ELECTROMAGNETIC ENGINEERING EE325
INDEX
Ampere's circuital law.....11 duality of J and D............. 8 magnetic energy ..............12 shunt stub ........................ 4
Ampere's law ................... 6 E electric field................. 5 magnetic field .................11 single-stub tuning............. 4
angstrom .......................... 2 electric field..................... 5 at the center of a circular Smith chart ...................... 4
Avogadro's number........... 2 electron mass ................... 2 wire........................11 Smith charts..................... 4
B Ampere's circuital law 11 electron volt ..................... 2 central axis of a solenoid space derivative ............... 8
Biot-Savart law ...............11 electrostatic ...............................11 sphere .............................. 8
Boltzmann's constant........ 2 force ............................ 5 due to a finite straight standing wave ratio .......... 4
capacitance ...................7, 8 potential ...................... 5 conductor................11 static magnetic field ........11
between coaxial cylinders electrostatics .................... 5 due to an infinite straight stub length ....................... 4
................................ 7 elipse ............................... 8 conductor................11 surface charge density ...... 6
between concentric Faraday's law ..............6, 12 magnetic field intensity...12 time average power .......... 5
spheres .................... 7 flux density...................... 6 magnetic flux ..................12 vector differential equation8
between parallel plates. 7 force magnetic force.................11 volume energy density...... 7
between two conductors 7 electrostatic ................. 5 magnetization..................13 wave
characteristic impedance .. 2 magnetic.....................11 matching transformer forward-traveling ......... 5
complex conjugate............ 1 Gauss' law........................ 6 inline  reactive load.... 3 wave equation .................. 2
complex notation.............. 1 geometry.......................... 8 inline  resistive load... 3 wavelength....................... 2
conductance ..................... 8 grad operator.................... 8 mathematics..................... 8 W potential energy......... 7
e
conductivity ..................... 8 H magnetic field intensity12 Maxwell's equations......... 6 w volume energy density 7
e
semiconductor.............. 8 impedance mutual inductance ...........12 X reactance ..................... 3
conservative field law....... 6 short-circuit ................. 2 nabla operator .................. 8
Z line impedance........... 3
in
constants.......................... 2 induced voltage permeability..................... 2
Åš electrostatic
continuity equation........... 8 due to changing magnetic permittivity...................... 2
potential ...................... 5
coordinate systems ..........10 field........................13 phase constant.................. 2
“ reflection coefficient .... 2
coordinate transformations10 due to conductor motion13 Planck's constant .............. 2
magnetic flux .............12
coulomb ........................... 1 Faraday's law ..............12 Poisson's equation ............ 6
 wavelength................... 2
Coulomb's law.................. 7 slider problem.............13 potential energy................ 7
Á surface charge density. 6
s
cross product...................10 inductance.......................12 power
à conductivity ................. 8
curl .................................. 9 J current density.............. 7 with phasor notation..... 5
" del............................... 8
current ............................. 8 joule ................................ 2 reactance.......................... 3
"× curl ........................... 9
current density ................. 7 Laplacian ......................... 9 reflection coefficient......... 2
"· divergence................ 9
D flux density ................. 6 Lenz's law .......................12 resistance ......................... 8
"2 Laplacian ................... 9
del ................................... 8 light, speed of .................. 2 Rydberg constant.............. 2
divergence........................ 9 line impedance................. 3 self-inductance ................12
dot product....................... 9 linkage............................12 series stub........................ 4
COULOMB [C] COMPLEX CONJUGATES
A unit of electrical charge equal to one amp second, The complex conjugate of a number is simply that
the charge on 6.21×1018 electrons, or one joule per number with the sign changed on the imaginary part.
volt. This applies to both rectangular and polar notation.
When conjugates are multiplied, the result is a scalar.
(a + jb)(a - jb) = a2 + b2
COMPLEX NOTATION
(A "B°)(A " - B°) = A2
jb
ae = (a "b)
Other properties of conjugates:
where b may be in radians or degrees (if noted).
(ABC + DE + F)* = (A* B *C * +D * E * +F*)
(e- jB )* = e+ jB
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 1 of 13
TRANSMISSION LINES
SHORT-CIRCUIT IMPEDANCE [&!]
Zsc = jZ0 tan(²l)
REFLECTION COEFFICIENT [V/V]
L
The reflection coefficient is a value from  1 to +1
where: Z0 is the characteristic impedance
which, when multiplied by the wave voltage,
2Ä„
determines the amount of voltage reflected at one
² = É LC = is the phase constant
end of the transmission line.

l is the length of the line [m]
ZL - Z0
1+ “L
jÈ
“L = Áe = and ZL = Z0
ZL + Z0 1- “L
where: ZL is the load impedance
CONSTANTS
“L is the load reflection coefficient Avogadro s number
Á is the reflection coefficient magnitude
[molecules/mole] N = 6.02×1023
A
È is the reflection coefficient phase
Boltzmann s constant k = 1.38×10-23 J/K
L
= 8.62 ×10-5 eV/K
Z0 = is the characteristic impedance
C
Elementary charge q = 1.60 ×10-19 C
Electron mass m0 = 9.11×10-31 kg
THE COMPLEX WAVE EQUATION
The complex wave equation is applicable when the
Permittivity of free space µ0 = 8.85×10-12 F/m
excitation is sinusoidal and the circuit is under steady
state conditions.
