Useful concepts of Electrical Engineering

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Useful Concepts of Electrical Engineering
Francisco Maerle
November 14, 2016
Contents
1 History 5
2 Math 6
2.1 Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Special Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Circular Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.2 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Derivatives and Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.2 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5.1 Even and Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5.2 Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6.1 Expected value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6.3 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6.4 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 RMS value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.8 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.9 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.10 Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.10.1 Phase Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.11 Partial Fractions Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.12.1 Useful Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.13 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.13.1 Standard convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.14 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.14.1 Useful Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.15 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.15.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.15.2 Inverse Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.15.3 Useful Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.16 Z Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.16.1 Z Transform of Common Discrete Time Signals . . . . . . . . . . . . . . . . . 23
1
2.16.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.16.3 Inverse Z Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.16.4 Final Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.16.5 Soluo de Equaes de Diferenas pela Transformada Z . . . . . . . . . . . . . . . 27
2.16.6 Useful Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.17 Mapping between the s-plane and the z-plane . . . . . . . . . . . . . . . . . . . . . . 28
2.18 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.18.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.18.2 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.18.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.19 Simultaneous Equations Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.20 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.21 Math facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Physics 34
3.1 Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Field from a continuous charge distribution . . . . . . . . . . . . . . . . . . . 35
3.1.2 Gauss’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Voltage and Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Power and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Electrical Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.2 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.3 Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6.1 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.2 Magnetic Force in a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.3 Magnetic Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.4 Magnetically Coupled Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.7 Electric Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.8 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Power Systems 52
4.1 Wheatstone Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Techniques of Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Two-port Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Superposition theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.3 Source transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.4 Thevenin Equivalent Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 Equivalent Circuit for Practical Transformers . . . . . . . . . . . . . . . . . . 55
4.3.2 Reflected Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3 Calculus of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.4 Single phase-shifting Transformers . . . . . . . . . . . . . . . . . . . . . . . . 57
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4.3.5 Autotransformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Three-phase System . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.1 Unbalanced Three-phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.2 Symmetrical Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.3 Symmetrical Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5.4 Unsymmetrical Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Single Line Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.7 Per-Unit System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.8 Electric Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.8.1 Rotational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.8.2 DC Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8.3 Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8.4 Asynchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 Signal Processing Systems 76
5.1 Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.1 Continuous-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.2 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.3 Discrete-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.4 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Cutoff Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3.1 PN Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3.2 Rectifier Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3.3 Zener Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.1 BJT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.2 MOSFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5.1 Common configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5.2 Effect of Finite Open-Loop Gain . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 Communication Systems 96
6.1 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7 Computer Systems 97
7.1 Logic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Logic Functions Transistor Level Schematic . . . . . . . . . . . . . . . . . . . . . . . 97
7.3 Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.1 C/C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 Special nomenclatures and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8 Control Systems 102
8.1 Physical Systems Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.1.1 Linear State-Space Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.1.2 State space to Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.1.3 Differential equation to State Space . . . . . . . . . . . . . . . . . . . . . . . 104
3
8.2 Impulse response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4 PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.5 Poles and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.5.1 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.5.2 Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.6 Design of Digital Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.6.1 Linear State-Space Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.6.2 Sample Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.6.3 Zero-order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.6.4 Pulse Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.7 Sistemas com Atraso de Transporte . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.7.1 Funo de Tranferncia a partir das Equaes do Espao de Estado . . . . . . . . . 115
8.7.2 Stability of Closed-loop Systems . . . . . . . . . . . . . . . . . . . . . . . . . 115
9 Special nomenclatures and Acronyms 117
10 Bibliography 118
4
Chapter 1
History
1730 – Benjamin Franklin: Lightning is electricity. Use of terms “positives” and “negatives”
charges.
1800 – Alessandro Volta: Built of the first electrical pile, or battery: a series of metal disks of two
kinds, separated by cardboard disks soaked with acid or salt solutions.
1815 – Michael Faraday: Electrical magnetic induction.
1855 – James Maxwell: Mathematical expressions to Faraday’s theory.
1870 – Alexander Graham Bell:
1870 – Thomas Edison:
1870 – George Westinghouse:
1880 – Nikola Telsa: Energy transmission
1880 – Heinrich Hertz: Vacuum tube. Electric waves.
1885 – William Stanley: High voltage alternating current transmission. Generators, transformers
and high-voltage transmission lines. Incandescent electric lamps.
1905 – Albert Eisten: Small amount of mass to produce huge amount of energy. (E = mc2)
1948 – Bell Telephone Laboratories: Invention of transistor
5
Chapter 2
Math
2.1 Logarithm
The logarithm function of base a y = loga x is the inverse of the exponential function y = a
x.
For an a > 0 and a 6= 1 is given the following relationship:
y = loga x ⇔ x = ay
Notations
loge x = ln x
log10 x = log x
Properties
loga a
x = x, a > 0, a 6= 1, x > 0
ln ex = x, x > 0
loga xy = loga x+ loga y
loga
x
y
= loga x− loga y
loga x
y = y loga x
loga x =
ln x
ln a
2.2 Vectors
A vector ~V is described by
~V = Vx~ax + Vy~ay + Vz~az
Its magnitude |~V | is given by
|~V | =
√
V 2x + V
2
y + V
2
z
6
A unit vector (vector with magnitude —v— = 1) to represent only the direction of a vector is
given by
a~V =
~V√
V 2x + V
2
y + V
2
z
=
~V
|~V |
Dot Product
The dot product ~u · ~v of vectors ~u = (u1, u2, u3) and ~v = (v1, v2, v3) is
~u · ~v = u1v1 + u2v2 + u3v3
~u · ~v = u v cos θ
Cross Product
The cross product ~u× ~v of vectors ~u = u1~i+ u2~j + u3~k and ~v = v1~i+ v2~j + v3~k is
~u× ~v =
∣∣∣∣∣∣
~i ~j ~k
u1 u2 u3
v1 v2 v3
∣∣∣∣∣∣
~u× ~v = −~v × ~u
~u× ~v = (|~u||~v| sin θ) ~n
2.3 Special Coordinate Systems
Many problems possess a type of symmetry that pleads for a more logical treatment.
2.3.1 Circular Cylindrical Coordinates
The circular cylindrical coordinate system is the three-dimensional version of the polar coordi-
nates of analytic geometry.
In polar coordinates, a point is located in a plane by giving both its distance ρ from the origin
and the angle φ between the line from the point to the origin and an arbitrary radial line, taken as
φ = 0.
In circular cylindrical coordinates, we also specify the distance z of the point from an arbitrary
z = 0 reference plane that is perpendicular to the line ρ = 0.
7
x = ρ cosφ
y = ρ sinφ
z = z
ρ=
√
x2 + y2
φ = tan−1
y
z
z = z
2.3.2 Spherical Coordinates
x = r sin θ cosφ
y = r sin θ sinφ
z = rcosφ
ρ =
√
x2 + y2 + z2
θ = cos−1
z√
x2 + y2 + z2
φ = tan−1
y
z
2.4 Derivatives and Integrals
2.4.1 Derivatives
Derivatives are used to analyse the rate in which a variable changes its value related with
another, if it is fast, slow or non-existent. Some examples are: the body response due to a drug
8
dosage or the cost of a production due to the quantity of any special material used.
The derivative of a function f(x) related to the variable x is the function f ′ which value in x is
defined by:
f ′(x) =
d
dx
f(x) = lim
h→0
f(x+ h)− f(x)
h
One of the most important use of the derivatives is to know in what point c a function reach
its maximum and minimum values, which can be found by:
f ′(c) = 0
Functions of Several Variables
If w = f(x, y, z) is differentiable and x, y and z are differentiable functions of t
dw
dt
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
+
∂f
∂z
dz
dt
Useful Derivatives
(un)′ = n un−1 u′ (uv)′ = v uv−1 u′ + uv (lnu) v′
(uv)′ = vu′ + uv′ (sinu)′ = u′ cosu
(
u
v
)′ =
vu′ − uv′
v2
(cosu)′ = −u′ sinu
(au)′ = au (ln a) u′ (tanu)′ = u′ sec2 u
(loga u)
′ =
u′
u
logae (secu)
′ = u′ secu tanu
9
2.4.2 Integrals
A function F is the integral of f on an interval I if F ′(x) = f(x) for all x in I and is denoted
by:
F (x) =
∫
f(x)dx ⇐⇒ F ′(x) = f(x)
Integrals play a key role in the calculus of areas and volumes of general geometric shapes,
average values, forces, etc. The idea behind integration is that we can effectively compute many
quantities by breaking them into small pieces, and then summing the contributions from each small
part.
Area
The area of a region with a curved boundary can be approximated by summing the areas of
a collection of rectangles. The height of each rectangle is the maximum value of the function f
calculated into the interval [a, b] subdivided into n subintervals of equal width ∆x = (b− a)/n: c1
in the first subinterval, c2 in the second and so on. The finite sum takes the form:
f(c1)∆x+ f(c2)∆x+ f(c3)∆x+ · · ·+ f(cn)∆x
By taking more and more rectangles it appears that these finite sums give better and better
approximations to the true area of the region.
lim
n→∞
n∑
k=1
f(ck)∆x =
∫ b
a
f(x)dx
Average Value of a Nonnegative Function
The average value of a collection of n numbers is obtained by adding them together and dividing
by n. But what is the average value of a continuous function f on an interval [a, b]?
A function with constant value c on an interval [a, b] has average value c. When c is positive,
its graph over [a, b] gives a rectangle of height c. The average value of the function can then be
interpreted geometrically as the area of this rectangle divided by its width a− b.
Average =
1
a− b
∫ b
a
f(x)dx
Work and magnetic fluid
A constant force ~F applied over a straight displacement ~L does an amount of work FL cos θ
which is easily written ~F · ~L. If the force varies along the path, integration is necessary to find the
total work:
W =
∫
~F · d~L
The total flux Φ crossing a surface of area S is given by BS if the magnetic flux density B is
perpendicular to the surface and uniform over it.
