Foundations of Electromagnetic Compatibility (eBook)

with Practical Applications

Bogdan Adamczyk (Autor)

eBook Download: EPUB

2017
John Wiley & Sons (Verlag)
9781119120803 (ISBN)

Infos

There is currently no single book that covers the mathematics, circuits, and electromagnetics backgrounds needed for the study of electromagnetic compatibility (EMC). This book aims to redress the balance by focusing on EMC and providing the background in all three disciplines. This background is necessary for many EMC practitioners who have been out of study for some time and who are attempting to follow and confidently utilize more advanced EMC texts.

The book is split into three parts: Part 1 is the refresher course in the underlying mathematics; Part 2 is the foundational chapters in electrical circuit theory; Part 3 is the heart of the book: electric and magnetic fields, waves, transmission lines and antennas. Each part of the book provides an independent area of study, yet each is the logical step to the next area, providing a comprehensive course through each topic. Practical EMC applications at the end of each chapter illustrate the applicability of the chapter topics. The Appendix reviews the fundamentals of EMC testing and measurements. 

About the Author
Bogdan Adamczyk is Professor of Engineering and the founder and director of the EMC Center at Grand Valley State University, Grand Rapids, USA. He is also the founder and principal educator of EMC Educational Services LLC, which specializes in EMC courses for industry. Professor Adamczyk's area of expertise is EMC education and EMC pre-compliance testing.

He is an iNARTE-certified EMC Master Design Engineer, a founding member and the chair of the IEEE EMC Chapter of West Michigan, and a member of the IEEE EMC Society Education Committee. He was a 2016 IEEE EMC Symposium Global University and Fundamentals of EMC instructor. This book has evolved from his participation at several IEEE EMC Symposia, EMC pre-compliance testing at the EMC Center, and his teaching of the Foundations of Electromagnetic Compatibility certificate courses for industry.

Textauszug

1
Matrix and Vector Algebra

Matrices and determinants are very powerful tools in circuit analysis and electromagnetics. Matrices are useful because they enable us to replace an array of many entries as a single symbol and perform operations in a compact symbolic form.

We begin this chapter by defining a matrix, followed by the algebraic operations and properties. We will conclude this chapter by showing practical EMC‐related applications of matrix algebra.

1.1 Basic Concepts and Operations

A matrix is a mathematical structure consisting of rows and columns of elements (often numbers or functions) enclosed in brackets (Kreyszig, 1999, p. 305).

For example,

(1.1)

The entries in matrix A are real numbers. Matrices L and C in Eq. (1.2) are the matrices containing per‐unit‐length inductances and capacitances, respectively, representing a crosstalk model of transmission lines (Paul, 2006, p. 567). (We will discuss the details of this model later in this chapter.)

(1.2)

We denote matrices by capital boldface letters. It is often convenient, especially when discussing matric operations and properties, to represent a matrix in terms of its general entry in brackets:

(1.3)

Here, A is an m × n matrix; that is, a matrix with m rows and n columns.

In the double‐subscript notation for the entries, the first subscript always denotes the row and the second the column in which the given entry stands. Thus a23 is the entry in the second row and third column.

If m = n, we call A an n × n square matrix. Square matrices are particularly important, as we shall see.

A matrix that has only one column is often called a column vector. For example,

(1.4)

Here, V and I are the column vectors representing the voltages and currents, respectively, associated with the crosstalk model of transmission lines (Paul, 2006, p. 566).

Equality of Matrices

We say that two matrices have the same size if they are both m × n.

Two matrices A = [aij] and B = [bij] are equal, written A = B, if they are of the same size and the corresponding entries are equal; that is, a11 = b11, a12 = b12, and so on. For example, let

(1.5)

Then A = B implies that a11 = 7, a12 = −4, a21 = 2, and a22 = 8.

Matrix Addition and Scalar Multiplication

Just like the matrix equality, matrix addition and scalar multiplication are intuitive concepts, for they follow the laws of numbers. (We point this out because matrix multiplication, to be defined shortly, is not an intuitive operation.)

Addition is defined for matrices of the same size. The sum of two matrices, A and B, written, A + B, is a matrix whose entries are obtained by adding the corresponding entries of A and B. That is,

(1.6)

The product of any matrix A and any scalar k, written kA, is the matrix obtained by multiplying each element of A by k. That is,

(1.7)

From the familiar laws for numbers, we obtain similar laws for matrix addition and scalar multiplication.

