advanced electromagnetism ece 121 (tip reviewer)

7/23/2019 Advanced Electromagnetism ECE 121 (TIP Reviewer)

1/30

Overview:

*Scalar Quantities - can be completely specified by magnitude only such as temperature, speed, mass and electric current.

*Vector Quantities - specify both magnitude and direction such as velocity and weight.

*Vector Analysis - provides the mathematical tools necessary for expressing and manipulating vector quantities in an efficient

and convenient manner.

*Vector Algebra governs the laws of addition, subtraction and multiplication of vectors in any given coordinate system.

*Vector Calculus - encompasses the laws of differentiation and integration of vectors

2 - 1 Basic Laws of Algebra

*Unit vector - magnitude of unity (1)

*Base vectors (ax, ay, az) - three mutually perpendicular unit vectors

2-1.1 Equality of Two Vectors

*Two vectors A and B are said to be equal if they have equal magnitudes and identical unit vectors .

*Equality of vectors does not necessarily imply that they are identical. InCartesiancoordinates,twodisplacedparallelvectorsof

equalmagnitudeandpointinginthesamedirectionareequal,buttheyareidenticalonlyiftheylieontopofoneanother.

Equal if they are parallel, equal in magnitude and in the same direction, but identical if they are equal and lie on top of the other.

2-1.2 Vector Addition and Subtraction

*Vector Addition - either obtained by parallelogram rule or the head-to-tail rule.

2-1.3 Position and Distance Vectors

*Position Vector (OP)- is the vector from the origin to point P.

*Distance Vector (P1P2) - distance between two vectors

2-1.4 Vector Multiplication

*Three Types of Multiplication: (1) Simple Product, (2) Dot Product and (3) Cross Product.

SIMPLE PRODUCT:

*Simple product = scalar to vector multiplication

SCALAR OR DOT PRODUCT:

*Scalar or Dot Product = defined geometrically as the product of the magnitude of one of the vectors and the projection of the

other vector

Vector Algebra (Chapter 2 - Ulaby)

ECE 121 Page 1


2/30

other vector onto to the first one.

*The scalar product of two vectors yields a scalar whose magnitude is less than or equal to the products of the magnitudes of the

two vectors

*Dot product of unit vectors = if same unit vector, dot product is equal to 1. Otherwise, equal to zero.

*Dot product follows both commutative and distributive properties

*Angle is measured from

tail of A to tail of B and is less than or equal to 180 degrees.

VECTOR OR CROSS PRODUCT:

*vector n is the unit vector perpendicular to both A and B.

*magnitude of the cross product is equal to the area of the parallelogram

*cross product of unit vectors follow the right hand rule (xyzxyz)

*cross product is anticommutative (A x B = - B x A), but still distributive.

2-1.5 Scalar Vector Triple Products

Scalar Triple Product - results to a scalar value.

*Any of this pattern holds as long as the vectors follow cyclic patterns.

Vector Triple Product - results to a vector

*does not follow associative law

ECE 121 Page 2


3/30

2-2 Orthogonal Coordinate Systems

*Orthogonal coordinate systems (rectangular, cylindrical and spherical) - all coordinates are mutually perpendicular

*Non orthogonal coordinate systems - not all coordinates are mutually perpendicular

2-2.1 Cartesian Coordinates

*also called the rectangular coordinate

2-2.2 Cylindrical Coordinates

* = radial coordinate

*

(azimuthal angle) - measured from the positive x-axis (up to 360 degrees or 2 radians)

*When talking about position vectors, a is not used

2-2.3 Spherical Coordinates

*R is called the range or spherical radius

*

is called zenith or polar angle - measured from the positive z-axis and it describes a conical surface with its apex at the

origin

2.3 Transformations between Coordinate Systems

2.3.1 Cartesian to Cylindrical Transformations

Dot Product a

a

ax cos -sin

ay sin cos

Vertical - if rectangular to cylindrical

Horizontal - if cylindrical to rectangular

2.3.2 Cartesian to Spherical Transformation

ECE 121 Page 3


4/30

Dot Product aR a a

ax sincos -sin coscos

ay sinsin cos cossin

az cos 0 -sin

*Notes:Parallel = same slope, Perpendicular = negative reciprocal

ECE 121 Page 4


5/30

3.1 Gradient

*Gradient of a scalar function is a vector whose magnitude is equal to the maximum rate of increasing

change of the scalar function per unit distance, and its direction is along the direction of maximum

increase. (Scalar turned vector!)