Permeability constant µ0 = 4Ä„ ×10-7 H/m
2
d V (z)
Planck s constant h = 6.63×10-34 J-s
= -²2V (z)
dz2
= 4.14×10-15 cV-s
2Ä„
Rydberg constant R = 109,678 cm-1
where ²=É LC = is the phase constant

kT @ room temperature kT = 0.0259 eV
The complex wave equation above is a second-order
ordinary differential equation commonly found in the
Speed of light c = 2.998×108 m/s
analysis of physical systems. The general solution is:
1 Å (angstrom) 10-8 cm = 10-10 m
+ -
V (z) = V e- j²z + V e+ j²z
1 µm (micron) 10-4 cm
1 nm = 10Å = 10-7 cm
where e- j²z and e+ j²z represent wave propagation
1 eV = 1.6 × 10-19 J
in the +z and  z directions respectively.
1 V = 1 J/C 1 N/C = 1 V/m 1 J = 1 N· m = 1
The same equation applies to current:
C· V
+ -
I (z) = I e- j²z + I e+ j²z
and
+ -
V e- j²z + V e+ j²z
WAVELENGTH [m]
I(z) =
Z0
vp vp = velocity of propagation (2.998×108 m/s
for a line in air)
 =
where is the characteristic impedance f = frequency [Hz]
Z0 = L / C
f
of the line. These equations represent the voltage
and current phasors.
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 2 of 13
ź -WAVELENGTH INLINE MATCHING X REACTANCE [&!]
TRANSFORMER  resistive load
XC = reactance [&!]
For use with a purely resistive load that does not match the
XL = reactance [&!]
- j
line impedance. The load is matched to the line by
X =
C
j = -1
inserting a ź -wavelength segment having a characteristic
ÉC
impedance ZQ. É = frequency [radians]
 /4
C = capacitance [F]
X = jÉL
L
L = inductance [H]
Z0 ZQ RL
Z0 = characteristic impedance of the Zin LINE IMPEDANCE [&!]
transmission line [&!]
l = distance from load [m]
 = wavelength [meters]
j = -1
ZQ = Z0RL ZL + jZ0 tan²l
RL = resistance of the load [&!]
Zin = Z0
² = phase constant
ZQ = characteristic impedance of the
Z0 + jZL tan²l
Z0 = characteristic
ź
-wave matching segment [&!]
impedance [&!]
ZL = load impedance [&!]
ź -WAVELENGTH INLINE MATCHING
The line impedance of a ź -wavelength line is the inverse
of the load impedance.
TRANSFORMER  reactive load
For use with a reactive load. The load is matched to the
Impedance is a real value when its magnitude is
line by inserting a ź -wavelength segment having a
maximum or minimum.
characteristic impedance ZQ at a distance l from the load. l
is the length of transmission line required to produce the
Z0 = characteristic
1 + Á
first voltage maximum closest to the load. If the load is
Zmax = Z0S = Z0
impedance [&!]
inductive, the first voltage maximum will be closer than the
1 - Á
S = standing wave ratio
first voltage minimum, i.e. within ½ wavelength.
Á = magnitude of the
Z0 - Á
1
 /4 l
Zmin = = Z0
reflection coefficient
S 1 + Á
Z0 ZQ Zin Z0 ZL
First find the reflection coefficient in order to determine the
value of È. Then find the length l of the line that will
convert the load to a pure resistance, i.e. produces the first
voltage maximum. Find this resistance (Z ) using the line
in
impedance formula. Then determine the impedance Z of
Q
the ź -wavelength segment that will match the load to the
line.
“L is the load reflection
coefficient
ZL - Z0 È = phase of the reflection
jÈ
“L = Áe =
coefficient [radians]
ZL + Z0
Á = magnitude of the
i.e. “L = Á "È (radians)
reflection coefficient [&!]
Z0 = characteristic
È È
impedance [&!]
l = =
² = 2Ä„/ 
2² 4Ä„
 = vp/f wavelength [m]
ZL + jZ0 tan ²l
Zin = impedance (resistive)
Zin = Z0
of the load combined
Z0 + jZL tan ²l
with the l segment [&!]
ZQ = line impedance of the
ZQ = Z0Zin
ź -wave matching
segment [&!]
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 3 of 13
SMITH CHARTS FINDING A STUB LENGTH
Example: Find the lengths of open and shorted shunt
First normalize the load impedance by dividing by the
stubs to match an admittance of 1-j0.5. The admittance
characteristic impedance, and find this point on the chart.
of an open shunt (zero length) is Y=0; this point is
An inductive load will be located on the top half of the
located at the left end of the Smith Chart x-axis. We
chart, a capacitive load on the bottom half.
proceed clockwise around the Smith chart, i.e. away
Draw a straight line from the center of the chart through
from the end of the stub, to the +j0.5 arc (the value
the normalized load impedance point to the edge of the
needed to match  j0.5). The difference in the starting
chart.
point and the end point on the wavelength scale is the
Anchor a compass at the center of the chart and draw an
length of the stub in wavelengths. The length of a
arc through the normalized load impedance point. Points
shorted-type stub is found in the same manner but
along this arc represent the normalized impedance at
with the starting point at Y= .
various points along the transmission line. Clockwise
Open stub of j.5
length .074 
ge
movement along the arc represents movement from the
matches an
admittance
load toward the source with one full revolution representing
of 1-j.5
1/2 wavelength as marked on the outer circle. The two
Admittance
points where the arc intersects the horizontal axis are the
(short)
Y = "
voltage maxima (right) and the voltage minima (left).
