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Appendix A Matrices and

A.I DEFINITIONS A rectangular array of ordered elements (, functions or just symbols) is known as a . The literal form of a matrix in general is written as

A=

We use boldface type to represent a matrix, and we enclose the array itself in square . The horizontal lines are called rows and the vertical lines are called columns. Each element is associated with its location in the matrix. Thus the element a;j is defined as the element located in the ith row and the jth column. Using this notation, we may also use the notation [a;jJmxn to identify a matrix of order m x n, i.e. a matrix having m rows (the of rows is given first) and n columns. Some frequently used matrices have special names. A matrix of one column but any number of rows is known as a column matrix or a column vector. Frequently, for such a matrix, only a single subscript is used for the elements of the array. Another type of matrix which is given a special name is one which contains only a single row. This is called a row matrix, or a row vector. A matrix which has the same nu.mber of rows and columns, i.e. m = n, is a of order (n x n) or just of order n. The main or prin• ciple of a square matrix consists of the elements all. a22 • .•.• ann. A square matrix in which all elements except those of the principal diagonal 688 Appendix A are zero is known as a . If, in , all elements of a diag• onal matrix are unity, the matrix is known as a or , U. If all elements of a matrix are zero, aij = 0, the matrix is called a , O. A subclass of a square matrix which is frequently encountered in circuit analysis is a . The elements of such a matrix satisfy the equality au = aji for all values of i and j, or in other words, this matrix is symmetrical about the main diagonal. For instance, the matrices

A-_[ 2 -1 IJ , -13 04] , -1 0 3 o 2 are of orders (2 x 3) for A, (3 x 3) for Band (3 x 1) for C. Matrix B is a square and symmetric matrix, and matrix C is a column matrix or vector. The matrix

is a unit matrix (also square, diagonal and symmetric). Finally, it should be noted that a matrix of the order (1 x 1), which consists of a single element, is simply refered to as a .

A.2 MATRIXEQUALITY Two matrixes are equal if and only if (1) they are of the same order, and (2) each element of one matrix is equal to its associated (placed in the row of the same number and the colunm of the same number) element in the other matrix. Thus, for two matrices, A and B, of the same order and with elements aij and bi}> respectively, if A = B, then all the elements have to be equal, i.e. aij = bij for all values of i andj. Thus, for example, for the matrices

a12 b12 [al1 al3 J and [bl1 bl3 J a21 a22 a23 b21 b22 b23 to be equal, we must have

al1 = bll, al2 = b12 , a13 = b13

a21 = b2!. a22 = b22 , a23 = b23 · Appendix A 689

On the other hand,

and

can never be equal no matter what the relationships between aij and bij are, as condition (1) above is not satisfied.

A.3 MATRIX The definition of equality of matrices allows us to express sets of equations in a compact form. Thus, for instance, ifF denotes a column matrix

alli! + a12i2 + a\3i3] F = [ a21~1 + a22~2 + a23~3 a31 11 + a32 12 + a33 13

(note that the sum of three addends in each row is actually one matrix element) and V denotes a vector

then the matrix

all il + a12i2 + a\3i3 ] [VI] [ a21~1 + a22~2 + a23~3 = V2 (A.la) a31 11 + a3212 + a33 13 V3 may be written simply as

F=V.

Moreover, either form of this matrix equation is equivalent to the three equations

allil + a22i2 + al3i3 = VI a21 il + a22i2 + a23 i3 = V2 (A.lb) a3li l + a32 i2 + a33 i3 = V3 690 Appendix A

A.4 ADDITION AND OF MATRICES If two matrices A and B are of the same order, i.e. have the same number of rows and the same number of columns, we may determine their sum by adding the corresponding elements. Thus if the elements of A are aij and those ofB are bij, then the elements of the resulting matrix Care

for all i andj , (A.2a) and (A.2b)

Clearly A + B = B + A for matrices. Note that if two matrices differ in their number of rows or their number of columns, their sum is not defined. As an example of , consider the following 1 2 5] [2 3 -1] [3 5 4] [3 4 6 + 7 -8 1 = 10 -4 7 .

Subtraction is similarly defined, i.e.

for all i andj, (A.3a) and C=A-B (A.3b)

For example, A-B=[ 1 20]_[2 1]=[-1 1 -1]. -1 3 4 3 2 -1 -4 1 5

A.5 BY A SCALAR A real or is referred to as a scalar to distinguish it from a matrix. The multiplication of a matrix by a scalar means that every element of the matrix is multiplied by the scalar. Thus, if k is a scalar and A is a matrix with elements ai}, the elements of the matrix kA are kai}:

kall ka12 kal n ka21 ka22 ka2n kA= (A.4) Appendix A 691

Consider the following example:

+Abl1 o 0] [all al3 ] b22 0 = a21 a23 ,

o b33 a31 a33 +A.b33 which in short form is A + A.B.