3.2 Divergence

*positive point charge - outward flux / negative point charge - inward flux

*Flux Density - amount of outward flux crossing a unit surface ds

*Flux lines - field lines that comes out of the surface

*divergence of a vector field - measure of the net outward flux per unit volume through a closed surface

surrounding the unit volume (Vector turned Scalar )

Positive divergence Source of flux

Negative divergence Sink of flux

Zero divergence Divergenceless or Solenoidal

Vector Calculus (Chapter 3 - Ulaby)

ECE 121 Page 5


6/30

3.3 Curl

*Curl = describes the rotational property or circulation

*It is the circulation per unit area.

*If curl is zero, the field B is said to be conservative or irrotational.

ECE 121 Page 6


7/30

3.4 Laplacian Operator

*Divergence of a gradient.

*Kapag horizontal, walang change in

ECE 121 Page 7


8/30

*Maxwell's Equations - fundamental relations where modern electromagnetism is based.

*Not only encapsulates the connection between the electric field and electric charge and between the

magnetic field and electric current, but they also define the bilateral coupling between the electric and

magnetic field quantities.

*These equations hold in any material, including free space (vacuum) and at any spatial location (x,y,z).

*Formulated by James Clerk Maxwell in 1873, he established the first unified theory of electricity and

magnetism.

*Deduced from experimental observations reported by Gauss, Ampere, Faraday, etc.

* * *

*

*

*It is no longer a function of time.

Static Case: All charges are permanently fixed in space, or, if they move, they do so at a steady rate so

that pv and J are constant in time.

Maxwell's Equations (Chapter 4.1 - Ulaby)

ECE 121 Page 8


9/30

*The electric fields are no longer interconnected in this case.

*Allows us to study electricity and magnetism as two distinct and separate phenomena, as long as

the spatial distributions of charge and current flow remain constant in time.

Electrostatics:

Magnetostatics

ECE 121 Page 9


10/30

*Capacitor - any two conducting bodies separated by an insulating (dielectric)

medium.

The conductor connected to the positive side will accumulate positive

charges, while the conductor connected to the negative side will

accumulate negative charges.

1.

When a conductor has excess charge, it distributes the charge on its surface

in such a manner as to maintain zero electric field everywhere within the

conductor. This ensures that it is an equipotential body (electric potential is

the same at every point.)

2.

Capacitance of Parallel-Plate Capacitor

Breakdown Voltage of a Parallel-Plate Capacitor

Capacitance of a Coaxial Line:

Capacitance (Chapter 4.10 - Ulaby)

ECE 121 Page 10


11/30

l p

ECE 121 Page 11


12/30

6.2 Parallel Plate Capacitor

Parallel Plate Capacitor - two conductor system in which the conductors are identical, infinite parallel planes with

separation d.

Main Formulas:

Diagram of a parallel plate conductor:

Parallel Plate Capacitor (Chapter 6.2 - Hayt)

ECE 121 Page 12


13/30

ECE 121 Page 13


14/30

6.4 Capacitance of a Two-Wire Line

Two-wire line - a configuration consisting of two parallel conducting cylinders, each of circular cross section, whic

the arrangement of an important type of transmission line, the coaxial cable.

Capacitance of a Two Wire Line (Chapter 6.4 - Hayt)

ECE 121 Page 14


15/30

p

ECE 121 Page 15


16/30

Complex Numbers (Chapter 7-1.3 - Ulaby)

ECE 121 Page 16


17/30

Poisson's and Laplace's Equation (Chapter 6.6 - Hayt)

ECE 121 Page 17


18/30

Introduction:

*Charges induce electric fields and currents induce magnetic fields.

*Electromagnetic waves are produced when magnetic and electric fields couple, which is normally caused when

charge and current sources were to vary with time.

*Faraday's Law - time-varying magnetic field gives rise to an electric field

*Ampere's Law - time-varying electric field gives rise to a magnetic field

6.1 Faraday's Law:

*Oersted - established that electricity and magnetism are closely connected.

*He showed that a wire carrying an electric current exerts a force on a compass needleand that needle always

turns so as to point the

direction when the current is along the z-direction

*Faraday - "if a current can produce a magnetic field, then the converse should also be true!"*In 1831, Michael Faraday (in London) and Joseph Henry (in Albany, New York) - "magnetic fields can

produce an electric current in a closed loop, but only if the magnetic flux linking the surface area of the loop

changes with time."