We define a vector surface S as having area for its magnitude and having a direction normal to
the surface. The flux crossing the surface is then ~B · ~S.
10
If the flux density is not constant over the surface, the total flux is
Φ =
∫
~B · d~S
Useful Integrals
∫
du = u+ C
∫
sinu du = − cosu+ C∫
un du =
un+1
n+ 1
+ C
∫
cosu du = sinu+ C∫
du
u
= ln|u|+ C
∫
tanu du = −ln|cos u|+ C = ln |sec u|+ C∫
au du =
1
u′
au
ln a
+ C
∫
u dv = uv −
∫
v du
2.5 Functions
2.5.1 Even and Odd Functions
Even Function
f(x) = f(−x) ∀ x
∫ a
−a
f(x) dx = 2
∫ a
0
f(x) dx
(Remember x2, x4, x6, cosx)
Odd Function
f(x) = −f(−x) ∀ x
∫ a
−a
f(x) dx = 0
(Remember x, x3, x5, senx)
Special Properties:
1. Adding
(a) The sum of two odd functions is odd.
(b) The sum of an even and odd function is neither even nor odd.
(c) The sum of two even functions is even.
2. Product
(a) The product of two even functions is an even function.
(b) The product of two odd functions is an even function.
(c) The product of an even function and an odd function is an odd function.
11
2.5.2 Orthogonal Functions
∫ b
a
f(x)g(x)w(x)dx = 0
2.6 Statistics
2.6.1 Expected value
E[X] = µ
2.6.2 Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a
random variable from its mean, and it informally measures how far a set of (random) numbers are
spread out from their mean.
V ar(X) = E[(X − µ)2]
The variance is also the second central moment of a distribution, and the covariance of the
random variable with itself.
2.6.3 Covariance
2.6.4 Standard Deviation
In statistics, the standard deviation (SD, or s) is a measure that is used to quantify the amount
of variation or dispersion of a set of data values.
σ =
√
E[(X − µ)2] =
√
V ar(X)
A low standard deviation indicates that the data points tend to be close to the mean (also called
the expected value) of the set, while a high standard deviation indicates that the data points are
spread out over a wider range of values.
A useful property of the standard deviation is that, unlike the variance, it is expressed in the
same units as the data.
2.7 RMS value
The root mean square, also known as the quadratic mean, is a measure of the magnitude of a
varying quantity. It is especially useful when variates are positive and negative.
Vrms ,
√
1
T
∫ τ+T
τ
v2(t) dt
In the field of electricity, The idea of effective value arises from the need to measure the effec-
tiveness of a voltage or current source in delivering power to a resistive load.
The effective value of a periodic current is the dc current that delivers the same average power
to a resistor as the periodic current. The same is true for the voltage.
12
+
−
v(t)
i(t)
R
−
+Vrms
Irms
R
P =
1
T
∫ T
0
i2R dt P = RI2rms
Ieff =
√
1
T
∫ T
0
i2 dt
2.8 Differential Equations
Linear differential equations model a number of real-world phenomena, including electrical
circuits problems.
A first-order linear differential equation is one that can be written in the form
dy
dx
+ P (x)y = Q(x).
The equation is linear in y because y and its derivative occur only to the first power, are not
multiplied together, nor they appear as the argument of a function such as sin y, ey or
√
dy/dx.
We solve the equation by multiplying both sides by a positive function v(x) that transforms the
left side into the derivative of the product vy
v
dy
dx
+ vPy =
d
dx
(vy) = v
dy
dx
+ y
dv
dx
⇒ v = e
∫
P (x)dx
d
dx
(vy) = vQ ⇒ y = 1
v
∫
vQ dx
We call v(x) an integrating factor because its presence makes the equation integrable.
Example: RL circuit
−+
V
i
LR
⇔ Ldi
dt
+Ri = V (switch closed)
Equation in the standard form:
di
dt
+
Ri
L
=
V
L
13
P (t) =
R
L
⇒ v(t) = e
∫
R
L
dt = e
R
L
t
i(t) = e−
R
L
t
∫
e
R
L
tV
L
dt = e−
R
L
tV
L
(
L
R
e
R
L
t + C
)
i(0) = 0 ⇒ 0 = V
L
(
L
R
+ C
)
⇒ C = −L
R
i(t) =
V
R
− V
R
e−(
R
L
t) ⇒ i(t) = V
R
(1− e−RL t)
2.9 Complex Numbers
General representation of a complex number:
z = a+ jb |z| =
√
a2 + b2 j =
√−1
a = |z| cos θ b = |z| sin θ θ = tan−1(b/a)
The Euler’s identity, proved by Taylor series expansion, states that:
ejθ = cos θ + j sinθ
so,
z = a+ jb = |z| ejθ
Equivalent forms of writing complex unities:
j = ejpi/2 − j = ej3pi/4 1 = ej2pi − 1 = ejpi
Some care when calculating the argument of z:
a > 0 and b > 0 ⇒ θ = tan−1(b/a)
a < 0 and b < 0 ⇒ θ = pi + tan−1(b/a)
a < 0 and b > 0 ⇒ θ = pi − tan−1(|b/a|)
a > 0 and b < 0 ⇒ θ = 2pi − tan−1(|b/a|)
Absolut value and argument in some operations with complex numbers:
z1 = |z1|eθ1 z2 = |z2|eθ2
z =
z1
z2
=
|z1|
|z2|e
θ1−θ2 z = z1 · z2 = |z1||z2|eθ1+θ2
14
2.10 Phasors
A phasor is a complex number Fˆ (F) representing a sinusoidal function whose amplitude (F),
frequency (w), and phase (φ) don’t change in the time. This is useful because the frequency is often
common to all the signals (u(t) and i(t)) in the components of a circuit. In those situations, phasors
allow this common feature to be factored out, leaving just the amplitude and phase features.
Definition
Given a sinusoidal function
f(t) = F cos(wt+ φ)
we can see that
f(t) = Re{F cos(wt+ φ) + jF sin(wt+ φ)} = Re{F ej(wt+φ)}
Re{F ej(wt+φ)} = Re{F ejφejwt}
being the phasor
Fˆ = Fejφ = F∠φ
which doesn’t care about the angular frequency information.
Differentiation and Integration
An important feature of the phasor transform is that differentiation and integration of sinusoidal
signals (having constant amplitude, period and phase) corresponds to simple algebraic operations
on the phasors; the phasor transform thus allows the analysis (calculation) of the AC steady state of
RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor
domain instead of solving differential equations (with real coefficients) in the time domain.
d
dt
F cos(wt+ φ) =
d
dt
Re{Fejφejwt} = Re{jw Fejφejwt}
Fˆd = jwFe
jφ = jwFˆ∫
F cos(wt+ φ) dt =
∫
Re{Fejφejwt} dt = Re{ 1
jw
Fejφejwt}
Fˆi =
1
jw
Fejφ =
1
jw
Fˆ
Example
Solve this second order differential equation:
4i+ 8
∫
i dt− 3di
dt
= 50 cos(2t+ 75◦) ↔ 4Iˆ + 8 Iˆ
jw
− 3jwIˆ = 50∠75◦
4Iˆ − 4jIˆ − 6jIˆ = 50∠75◦ ⇒ Iˆ(4− j10) = 50∠75◦
Iˆ =
50∠75◦
10.77∠− 68◦ = 4.6∠143
◦A ↔ i(t) = 4.6 cos(2t+ 143◦)A
15
2.10.1 Phase Difference
Not all sinusoidal waveforms will pass exactly through the zero axis point at the same time, but
may be ”shifted” to the left by φ = θv − θi.
Below are some distinct representations of phase difference. To check who is leading or lagging
you must check the time representation. The closest wave to reach its maximum value is leading
relating to the other wave.
pi
2
pi
2
pi 3pi
2
2pi
Vm
Im
0
−Im
−Vm
φ
i(t)
v(t)
θ
⇔
Iˆ
Vˆ Re
Im
φ
v(t) = Vm sin(wt) i(t) = Im sin(wt− φ)
The current i is lagging the voltage v by angle φ. Or one can say the voltage v is leading the
current i by angle φ (degrees or radians/s).
pi
2
pi
2
pi 3pi
2
2pi
Vm
Im
0
−Im
−Vm
φ
i(t)
v(t)
θ
⇔
Iˆ
Vˆ Re
Im
θi
v(t) = Vm sin(wt) i(t) = Im sin(wt+ φ)
The current i is leading the voltage v by angle φ. Or one can say the voltage v is lagging the
current i by angle φ (degrees or radians/s).
Make an example showing that a current i = 5sin(377t+50) is leading the voltage v = 4sin(377
+ 205).
16
2.11 Partial Fractions Expansion
Given a rational function
N(s)
D(s)
Case 1 : N(s) < D(s)
a) D(s) has distinct roots
s+ 1
s3 + s2 − 6s =
s+ 1
s(s− 2)(s+ 3) =
A
s
+
B
s− 2 +
C
s+ 3
A = s
N(s)
D(s)
∣∣∣∣
s=0
= −1
6
B = (s− 2)N(s)
D(s)
∣∣∣∣
s=2
=
3
10
C = (s+ 3)
N(s)
D(s)
∣∣∣∣
s=−3
= − 2
15
b) D(s) has repeated roots
s+ 1
s(s− 2)3 =
A
s
+
B
s− 2 +
C
(s− 2)2 +
D
(s− 2)3
A = s
N(s)
D(s)
∣∣∣∣
s=0
= −1
8
D =
1
0!
