Applications of Linear Algebra in Electrical Engineering

In Engineering, Linear algebra has an important role but most of people whom work in science and technology don’t have sufficient information about using it in field, in this report try somewhat introduce applications of linear algebra in electrical engineering, predominantly in electrical networks, communication and relevancy. Some other mathematical subjects that are significant for electrical engineering include statistics, and discrete mathematics, probability and stochastic processes.

Introduction

Linear algebra is indispensable to every sub-class of Electrical Engineer. It is commonly linked with vector spaces but is more simply a means to solving systems of linear equations. Linear algebra allows us to transform the system of equations into a matrix and using Gaussian Elimination to solve for each unknown. This can be done by hand or more expediently using a computer. The understanding of vector mathematics is important to the application of rotating magnetic fields. Allusion to real and apparent power can be represented as vector quantities mathematically. In order to solve complex systems of vectors, linear algebra is rather handy.

Electrical Networks

When dealing with electrical engineering and circuit analysis, matrices and linear algebra are used as a way of organizing and simplifying calculations. However, using Gaussian Elimination along with computers, engineers are able to efficiently calculate unknown values of extremely large and complex systems without performing hundreds of calculations and exhaustive bookkeeping of values. Today more than ever, electronics are an integral part of our everyday lives. They contribute to every aspect of our way of life from lighting the space around our work environments, to exploring uncharted territories. But behind each and every electrical appliance or device, no matter what task it was designed for, lies a vast system of electrical components that must function as a whole. Each component (resistors, capacitors, inductors, etc.) has specifications of their own, as does the final product that they are a part of, so engineers must design their devices to meet not only their intended purpose, but so that the individual components are within their tolerances

Vital to this is the analysis of currents and voltages throughout the electrical circuit..A simple electric circuit consists of voltage sources, wire with current running through it, and various types of resistors (including capacitors, inductors, etc) that cause a voltage drop in the circuit. In order to tell how much the voltage drops across a resistor, one must use Ohm’s law. This law simply states that V = IR, where V is voltage across the resistor (volts), I is the current through the resistor (amperes), and R is the resistance of the resistor (ohms). When analyzing an electric circuit, there are many techniques that may be used. One simple technique uses the idea of Kirchhoff’s voltage law (KVL). Kirchhoff’s voltage law states that: “The algebraic sum of the voltages around any closed path in a circuit is identically zero for all time”

Simply put, this means that around any path in a circuit, the sum of the voltage sources must equal the sum of the voltage drops. In order to use Kirchhoff’s voltage law, the method of mesh current analysis must be used to examine the circuit. A mesh is a single loop that doesn’t contain any other loops within it. Thus, the mesh current is the current that flows within each individual loop. The direction of current flow in each mesh is arbitrary, so it is easiest to choose the current flow in a clockwise direction.

Analyzing the circuit now involves finding out the current flow in each mesh given the values of the voltage sources and resistances. In order to analyze the circuit, Kirchhoff’s voltage law must be applied to each mesh. It is important to note that if a resistor is shared by two meshes, the voltage across this resistor is (current in mesh being analyzed – current in other mesh) x (resistance). For example, if we were analyzing mesh 1 below, the voltage across R2 would be (I1 – I2) x R2. After Putting these equations into matrices lets the engineer solve for each current by using the matrix equation RI = V. By solving the augmented matrix (reducing the matrix to reduced echelon form), values of I1, I2, and I3 can be found. The matrix equation can be very useful when large circuits with many loops are too be analyzed. Also, because of Kirchhoff’s voltage law, the matrix is consistent every time, so there will always be a solution. This is due to the fact that if there is any voltage drop in a mesh, there must be some sort of voltage source in the loop (in order to satisfy Kirchhoff’s voltage law).

Communication

The distance in signal space is measure by calculating the norm: Norm between two signals || ( ) ( )|| , d x t y t x y (3) We refer to the norm between two signals as the Euclidean distance between two signals. This is what Fourier transform does projects a function onto a infinite number of orthogonal basis functions: ejw or ej2πnθ, and adds the results up (to get an equivalent representation in the frequency domain). CDMA codes are orthogonal, and projecting the composite received signal on each code helps extract the symbol transmitted on that code. Each code is an orthogonal basis vector, then signals sent are orthogonal. In figure 1 we show orthogonal projections in CDMA Linear channel codes (e.g.: Hamming, Reed-Solomon, BCH) can be viewed as k-dimensional vector sub-spaces of a larger N-dimensional space. Then k-data bits can therefore be protected with N-k parity bits. This is linear algebra in action: design an appropriate k-dimensional vector sub-space out of an N-dimensional vector space.

Determinant – criterion for space – time code design good code exploiting time diversity should maximize the minimum product distance between code words. Coding gain determined by min of determinant over code words. The error vector e of x is orthogonal to x. i.e. Inner product of them is zero, e T x = 0. From this idea use to: ea) Find a least-squares line that minimizes the sum of squared error (i.e. min T e) b) Detection under AWGN noise to get the test statistic. Matched filter is the filter that maximizes the signal-to-noise ratio it can be shown that it also minimizes the BER: it is a simple projection operation. Now show Matched Filter (MF) receiver as pictorially if the receiver were to sample this signal at the correct times, the resulting binary message would have a lot of bit errors. a) Consider the received signal as a vector r, and the transmitted signal vector as s b) MF ―projects‖ the r onto signal space spanned by s (―matches it).

Filtered signal can now be safely sampled by the receiver at the correct sampling instants, resulting in a correct interpretation of the binary message. Gaussian vector formula has a quadratic form term in its exponent: exp [-0.5 (x - T K -1 (x - )] )a) Similar to 1-variable gaussian: exp(-0.5(x - 2 / 2 ) b) K -1 (inverse covariance matrix) instead of 1/ 2 )) instead of (x -c) Quadratic form involving (x – 2(A TA) or (A *A) will appear often in communications math (MIMO). They will also appear in SVD (singular value decomposition). The pseudo inverse (A TA) -1 A T will appear in decorrelator receivers for MIMO.

In this paper, we provided a comprehensive overview of linear algebra, and its applications in electrical engineering. We explained the fundamental concepts and properties, and provided a detailed discussion about the methodologies on how to apply the linear algebra in communication as MIMO, coding, and other fields. In the future work we want to investigate roles of LA in other fields of electrical engineering as control engineering and power engineering.

References

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