(1.8a)

(1.8b)

(1.8c)

(1.8d)

(1.8e)

(1.8f)

There is one more algebraic operation: the multiplication of matrices by matrices. Since this operation does not follow the familiar rule of number multiplication we devote a separate section to it.

1.2 Matrix Multiplication

Matrix multiplication means multiplying matrices by matrices. Recall: matrices are added by adding corresponding entries, as shown in Eq. (1.6). Matrix multiplication could be defined in a similar manner:

(1.9)

But it is not. Why? Because it is not useful.

The definition of multiplication seems artificial, but it is motivated by the use of matrices in solving the systems of equations.

Matrix Multiplication

If is an m × n matrix and is an n × p matrix, then the product of A and B, , is an m × p matrix defined by

(1.10)

Note that AB is defined only when the number of columns of A is the same as the number of rows of B. Therefore, while in some cases we can calculate the product AB, of matrix A by matrix B, the product BA, of matrix B by matrix A, may not be defined.

We also observe that the (i,j) entry in C is obtained by using the ith row of A and the jth column of B.

(1.11)

Example 1.1 Matrix multiplication

Example 1.2 Multiplication of a matrix and a vector

whereas is undefined.

It is important to note that unlike number multiplication, multiplication of two square matrices is not, in general, commutative. That is, in general, AB ≠ AB

Example 1.3 Multiplication of matrices in a reverse order

Using the matrices from Example 1.1, but multiplying them in a reverse order, we get

which differs from the result obtained in Example 1.1.

1.3 Special Matrices

The most important special matrices are the diagonal matrix, the identity matrix, and the inverse of a given matrix.

Diagonal Matrix

A diagonal matrix is a square matrix that can have non‐zero entries only on the main diagonal. Any entry above or below the main diagonal must be zero.

For example,

(1.12)

Identity Matrix

A diagonal matrix whose entries on the main diagonal are all 1 is called an identity matrix and is denoted by In or simply I.

For example,

(1.13)

The identity matrix has the following important property

(1.14)

where A and I are square matrices of the same size.

Also, for any vector b we have

(1.15)

where the identity matrix is of the appropriate size.

1.4 Matrices and Determinants

If we were to associate a single number with a square matrix, what would it be? The largest element, the sum of all elements, or maybe the product? It turns out that there is one very useful single number called the determinant.

For a 2 × 2 matrix, we can obtain its determinant using the following approach:

(1.16)

Note that we denote determinant by using bars (whereas we denote the matrices by using brackets).

Example 1.4 Determinant of a 2 × 2 matrix

The procedure for obtaining the determinant for a 3 × 3 matrix is a bit more involved.

Let the matrix A be specified as

(1.17)

Its determinant

(1.18)

can be obtained using the following procedure. Let’s create an augmented “determinant” by rewriting the first two rows underneath the original ones:

(1.19)

then the value of det A can be obtained by adding and subtracting the triples of numbers from the augmented determinant as follows:

(1.20)

Example 1.5 Determinant of a 3 × 3 matrix

Calculate determinant of a matrix A given by

Solution: 

Create and evaluate the augmented determinant.

Why do we need to know how to obtain a second‐ or third‐order determinant? Obviously, we could use a calculator or a software program to do that for us. There are numerous occasions when the software or a calculator would not be able to handle the calculations.

As we will later see, when discussing capacitive termination to a transmission line, we will need to obtain a symbolic solution in a proper form; even if we had access to a symbolic‐calculation software, its output, in most cases, would not be in a useful form.

When discussing Maxwell’s equations, we will need to evaluate a third‐order determinant whose entries are vectors, vector components, and differential operators. This can only be done by hand.

1.5 Inverse of a Matrix

An inverse of a square matrix A (when it exists) is another matrix of the same size, denoted...

Produkt-Details

• Erscheint lt. Verlag: 27.2.2017
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Nachrichtentechnik
• Schlagworte: beginning electromagnetic theory • Circuit Theory & Design • Electrical & Electronics Engineering • Electrical Circuit Theory • electrical engineering text • Electric and Magnetic fields • Electromagnetic Compatibility • Elektromagnetische Verträglichkeit • Elektrotechnik u. Elektronik • (EMC) • IEEE EMC • mathematics for electricity and electronics • Numerical Methods & Algorithms • Numerische Methoden u. Algorithmen • Schaltkreise - Theorie u. Entwurf • Schaltkreistechnik
• ISBN-13: 9781119120803 / 9781119120803