(s− 2)3N(s)
D(s)
∣∣∣∣
s=2
=
3
2
C =
1
1!
d
ds
[
(s− 2)3N(s)
D(s)
] ∣∣∣∣
s=2
= − 1
s2
∣∣∣∣
s=2
= −1
4
B =
1
2!
d2
ds2
[
(s− 2)3N(s)
D(s)
] ∣∣∣∣
s=2
= − 2
s3
∣∣∣∣
s=2
=
1
4
Case 2 : N(s) > D(s)
(x+ 2)3
(x+ 1)
x2 + 5x+ 7
x+ 1
)
x3 + 6x2 + 12x+ 8
− x3 − x2
5x2 + 12x
− 5x2 − 5x
7x+ 8
− 7x− 7
1
17
N
D
= Q+
R
D
⇒ (x+ 2)
3
(x+ 1)
= x2 + 5x+ 7 +
1
x+ 1
Case 3 : Irreducible quadratic factor
−2x+ 4
(x2 + 1)(x− 1)2 =
Ax+B
x2 + 1
+
C
x− 1 +
D
(x− 1)2
−2x+ 4 = (Ax+B)(x− 1)2 + C(x− 1)(x2 + 1) +D(x2 + 1)
= (A+ C)x3 + (−2A+B − C +D)x2
+ (A− 2B + C)x+ (B − C +D)
Equating coefficients of like terms and solving equations simultaneously
−2x+ 4
(x2 + 1)(x− 1)2 =
2x+ 1
x2 + 1
− 2
x− 1 +
1
(x− 1)2
Case 4 : Repeated irreducible quadratic factor
1
x(x2 + 1)2
=
A
x
+
Bx+ C
x2 + 1
+
Dx+ E
(x2 + 1)2
1 = A(x2 + 1)2 + (Bx+ C)x(x2 + 1) + (Dx+ E)x
= (A+B)x4 + Cx3 + (2A+B +D)x2 + (C + E)x+A
Equating coefficients of like terms and solving equations simultaneously
1
x(x2 + 1)2
=
1
x
− x
x2 + 1
− x
(x2 + 1)2
2.12 Laplace Transform
The Laplace transform is a method that can be used for solving ordinary differential equations
by reducing a differential equation to an algebraic equation.
The Laplace transform of a function f(t) is defined by
F (s) = L(f(t)) =
∫ ∞
0
e−stf(t) dt
18
2.12.1 Useful Table
Signal L{f(t)} Signal L{f(t)}
δ(t) 1 u(t)
1
s
r(t)
1
s2
e−at
1
s+ a
sinwt
w
s2 + w2
coswt
s
s2 + w2
e−at
1
(s+ a)2
e−at sin t
w
(s+ a)2 + w2
e−at cos t
s+ a
(s+ a)2 + w2
Kf(t) KF (s)
df(t)
dt
sF (s)− f(0)
∫ t
0
f(x)dx
F (s)
s
e−atf(t) F (s+ a) tnf(t) (−1)nd
nF (s)
dsn
f(t− a)u(t− a) e−asF (s) f(at) 1
a
F
(s
a
)
2.13 Convolution
Convolution is a mathematical operation on two functions f and g, producing a third function
that is typically viewed as a modified version of one of the original functions, giving the area overlap
between the two functions.
It is defined as the integral of the product of the two functions after one is reversed and shifted.
As such, it is a particular kind of integral transform:
f(t) ∗ g(t) ,
∫ ∞
−∞
f(t− τ)g(τ)dτ =
∫ ∞
−∞
f(τ)g(t− τ)dτ
In that context, the convolution formula can be described as a weighted average of the function
f(τ) at the moment t where the weighting is given by g(−τ) simply shifted by amount t. As t
changes, the weighting function emphasizes different parts of the input function.
Discrete convolution:
f [n] ∗ g[n] ,
∞∑
m=−∞
f [n−m]g[m] =
∞∑
m=−∞
f [m]g[n−m]
2.13.1 Standard convolutions
An exponential and a step function
19
e−αt ∗ s(t) =
∫ t
0
e−α(t−τ)s(τ)dτ =
∫ t
0
e−α(t−τ)dτ
= e−αt
∫ t
0
eατdτ = e−αt
eατ
α
∣∣∣∣t
0
=
e−αt
α
(eαt − eα0) = e
−αt
α
(eαt − 1)
=
1− e−αt
α
2.14 Fourier Series
Definition
Almost all periodic waveforms or functions satisfying certain conditions can be composed in a
sum of cosines and sines. (Phenomenon known as Fourier Series Expansion).
vac(t) = V0 +
∞∑
n=1
Vn cos(nwt+ θn)
V0: DC component. Simply the average value of the signal:
V0 = < vac(t) > =
1
Ts
∫ T+Ts
T
vac(t) dt
Vn: Amplitude of the nth harmonic:
Vn =
√
a2n + b
2
n
an =
2
Ts
∫ T+Ts
T
vac(t) cos(nwt) dt
bn =
2
Ts
∫ T+Ts
T
vac(t) sin(nwt) dt
θn: Phase shift of the nth harmonic:
θn = tan
−1
(
− bn
an
)
Definition II???
The Fourier Series decomposition allows us to express any periodic signal x(t) with period T
an a linear combination (or weighted some) of a countable set of frequencies:
x(t) =
∞∑
k=−∞
cke
jkw0t, k = 0,±1,±2, ...
ck =
1
T
∫ τ+T
τ
x(t)e−jkw0tdt
20
Example
Calculate the Fourier Series Expansion for a Pulse Train Signal
t
1
−T−2T−3T 0 T 2T 3T
·· ·· · ·
δT (t) =
∞∑
k=−∞
δ(t− kT )
Applying the definition
δT (t) =
∞∑
n=−∞
cne
jnw0t, n = 0,±1,±2, ...
where
cn =
1
T
∫ T/2
−T/2
∞∑
k=−∞
δ(t− kT )e−jnw0tdt
cn =
1
T
∞∑
k=−∞
∫ T/2
−T/2
δ(t− kT )e−jnw0tdt
By the blending property
cn =
1
T
∞∑
k=−∞
e−jnw0kT , ∀ − T/2 0 kT 0 T/2
cn =
1
T
1/2∑
k=−1/2
e−jnw0kT
Since k is only represented by integer numbers, the only possible value for it is k = 0
cn =
1
T
Thus, the Fourier series expansion for the impulse train is:
δT (t) =
1
T
∞∑
n=−∞
ejnw0t
2.14.1 Useful Table
2.15 Fourier Transform
F (jw) =
∫ +∞
−∞
f(t)e−jwtdt
21
2.15.1 Properties
Complex domain shift
Le−atf(t) = F (s+ a)
2.15.2 Inverse Fourier Transform
x(t) =
1
2pi
∫ +∞
−∞
X(jw)ejwtdw
2.15.3 Useful Table
f(t) F (jw)
Unit impulse train δT (t) =
∞∑
k=−∞
δ(t− kT ) 2pi
T
∞∑
n=−∞
δ(w − nws)
= f
= h
2.16 Z Transform
The Z transform is an important tool in the analysis and project of discrete time systems. She
simplifies the solution of problems of discrete time, converting equations of difference and time
invariant equations to algebric equations and convolutions to multiplications.
Definition
Given a sequence {u0, u1, u2, ..., uk, ...}, its Z transform is defined as
U(z) , u0 + u1z−1 + u2z−2 + · · ·+ ukz−k =
∞∑
k=0
ukz
−k
Relation between the Laplace transform and the Z transform
Given the represenation of the pulse train of a discrete time signal
u∗(t) , u0δ(t) + u1δ(t− T ) + u2δ(t− 2T ) + · · ·+ ukδ(t− kT ) + · · ·
=
∞∑
k=0
ukδ(t− kT )
Its LaPlace transform is
22
U∗(s) = u0 + u1e−sT + u2e−2sT + · · ·+ uke−KsT + · · ·
=
∞∑
k=0
uk(e
−sT )k
being z defined as
z = esT
2.16.1 Z Transform of Common Discrete Time Signals
2.16.1.1 Unit Pulse
Considering the discrete time Unit Pulse definition:
u(k) = δ(k) ,
{
1, k = 0
0, k 6= 0
k
1
0−1 1 2
By applying the definition one have
U(z) = 1
2.16.1.2 Discrete Unit Step
Considere a sequncia {uk}∞k=0 = {1, 1, 1, 1, 1, ...}
k
1
0−1 1 2 3 4
· · ·
Aplicando a definio temos
U(z) = 1 + z−1 + z−2 + z−3 + ...+ z−k + ... =
∞∑
k=0
z−k
Sabendo que
n∑
k=0
ak =
1− an+1
1− a , a 6= 1
U(z) =
∞∑
k=0
z−k =
1
1− z−1 =
z
z − 1
23
2.16.1.3 Exponencial Discreta
Considerando a exponencial discreta
u(k) =
{
ak, k > 0
0, k < 0
k
1
0−1 1 2 3
· · ·a
a2
a3
Aplicando a definio temos
U(z) = 1 + az−1 + a2z−2 + ...+ akz−k + ... =
∞∑
k=0
akz−k =
∞∑
k=0
(a
z
)k
Sabendo que
n∑
k=0
ak =
1
1− a, |a| < 1
U(z) =
∞∑
k=0
(a
z
)k
=
1
1− (a/z) =
z
z − a
2.16.2 Properties
1. Linearity
Z{αf1(k) + βf2(k)} = αF1(z) + βF2(z)
2. Right time shift (delay)
Z{f(k − n)} = z−nF (z)
3. Left time shift (advance)
Z{f(k + 1)} = zF (z)− zf(0)
Z{f(k + n)} = znF (z)− znf(0)− zn−1f(1)− ...− zf(n− 1)
4. Exponential multiplication
Z{a−kf(k)} = F (az)
24
5. Differentiation in the Z domain
Z{kmf(k)} =
(
−z d
dz
)m
F (z)
6. Convolution
Z{f1(k) ∗ f2(k)} = Z
{ ∞∑
i=0
f1(k − i)f2(k)
}
= F1(z)F2(z)
2.16.2.0.1 Example Z transform of the discrete ramp
f(k) = k, k = 0, 1, 2...
k
0−1 1 2 3 4
· · ·
1
2
3
4
Having
f(k) = k × 1(k)
and that for the step function
Fs(z) =
z
z − 1
Applying the property of differentiation in the Z domain, one has that
Z{k × 1} =
(
−z d
dz
)(
z
z − 1
)
= (−z)(z − 1)− z
(z − 1)2 =
z
(z − 1)2
2.16.3 Inverse Z Transform
Como o propsito da transformada Z simplificar solues no domnio do tempo, essencial fazer a
transformada inversa de funes no domnio de z.