.074 
Points opposite the impedance (180° around the arc) are
admittance. The reason admittance is useful is because
admittances in parallel are simply added.
j2²z
z = distance from load
“(z) = “Le
[m]
j 2²z Admittance
e = 1 "2²z j = -1 (open)
Y = 0
Á = magnitude of the
Z(z) -1
“(z) = reflection coefficient
.324 
Z(z) +1
² = phase constant
“L -1 ZL “ = reflection coefficient
ZL = Z =
Z = normalized
“L +1 Z0
impedance [&!]
Shorted stub of
length .324 
matches an
admittance
of 1-j.5
SINGLE-STUB TUNING
In this example, all values were in units of admittance.
The basic idea is to connect a line stub in parallel
If we were interested in finding a stub length for a
(shunt) or series a distance d from the load so
series stub problem, the units would be in impedance.
that the imaginary part of the load impedance will The problem would be worked in exactly the same way.
Of course in impedance, an open shunt (zero length)
be canceled.
would have the value Z= , representing a point at the
d
Shunt-stub: Select d
right end of the x-axis.
so that the
Y0 Y0 YL
admittance Y looking
toward the load from
Open
SWR STANDING WAVE RATIO [V/V]
or
a distance d is of the
short
Y0
form Y0 + jB. Then
V I
1+ Á
max max
l
the stub SWR = = =
V I 1- Á
susceptance is
min min
chosen as  jB,
resulting in a
matched condition.
d
Series-stub: Select d
so that the admittance Z0 Z0 ZL
Z looking toward the
load from a distance d
is of the form Z + jX.
0
Z0
Then the stub
l
susceptance is chosen
Open
as -jX, resulting in a
or
short
matched condition.
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 4 of 13
.2
5
0 . 1
2
5
0
5 . 0
0 . 1
1 . 0
0 . 1
n
e
d
r
r
a
a
t
o
w
r
o
T
.13
.12
.14
.11
.15
.1
.16
.09
.17
.08
.18
.07
0.5
.19
.06
2.0
.2
.05
.21
.04
.22
.03
5.0
.23
.02
.24
.01
.26
.49
.27
5.0
.48
.28
.47
.29
.46
.3
.45
2.0
.31
.44
0.5
.32
.43
.33
.42
.34
.41
.35
.4
.36
.39
.37
.38
P(z) TIME-AVERAGE POWER ON A E ELECTRIC FIELD
LOSSLESS TRANSMISSION LINE [W]
Ep = electric field at point
p due to a charge Q
Equal to the power delivered to the load. The power
or charge density Á
delivered to the load is maximized under matched
n [V/m]
conditions, i.e. Á = 0, otherwise part of the power is
2
1 r - rk
Ep = dE = an increment of
reflected back to the source. To calculate power, it
"Qk
4Ä„µ0 k =1 r - rk 3 electric field [V/m]
2
may be simpler to find the input impedance and use
Q = electric charge [C]
P = I2R or P = IV.
2
1 Ál(r )
V+ = the voltage of the µ0 = permittivity of free
2 Ć
2
+ dE = R dl
V forward-traveling space 8.85 × 10-12
4Ä„µ0 r - r 2
2
F/m
P(z) = (1- Á2)
wave [V]
2Z0
Ál = charge density;
Z0 = characteristic
2
1 Ál(r )
Ć
impedance [&!]
E =
1
+"R 2 dl2 charge per unit
length* [C/m]
4Ä„µ0 r - r 2
P(z) = Re V (z) I (z) * Á = magnitude of the
{ [ ] }
dl' = a small segment of
2
reflection coefficient
line l*
Re = "the real part"
Electric field from a potential: Ć
R = unit vector pointing
from r' to r , i.e. in
E = -"Åš
POWER USING PHASOR NOTATION [W]
the direction of r - r'.
refer to the NABLA notes on
r' = vector location of the
S = power [W]
page 8.
1
source charge in
S = VI * V = volts [V]
2 relation to the origin
I* = complex conjugate of current [A]
*NOTE: The l symbols could
r = vector location of
be replaced by a symbol for
the point at which
area or volume. See Working
V+ FORWARD-TRAVELING WAVE
the value of Ep is
With & on page 9.
observed
ZinV0
+
" = Del, Grad, or Nabla
V =
j²l
operator
(Zin + ZS )e (1+ “Le- j 2²l )
V+ = the voltage of the forward- ² = phase constant
traveling wave [V] l = length of the line [m]
ELECTROSTATIC POTENTIAL [V]
V0 = source voltage [V]
“L = load reflection
n
coefficient 1 Qk Åš = the potential [V]
Zin = input impedance [&!]
Åš =
"
ZS = source impedance [&!]
