Electromagnetic II - EE 234
Introduction
Dr. Abdullah Al-Ahmadi
Electrical Engineering Department
College of Engineering
Majmaah University
1
Introduction
Electric Fields
• The electromagnetic force consists of an electrical component
Fe and a magnetic component Fm .
• All matter contains a mixture of:
• neutrons,
• positively charged protons,
• and negatively charged electrons,
• Electric charge is measured is the coulomb (C).
2
• The magnitude of e is
e = 1.6 × 10−19
(C)
(1)
• The charge of a single electron is qe = −e and that of a
proton is equal in magnitude but opposite in polarity: qp = e
3
Coulomb’s law
• Coulomb’s experiments:
• Two like charges repel one another, whereas two charges of
opposite polarity attract.
• the force acts along the line joining the charges, and
• its strength is proportional to the product of the magnitudes of
the two charges and inversely proportional to the square of the
distance between them.
4
Coulomb’s law
Fe21 = R̂12
q1 q2
4πϵ0 R212
(N)
(2)
Figure 1: Electric forces on two positive point charges in free space.
5
• From Equation 2, Fe21 is the electrical force acting on charge
q2 due to charge q1 when both are in free space (vacuum),
• R21 is the distance between the two charges,
• R̂21 is a unit vector pointing from charge q1 to charge q2 .
• ϵ0 is a universal constant called the electrical permittivity of
free space. ϵ0 = 8.854 × 10−12 farad per meter (F/m).
• The force Fe12 acting on charge q1 due to charge q2 is equal
to force Fe21 in magnitude, but opposite in direction:
Fe12 = −Fe21
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Electric Field Intensity
E = R̂
q
4πϵ0 R2
(V/m)
(3)
Figure 2: Electric field E due to charge q
7
• To extend Equation 3 from the free-space case to any medium,
E = R̂
q
4πϵR2
(V/m)
(4)
where R is the distance between the charge and the
observation point
• Often, ϵ is expressed in the form
ϵ = ϵr ϵ0
(V/m)
(5)
• ϵr is a dimensionless quantity called the relative permittivity
or dielectric constant of the material. For vacuum, ϵr = 1
8
Electric Flux Density
D = ϵE (C/m2 )
(6)
9
Magnetic Fields
Fact
Electric charges can be isolated, but magnetic poles always exist
in pairs.
10
Figure 3: Pattern of magnetic field lines around a bar magnet
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• The magnetic lines surrounding a magnet represent the
magnetic flux density B.
• A magnetic field can also be created by electric current.
• Danish scientist Hans Oersted (1777–1851),
• observed that an electric current in a wire caused a compass
needle to deflect.
• direction was always perpendicular to the wire and to the
radial line connecting the wire to the needle.
12
Figure 4: The magnetic field induced by a steady current flowing in the
z direction
13
Biot–Savart law
• Relates the magnetic flux density B at a point in space to the
current I in the conductor.
• The magnetic flux density B induced by a constant current I
flowing in the z direction is given by:
B = Φ̂
µ0 I
2πr
(T)
(7)
• The magnetic field is measured in tesla (T),
• µ0 is called the magnetic permeability of free space
= 4π × 10−7 henry per meter (H/m)
• Φ̂is an azimuthal unit vector
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• the product of ϵ0 and µ0 specifies c, the velocity of light in
free space:
1
c= √
= 3 × 108 (m/s)
(8)
µ0 ϵ0
15
• To extend Equation 7 from the free-space case to any medium,
B = Φ̂
µI
2πr
(T)
(9)
• µ is expressed in the form
µ = µr µ0
(H/m)
(10)
• µr is a dimensionless quantity called the relative magnetic
permeability of the material.
16
Magnetic Flux Density
B = µH
(11)
where H is the magnetic field intensity
17
Static and Dynamic Fields
Static
describes a quantity that does not change with time.
Dynamic
refers to a quantity that does vary with time.
18
Table 1: The three branches of electromagnetics.
19
• Electrostatics and Magnetostatics: stationary charges and
dc currents.
• Dynamics: time-varying fields induced by time-varying
sources.
Fact
A time-varying electric field generates a time-varying magnetic
field, and vice versa.
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Conductivity
The conductivity (σ) characterizes the ease with which charges
(electrons) can move freely in a material.
• If σ = 0, the charges do not move more than atomic distances
and the material is said to be a perfect dielectric.
• if σ = ∞, the charges can move very freely throughout the
material, which is then called a perfect conductor.
• measured in siemens per meter (S/m).
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• The parameters ϵ, µ and σ are often referred to as the
constitutive parameters of a material (Table 2).
• A medium is said to be homogeneous if its constitutive
parameters are constant throughout the medium.
Table 2: Constitutive parameters of materials
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Traveling Waves
Traveling Waves
• Waves are of two types:
1. Transient waves caused by sudden disturbances.
2. Continuous periodic waves generated by a repetitive source.
• one-dimensional wave:
• If this disturbance varies as a function of one space variable.
Figure 5.
• A two-dimensional wave propagates out across a surface.
Figure 6.
• A three-dimensional wave propagates through a volume.
Figure 7.
• plane waves, cylindrical waves, and spherical waves.
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Figure 5: A one-dimensional wave traveling on a string.
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Figure 6: Circular waves
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(a) Plane and cylindrical waves
(b) Spherical wave
Figure 7: Examples of two-dimensional and three-dimensional waves
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• A traveling wave can be expressed as:
(
)
2πt 2πx
−
+ ϕ0
y (x, t) = A cos
T
λ
(m)
(12)
where:
•
•
•
•
A is the amplitude of the wave.