For that there are three methods:
2.16.3.1 Direct division
This approach consist in directly relate the digital sequence making a simple division. It is
useful only to find the first few terms of f(kT).
25
1. Dividing the polynomium, one will have the folowing sequence which is already similar with
the Z transform definition:
Ft(z) = f0 + f1z
−1 + ...+ ftz−t =
t∑
k=0
fkz
−k
2. The inverse Z transform is written as the sequence
{f0, f1, ..., ft, ...}
2.16.3.2 Partial fraction expansion
Assim como nas transformadas de LaPlace, expanso em fraes parciais torna possvel escrever
uma funo como a soma de outras funes mais simples que so trasnformadas Z de funes discretas j
conhecidas.
1. Encontre a expanso em fraes parciais de F (z)/z ou de F (z)
2. Obtenha a transformada inversa f(k) usando tabelas de transformada Z.
2.16.3.3 Computational method
For this method, it is applied the input signal Delta-function (Unit Pulse) into the system which
basically just multiplies a signal per 1 by a regular interval of time.
F (z) =
N(z)
D(z)
U(z)
With this operation we end up with a difference equation, which the values of the sequence
{f(kT )} can be recursively determined
2.16.3.3.1 Example
F (z) =
z2 + 2z
z3 − 4.5z2 + 6.5z − 3U(z) =
z−1 + 2z−2
1− 4.5z−1 + 6.5z−2 − 3z−3U(z)
f(k)− 4.5f(k − 1) + 6.5f(k − 2)− 3f(k − 3) = u(k − 1) + 2u(k − 2)
with
u(0) = 1, u(k) = 0, k = 1, 2, 3, ... f(k) = 0, k < 0
k = 0 : f(0) = 4.5f(−1)− 6.5f(−2) + 3f(−3) + u(−1) + 2u(−2) f(0) = 0
k = 1 : f(1) = 4.5f(0)− 6.5f(−1) + 3f(−2) + u(0) + 2u(−1) f(1) = 1
k = 2 : f(2) = 4.5f(1)− 6.5f(0) + 3f(−1) + u(1) + 2u(0) f(2) = 6.5
k = 3 : f(3) = 4.5f(2)− 6.5f(1) + 3f(0) + u(2) + 2u(1) f(3) = 22.75
k = 4 : f(4) = 4.5f(3)− 6.5f(2) + 3f(1) + u(3) + 2u(2) f(4) = 63.125
...
...
...
{f(kT )} = {0, 1, 6.5, 22.75, ...}
26
2.16.4 Final Value Theorem
Se uma sequncia converge a um valor constante quando k tende ao infinito, o limite/valor
encontrado por
f(∞) = lim
k→∞
f(k) = lim
z→1
(
z − 1
z
)
F (z) = lim
z→1
(z − 1)F (z)
2.16.5 Soluo de Equaes de Diferenas pela Transformada Z
Finalmente, para a solucionamento de equaes de diferenas usamos a trasnformada Z passando
a equao em estudo para o domnio de z, usando geralmente a propriedade de atraso ou avano de
tempo. Realiza-se a manipulao necessria e por ltimo obtida a resposta por sua transformada
inversa.
Example
Solucione a equao de diferena linear
x(k + 2)− (3/2)x(k + 1) + (1/2)x(k) = 1(k)
com as condies iniciais x(0) = 1, x(1) = 5/2.
1. Transformada Z
[z2X(z)− z2x(0)− zx(1)]− (3/2)[zX(z)− zx(0)] + (1/2)X(z) = z/(z − 1)
2. Rearranjando os termos
X(z) =
z[1 + (z + 1)(z − 1)]
(z − 1)(z − 1)(z − 0.5) =
z3
(z − 1)2(z − 0.5)
3. Expanso em fraes parciais
X(z)
z
=
z2
(z − 1)2(z − 0.5) =
A
(z − 1)2 +
B
z − 1 +
C
z − 0.5
onde
A = (z − 1)2X(z)
z
∣∣∣∣
z=1
=
z2
z − 0.5
∣∣∣∣
z=1
= 2
B = (z − 1)X(z)
z
∣∣∣∣
z=1
=
z2
(z − 1)(z − 0.5)
∣∣∣∣
z=1
⇒ indeterminado
C = (z − 0.5)X(z)
z
∣∣∣∣
z=0.5
=
z2
(z − 1)2
∣∣∣∣
z=0.5
= 1
mas para achar B
27
z2 = A(z − 0.5) +B(z − 0.5)(z − 1) + C(z − 1)2 ⇒ 1 = (B + 1)⇒ B = 0
ento
X(z) =
2z
(z − 1)2 +
z
z − 0.5
4. Transformada Z inversa
x(k) = 2k + (0.5)k
2.16.6 Useful Table
u(k) U(z)
δ(k) 1
1(k)
z
z − 1
ak
z
z − a
f(k − n) z−nF (z)
f(k + n) znF (z)− znf(0)− zn−1f(1)− ...− zf(n− 1)
a−kf(k) F (az)
kmf(k)
(
−z d
dz
)m
F (z)
k · 1(k) z
(z − 1)2
2.17 Mapping between the s-plane and the z-plane
Being the relationship between the Laplace transform and the Z transform
z = eTs = eTσej(Tw+2pik); k = 0,±1,±2, · · ·
|z| = eTσ arg(z) = Tw + 2pik
The frequency ws/2 = pi/T plays an important role because it separates the primary strip from
the complementary strips. These points are on the s-plane imaginary axis are exactly in the point
(1± 0j) in the z plane. The mapping is unique only for points within the primary strip.
28
2.18Matrices
2.18.1 Determinants
Examples on how to calculate determinants of matrices with order 4 or more:∣∣∣∣∣∣
a b c
d f g
h i j
∣∣∣∣∣∣ = a
∣∣∣∣f gi j
∣∣∣∣− b ∣∣∣∣d gh j
∣∣∣∣+ c ∣∣∣∣d fh i
∣∣∣∣
below is where to put + or - signs:
+ − + · · ·
− + − · · ·
+ − + · · ·
...
...
...
. . .

n×m
⇔ (−1)n+m
Complementary one:
detA =
∣∣∣∣∣∣∣∣
1 0 2 −1
3 0 0 5
2 1 4 −3
1 0 5 0
∣∣∣∣∣∣∣∣ = −0
∣∣∣∣∣∣
3 0 5
2 4 −3
1 5 0
∣∣∣∣∣∣+ 0
∣∣∣∣∣∣
1 2 −1
2 4 −3
1 5 0
∣∣∣∣∣∣− 1
∣∣∣∣∣∣
1 2 −1
3 0 5
1 5 0
∣∣∣∣∣∣+ 0
∣∣∣∣∣∣
1 2 −1
3 0 5
2 4 −3
∣∣∣∣∣∣
2.18.2 Inverse Matrix
A ·A−1 = A−1 ·A = I
A−1 =
A∗T
|A|
A∗: Adjoint Matrix
AT : Transposed Matrix
|A|: detA
How to calculate the Adjoint Matrix of a 3x3 one:
A =
a11 a12 a13a21 a22 a23
a31 a32 a33

A∗ =

∣∣∣∣a22 a23a32 a33
∣∣∣∣ − ∣∣∣∣a21 a23a31 a33
∣∣∣∣ ∣∣∣∣a21 a22a31 a32
∣∣∣∣
−
∣∣∣∣a12 a13a32 a33
∣∣∣∣ ∣∣∣∣a11 a13a31 a33
∣∣∣∣ − ∣∣∣∣a11 a12a31 a32
∣∣∣∣∣∣∣∣a12 a13a22 a23
∣∣∣∣ − ∣∣∣∣a11 a13a21 a23
∣∣∣∣ ∣∣∣∣a11 a11a21 a22
∣∣∣∣

which signs follow the same rule mentioned before: (−1)n+m
29
2.18.3 Diagonalization
Find the eigenvalues of a matrix by:
det(λI −A) = 0
Given Av = λv ⇒ Av − λv = 0, find its eigenvectors by:
(A− λI)v = 0
Let P be the diagonalizing matrix with these eigenvectors as its columns:
P =
[
v1 v2 · · · vn
]
Now just calculate P−1AP and a diagonal matrix Λ will result with the eigenvalues in it.
Example 1:
A =
[
5 3
−6 −4
]
det(λI −A) =
∣∣∣∣λ− 5 −36 λ+ 4
∣∣∣∣ = λ2 − λ− 2 = 0
λ1 = 2 λ2 = −1
for λ = 2 and (A− λI)v = 0 :
[
5− 2 3
−6 −4− 2
] [
v11
v12
]
= 0 ⇒
{
3v11 + 3v12 = 0
−6v11 − 6v12 = 0
⇒ v1 =
[
1
−1
]
for λ = −1 :
[
5 + 1 3
−6 −4 + 1
] [
v21
v22
]
= 0 ⇒
{
6v21 + 3v22 = 0
−6v21 − 3v22 = 0
⇒ v2 =
[−1
2
]
P =
[
v1 v2
]
=
[
1 −1
−1 2
]
⇒ P−1 = P
∗T
|P | =
[
2 1
1 1
]
Λ = P−1AP =
[
2 0
0 −1
]
30
2.19 Simultaneous Equations Solution
In circuit analysis, we often encounter a set of simultaneous equations having the form
a11x1 + a12x2 + · · ·+ a1nxn = b1
a21x1 + a22x2 + · · ·+ a2nxn = b2
...
