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