*Galvanometer - sensitive instrument used in the 1800s to detect the flow of current in a circuit / predecessor

of voltmeter and ammeter

Electromotive Inducgtion can be done in three ways:

Type Magnetic Field Loop

1. Transformer EMF Time-varying Stationary

2. Motional EMF Static Moving loop with time-varying area

3. Combined Time-varying Moving loop

Maxwell's Equations in Time-Varying Fields (Chapter 6)

ECE 121 Page 18


19/30

6.2 Stationary Loop in a Time Varying Magnetic Field

*R1 is usually very small and it may be ignored.

*Lenz Law - the current in the loop is always in such a direction as to oppose the change of magnetic flux(t)

that produced it

*Faraday's Law - a voltage is induced across the terminals of a loop if the magnetic flux linking its surface

changes with time.

B (Magnetic Flux Density) = T1.

(Magnetic Flux) = Wb2.

Units:

ECE 121 Page 19


20/30

Introduction:

*Electric field produces magnetic field and vice versa = this cyclic pattern produces Electromagnetic Waves.

Guided Medium - a material structure that guides EM wave propagation such as transmission line, Earth's s

and ionosphere

1.

*Plane wave propagation - easier to accommodate in rectangular than spherical coordinates

Unbounded Media - light waves emitted by the sun and radio transmissions by antennas2.

When energy is emitted by a source, it expands outwardly from the source spherically in all directions with

same speed. This spherical emission is the wavefront.

1.

To an observer very far away, this wavefront appears planar as if it were a part of a uniform plane wave. Th

because the observer's aperture (a hole or an opening through which light travels.) appears approximately p

2.

Process:

7-1.1 Sinusoidal Wave in a Lossless Medium

Frictional forces are also ignored.

Allows a wave generated to travel indefinitely with no loss in energy.

*Lossless medium - a medium which does not attenuate the amplitude of the wave travelling within it or on its su

x = distance of wave travel (in m)

A = wave amplitude (in m)

T = time or temporal period = repeating period

t = time = spatial period or wavelength

0 = reference phase (in rads)= constant with respect to both time and space

Where y = height of the water (in m)

Peak - can be achieved when the wave phase is 0 or multiples of 2pi

the velocity of the wave pattern as it moves across the water surface-

Phase Velocity (or propagation velocity) - apparent velocity of the fixed height

Waves and Phasors (Chapter7.1 - Ulaby)

ECE 121 Page 20


21/30

direction? Negative x-direction -> x and t have the same signs-

Positive x-direction -> x and t have opposite signs.-

*Constant phase reference0 has no influence on either the speed or the direction of the wave propagation.

*Frequency (Hz or cycles per second)- reciprocal of the time period / Heinrich Hertz

+x direction -x direction

Positive0 Negative0

Phase Lead Phase Lag

Shift to the left Shift to the right

7-1.2 Sinusoidal Signal in a Lossy Medium

*the real unit of the attenuation constant is 1/m

*Neper = dimensionless, artificial adjective traditionally used as a reminder that the unit Np/m refers to the

attenuation constant.*Similar to constant where the unit is assigned as rad/m instead of just 1/m.

*attenuation constant: (in Nepers per meter -> Np/m)

ECE 121 Page 21


22/30

Use of phasor notation to represent time-dependent variables allows us to convert the integro-differentia

equation into a linear equation with no sinusoidal functions.

-

Phasor Analysis - useful mathematical tool for solving problems involving linear systems in which the excitation

periodic time function.

Excitation (also known as forcing function)

Sin(x) = cos(x - /2)a.

Cos(x) = cos(-x)b.

Adopt a cosine reference.- express the forcing function as acosine.1.

~ (tilde)a.

Express time-dependent variables as phasors.2.

Recast the differential / integral equation in phasor form.3.

Solve the phasor-domain equation.4.

Find the instantaneous values.5.

Phasor Analysis:

Notes:

*Before turning it into a phasor. Convert it first to cosinusoidal form.

Sound:

Speed of light: C0

No negative polar form.

Phasors

ECE 121 Page 22


23/30

Problem 1.6 (Chapter 7)





ECE 121 Page 23


24/30

*gets its name from the fact that it is used to transform currents, voltages and impedances between its primary and

secondary circuits

*Primary Coil - connected to an AC Voltage source

*Secondary Coil - connected to a load resistor RL

*The directions of I1 and I2 are such that the flux generated by one of them is opposite that generated by the other.

*Ideal lossless transformer - all the instantaneous power supplied by the source connected to the primary coil is deliv

the load on the secondary side. No power is lost in the core.

*P1 = P2

*Transformer - consists of coils wounded around a common magnetic core with infinite permeability ( =) and magnetic

within the core

Formula:

*It only differs in the number of turns.