2
4Ä„µ0 k =1 r - rk dÅš = an increment of potential
[V]
2 Åšab = the potential difference
1 Áldl
dÅš =
ELECTROSTATICS
between points a and b [V]
2
4Ä„µ0 r - r
E = electric field
1 Ál 2
F ELECTROSTATIC FORCE dl' = a small segment of line l*
Åš = dl
+"
dl = the differential vector
2
4Ä„µ0 r - r
1 (r2 - r1)
1
displacement along the
F12 = Q1Q2 = 9×109
path from a to b
4Ä„µ0 r2 - r1 3 4Ä„µ0
Potential due to an
µ0 = permittivity of free space
electric field:
F12 = the force exerted by charge Q on Q . [N] 8.85 × 10-12 F/m
1 2
b
r1 = vector from the origin to Q Q = electric charge [C]
1
Åšab = - E·dl
+"
a
r2 = vector from the origin to Q . Ál = charge density along a
2
line* [C/m]
When finding the force on one charge due to multiple
To evaluate voltage at
rk' = vector location of source
charges, the result can be found by summing the
all points.
charge Qk
effects of each charge separately or by converting the r
Åš(r) = - E·dl
multiple charges to a single equivalent charge and r' = vector location of the
+"
"
source charge in relation
solving as a 2-charge problem.
*NOTE: The l symbols
to the origin
could be replaced by a
r = vector location of
symbol for area or
electrostatic potential Åš
volume. See Working
in relation to the origin
With & on page 9.
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 5 of 13
MAXWELL'S EQUATIONS GAUSS'S LAW
The net flux passing through a surface enclosing a charge
Maxwell's equations govern the principles of guiding
is equal to the charge. Careful, what this first integral really
and propagation of electromagnetic energy and
means is the surface area multiplied by the perpendicular
provide the foundations of all electromagnetic
electric field. There may not be any integration involved.
phenomena and their applications.
"B
µ0E·ds = Qenc
Á dv = Qenc
+"
"× E =- Faraday's Law +"D·ds = +"
S
S V
"t
µ0 = permittivity of free space 8.85 × 10-12 F/m
E = electric field [V/m]
"Å"D = Á Gauss' Law
D = electric flux density vector [C/m2]
ds = a small increment of surface S
" D
Á = volume charge density [C/m3]
"× H = J + Ampere's Law*
" t dv = a small increment of volume V
Qenc = total electric charge enclosed by the Gaussian
surface [S]
"Å"B =0 no name law, where:
E = electric field [V/m]
The differential version of Gauss's law is:
B = magnetic field [T]
"·D = Á div(µ0 ·E) = Á
or
t = time [s]
D = electric flux density [C/m2]
Á = volume charge density [C/m3]
GAUSS'S LAW  an example problem
H = magnetic field intensity [A/m]
Find the intensity of the electric field at distance r from a
J = current density [A/m2]
straight conductor having a voltage V.
" D
*Maxwell added the term to Ampere's Law.
Consider a cylindrical surface of length l and radius r
" t
enclosing a portion of the conductor. The electric field
passes through the curved surface of the cylinder but not
the ends. Gauss's law says that the electric flux passing
POISSON'S EQUATION
through this curved surface is equal to the charge enclosed.
Á
2Ä„
"2Åš = -
µ0 µ0E·ds = µ0 0 Erlr dĆ = Qenc = Áll = ClVl
+" +"
S
2Ä„ ClV
Er =
µ0Err dĆ = ClV
SURFACE CHARGE DENSITY [C/m2] so and
s
+"
0
2Ä„µ0r
12
Ás = µ0En µ0 = permittivity of free space 8.85 × 10-
Er = electric field at distance r from the conductor [V/m]
F/m
l = length [m]
En = electric field normal to the
Ć
En = n·E r d = a small increment of the cylindrical surface S [m2]
surface [V/m]
Ál = charge density per unit length [C/m]
Cl = capacitance per unit length [F/m]
V = voltage on the line [V]
D FLUX DENSITY [C/m2]
or ELECTRIC DISPLACEMENT PER UNIT AREA
Q = electric charge [C]
Q
CONSERVATIVE FIELD LAW
Ć
D a" r
4Ä„r2 µ = dielectric constant µ = µ0µr
"×E = 0
+"E·dl = 0
E = electric field [V/m]
S
D = µE
E = electric field [V/m]
ds = a small increment of length
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 6 of 13
COULOMB'S LAW C CAPACITANCE BETWEEN TWO
PARALLEL SOLID CYLINDRICAL
"·D = Á
Ádv
+"D·ds = +"
S V
CONDUCTORS
D = electric flux density vector [C/m2]
This also applies to a single conductor above ground,
Á = volume charge density [C/m3]
where the height above ground is d/2.
ds = a small increment of surface S
C = capacitance
Ä„µ
[F/m]
C =
, where d a
µ = permittivity of
ln(d /a)
We POTENTIAL ENERGY [J]
the material
The energy required to bring charge q from infinity to
d = separation
Ä„µ
a distance R from charge Q.