T is its time period
λ is its spatial wavelength
ϕ0 is a reference phase
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• The phase velocity, also called the propagation velocity
• The phase velocity up :
up =
λ
T
(m/s)
(13)
• The frequency of a sinusoidal wave, f , is the reciprocal of its
time period T :
1
f=
(Hz)
(14)
T
• Combining Equations 13 and 14 yields:
up = fλ
(m/s)
(15)
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• Equation 12 can be written as:
(
)
2πx
y (x, t) = A cos 2πft −
+ ϕ0
λ
(m)
= A cos (ωt − βx)
(16)
where ω is the angular velocity of the wave, and β is its
phase constant (or wavenumber)
ω = 2πf (rad/s)
2π
β=
(rad/m)
λ
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• The direction of wave propagation is determined by the signs
of the t and x:
• if one of the signs is positive and the other is negative, then
the wave is traveling in the positive x direction.
• if both signs are positive or both are negative, then the wave is
traveling in the negative x direction.
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Example
The electric field of a traveling electromagnetic wave is given by
(
)
E (z, t) = 10 cos π × 107 t + πz/15 + π/6
(V/m)
Determine:
1. the direction of wave propagation,
2. the wave frequency f,
3. its wavelength λ
4. its phase velocity up .
Answer
1. −z direction, 2. f = 5 MHz, 3.λ = 30 m, 4.
up = 1.5 × 108 m/s.
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Review of Complex Numbers
Complex Numbers
• Any complex number z can be expressed in rectangular form
as:
z = x + jy
(17)
• Alternatively, z may be cast in polar form as:
z = |z| ejθ = |z| ∠θ
(18)
where |z| is the magnitude of zand θ is its phase angle
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Euler’s identity
ejθ = cos θ + j sin θ
(19)
• Applying Euler’s identity, we can convert z from polar form
into rectangular form,
z = |z| ejθ = |z| cos θ + j |z| sin θ
(20)
• This leads to the relations:
x = |z| cos θ,
y = |z| sin θ
√
+
2
2
|z| =
x + y , θ = tan−1 (y/x)
(21)
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• The complex conjugate of z
z∗ = (x + jy)∗ = x − jy = |z| e−jθ = |z| ∠ − θ
(22)
• The magnitude |z| is equal to:
|z| =
√
+
zz∗
(23)
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Complex Algebra
Equality
If two complex numbers z1 and z2 are given by:
z1 = x1 + jy1 = |z1 | ejθ1
z2 = x2 + jy2 = |z2 | ejθ2
then z1 = z2 if and only if x1 = x2 and y1 = y2 or, equivalently,
|z1 | = |z2 | and θ1 = θ2
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Addition
z1 + z2 = (x1 + x2 ) + j (y1 + y2 )
Multiplication
z1 z2 = (x1 x2 − y1 y2 ) + j (x1 y2 + x2 y1 )
or
z1 z2 = |z1 | |z2 | ej(θ1 +θ2 )
= |z1 | |z2 | [cos (θ1 + θ2 ) + j sin (θ1 + θ2 )]
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Division
For z2 ̸= 0
(x1 + jy1 ) (x2 − jy2 )
z1
=
.
z2
(x2 + jy2 ) (x2 − jy2 )
(x1 x2 + y1 y2 ) + j (x2 y1 − x1 y2 )
=
x22 + y22
or
z1
|z1 | j(θ1 −θ2 )
=
e
z2
|z2 |
|z1 |
[cos (θ1 − θ2 ) + j sin (θ1 − θ2 )]
=
|z2 |
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Powers
For any positive integer n,
(
)n
zn = |z| ejθ
= |z|n ejnθ = |z|n (cos nθ + j sin nθ)
z1/2 = ± |z|1/2 ejθ/2
= ± |z|1/2 [cos (θ/2) + j sin (θ/2)]
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Useful Relations
−1 = ejπ = e−jπ = 1∠180◦
j = ejπ/2 = 1∠90◦
−j = −ejπ/2 = e−jπ/2 = 1∠ − 90◦
)1/2
(
√
± (1 + j)
√
j = ejπ/2
= ±ejπ/4 =
2
√
±
(1
−
j)
√
−j = ±e−jπ/4 =
2
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Example
Given two complex numbers
V = 3 − j4
I = − (2 + j3)
1. Express V and I in polar form
2. Find VI
3. Find VI∗
4. Find V/I
√
5. Find I
Solution
◦
1. V = 5∠ − 53◦ , I = 3.61∠ − 123.69◦ 2. VI = 18.03ej183.2 3.
√
◦
◦
◦
VI∗ = 18.03ej70.6 4. V/I = 1.39ej70.6 5. I = ±1.90ej118.15
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Review of Phasors
Solution Procedure
Step 1: Adopt a cosine reference. Table 3.
Step 2: Express time-dependent variables as phasors. Table 4
Step 3: Recast the differential / integral equation in phasor
form
Step 4: Solve the phasor-domain equation
Step 5: Find the instantaneous value
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Table 3: Trigonometric Relations
42
Table 4: Time-domain sinusoidal functions z (t) and their
e
cosine-reference phasor-domain counterparts Z
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Example
The voltage source of the circuit shown in figure below is given
by:
(
)
vs (t) = 5 sin 4 × 104 t − 30◦
(V)
Obtain an expression for the voltage across the inductor.
Solution
(
)
vL (t) = 4 cos 4 × 104 t − 83.1◦
(V)
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