy  bys  dt  sy ]
E
tt  ss  st  sy
Conclusions 3/3
One equation can describe all
electromagnetic analyses
Complete model includes fields
and dispersion in space
When distances are resolved, the
relationship solves to a circuit
problem
Conclusively
By using poles, charge, & time,
with direction, application engineers can
* define any problem ,
* read complex math articles
* with algebra
* without calculus