(center-to-
C =
or
center) [m]
d
Qq
cosh-1
a = conductor
We = qÅš =
2a
4Ä„µR radius [m]
1 1
We = ÁÅš dv = D·E dv
+" +"
V V
2 2 C CAPACITANCE BETWEEN PARALLEL
Åš = the potential between q and Q [V]
PLATES
q,Q = electric charges [C]
C = capacitance [F]
µ = permittivity of the material
µA
µ = permittivity of the material
C =
R = distance [m]
d = separation of the plates [m]
d
Á = volume charge density [C/m3]
A = area of one plate [m2]
E = electric field [V/m]
D = electric flux density vector [C/m2]
C CAPACITANCE BETWEEN COAXIAL
CYLINDERS
we VOLUME ENERGY DENSITY [J/m3]
C = capacitance [F/m]
for the Electrostatic Field
µ = permittivity of the material
2Ä„µ
b = radius of the outer cylinder
1 1
C =
we = D·E = µE2
[m]
ln b/ a
( )
2 2
a = radius of the inner cylinder
Åš = the potential between q and Q [V]
[m]
µ = permittivity of the material
R = distance [m]
E = electric field [V/m] C CAPACITANCE OF CONCENTRIC
D = electric flux density vector [C/m2]
SPHERES
C = capacitance [F/m]
4Ä„µab
µ = permittivity of the material
CAPACITANCE
C =
b = radius of the outer sphere [m]
b-a
a = radius of the inner sphere [m]
C CAPACITANCE [F]
Q = total electric charge [C]
Q
C =
Åš = the potential between q and Q [V]
J CURRENT DENSITY
Åš
Cl = capacitance per unit length [F/m]
Ál Ál = charge density per unit length [C/m] The amount of current flowing perpendicularly
Cl =
through a unit area [A/m2]
V
V = voltage on the line [V]
à = conductivity of the material [S/m]
J = ÃE
E = electric field [V/m]
I =
+"J·ds I = current [A]
S
ds = a small increment of surface S
In
nc = the number of conduction band
semiconductor
electrons
material:
qe = electron charge -1.602×10-19 C
J = ncqevd
vd = a small increment of surface S
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 7 of 13
MATHEMATICS
CONTINUITY EQUATION
"Á
WORKING WITH LINES, SURFACES, AND
"·J + = 0
VOLUMES
"t
Ál(r') means "the charge density along line l as a
J = current density [A/m2] J = ÃE
function of r'." This might be a value in C/m or it
Á = volume charge density [C/m3]
could be a function. Similarly, Ás(r') would be the
charge density of a surface and Áv(r') is the
charge density of a volume.
DUALITY RELATIONSHIP of J and D
For example, a disk of radius a having a uniform
RESISTANCE, CAPACITANCE, CURRENT,
charge density of Á C/m2, would have a total
CONDUCTIVITY
charge of ÁÄ„a2, but to find its influence on points
Where current enters and leaves a conducting
along the central axis we might consider
medium via two perfect conductors (electrodes) we
incremental rings of the charged surface as
have:
Ás(r') dr'= Ás2Ä„r' dr'.
à ÃQ
I = = Ã = If dl' refers to an incremental distance along a circular
+"J·ds +"E·ds µ +"D·ds = µ
S S S
contour C, the expression is r'd , where r' is the
J = current density [A/m2] J = ÃE
radius and d is the incremental angle.
E = electric field [V/m]
D = electric flux density vector [C/m2] D = µE
GEOMETRY
As a result of this, we have the following relation,
useful in finding the resistance between two
SPHERE ELLIPSE
conductors:
2
Area A = Ä„AB
Area A = 4Ä„r
R = resistance [&!]
4 Circumference
3
µ
C = capacitance [F] Volume
V = Ä„r
RC = 2
3
a + b2
µ = permittivity of the material
Ã
L H" 2Ä„
à = conductivity of the material [S/m]
2
G CONDUCTANCE [&!-1]
NABLA, DEL OR GRAD OPERATOR
R = resistance [&!]
[+ m-1]
1 I
G = =
I = current [A]
R "Åš Compare the " operation to taking the time
"Åš = voltage potential [V]
derivative. Where "/"t means to take the derivative
à = conductivity of the material
Ã
with respect to time and introduces a s-1 component to
+"E·ds
S
[S/m]
=
- the units of the result, the " operation means to take
E·dl
the derivative with respect to distance (in 3
+"
+
dimensions) and introduces a m-1 component to the
units of the result. " terms may be called space
SEMICONDUCTOR CONDUCTIVITY
derivatives and an equation which contains the "
operator may be called a vector differential
[&!-1]
equation. In other words "A is how fast A changes
à = conductivity of the material
as you move through space.