...
...
...
an1x1 + an2x2 + · · ·+ annxn = bn
⇔
a11 a12 · · · a1n
a21 a22 · · · a2n
...
...
. . .
...
an1 an2 · · · ann


x1
x2
...
xn
 =

b1
b2
...
bn

Cramer’s Rule
x1 =
∆1
∆
x2 =
∆2
∆
· · · xn = ∆n
∆
where
∆ =
∣∣∣∣∣∣∣∣∣
a11 a12 · · · a1n
a21 a22 · · · a2n
...
...
. . .
...
an1 an2 · · · ann
∣∣∣∣∣∣∣∣∣ ∆1 =
∣∣∣∣∣∣∣∣∣
b1 a12 · · · a1n
b2 a22 · · · a2n
...
...
. . .
...
bn an2 · · · ann
∣∣∣∣∣∣∣∣∣
∆2 =
∣∣∣∣∣∣∣∣∣
a11 b1 · · · a1n
a21 b2 · · · a2n
...
...
. . .
...
an1 bn · · · ann
∣∣∣∣∣∣∣∣∣ · · · ∆n =
∣∣∣∣∣∣∣∣∣
a11 a12 · · · b1
a21 a22 · · · b2
...
...
. . .
...
an1 an2 · · · bn
∣∣∣∣∣∣∣∣∣
Matrix Inversion
X = A−1B
2.20 Trigonometric Identities
sin2(x) + cos2(x) = 1
sin2(x) =
1− cos(2x)
2
cos2(x) =
1 + cos(2x)
2
sin(x± y) = sin(x) cos(y)± cos(x) sin(y)
cos(x± y) = cos(x) cos(y)∓ sin(x) sin(y)
31
2.21 Math facts
1. Letters in math
a, b, c ⇒ constants
d, e ⇒ derivative and exponential
f, g, h ⇒ functions
i, j, k, l ⇒ indexes
m,n ⇒ quantities
o ⇒ too similiar with 0 to be used
p, q, r, s ⇒ polynomials
t ⇒ time
u, v, w ⇒ vectors
x, y, z ⇒ variables or coordinates
2. Being n an integer and a positive number:
an − bn = (a− b)(an−1 + an−2b+ · · ·+ abn−2 + bn−1)
3. The geometric series (also applicable for matrices)
n∑
k=0
ak =
1− an+1
1− a , a 6= 1
4. Approximations from exponential to algebraic representation when b >> a:
e−a/b ' 1− a
b
ea/b ' 1 + a
b
5. Taylor series expansion of a function f(x) about a point a.
f(x) = f(a) + f ′(a)(x− a) + f
′′(a)
2!
(x− a)2 + f
(3)(a)
3!
(x− a)3 + · · ·+ f
(n)(a)
n!
(x− a)n + · · ·
Example with Maclaurin Series (Taylor Series with a = 0) to proof the Euler Identity:
ejθ = 1 + jθ +
(jθ)2
2!
+
(jθ)3
3!
+ · · ·+ (jθ)
n
n!
+ · · · =
∞∑
n=0
(jθ)n
n!
cos(θ) = 1− 0− θ
2
2!
+ 0 +
θ4
4!
+ · · ·+ cos
(n)(θ)θn
n!
+ · · ·
jsin(θ) = 0 + jθ − 0− j θ
3
3!
+ 0 + · · ·+ j sin
(n)(θ)θn
n!
+ · · · = j
∞∑
n=0
(−1)nθ2n+1
(2n+ 1)!
ejθ = cos θ + j sin θ
32
6. Blending Property
∫ t
0
f(τ)δ(τ − hT )dτ =
{
f(hT ) ∀ 0 6 hT 6 t
0 ∀ hT > t
33
Chapter 3
Physics
3.1 Electric Field
An electric charge q has an electric field ~E surrounding itself which exert one force to other
charge q0 close to it. This relationship can be described as
~E ,
~F
q0
It is important to notice that this is a formula for the interaction between only two charges
(The world is much more complex than that!).
Since we have the experimental Coulomb’s Law:
~F , 1
4piε0
qq0
r2
rˆ ⇒ ~E = 1
4piε0
q
r2
rˆ
Electric field lines for a positive and a negative charge:
When we have a capacitor, we have static positive charges in one plate and static negative
charges in the other one. By this configuration they create an electric field between them directed
from the positive to negative charges. A decreasing electric field on that could be defined as some
lost of charges in both sides of the plate.
Another curiosity is that near Earth’s surface the electric field has a magnitude of 150 N/C and
is directed downward, behaving this way as an a big electron.
34
3.1.1 Field from a continuous charge distribution
First of all we must know how to calculate the amount of a charge in a volume. For that it is
important to know the volume charge density rhov (C/m
3).
Q =
∫
v
ρvdv
3.1.1.1 Cylinder
Q =
∫ z2
z1
∫ 2pi
0
∫ rho2
0
ρvρ dρdφdz
3.1.2 Gauss’s Law
∫
D · ds = Qenv
3.2 Voltage and Current
Electrical effects are attributed to both the separation of charge and charges in motion. In
circuit theory, the separation of charge creates an electric force (voltage), and the motion of charge
creates and electric fluid (current).
3.2.1 Voltage
Whenever positive or negative charges move in the direction of each other, energy is expended.
For either attracting or repelling, the change in the electrical potential energy is defined as being
equal to the negative of the work done.
∆U = −W
Imagine an electron (q0 = −1.6×10−19C) in the air. As the electric field of the Earth is directed
downward, the work for lifting this electron would be facilitated (W = q ~E · ~d), which has positive
signal in contrast with the new and weaker electrical potential energy of the single electron.
One physical fact is that the potential energy per unit charge at one specific point in an electric
field (static charges) has a unique value, we can define this ratio as electric potential V:
V =
U
q
However, what we define as voltage is the potential difference ∆V between two points i and f
in an electric field
∆V = Vf − Vi = ∆U
q
= −W
q
35
3.2.1.1 Electric Field and Voltage Relationship
Until now, we talked about the work done by the electric field. Suppose now, we move a particle
by applying an external force to it. The applied force does work Wappl on the charge while the
electric field does W on it. The kinetic energy change of the particle is
∆K = Ki −Kf = Wappl −W
If the particle is stationary before and after the move (”If it was standing, moved and then
standed again?”).
Wappl = −W
We can relate the work done by our applied force to the change in the potential energy of the
particle during the move
∆U = Wappl ⇒ ∆V = Wappl
q
So if we wish to move a charge q a distance d~l against to an electric field ~E, the force on q0
arising from the electric field is~F = q0 ~E
but the force we must apply must be opposite to the force associated with the field
~Fappl = −q0 ~E
The energy expended is
dWappl = ~Fappl ·~l = −q0 ~E · d~l
Wappl = − q0
∫ f
i
~E · d~l ⇒ ∆V = −
∫ f
i
~E · d~l
3.2.2 Current
The rate of positive charge flow is know as the electric current. Unity: Ampere (A).
i(t) =
dq(t)
dt
3.2.2.1 Current density
Current density is a measure of the density of an electric current. It is defined as a vector whose
magnitude is the electric current per cross-sectional area. Unity: (A/m2).
∆I = ~J ·∆~S I =
∫
S
~J · d~S
36
3.3 Power and Energy
The useful output of a system is often non-electrical, which is conventionally expressed in terms
of power and energy, not in voltage and current. Also, practical devices have limitations on the
amount of power they can handle.
3.3.1 Power
Time rate expending or absorbing energy.
p(t) =
dw(t)
dt
=
dw(t)
dq(t)
dq(t)
dt
= v(t)i(t)
Consider
v(t) = Vp coswt i(t) = Ip cos(wt− φ)
p(t) = v(t)i(t) = VpIp coswt cos(wt− φ)
= VpIp
cos(2wt− φ) + cos(φ)
2
= VrmsIrms [cos(2wt− φ) + cosφ]
= VrmsIrms [cos 2wt cosφ+ sin 2wt sinφ+ cosφ]
= VrmsIrms cosφ(1 + cos 2wt) + VrmsIrms sinφ sin 2wt
= P (1 + cos 2wt) +Q sin 2wt
3.3.1.1 Average or Active Power
The average power, P, is also called real power or active power. It corresponds to the effective
energy consumed by the load. Its unity is Watts(W ).
P =
1
T
∫ T
0
p(t) dt = VrmsIrms cosφ
3.3.1.2 Reactive Power
Q = VrmsIrms sinφ
Maximum value of the componentQ sin 2wt, from the total power expression, which is associated
with the energy storage elements (capacitors and inductors). Its unity is var.
37
3.3.1.3 Complex Power
To work in power calculations with phasors, rms values of Vˆ and Iˆ are the best choice. Using
peak values is also correct, but the two types may not be mixed in one problem.
Vˆ = Vrms∠0 Iˆ = Irms∠− φ
Sˆ = Vˆ Iˆ∗ = VrmsIrms∠φ
Sˆ = VrmsIrms cosφ+ VrmsIrms sinφ j = P +Qj
Sˆ = ZˆI2 Sˆ = V 2/Zˆ∗
3.3.1.4 Apparent Power
It corresponds the maximum oscillation of instantaneous power around the average power.
Quantity used in the specifications of electric equipments as generators and transformers. Its unity
is VA.
|Sˆ| = Pmax − P
|Sˆ| = VrmsIrms =
√
P 2 +Q2
3.3.1.5 Physical Meaning
To know if a machine is generating or absorbing power one can use the following convention:
Supposing the current in being introduced in the positive terminal, for P > 0 or Q > 0,the
respective power is being absorbed. If current is supposed to be going out, for P > 0 or Q > 0, the
respective power is being generated.
