Vector Algebra in Electrical Engineering

Vector algebra is a fundamental tool in electrical engineering, particularly in the analysis of circuits and protection systems. It allows for the representation of physical quantities, such as current or voltage, not only by their magnitude but also by their direction.

This article highlights the importance of vector representation in simplifying calculations involving complex quantities, while providing a clearer and more practical understanding of their applications.

Content of the Article

Definition of a Vector

A vector represents a quantity that has both a magnitude (or modulus) and a direction. For example, consider a vector OP with a magnitude $∣Z∣$ forming an angle $\theta$ with respect to a reference axis OX. This vector can be decomposed into two orthogonal components: a horizontal component $x$ and a vertical component $y$ .

Graphically, this is represented on a Cartesian plane where:

The modulus $∣Z∣$ is the length of the vector.
The angle $\theta$ is the argument, denoted $argZ$ .

The conventional notation for expressing a vector $Z$ is $∣Z∣∠\theta$ , which fully specifies its graphical representation or conversion to other forms.

Algebraic Representation

To use vectors in calculations, it is necessary to express them algebraically. The vector $Z$ , resulting from the vector addition of its components $x$ and $y$ , is written as:

Z = x + jy

Here, the operator $j$ indicates that the component $y$ is perpendicular to $x$ (a 90° counterclockwise rotation). In electrical nomenclature:

The OX axis is the “real” or “in-phase” axis.
The OY axis is the “imaginary” or “quadrature” axis.

The operator $j$ satisfies the algebraic properties: $j^2=-1$ .

This form $x + jy$ is called a complex quantity, representing the vector in rectangular coordinates.

From this, we deduce:

|Z| = \sqrt{x^2 + y^2}

And:

\theta = \arctan\left(\frac{y}{x}\right)= \tan^{-1}\left(\frac{y}{x}\right)

With:

$x = |Z| \cos(\theta)$
$y = |Z| \sin(\theta)$

Trigonometric and Exponential Forms

The vector can also be expressed trigonometrically:

Z = |Z| \big( \cos(\theta) + j \sin(\theta) \big)

Using Euler’s identities:

$\cos(\theta) = \frac{e^{j\theta} + e^{-j\theta}}{2}$
$\sin(\theta) = \frac{e^{j\theta} - e^{-j\theta}}{2j}$

We obtain the exponential form:

Z = |Z| e^{j\theta}

These forms facilitate manipulations, especially for time-varying quantities, such as in alternating current circuits.

Operations on Complex Numbers

Complex numbers can be manipulated algebraically or in polar/exponential form. Here are the main operations:

Addition and Subtraction

For two complex numbers $Z_1 = x_1 + jy_1$ and $Z_2 = x_2 + jy_2$ :

$Z_1 + Z_2 = (x_1 + x_2) + j(y_1 + y_2)$
$Z_1 - Z_2 = (x_1 - x_2) + j(y_1 - y_2)$

Components are added or subtracted separately (real with real, imaginary with imaginary).

Multiplication

Algebraic form:

Z_1 \cdot Z_2 = (x_1 + jy_1)(x_2 + jy_2) = (x_1 x_2 - y_1 y_2) + j(x_1 y_2 + x_2 y_1)

Polar/exponential form:

If $Z_1 = |Z_1| e^{j\theta_1}$ and $Z_2 = |Z_2| e^{j\theta_2}$ , then:

Z_1 \cdot Z_2 = |Z_1||Z_2| e^{j(\theta_1 + \theta_2)}

In polar form, magnitudes are multiplied and arguments are added.

Division

Algebraic form:

$\frac{Z_1}{Z_2} = \frac{x_1 + jy_1}{x_2 + jy_2} = \frac{(x_1 + jy_1)(x_2 - jy_2)}{x_2^2 + y_2^2} = \frac{x_1 x_2 + y_1 y_2}{x_2^2 + y_2^2} + j \frac{y_1 x_2 - x_1 y_2}{x_2^2 + y_2^2}$

Polar/exponential form:

\frac{Z_1}{Z_2} = \frac{|Z_1|}{|Z_2|} e^{j(\theta_1 - \theta_2)}

In polar form, magnitudes are divided and arguments are subtracted.

Applications in Electricity

Calculation of the Zero Component $V_0$

In a three-phase system, the zero component $V_0$ represents the common part of the phase voltages, useful for analyzing imbalances or ground faults.

For a three-phase system with voltages $V_a$ , $V_b$ , and $V_c$ , the zero component is calculated by:

V_0 = \frac{V_a + V_b + V_c}{3}

$V_a$ , $V_b$ , $V_c$ : voltages of each phase, expressed in complex form if the phase angle is considered.
$V_0$ : zero component.

Suppose a balanced three-phase system with:

$V_a = 230 \angle 0^\circ = 230 e^{j 0}$
$V_b = 230 \angle -120^\circ = 230 e^{-j 120^\circ}$
$V_c = 230 \angle 120^\circ = 230 e^{j 120^\circ}$

Expanding with Euler ( $e^{j\theta}=\cos\theta + j\sin\theta$ ):

V_a + V_b + V_c = V\Big(\cos 0 + j\sin 0\Big) + V\Big(\cos(-120^\circ) + j\sin(-120^\circ)\Big) + V\Big(\cos 120^\circ + j\sin 120^\circ\Big) = V\Big(1 + \cos 120^\circ + \cos 120^\circ\Big) + jV\Big(0 + \sin(-120^\circ) + \sin(120^\circ)\Big)

With:

$\cos 120^\circ = -\tfrac{1}{2}$ , $\sin 120^\circ = +\tfrac{\sqrt{3}}{2}$ , $\sin(-120^\circ)=-\tfrac{\sqrt{3}}{2}$

Thus:

$V_a + V_b + V_c = V\big(1 - \tfrac{1}{2} - \tfrac{1}{2}\big) + jV\big(0 - \tfrac{\sqrt{3}}{2} + \tfrac{\sqrt{3}}{2}\big) = 0 \;\;\Rightarrow\;\; V_0 = 0$ .

The zero component is obtained by applying the formula:

V_0 = \frac{V_a + V_b + V_c}{3} = \frac{0}{3} = 0

In this case, a perfectly balanced system has a zero component of zero.

In the presence of an imbalance or ground fault, V₀ will be non-zero, allowing for the detection and analysis of these situations.

Conclusion

Vector algebra, as described in this article, transforms abstract concepts into practical tools for electrical analysis. It connects rectangular, polar, trigonometric, and exponential forms, offering flexibility for complex calculations.

For engineers in network protection, mastering these concepts is essential to limit the effects of disturbances and ensure the stability of electrical systems.