[S/m]G = conductance [&!-1]
"A "A "A
in rectangular
Ć
"A = x + w
+ Ä™
coordinates:
à H" q µeNd q = electron charge -1.602×10-19 C "x "y "z
µe = electron mobility [m2/(V-s)]
in cylindrical
"Â 1 "A "A
Nd = concentration of donors, and
Ć
"A = r +Ć + ę
coordinates:
thereby the electron concentration "r r "Ć "z
in the transition region [m-3]
in spherical
"A 1 "Â 1 "A
Ć
Ć
"A = r +¸ +Ć
coordinates:
"r r "¸ r sin ¸ "Ć
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 8 of 13
2
THE LAPLACIAN [+ m-2] × CURL [+ m-1]
in rectangular The circulation around an enclosed area. The curl of
Ć
"2A = x"2Ax + w
"2 Ay + Ä™"2 Az = 0
coordinates: vector B is
"2 "2 "2
in rectangular coordinates:
"2 a" + + = 0
"x2 "y2 "z2 curl B ="×B =
in spherical and
"2A a" "("·A)- " × " × A
ëÅ‚öÅ‚ëÅ‚öÅ‚
"Bz "By "Bx "Bz "By "Bx
ëÅ‚öÅ‚
cylindrical x - + w
- + Ä™ -
Ć
ìÅ‚÷Å‚ ìÅ‚÷Å‚ ìÅ‚÷Å‚
= grad(div A)- curl(curl A)
"y "z "z "x "x "y
íÅ‚Å‚Å‚
coordinates: íÅ‚Å‚Å‚íÅ‚Å‚Å‚
for example, the
"2Åš "2Åš "2Åš
in cylindrical coordinates:
Laplacian of electro- "2Åš = + + = 0
"x2 "y2 "z2
curl B ="×B =
static potential:
îÅ‚Å‚Å‚
" rBĆ "Br
îÅ‚1 "Bz "BĆ Å‚Å‚ "Br "Bz 1 ( )
îÅ‚Å‚Å‚
Ć
Ć
r - +Ć - + ę
ïÅ‚ - śł
ïłśł
ïłśł
· DIVERGENCE [+ m-1]
r "Ć "z "z "r r "r "Ć
ðÅ‚ûÅ‚
ðÅ‚ûÅ‚ ïłśł
ðÅ‚ûÅ‚
The del operator followed by the dot product operator
is read as "the divergence of" and is an operation
in spherical coordinates:
performed on a vector. In rectangular coordinates, "Å"
means the sum of the partial derivatives of the
îÅ‚" BĆ sin ¸ "B¸ Å‚Å‚
1 ( )
Ć
magnitudes in the x, y, and z directions with respect to curl B ="× B = r ïÅ‚ - śł +
r sin ¸ "¸ "Ć
ïłśł
the x, y, and z variables. The result is a scalar, and a
ðÅ‚ûÅ‚
factor of m-1 is contributed to the units of the result.
îÅ‚
( )Å‚Å‚ +Ć îÅ‚" rB¸ Å‚Å‚
1 1 "Br " rBĆ Ć 1 ( ) "Br
Ć
¸- śł
For example, in this form of Gauss' law, where D is a ïÅ‚ -
ïłśł
r ¸ "Ć "r r "r "¸
ïÅ‚sin ðÅ‚
density per unit area, "Å"D becomes a density per unit
ðłśł ûÅ‚
ûÅ‚
volume.
The divergence of a curl is always zero:
"·(" × H) = 0
" Dx " Dy " Dz
div D ="Å" D = + + =Á
" x " y " z
DOT PRODUCT [= units2]
D = electric flux density vector D = µE [C/m2]
Á = source charge density [C/m3]
The dot product is a scalar value.
Ć Ć
A " B = (xAx + w
Ay + ęAz )" (xBx + w
By + ęBz )= Ax Bx + Ay By + Az Bz
In the electrostatic context, the divergence of D is the
total outward flux per unit volume due to a source B
A " B = A B cosÈAB
charge. The divergence of vector D is:
È A
in rectangular
Ć Ć Ć
"Dx "Dy "Dz x " w
= 0 , x " x = 1
div D = + +
coordinates:
Ć
"x "y "z B " w
= (xBx + w
By + ęBz)" w
= By
A" B
Projection of B
in cylindrical
"DĆ
1 " 1 "Dz
B
div D = (rDr )+ +
coordinates: along â:
B
r "r r "Ć "z
È
(B " â)â
in spherical coordinates:
â
È
â
"DĆ
1 "(r2Dr)+ 1 "(sin ¸D¸)+ 1
The dot product of 90° vectors is zero.
div D =
The dot product is commutative and distributive:
r2 "r r sin ¸ "¸ r sin ¸ "Ć
A " B = B " A A " (B + C) = A " B + A " C
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 9 of 13
CROSS PRODUCT
COORDINATE TRANSFORMATIONS
Ć Ć
A × B = (xAx + w
Ay + Ä™Az)× (xBx + w
By + ęBz)
Rectangular to Cylindrical:
Ć
= x(AyBz - Az By)+ w
(Az Bx - AxBz )+ Ä™(AxBy - AyBx)
Ć
To obtain: A(r,Ć, z) = rÂr +ĆAĆ + Ä™Az
Ć
A × B = n A B sin ÈAB A×B
Ć
Ar = x2 + y2 r = x
osĆ+ y %5ńin Ć
Ć
where n is the unit vector normal to
n
y
Ć
both A and B (thumb of right-hand rule).
Ć = tan-1 Ć=-x%5ńin Ć+ y
os Ć
A
x
B × A = -A × B
È
z = z Ä™ = Ä™
x × y = z y × x = -z x× x = 0
Cylindrical to Rectangular:
B
Ć× z = r Ć×r =-z
To obtain: r(x, y, z) = xx + w
w
+ zÄ™
The cross product is distributive:
Ć
x = r cosĆ x = r
o%5ńĆ-ĆcosĆ
A ×(B + C) = A × B + A × C
Ć
y = r sin Ć Ć= r %5ńin Ć+ y
os Ć
Also, we have:
A× B×C = AÅ"C B -( ) z = z Ä™ = Ä™
A Å"B C
( ) ( )
Rectangular to Spherical:
Ć
To obtain: A(r,¸,Ć) = rAr +¸¸ +ĆÂĆ
COORDINATE SYSTEMS
Cartesian or Rectangular Coordinates:
Ar = x2 + y2 + z2
Ć
r(x, y, z) = xx + w
w
+ zÄ™ x is a unit vector
Ć
r = x%5Å„in ¸cos Ć+ y %5Å„in ¸sin Ć+ Ä™ cos ¸
2
r = x2 + y2 + z z cos-1
¸ =
Spherical Coordinates: x2 + y2 + z2
P(r,¸,Ć) r is distance from center Ć
¸= x
os ¸cos Ć+ y
os ¸sin Ć- Ä™ sin ¸
¸ is angle from vertical
y
Ć
Ć = tan-1 Ć=-x%5ńin Ć+ y
os Ć
Ć is the CCW angle from the x-axis
x
Ć
Ć
Ć
r , , and Ć are unit vectores and are functions of
¸
Spherical to Rectangular:
position their orientation depends on where they
To obtain: r(x, y, z) = xx + w
w
+ zÄ™
are located.