3.3.1.6 Power Factor
The power factor of an AC electrical power system is defined as the ratio of the real power
flowing to the load, to the apparent power in the circuit.
pf =
P
S
=
VrmsIrms cosφ
VrmsIrms
= cosφ
It helps to evaluate the performance of electric loads. In an electric power system, a load with a
low power factor draws more current than a load with a high power factor for the same amount of
useful power transferred. The higher currents increase the energy lost in the distribution system,
and require larger wires and other equipment.
3.3.1.7 Power Triangle
The graphic representation of power allows sorts of problems of laborious analytic solutions be
more easily resolved through the means of Euclidean geometry. The power triangle has four items
- the apparent power, real power, reactive power and the power factor angle. when S lies in the
first quadrant, we have an inductive load and a lagging pf. When S lies in the fourth quadrant,
the load is capacitive and the pf is leading. It is also possible for the complex power to lie in the
second or third quadrant. This requires that the load impedance have a negative resistance, which
is possible with active circuits.
38
φ
P
Q
S
Inductive Power Factor
φ
P
S
Q
Capacitive Power Factor
3.4 Electrical Components
3.4.1 Resistors
iR(t)
+ vR(t) −
vR(t) , R iR(t)
Zˆ =
Vˆ
Iˆ
=
RIˆ
Iˆ
= R
Resistance of a material
R , ρ l
A
ρ: electrical resistivity of the material;
l: lenght of the piece of the material;
A: cross-sectional area of the specimen.
Equivalent circuits
Series connection
R1 R2 R3
⇔
Req
Req = R1 +R2 +R3
Parallel connection
R3R1 R2 ⇔ Req
1
Req
=
1
R1
+
1
R2
+
1
R3
∆-to-Y and Y-to-∆
39
Rc
RaRb
a b
c
⇔
R1 R2
R3
a b
c
Rab =
Rc(Ra +Rb)
Rc +Ra +Rb
= R1 +R2
Rbc =
Ra(Rb +Rc)
Ra +Rb +Rc
= R2 +R3 Rca =
Rb(Rc +Ra)
Rb +Rc +Ra
= R1 +R3
⇔
R1 =
RbRc
Ra +Rb +Rc
R2 =
RcRa
Ra +Rb +Rc
R3 =
RaRb
Ra +Rb +Rc
Ra =
R1R2 +R2R3 +R3R1
R1
Rb =
R1R2 +R2R3 +R3R1
R2
Rc =
R1R2 +R2R3 +R3R1
R3
3.4.2 Capacitors
iC(t)
+ vC(t) −
iC(t) =
dq(t)
dt
= C
dvC(t)
dt
Zˆ =
Vˆ
Iˆ
=
Vˆ
jwCVˆ
=
1
jwC
= − j
wC
q(t): electric charge accumulated on the capacitor plates.
q(t) = Cv(t)
C: capacitance of a material
C = ε
A
d
ε: dielectric constant or static permittivity of the material between the plates;
A: area of overlap of the two plates;
d: distance between the plates.
40
Because the current is the derivative of the voltage, current will be 90◦ ahead the voltage.
v = sin(t) ⇒ i = C cos(t) = C sin(t+ pi/2)
Iˆ
Vˆ
Re
Im
φ
θ = φ+ 90°
Voltage and current phasors of a capacitor
3.4.3 Inductors
Inductance is the property of a given conductor configuration produce a magnetic field when
traversed by a current.
iL(t)
+ vL(t) −
vL(t) = L
diL(t)
dt
Zˆ =
Vˆ
Iˆ
=
jwLIˆ
Iˆ
= jwL
Because the voltage is the derivative of the current, current will be 90◦ behind the voltage.
i = sin(t) ⇒ v = L cos(t) = L sin(t+ pi/2)
Iˆ
Vˆ
Re
Im
φ
θ = φ− 90°
Voltage and current phasors of an inductor
41
3.5 Flux
In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component
of a vector field perpendicular to the surface at each point.
Φ =
∫ ∫
A
~V · d ~A
where ~V is a vector field and d ~A is the vector area of the surface A, directed as the surface
normal.
3.6 Magnetic Field
In everyday life, magnetic fields are most often encountered as a force created by permanent
magnets, which pull on ferromagnetic materials such as iron, cobalt, or nickel, and attract or repel
other magnets. The Earth produces its own magnetic field, which is important in navigation, and
it shields the Earth’s atmosphere from solar wind.
3.6.0.1 Sources
There are two ways to produce magnetic fields. One way is to use moving electrically charged
particles, such as a current in a wire, to make an electromagnet. The other way is by means of
elementary particles such as electrons which have an intrinsic magnetic field around them.
CASE 1
×
dl
id~l
rˆ
d ~B
P
r
θ
i ~B
~B
wire with current
out of the page
We mentally divide the wire into differential elements dl and then define for each element a
length vector d~l that has length dl and whose direction is the direction of the current in dl. It
is defined a current-length element id~l which produces the field d ~B, magnetic flux density, at P.
Unity: [T] Tesla.
d ~B , µ0
4pi
id~l × rˆ
r2
(Biot− Savart Law)
Thus we can calculate the net field ~B at P by summing, via integration, the contributions d ~B
from all the current-length elements. Symbol µ0 is a constant called permeability whose value is
defined to be exactly 4pi × 10−7 T · m/A for a free space.
Here a simple right-hand rule for finding the direction of the magnetic field set up by a current-
lenght element, such as a section of a long wire:
42
From this rule, one can see that the magnetic fielddistribution will depend on the wire config-
uration. Some examples:
x
y
-a a
P
θ1
θ2
φ
rˆ
id~l
⇔ B = µ0i
4piy
(sin θ1 + sin θ2)
rˆ id~l
P θ
R
⇔ dB = µ0
4pi
idl sin 90°
R2
=
µ0
4pi
iRdθ
R2
B =
∫ θ
0
dB =
µ0iθ
4piR
B =?
Just as Gauss’s Law for electric fields, a magnetic field which has symmetry in its distribution
can be easily calculated applying Ampere’s law.
×
d~l
d ~B
i1
i2
i3
⇔
∮
d ~B · d~l , µ0ienc = µ0(i1 − i2)
43
To apply Ampere’s law, we mentally divide the loop into differential vector elements d~l directed
along the tangent to the loop in the direction of the integration. d ~B is the magnetic field due to
the three currents located at the element dl.
Example
For an ideal solenoid (infinite length):
××××××××××××××××××××
i
d ~B a b
d c
h
∮
d ~B · d~l =
∫ b
a
d ~B · d~l +
∫ c
b
d ~B · d~l +
∫ d
c
d ~B · d~l +
∫ a
d
d ~B · d~l = Bh+ 0 + 0 + 0
Being n the number of turns per unit length of the solenoid, the net current encircled by the
determined Amperian loop is defined by
ienc = i(nh)
Bh = µ0inh ⇒ B = µ0in
CASE 2
Nature, Earth, magnets, atoms.
3.6.1 Induction
As we’ve seen before, one current can produce a magnetic field. However, one even more
surprising discovery was the reverse effect: a magnetic field can produce an electric field that can
drive a current. In the following it is shown two experiments that can illustrate better how this
phenomenon occurs.
Experiment 1
With a circuit without any source, a current suddenly appears in the circuit when a magnet
bar is moved toward or away the loop. The current disappears when the magnet stops.
44
If moving the magnet’s north pole toward the loop causes, say, clockwise current, then moving
the north pole away causes counter-clockwise current. Moving the south also causes current but in
reversed directions.
Experiment 2
With two conducting loops close to each other but not touching, if we close switch S to turn on
a current in the right-hand loop, a current suddenly and briefly appears in the left-hand circuit. If
we open the switch another sudden and brief induced current appears in the left-hand circuit but
in opposite direction.
The current produced in the loop is called an induced current; the work done per unit charge
to produce that current is called an induced emf (electromagnetic force).
Faraday’s Law
From this observations one can see that by changing the amount of magnetic field passing
through the loop, is induced an emf and a current in itself.
Suppose a loop enclosing an area S is placed in a magnetic field density ~B, then the magnetic
flux through the loop will be:
ΦB ,
∫
~B · d~S
E , −dΦB
dt
Lenz’s Law
The produced magnetic field in the loop opposes the magnetic field that induces the current.
45
Generation of alternating voltage
A spiral with constant area A and a constant magnetic field density B, produced by the magnets,
there will be a magnetic flux Φ
ΦB ,
∫
~B · d~S = BA cos θ
If this spiral rotates with an angular velocity w. (Water in hydroelectric plants or by wind in
wind turbines)
ΦB = BA cos(θ0 + wt)
By Faraday’s Law:
E , −dΦB
dt
= wBA sin(θ0 + wt) = Vm sin(θ0 + wt)
A Closer Look at Faraday’s Law
Let us place o cooper ring in an uniform magnetic field, filling a cylindrical volume of radius
R. Suppose the strength of this field is increased. By Faraday’s law, an induced emf and thus an
induced current will appear in the ring, also by Lenz’s law we can deduce that the direction of the
induced current in counter-clockwise.
As a conclusion, if there is a current in the cooper ring, an electric field must be present along
the ring because an electric field is needed to do the work of moving the conduction electrons.
This induced electric field ~E is just as real as an electric field produced by static charges; either
field will exert a force q0 ~E on a particle of charge q0. By this line of reasoning one can restate
Faraday’s law as: A changing magnetic field produces an electric field.
46
The striking feature of this statement is that the electric field is induced even if there is no
cooper ring. Thus the electric field would appear even if the changing magnetic field were in a
vacuum. If the magnetic field is decreasing with time, the electric field lines will still be concentric
circles, but they will have now the opposite direction.
So considering the particle of charge q0 moving around the circular path, the work W done on
it in one revolution by the induced electric field is defined by W = E q0, where E is the induced
emf.
