Applications Engineer Approach to Maxwell and Other Mathematically Intense Problems
description
Transcript of Applications Engineer Approach to Maxwell and Other Mathematically Intense Problems
Applications Engineer Approach to Maxwell and Other Mathematically Intense Problems
OrApplications Engineers Don’t Do
Hairy Math
Marcus O Durham, PhD, PEFellow, IEEETheway Corp
Robert A Durham, PEMember, IEEED2 Tech Solutions, Inc.
Karen D Durham, EIMember, NSPENATCO
ObjectiveObjective
Develop a structure for app engineers to use when reading or working with complex math concepts
AbstractAbstract
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 totwo simple equations
Abstract tooAbstract too
Single Unified Equation (SUE) for circuits
Add distance to encompass Maxwell’s suite
Math is vector algebra
NO CALCULUS YXy
y
Any Questions?
?
But First…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 BeliefCore 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 BeliefCore 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
MATH
Are we together so far?
Okay then . . .
P’s and Q’sP’s and Q’s
Three elements of matter
- Mass (m)
- Magnetic Pole (p or φ)
- Charge (q)
Equations use elementalrather than derived
Time is on Our SideTime 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 PointTo 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
ttpqE
E = [p q] / tE = [p q] / ttt
Equation is for point conditions (node)
Concept so fundamental and inclusive appears intuitively obvious
However, NO previous references
What are Measurables?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…Calculating… Can only calculate three things Measured components are
unique, so can’t add or subtract Leaves multiplication and
division
Calculating…Calculating…
ProductS = V*I = [p/tr] * [q/tt] = [E]/tr
RatioZ = V/I= [p/tr] / [q/tt]
Delay or phase shifttd = tr – tt
Anything else?
Are the Laws LegalLegal?
Concepts embedded in SUE are staggering- KVL- KCL- Faraday- Definitions of “Measurables”
At a node, this is all encompassing No more complex than algebra
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?”
E = [p q] / tE = [p q] / ttt
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 FrontierSpace, 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 CoordinatesSpherical Coordinates
Corresponds to navigation coordinatest ~ latitudes ~ longitudey ~ altitude
y
sx t
Spherical CoordinatesSpherical Coordinates
bys defines a point on the surface relative to the origin
dt defines the distance aroundthe sphere for a given “parallel”
y
zdt
bysbs
θ
ssx st
sy
Moving and MooningMoving 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
Consider the magnetic rotation of a motor
How does the motor work?- Rotational (shaft)- Orbital (coils)
The combination creates sinusoidal motion
OrOr
Crank out the VolumeCrank 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) describesthe orbital motion
If you build it…
So, what’s the deal withspheres and volumes?
The Simplified Unified Equation Multiplied by the ratio of Operational Volume to Surface
Volume Yields electromagnetic field energy
Here’s the pitch
surface
loperationa
VV
tpqE *][
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
yts
ytys
t
yz
ssssdb
tqpE
][
DensityDensity
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 DensityIntensity 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
Electric Intensity
Magnetic Intensity
ytst
ytysyz
ssstsdbqpE
][
tstp
tr
z
E
sstq
st
y
H
y
z
dtbys bz
bs
by
θ
ss
Electric Density
Magnetic Density
ytst
ytysyz
ssstsdbqpE
][
yAq
y
y
D
zAp
z
z
B
y
z
dtbys bz
bs
by
θ
ss
The Bottom LineThe Bottom Line
All four relationships, which are the basis of all field analysis, can be extracted from the single e-m field relationship
E-MEquationH E
D
B
For the DetailsFor the Details
Correspondence to Maxwell is straightforward, if ever needed
Check Appendix for details
The suite of equations developed by Maxwell contains four relationships. X E = - dB/dt Volt/m2
X H= J+ dD /dt Amp/m2 D = Cb/m3
B = 0Using 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/s y’. 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 ‘s t, 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 intensityEt = [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
= x E Equation of magnetic intensityHs = [qy / tt ss]s Amp/m (1/sy)Hs = [qy / tt sy ss ] Amp/m2
= i / At
= Jt
= x H Equation of charge densityDy = [qy/Ay] Cb/m2
[Dy / tt] = [qy / tt Ay] Amp/m2
= i / Ay
= J= x H
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 densityDy = [qy/Ay] Cb/m2
(1/sy)Dy = qy/Ay sy Cb/m3
= qy / Vy
= y
= D Equation of magnetic densityBz = [pz/Az] Wb/m2
Bz / sy = pz/Azsy Wb/m3
= 0= B
It is fascinating that all the action is on the radius axis ‘s y’. However, it is the understanding of physical relationships that make the unified electric-magnetic equations possible.
Conclusions 1/3Conclusions 1/3
Electro-magnetics is made up of electrical charges and magnetic poles moving in some time frame
E = [p q] / tt
Conclusions 2/3Conclusions 2/3
The circuit, or rotational motion, makes a sphere
By maintaining directional orientation, all fields, one equation
ytst
ytysyz
ssstsdbqpE
][
Conclusions 3/3Conclusions 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
By using poles, charge, & time,with direction, application engineers can
* define any problem ,* read complex math articles * with algebra* without calculus
ConclusivelyConclusively