x = r sin ¸cosĆ
Cylindrical Coordinates:
Ć
x = r %5Å„in ¸cos Ć-¸
os¸cosĆ-Ć%5Å„in Ć
C(r,Ć, z) r is distance from the vertical (z) axis
y = r sin ¸sin Ć
Ć is the CCW angle from the x-axis
z is the vertical distance from origin
w
= r %5Å„in ¸sin Ć+¸
os¸sin Ć+Ć
osĆ
Ć
z = r cos¸ Ä™ = r co%5Å„ ¸-¸sin ¸
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 10 of 13
THE STATIC MAGNETIC FIELD
B MAGNETIC FIELD [T or A/m]
due to an infinite straight conductor
F F12 MAGNETIC FORCE [N/m]
May also be applied to the magnetic field close to a
due to a conductor conductor of finite length.
BP = magnetic field vector [T]
If the current in the two wires travels in opposite
directions, the force will be attractive. µ0 = permeability constant 4Ä„×10-7 [H/m]
µ0I
Ć
F12 = the force exerted by conductor 1 I = current [A]
BP =Ć
2Ä„r
carrying current I on conductor 2. r = perpendicular distance from the
Ć
xµ0I1I2 [N/m] conductor [m]
F12 =
µ0 = permeability constant 4Ä„×10-7
2Ä„d
[H/m]
I = current [A] B MAGNETIC FIELD [T]
d = distance between conductors [m] due to a finite straight conductor at a point
perpendicular to the midpoint
BP = magnetic field vector [T]
µ0Ia
Ć
BP =Ć
BP BIOT-SAVART LAW
µ0 = permeability constant
2Ä„r r2 + a2 4Ä„×10-7 [H/m]
Determines the B field vector at any point P identified
I = current [A]
by the position vector r, due to a differential current
a
a = half the length of the
element I dl' located at vector r'.
r
conductor [m]
BP = magnetic field vector
r = perpendicular distance
Ć
[T] I
µ0I dl'×R
from the conductor [m]
dBP =
µ0 = permeability constant
4Ä„R2
4Ä„×10-7 [H/m]
I dl' = current element [A]
µ0I dl'×(r - r')
B MAGNETIC FIELD [T]
BP =
Ć
+" 3 R = unit vector pointing
C
4Ä„
r - r' at the center of a circular wire of N turns
from the current
B = magnetic field [T]
element to point P
r - r'
µ0 = permeability const. 4Ä„×10-7 [H/m]
Ć
R = distance between the
R =
µ0 NI
N = number of turns of the coil
r - r' current element and
Bctr = Ä™
2a
I = current [A]
point P [m]
a = radius [m]
B AMPERE'S CIRCUITAL LAW
B MAGNETIC FIELD [T]
Ampere's law is a consequence of the Biot-Savart
along the central axis of a solenoid
law and serves the same purpose as Gauss's law.
îÅ‚ Å‚Å‚
Ampere's law states that the line integral of B around
µ0NI (z + l / 2) (z - l / 2)
B(z)= Ä™ ïÅ‚ - śł
any closed contour is equal to µ0 times the total net
2 2
2l
ïÅ‚ śł
a2 + (z + l / 2) a2 + (z - l / 2)
ðÅ‚ ûÅ‚
current I passing through the surface S enclosed by
µ0NI
the contour C. This law is useful in solving
and at the center of the coil:
Bctr H" Ä™
magnetostatic problems having some degree of
l
symmetry.
B = magnetic field [T] l = length of the solenoid [m]
B = magnetic field vector, equal to
µ0 = permeability constant z = distance from center of
B times the appropriate unit
4Ä„×10-7 [H/m] the coil [m]
vector [T]
N = number of turns a = coil radius [m]
µ0 = permeability constant 4Ä„×10-7
I = current [A]
µ0J·ds
[H/m]
+"B·dl = +"
C S
dl = an increment of the line which
= µ0I
is the perimeter of contour C
[m]
J = current density [A/m2]
J = ÃE
ds = an increment of surface [m2]
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 11 of 13
H MAGNETIC FIELD INTENSITY [A/m]
L11 SELF-INDUCTANCE [H]
The magnetic field intensity vector is directly
When a current in coil 1 induces a current in coil 2,
analogous to the electric flux density vector D in
the induced current in coil 2 induces a current back in
electrostatics in that both D and H are medium-
coil 1. This is self-inductance.
independent and are directly related to their sources.