W =
∮
~F · d~s = q0
∮
~E · d~s
E =
∮
~E · d~s = −dΦB
dt
Although it was seen that induced electric fields are produced not by static charges but by a
changing magnetic flux with both exerting electric forces on charged particles, there is an important
difference between them. The simplest evidence is that the field lines of induced electric fields form
closed loops, when field lines produced by static charges never do so but must start on positive
charges and end on negative ones as
Vf − Vi = −
∫ f
i
~E · d~s
In another sense, consider what happens to a charged particle that makes a journey around a
single path: It starts at a certain point and, on its return to the same point, has experienced an emf
E of, let say, 5V; that is, work of 5 J/C has been done on the particle, and thus the particle should
be at a point that is 5 V greater in potential. However that is impossible because the particle is
back at the same point, which cannot have two different values of potential. If i and f in the above
equation are at the same point , the path connecting them is a closed loop, Vi and Vf are identical
and the equation will reduce to ∮
~E · d~s = 0
which is not true when a changing magnetic flux is present, leading us to a contradiction. This
way, as a unique conclusion, one must see that electric potential has no meaning for electric
fields that are set up by changing magnetic fields.
47
3.6.2 Magnetic Force in a Particle
After knowing how a magnetic field can be created, it is interesting to study how this field exert
a force in an electric charge. With some experiments realized, one could mathematically describe
this phenomenon by
~F , q ~v × ~B ⇔ F = qvB sin θ
If the magnetic field is produced by an electric current in a squared coil with side L, this equation
can also be written in the following way:
~F = i ~L× ~B, dq = i dt ~v = d
~L
dt
3.6.3 Magnetic Circuit
Simple magnetic circuit:
The relationship between magnetic field intensity ~H and the magnetic flux density ~B is a
property of the material in which the field exists. In case of diamagnetism or paramagnetism,
constant core permeability
~B = µ ~H
The source of the magnetic field in the core is the ampere-turn product Ni, as known as
magnetomotive force (mmf) F and its relationship with the magnetic field intensity ~H is determined
by Ampere’s Law (the path of the mean core lenght being the Amperian loop)
~H
~H
~l
~l
×
×
×
×
i
i
i
i
...
F , Ni =
∮
~H · ~dl = Hl = lB
µ
=
l
µ
Φ
A
48
where we define now the core reluctance as
Rc =
l
µA
F = RcΦ ⇔ V = RI
3.6.4 Magnetically Coupled Coils
Mutually couple coils have the following configuration
i2i1
φ21
e2e1
+
−
+
−
The direction of current i2 is determined in order to produce an opposing magnetic flux related
to the coil that produces it. (use right hand rule); In practice, the winding directions are not visible.
Then it is used a dot convention where it is said where the current i1 enters and i2 leaves or it says
where is the + polarity on the coil.
Flux linkage: Resultant magnetic fluxdue to current flowing in a coil and is represented by
λ = Nφ = Li
In the magnetically coupled case, the flux linkage in the the coil 1 is the sum of the following
fluxes
λ1 = λ11 + λ12 = L11i1 + L12i2
The indices ‘mn’ means the flux linkage in coil ‘m’ due to coil ‘n’. Remembering that each coil
has an own component (leakage) and other that goes to the other.
λ11 = N1φ11 = L11i1 λ2l = N1φ21 = L21i1
where L11 is called self-inductance and L21 mutual inductance between the coils.
The voltage drop in coils will then follow the Faraday’s Law
e1 =
λ1
dt
= L11
i1
dt
+ L12
di2
dt
e2 =
λ2
dt
= L22
i1
dt
+ L22
di2
dt
Vˆ1 = (r1 + jwL11)I1 + (jwL12)I2 Vˆ2 = (jwL21)I1 + (r2 + jwL22)I2
An equivalent circuit, using two port circuits for a T model and a = N1/N2, is given by
49
I1 I2/a
+
Vˆ1
−
r1
wL1l a2wL2l
a2r2
+
aVˆ2
−
awL21
where the inductances in the superior part are the leakage inductances L1l and L2l.
L1l , L11 − aL21 L2l , L22 − L12/a
3.7 Electric Flux Density
This topic has to do with static electric fields and the effect of various insulating materials on
these fields.
Beginning as an experiment with a pair of concentric metallic spheres and shells of insulating
material to occupy the entire volume between them, one can do the following steps:
1. It is given a know positive charge to the inner sphere.
2. It is added the dielectric material between the spheres.
3. The outer sphere is discharged by connecting it to ground.
4. The negative induced charge on each hemisphere is measured.
It was found that the total charge on the outer sphere was equal in magnitude to the original
charge placed on the inner sphere and that this was true regardless of the dielectric material
separating the two spheres.
There is some sort of displacement from the inner sphere to the outer which is independent of
the medium, a flux that is referred as displacement flux or simply electric flux.
It is also showed that a larger positive charge on the inner sphere induced a correspondingly
larger negative charge on the outer sphere, leading to a direct proportionality between the electric
flux and the charge on the inner sphere.
The constant of proportionality is dependent on the system of units involved. If electric flux is
denoted by Ψ and the total charge on the inner sphere by Q, then for Faradays experiment
Ψ = Q
50
and the electric flux is measured in coulombs.
The paths of electric flux Ψ extending from the inner sphere to the outer sphere are indicated
by the symmetrically distributed streamlines drawn radially from one sphere to the other.
At the surface of the inner sphere, electric flux are produced by the charge Q distributed
uniformly over a surface having an area of 4pia2m2.
The density of the flux at this surface is Ψ/4pia2 C/m2 and this is the new quantity called
Electric flux density, sometimes described as lines per square meter, which is given the letter D,
originally chosen because of the alternate names of displacement flux density.
~D =
Q
4pir2m2
~ar
If we compare this result with the electric field intensity in free space we have
~E =
Q
4pi�0r2m2
~ar =
~D
�0
Although this relationship is applicable only to a vacuum, it is not restricted solely to the field
of a point charge, but for a general volume charge distribution in free space.
3.8 Maxwell’s Equations
Maxwell’s equations describe how electric and magnetic fields are generated and altered by each
other and by charges and currents.
51
Chapter 4
Power Systems
4.1 Wheatstone Bridge
The Wheatstone bridge can be used to measure resistance, capacitance, inductance, impedance
and other quantities, in many cases related to measuring the impact of some physical phenomenon
(such as force, temperature, pressure, etc.) indirectly.
+ Vo −
+
VEX
−
R1
R2
R3
R4
Vo =
(
R3
R3 +R4
− R1
R1 +R2
)
VEX
At the point of balance, Vo = 0,
R3
R3 +R4
− R1
R1 +R2
= 0
R3 +R4
R3
=
R1 +R2
R1
⇒ R1
R2
=
R3
R4
4.2 Techniques of Circuit Analysis
4.2.1 Two-port Circuits
Vi Vo
Ii Io
+
−
+
−−
Two-port
Network
52
[
Vi
Vo
]
=
[
z11 z12
z21 z22
] [
Ii
Io
] [
Ii
Io
]
=
[
y11 y12
y21 y22
] [
Vi
Vo
]
Which elements, also called open circuit parameters, are given by the following expressions:
z11 =
Vi
Ii
∣∣∣∣
Io=0
z12 =
Vi
Io
∣∣∣∣
Ii=0
z21 =
Vo
Ii
∣∣∣∣
Io=0
z22 =
Vo
Io
∣∣∣∣
Ii=0
y11 =
Ii
Vi
∣∣∣∣
Vo=0
y12 =
Ii
Vo
∣∣∣∣
Vi=0
y21 =
Io
Vi
∣∣∣∣
Vo=0
y22 =
Io
Vo
∣∣∣∣
Vi=0
4.2.2 Superposition theorem
The response (voltage or current) in any branch of a linear circuit having more than one inde-
pendent source equals the algebraic sum of the responses caused by each independent source acting
alone.
During the analysis one must consider one independent source at a time while all other inde-
pendent sources are turned off. This implies that we replace every voltage source by 0 V (or a short
circuit), and every current source by 0 A (or an open circuit). This way we obtain a simpler and
more manageable circuit. Dependent sources are left intact because they are controlled by circuit
variables.
The theorem becomes important to ac circuits if it has sources operating at different frequencies.
The total response must be obtained by adding the individual responses in the time domain since
impedance depends on frequency.
4.2.3 Source transformation
Recalling that an equivalent circuit is one whose v-i characteristics are identical with the original
circuit, a source transformation is the process of replacing a voltage source vS in series with an
impedance Zˆ by a current source iS in parallel with an impedance Zˆ, or vice versa.
−
+
VˆS
ZˆS
a
b
ZˆSIˆS
a
b
VˆS = ZˆS IˆS ⇔ IˆS = VˆS
ZˆS
4.2.4 Thevenin Equivalent Circuits
It often occurs that a particular element in a circuit is variable (load) while other elements are
fixed. A household outlet terminal, for example, is connected to different appliances all the time.
Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent
circuit consisting of a voltage source VTh in series with a resistor RTh, where VTh is the open-circuit
voltage at the terminals and RTh is the input or equivalent resistance at the terminals when the
independent sources are turned off.
53
If the circuit has sources operating at different frequencies, the Thevenin equivalent circuits must
be determined at each frequency. This leads to entirely different circuits, not only one equivalent.
−
+VTh
RTh
I a
+
b
−
V Load
To apply this idea in finding the Thevenin resistance, we need to consider two cases.
CASE 1: If the network has no dependent sources, we turn off all independent sources. RTh is
the input resistance of the network looking between terminals a and b.
CASE 2: If the network has dependent sources, we turn off all independent sources. As with
superposition, dependent sources are not to be turned off because they are controlled by circuit
variables. We apply a voltage source vo at terminals a and b and determine the resulting current io.
