added power factor and correction subsection

problemexpres.tex

@@ -1,5 +1,5 @@
 \section{Electric power fundamentals}
-jIn general physics terms, power is defined as the rate at which energy is transferred (or transformed). Electric energy in particular, begins as electric potential energy – what we commonly refer to as voltage. When electrons flow through that potential energy, it turns into electric energy. In most useful circuits, that electric energy transforms into some other form of energy. Electric power is measured by combining both how much electric energy is transferred, and how fast that transfer happens.
+In general physics terms, power is defined as the rate at which energy is transferred (or transformed). Electric energy in particular, begins as electric potential energy – what we commonly refer to as voltage. When electrons flow through that potential energy, it turns into electric energy. In most useful circuits, that electric energy transforms into some other form of energy. Electric power is measured by combining both how much electric energy is transferred, and how fast that transfer happens.
 
 The electric power P is equal to the energy consumption E divided by the consumption time t
 $$P = \frac Et$$
@@ -33,7 +33,7 @@ $$A(t) = A_{max} \cdot sin(2 \pi f t)$$
 
 The term AC or to give it its full description of Alternating Current, generally refers to a time-varying waveform with the most common of all being called a \textbf{Sinusoid} better known as a \textbf{Sinusoidal Waveform}. Sinusoidal waveforms are more generally called by their short description as \textbf{Sine Waves}. Sine waves are by far one of the most important types of AC waveform used in electrical engineering.
 
-This means then that the AC waveform is a “time-dependent signal” with the most common type of time-dependant signal being that of the Periodic Waveform. The periodic or AC waveform is the resulting product of a rotating electrical generator. Generally, the shape of any periodic waveform can be generated using a fundamental frequency and superimposing it with harmonic signals of varying frequencies and amplitudes but that is out of the waveform fundamentals theory
+This means then that the AC waveform is a “time-dependent signal” with the most common type of time-dependant signal being that of the Periodic Waveform. The periodic or AC waveform is the resulting product of a rotating electrical generator. Generally, the shape of any periodic waveform can be generated using a fundamental frequency and superimposing it with harmonic signals of varying frequencies and amplitudes but that is out of the waveform fundamentals theory.
 
 Alternating voltages and currents can not be stored in batteries or cells like direct current (DC) can, it is much easier and cheaper to generate these quantities using alternators or waveform generators when they are needed. The type and shape of an AC waveform depends upon the generator or device producing them, but all AC waveforms consist of a zero voltage line that divides the waveform into two symmetrical halves. The main characteristics of an AC waveform are defined as:
 
@@ -63,7 +63,7 @@ The relation all these three quantities are in is defined as
 $$ P^2 + Q^2 = S^2 $$
 however, again, nothing in the real world is perfect, and this relation only applies for a perfectly \textbf{sinusoidal waveforms}!
 
-\subsection{Phasor and phase difference}
+\subsection{Phasor and phase shift}
 A phasor is a constant complex number representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. It is usually expressed in exponential form. Phasors are used in engineering to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one. The origin of the word phasor comes from phase + vector.
 
 Phasor is a vector that represents a sinusoidally varying quantity, as a current or voltage, by means of a line rotating about a point in a plane, the magnitude of the quantity being proportional to the length of the line and the phase of the quantity being equal to the angle between the line and a reference line.
@@ -76,10 +76,18 @@ Phasor is a vector that represents a sinusoidally varying quantity, as a current
 
 Considering the figure \ref{f:ph_diff}, the voltage waveform above starts at zero along the horizontal reference axis, but at that same instant of time the current waveform is still negative in value and does not cross this reference axis until 30\textdegree  later. Then there exists a Phase difference between the two waveforms as the current cross the horizontal reference axis reaching its maximum peak and zero values after the voltage waveform.
 
-As the two waveforms are no longer “in-phase”, they must therefore be “out-of-phase” by an amount determined by phi, $\varphi$ and in our example this is 30\textdegree. It can now be said, that the two waveforms are now 30\textdegree  out-of phase. The current waveform can also be said to be “lagging” behind the voltage waveform by the phase angle $\varphi$  \cite{maxfield2011electrical}.
+As the two waveforms are no longer \textit{in-phase}, they must therefore be \textit{out-of-phase} by an amount determined by phi, $\varphi$ and in our example this is 30\textdegree. It can now be said, that the two waveforms are now 30\textdegree  out-of phase. The current waveform can also be said to be \textit{lagging} behind the voltage waveform by the phase angle $\varphi$  \cite{maxfield2011electrical}. This angle represents the phase shift (also called phase difference) between two sinusoids.
+
+\subsection{Power factor and power factor correction}
+The power factor is just a specific name for a phase shift between the sinusoids of a current and voltage. So the figure \ref{f:ph_diff} in fact shows the power factor. However, it is not expressed in a plane angle, but rather as a dimensionless number between -1 and 1.
+
+The power factor is defined as $\frac{P}{S}$, as a ratio of the real power over the apparent power. If $\varphi$ is the phase angle between the current and voltage, then the power factor is equal to the cosine of the angle, $cos\,\varphi$:
+$$|P| = |S|\,\cdot\,cos\,\varphi$$
+If the power factor is 1, it means that current flows only through purely resistive components. This is the best possible outcome. A positive power factor indicates that the current flow is altered by a reactive components. The lower the factor, the higher the effect. A negative power factor means that the device, considered to be power load is in fact a power source (produces more power than consumes).
+
+Now why is power factor important? Every device with a power factor other than 1 returns some power back to the transmission line. Since the transmission lines does have some resistance, this returned power translates to some wasted power in a form of heat. Energetic companies want to minimise the power wasted in the transmission lines to increase their profit, so numerous laws are coming into effect to correct (increase) the power factor.
+
 
-\subsection{Power factor}
-also power factor correction
 
 
 \subsection{Electric power measurement}
@@ -122,7 +130,7 @@ If not handled with care, operating or manipulating with voltage can cause perma
 %some words about sampling too
 
 
-\subsection{Power measuring Integrated Circuits}
+\subsection{Power measuring integrated circuits}
 
 Although it is possible to construct a circuit out of discrete components that would measure the mentioned quantities, and such a solution would probably be the cheapest solution out there, it would be highly impractical due to multiple reasons.