N1›11 N = number of turns of the coil
H = magnetic field [A/m]
L11 =
›11 = the total flux linked by a single
B
B = magnetic field vector [T] I1
H a" - M
turn of coil 1 [Wb]
µ0
µ0 = permeability const. 4Ä„×10-7 [H/m]
N12¨11 I1 = current in coil 1 [A]
M = magnetization [A/m]
"D = ¨11 = the magnetic flux produced by
" × H = J +
J = current density [A/m2] I1
J = ÃE
a single turn of coil 1 and linked
" t
D = electric flux density vector
by a single turn of coil 1 [Wb]
[C/m2]
L12 MUTUAL INDUCTANCE [H]
(,lambda) MAGNETIC FLUX, LINKAGE
The mutual inductance between two coils.
Flux linkage › is the ability of a closed circuit to store
N2›12 N2 N1¨12 N = number of turns of
L12 = = the coil
magnetic energy. It depends, in part, on the
I1 I1
physical layout of the conductors. It is the total › = flux linkage [Wb]
magnetic field due to circuit #1 passing through the I = current [A]
Neumann formula:
area enclosed by the conductors of circuit #2. The
¨ = magnetic flux [Wb]
µ0 N1N2 dl1·dl2 r = vector to the point
text seemed to describe ¨ as the flux due to one turn
L12 =
+" +"
C1 C2
and › as the flux due to all of the turns of the coil, but of observation
4Ä„ r - r'
was not consistent so be careful. r' = vector to source
¨12 = the magnetic flux passing
through coil 2 that is produced
¨12 = B1·ds2
+"
S2
by a current in coil 1 [Wb]
Wm MAGNETIC ENERGY [J]
› = total flux linkage [Wb]
›12 = N1¨12
Energy stored in a magnetic field [Joules].
B = magnetic field vector [T]
Wm = energy stored in a magnetic
› = N
+"B·ds N = number of turns of the coil
S
field [J]
ds = an increment of surface [m2]
1
µ0 = permeability constant
Wm = B2dv'
+"
4Ä„×10-7 [H/m]
2µ0 V
B = magnetic field [T]
LENZ'S LAW
Induced voltage causes current to flow in the direction
that produces a magnetic flux which opposes the flux
FARADAY'S LAW
that induced the voltage in the first place. This law is
useful in checking or determining the sign or polarity
When the magnetic flux enclosed by a loop of wire
of a result.
changes with time, a current is produced in the loop.
The variation of the magnetic flux can result from a
time-varying magnetic field, a coil in motion, or both.
L INDUCTANCE [H]
"B "×E = the curl of the electric field
" × E = -
Inductance is the ability of a conductor configuration
B = magnetic field vector [T]
" t
to "link magnetic flux", i.e. store magnetic energy.
Two methods of calculating inductance are given Another way of expressing Faraday's law is that a
below. changing magnetic field induces an electric field.
› where S is the surface
d
› = flux linkage [Wb]
L =
Vind = E·dl =- B·ds
enclosed by contour

+"+"
I I = current [A] CS
dt
C.
2Wm Wm = energy stored in a magnetic field
(see also Induced Voltage below)
L =
[J]
2
I
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 12 of 13
Vind INDUCED VOLTAGE M MAGNETIZATION [A/m]
The voltage induced in a coil due to a changing The induced magnetic dipole moment per unit
magnetic field is equal to the number of turns in the volume.
2
coil times the rate at which the magnetic field is N = number of turns of the coil
Nqe a2B
changing (could be a change in field strength or coil M = - qe = electron charge -
4me
area normal to the field). 1.602×10-19 C
N = number of turns of the coil
a = orbit radius of an electron [m]
d¨
ÇmB
Vind = -N
or M =
¨ = the magnetic flux produced by B = magnetic field vector [T]
dt
µ0 µ0 = permeability constant 4Ä„×10-7
a single turn of the coil [Wb]
Vind =
where
[H/m]
+"E·dl
C
2
Nqe a2µ0 me = who knows?
Çm = -
4me Çm = magnetic susceptibility
Vind INDUCED VOLTAGE DUE TO
MOTION
When conductors move in the presence of magnetic
fields, an induced voltage is produced even if the
magnetic fields do not vary in time. For the voltage
produced due to both a changing magnetic field and a
conductor in motion:
"B
Vind =- ·ds + v ×B ·dl
( )
+"
+"
SC
"t
B = magnetic field vector [T]
v = velocity vector of the conductor [m/s]
ds = increment of the surface normal to the magnetic field
vector [m2]
dl = incremental length of conductor [m]
INDUCED VOLTAGE  SLIDER PROBLEM
A frictionless conducting bar moves to the right at
velocity v produces a current I.
d
I
B0 v
R h
An expanding magnetic field area having a static
magnetic field directed into the page produces a
CCW current.
Vind = induced voltage [V]
B = static magnetic field [T]
0
h = distance between the conductor rails
[T]
v = velocity of the conductor [m/s]
Vind = B0hv
Fmag = magnetic force opposing slider
[N]
Ć
Fmag = xB0Ih
Ć
x = unit vector in the direction against
conductor movement [m/s]
d
2
E = I R
I = current [A]
v
E = energy produced [J or W/s]
R = circuit resistance [&!]
d = distance the conductor moves
[m]
Tom Penick [e-mail] www.teicontrols.com/notes 10/22/2000 Page 13 of 13