Phasors from the linear algebra perspective applied to RLC circuits

Phasors provide a simple way to analyze sinusoidally excited linear circuits. Solutions of those circuits would be undoable otherwise. Besides, measurement units in phasors are a technological resource that powers with precision the observation of the electric power system dynamic state. In the last technological research in electronics, through this units tension phasors and current phasors are obtained in a synchronized way. Since the analysis of complex circuits with resistors, inductors and capacitors for sinusoidal entry types is time consuming, the sinusoidal analysis by phasors is a simple way to analyze such circuits without solving the differential equations. This applies to the case of the sinusoidal entries to a given frequency and once the system is in a stable state. This analysis is shown in this article.


INTRODUCTION
In the first engineering classes, sometimes linear algebra (Grossman, 2005) is presented without applications in engineering of vector spaces. On the other hand, in these semesters, in the electricity and magnetism physics (Steinmetz, 1983), the concept of phasors is introduced, sometimes without a rigorous foundation.
In Araújo and Tonidandel (2013), it is said that the history of phasors starts in 1868 with the RLC circuits studies by Maxwell (1868), but that the phasor concept was introduced by Steinmetz in 1889 (Serway, 2009), In addition, phasor measurement units are a technological resource that enables very accurately the dynamic state of the electric power system (Lozano and Castro, 2012).
Phasors are widely used in the analysis of electrical circuits. For instance, in the electrical circuits book by Nilson and Riedel (2005), they said that a phasor is a complex number that provides information of amplitude and phase angle of a sinusoidal function. The concept of phasor is based on the identity of Euler that relates the exponential function with the trigonometrical function (Zhang et al., 2010).
In this paper, the space of the will be tackled first, then the space of complex numbers, later an example of RLC circuits will be shown; the article ends with impedance.

THE Acos(+) FUNCTION TYPE SPACE
The set of functions in With appropriate units, an element of the set could represent a sound similar to the sound of a flute (Dagle, 2010). The amplitude A represents the volume of the sound or how close the instrument is and the phase represents the direction in which the instrument is placed.
In principle, the set does not have the appearance of a vector space. However, using the following identity, another perspective of the same set is gained,  Figure 1).

The complex plane
The complex plane is the set that with the usual operations form a vector space, which has the canonical base {1, j}. This implies that the complex plane has dimension 2 and, therefore, is isomorphic with . Besides, with this canonical base, a complex number corresponds to the vector of coordinates , . Due to the fact that the set of the previous section is isomorphic with , then it is also isomorphic with set C and, therefore, the complex number can be represented by the function .

Application in RLC circuits
It is known that when a current i passes through a resistor (R), an inductor (L) and a capacitor (C), the following potential differences are generated, respectively, If we have a power source , then the respective voltages are: and .
According to Acos(+) function type space, the coordinates of each one of those functions in the space are: , which correspond to the following complex numbers: If the resistor, the inductor and the capacitor are in series, then the same current passes through every one of them and the total voltage is the sum of each one of the voltages (Figure 2).

Series Circuit R-L-C
As the isomorphisms between with and C preserve the sum, then the total voltage can also be represented in such spaces as the sum of partial voltages.

Impedance
Impedance is the relation between voltage and current (Maxwell, 1868). If we assume a current that varies in time like a function of space , which has a coordinates , where and are constant, then this current is (11) and, therefore, the respective voltages are: When writing the coordinates in of the voltages, we obtain: The previous expressions allow to represent the impedances as matrixes of (18) Next, we present how the relation would be between current and voltage in complex numbers. Given the following complex numbers that represent voltage, impedance and current, respectively. These quantities should relate in the following manner: These complex numbers can be represented in by the following coordinates: which can represent impedances also as matrices: Then, it allows to represent the impedances of the resistor, the inductor and the capacitor, which will represented in the following manner: . With the complex number notation, linear equations systems can be posed and solved in an analogous way to resistive equations systems, which simplifies the solution of differential equations produced by the inductors and capacitors for the concrete case of sinusoidal signals.

CONCLUSIONS
In this article, we present another way, from linear algebra, that do not require the Euler identity, to show the relationship between the sinusoidal function, complex numbers and plane . This approach allows the student to have another perspective of phasors, bettering her or his understanding of this theme.
The proposed path is an application of the isomorphism between vector spaces, which illustrates the importance of this concept.