*Magkabaliktad ang current and voltage.

*Simple ohm's law.

*Resistance is also applicable in impedance.

Ideal Transformer (Chapter 6.3 - Ulaby)

ECE 121 Page 24


25/30

Formula Needed:

Formula Variables Comment

Magnetic Force:

q = charged particle

u = velocity B = magnetic field in Tesla ( 1 T

= 10, 000 Gauss)

Where Fm = magnetic force

Motional Electric Field Motional Electric Field - electric field ge

by the motion of the charged particle

Motional emf qaaqa Motional emf - voltage induced by movin

- only those segments of the circuits that

magnetic field lines contribute to motion

Magnetic Field:

Velocity: =2pi*radius*angular velocity

Get u x B. This would result to a direction, which would generate the Vemf.1.

If it is a square or rectangular loop moving away, -Vemf.a.

If it is a spinning loop, it is a -Vemf sinusoidal equation.b.

Substitute to the Vemf equation.2.

To get the voltage difference, it is always V43 - V21.3.

Divide voltage difference with the value of R.4.

Steps:

If you see these figures, -Vemf.

`````````````` `If you see this one, the answer is in sinusoidal form.

Moving Conductor in Static Magnetic Field (Chapter 6.4 -

Ulaby)

ECE 121 Page 25


26/30

Schaums Outline: 31

ECE 121 Page 26


27/30

*In the time-varying case, magnetic and electric field variables are all a function of the spatial coordinates

(x,y,z) and the time variable t.

*By converting them to phasor form, they will only depend on (x,y,z) only.

Time Harmonic Fields (Chapter 7.2 - Ulaby)

ECE 121 Page 27


28/30

7-3 Plane Wave Propagation

*three constitutive parameters of the medium:, and

*Lossless: (1) nonconducting 7-3.1 Uniform Plane Waves

*E and H do not vary with x and y only, but varies with z.

*No electric and magnetic field components along its direction of propagation.

*Uniform Plane Wave - characterized by electric and magnetic fields that have uniform properties at all

points across an infinite plane.

*For a wave travelling from the source toward the load on a transmission line, the amplitudes of its

voltage and current phasors are related by the intrinsic impedance.

*Intrinsic Impedance (n)

*The electric and magnetic field are perpendicular to each other and both are perpendicular to the

direction of wave travel. These directional properties characterize a transverse electromagnetic (TEM

wave).

*In phase - exhibit the same functional dependence on z and t

*Phase Velocity of lossless medium:

7.3.2 General Relation between E and H

*Valid for both lossless and lossy media.

*Right hand rule: When we rotate the four fingers of the right hand from the direction of E toward the

direction of H, the thumb will point in the direction of wave travel k.

ECE 121 Page 28


29/30

*Polarization states = ellipse, circular or linear. Can be determined by tracing the tip of E as a

function of time in a plane orthogonal to the direction of the wave travel.

*Polarization of a uniform plane wave - describes the shape and locus of the tip of the E vector at a

given point in space as a function of time.

*Note that Ex0 and Eyo are complex quantities comprised with a magnitude and a phase angle.

*Negative exponent of e indicates that that the wave is travelling in the positive direction.

*Ex0 s phase would be the reference, thereby, its phase is zero degrees and = phasetv difference

7-4.1 Linear Polarization

*If ay=0 and angle is 0 degrees, wave is x-polarized. (positive axcoswt)

*If ax=0 and angle is 90 or -90 degrees, wave is y-polarized. (positive aycoswt)

*A wave is said to be linearly polarized if Ex(z,t) and Ey(z,t) are in phase or out of phase (=).

Example:

7-4.1 Circular Polarization

*left-handed circular when =90 degrees

*right-handed circular when =-90 degrees

*ax=ay, = 90 degrees

*To get the sign of y-component, find the rotation (clockwise or counter clockwise) and measured 90

degrees from there. If it falls on negative -y, make the y component, negative too.

Rotation Angle (Between-90 and 90 degrees) - angle between themajor axis of the ellipse1.

7.4.2 Elliptical Polarization

Wave Polarization (7.4 - Ulaby)

ECE 121 Page 29


30/30

and a reference direction.

Ellipticity Angle (Between -45 and 45 degrees) - characterizes the shape of the ellipse

andits handedness

2.

*Axial Ratio - between 1 (circular polarization) and infinity (linear polarization)

Auxiliary Angle 0 (Between 0 and 90 degrees)3.

Relationship between rotation and ellipticity angles

advanced electromagnetism ece 121 (tip reviewer)

Documents