Then RTh = vo/io, as shown in the figure below. Alternatively, we may insert a current source io at
terminals a-b and find the terminal voltage vo. Again RTh = vo/io. Either of the two approaches
will give the same result. In either approach we may assume any value of vo and io. For example,
we may use vo = 1V or io = 1A, or even use unspecified values of vo or io.
a
io
b
−
+
voCircuit
a
+
vo
b
−
ioCircuit
RTh =
vo
io
54
4.3 Transformers
Essentially, a transformer consists of two or more windings coupled by mutual magneticflux.
The mutual flux will link the other winding, the secondary, and will induce a voltage in it whose
value will depend on the number of secondary turns as well as the magnitude of the mutual flux
and the frequency.
Explained by Amperes’s Law and magnetic circuit representation (considering an infinity per-
meability of the coil and the defined dot marking) we have the following expressions:
~H
~H
~l
~l
×
×
×
×
×
i1
i1
i1
i1
i1
i2
i2
i2
i2
i2
Hl = N1i1 −N2i2 = l
µA
Φ = 0, µ =∞
N1i1 = N2i2
N1
N2
=
i2
i1
=
p2/v2
p1/v1
=
v1
v2
, p1 = p2
4.3.1 Equivalent Circuit for Practical Transformers
What we’ve seen before were equivalent circuits for ideal transformers, which
1. Leakage resistance: the windings had no resistance due to the loss on cooper (conductor
material).
2. Leakage reactance: the core permeability µ was infinity but there is a leakage of field lines in
the coil represented by an inductance.
3. no hysteresis in the core, Foucault active losses are represented by a conductance.
55
4. had no core magnetization that is represented by susceptance. (soft iron can be easily mag-
netized and desmagnetized)
Iˆ1 Iˆ2
+
Vˆ1
−
R1 jX1
N2
N1
Iˆ2 jX2 R2
+
Vˆ2
−
Gc Bm
+
Eˆ2
−Iˆc Iˆm
N1 N2
From now on a practical tranformer can be represented by the resistance of winding 1 R1,
leakage reactance of winding 1 X1, resistance of winding 2 R2, leakage reactance of winding 2 X2.
With finite core permeability
RˆcΦˆ = N1Iˆ1 −N2Iˆ2 ⇒ Iˆ1 − N2
N1
Iˆ2 =
RˆcΦˆ
N1
e1 = N1
dφ
dt
⇔ Eˆ1 = jwN1Φˆ ⇒ Iˆ1 − N2
N1
Iˆ2 =
Rˆc
N1
E1
jwN1
= −j Rc
wN21
Eˆ1
Iˆ1 − N2
N1
Iˆ2 = −jBmE1
we have a magnetizing current Iˆm with susceptance Bm that if we also include the core less
current Ic with conductance Gc (due to hysteresis and eddy current losses) we have the exciting
current
Iˆ1 − N2
N1
Iˆ2 = Iˆe = Iˆc + Iˆm = (Gc − jBm)Eˆ1
4.3.2 Reflected Impedances
Iˆi Iˆo
+
Vˆi
−
jX2
R2
+
Eˆ2
−
N1 N2
⇔
N1 N2
(
N1
N2
)2
jX2
(
N1
N2
)2
R2
e1 = N1
dφ1
dt
e2 = N2
dφ1
dt
Eˆ1
Eˆ2
=
N1
N2
Sˆ1 = Sˆ2 ⇒ Eˆ1Iˆ∗1 = Eˆ2Iˆ∗2 ⇒ N1Iˆ1
∗
= N2Iˆ2
∗
56
Eˆ1 =
N1
N2
Eˆ2 Iˆ1 =
N2
N1
Iˆ2 ⇒ Eˆ1
Iˆ1
=
(
N1
N2
)2 Eˆ2
Iˆ2
Z1 =
(
N1
N2
)2
Z2
Therefore, with a = N1/N2, the equivalent circuit of the transformer become
I1 I2
+
V1
−
r1 x1 a2x2 a
2r2
+
Eˆ2
−
Gc Bm
4.3.3 Calculus of Parameters
Parameters from the transformer are calculated by short and open circuit tests. Short circuit
test gives R and X, while open circuit Gc and jBm. Gc and jBm by default are already in the
primary of the circuit.
Example
Consider a single-phase transformer rated 480/120V, 20kVA yelding the following results:
Short-circuit test (Secondary shorted): V1 = 35V ; rated I1; P1 = 300W . Open-circuit test
(Primary open): rated V2; I2 = 12A; P2 = 200W .
I1
+
V1
−
R X
shunt admittance is neglected
4.3.4 Single phase-shifting Transformers
4.3.5 Autotransformers
Transformers that the primary and secondary windings are electrically and magnetically cou-
pled. They result in lower cost, smaller size and weight, but lose the loss of electrical isolation
between primary and secondary.
Ix = I1 + I2
E2
I1E1
I2
+
E1 + E2
−
57
4.4 Transmission lines
Equivalent Circuit
jXLRL
CL/2 CL/2
Line Inductance
When a phase has more then one conductor, the impedance of the line is considerably decreased.
[Material Prof. Washington, Chapter 3]
Line Capacitance
Voltage Regulation
Change in output voltage terminal when the load varies from one scenario without load for a
load level with particular power factor while maintaining the voltage on the input terminal constant.
4.5 Three-phase System
A three-phase system consists of a generator formed by three sources of same amplitude and
frequency but delayed one from each other by 120°. This generator is formed by a rotary magnet
(rotor) surrounded by fixed coils (estator). These coils are arranged physically separated by 120°
around the stator.
There are two possible configurations. The Y and ∆, respectively:
58
Van, Vbn e Vcn are the voltages between the phases (a,b e c) and the neutral terminal. They are
called phase voltages. Vab, Vbc e Vca are the voltages between the phases. They are called line
voltages.
Phase sequence
The order phase voltages happens is important in the circuit project. There are two possible
sequences for a three-phase system:
Positive Sequence (ABC, BCA, CAB)
Negative Sequence (ACB, CBA, BAC)
Relation between phase and line voltages (in the case of a positive sequence):
Where VP = |Van| = |Vbn| = |Vcn| and VL = |Vab| = |Vbc| = |Vca| for balanced voltages.
Balanced Three-phase Connections
Phase current is the current in each phase of the source or load. Line current is the current in
each line.
59
1. Y-Y
Line and phase current are the same
Iˆa =
Vˆan
ZY
“Per phase analysis”:
Vˆan ZY
Iˆa
2. Y-∆
Line currents:
Iˆa = IˆAB − IˆCA, Iˆb, Iˆc
supposing that in a positive sequence
IˆAB = IAB∠0°
Iˆa = IAB∠0°− IAB∠− 240° = IAB
√
3∠− 30°
“Per phase analysis”:
Vˆan
Z∆
3
Iˆa
60
3. ∆-∆
Line voltages:
Vˆab = VˆAB, Vˆbc, Vˆca
Line currents:
Iˆa = IˆAB − IˆCA, Iˆb, Iˆc
supposing that in a positive sequence
IˆAB = IAB∠0°
Iˆa = IAB∠0°− IAB∠− 240° = IAB
√
3∠− 30°
4. ∆-Y
Line voltages:
Vˆab = VˆAB, Vˆbc, Vˆca
We transform the source connection from ∆ to Y:
61
Vˆan =
Vab√
3
∠− 30°, Vˆan, Vˆcn
“Per phase analysis”
Vab√
3
∠− 30° ZY
Iˆa
Three-phase Power Measurement
Considering a balanced three-phase system of positive sequence ABC feeding a balanced load
impedance Z = |Z|∠θ° Ω /phase
P3φ = 3VP IP cosφ Q3φ = 3VP IP sinφ S3φ = 3VP IP
P3φ =
√
3VLIL cosφ Q3φ =
√
3VLILsinφ S3φ =
√
3VLIL
1. Three-wattmeter method
(a) Active Power
Z = |Z|∠θ°
Ω /phase
N
C
B
A
WA
WB
WC
±
±
±
±
±
±
62
PT = WA +WB +WC
WA = VanIa cos(θVˆan − θIˆa)
WB = VbnIb cos(θVˆbn − θIˆb), WC = VcnIc cos(θVˆcn − θIˆc)
(b) Reactive Power
Z = |Z|∠θ°
Ω /phase
N
C
B
A
WA
WB
WC
±
±
±
±
±
±
Q = (WA +WB +WC)/
√
3
WA = VbcIa cos(θVˆbc − θIˆa)
WB = VcaIb cos(θVˆca − θIˆb), WC = VabIc cos(θVˆab − θIˆc)
2. Two-wattmeter method
Z = |Z|∠θ°
Ω /phase
N
C
B
A
WA
WB
±
±
±
±
(a) Active Power
P = WA +WB
(b) Reactive Power
Q =
√
3(WA −WB)
WA = VacIa cos(θVˆac − θIˆa), WB = VbcIb cos(θVˆbc − θIˆb)
63
4.5.1 Unbalanced Three-phase Systems
A system which sources don’t have same amplitude or distinct phase shifts, or load impedances
are not equal.
4.5.2 Symmetrical Components
The method of symmetrical components is a powerful technique for analysing unbalanced three-
phase systems. Being defined a linear transformation from phase components to a new set of
components called symmetrical components.
The symmetrical component method is basically a modelling technique that permits systematic
analysis and design of three-phase systems. Decoupling a detailed three-phase network into three
simpler sequence networks reveals complicated phenomena in more simplistic terms. Sequence
network results can then be superposed to obtain three-phase network results.
Having a set of three-phase voltages Vˆa, Vˆb, and Vˆc, these phase voltages are resolved into the
following three sets of sequence components:
1. Zero-sequence components, consisting of three phasors with equal magnitudes and with zero
phase displacement:
Vˆa0 Vˆb0 Vˆc0 = Vˆ0
2. Positive-sequence components, consisting of three phasors with