A.6 MULTIPLICATION OF MATRICES We first consider multiplication of a square matrix by a column matrix. It was shown (see Section A.3) that the sets of simultaneous equations Al b might be represented by the matrix equation A.la. We ca.n next represent the three currents in equation A.l as a column matrix

Then the matrix equation Ala can be written, by analogue to the scalar equation ai = v, as AI=V, (A5) where A is a square matrix of the of the equation, aij' In order to use the above notation, the mUltiplication , indicated as Al, must be defined in order to produce the correct members for equations A.lb. Therefore the in equation A6 must be interpreted as requiring that each element of the first row of the first matrix A must be mUltiplied by the corresponding element of the second, column matrix I, and the sum of these products should be taken as a first: element of the resulting, i.e. of the , matrix. Similarly, the second element of the resulting matrix will be the sum of the products of each element of the second row of matrix A with the corresponding element of the matrix I, and so forth. For the case where A is an nth-order square matrix and Y and X are column matrices with n rows, the matrix equation Y=AX (A.6) is simply defined by the n Yi = Laikxk, i = 1,2, ... , n (A 7) k=l 692 Appendix A

It is very important to emphasize that the multiplication of an (n x n)• order square matrix by a column matrix may be performed if and only if the column matrix has n elements, or, which is the same, n rows. In other words the number of columns of the first matrix, A, has to be the same as the number of rows of the second matrix, X. Note, that the product matrix, Y, is also the column matrix of the same order, n. Note also, that if these two numbers (columns in the first matrix and rows in the second matrix) are different, the product matrix AX does not exist. We may now generalize the multiplication rule for any two matrices. Thus, the multiplication of two matrices A and B is defined only if the number of columns of A is equal to the number of rows ofB. If A is of order (m x n) and B is of order (n x p) (such a pair of matrices is said to be conform• able for multiplication), then the product AB is a matrix C of order (m x p):

(A. 8)

The elements of C are found from the elements of A and B by multiplying the ith row elements of A and the correspondingjth column elements of B and summing these products to give cij:

n cij = ailblj + ai2b2j + ... + ainbnj = Laikbkj, (A.9) k=l where i = 1,2, ... , m,j = 1,2, ... , p. By contrast with , where ab = ba, matrix multiplica• tion is not commutative in general, i.e.

AB#BA usually, even when BA is defined, which means that their orders satisfy equation A.8. Thus the order of matrix is of fundamental importance. Matrix C in equation A.8 is the product of matrix A into matrix B, i.e. B is said to be pre-multiplied by A, or A is said to be post-multi• plied byB. We illustrate matrix multiplication by the following simple example:

It is worth considering the multiplication of two vectors. If a row-vector and a column vector each have n elements, we then obtain the product of the row-vector into the column-vector as Appendix A 693

YI

Y2 = XIYl + X2Y2 + ... + Xn)'n· (A. 10)

Yn

Note that the result of this multiplication is simply a scalar, and is some• times referred to as the scalar product of a row- and column-vector. For example,

However, the product of a column- and row-vector is undefined. Using the above meaning of a scalar product, the elements of matrix C, which is the product of two conformable matrices A and B, can be defined as the scalar products of the rows of A into columns ofB. Finally, we note that

UA=AU=A (A.ll) if A is a square matrix and both matrices A and U are of the same order.

A.7 OF A MATRIX The transpose of matrix A is AT and is formed by interchanging the rows and columns of A. This operation is know as transposition. For example, if

then

or in general, if A = [aij ] then

(A. 12) 694 Appendix A

A.S THE INVERSE OF A MATRIX In the of scalars (numbers) b/a = ba-I = a-Ib i.e. by a is exactly equivalent to multiplication by a-I, the reciprocal or inverse of a. Unless a is zero, there is always a unique reciprocal of a. The essential property of the inverse is that

(A. 13 a) It will be shown in this section that for certain square matrices we can define an inverse which has analogous properties, i.e. (A. 13 b) At this point, we remind the reader of some terms and properties relating to determinants which are needed for the treatment of the determination of an inverse matrix.

A.S.l Determinants and cofactors A is a scalar defined in terms of the elements of a square matrix. The determinant of square matrix A, being of order n, may be indicated by one of the forms:

£lA = detA = \A\ = (A.14)

and is referred to as a determinant of order n. For low-order (n = 2 or 3) matrices, the value of the determinant may be found directly by simple rules. Thus, for a second-order determinant, it is well known that

all detA= I (A. 15) a21 which is the product of the elements on the main diagonal minus the product of the elements on the other diagonal. A similar procedure may be applied to calculate the determinant of a third-order array. In this case the products of three elements are taken so that each of them belongs to different rows and columns. Choosing the product elements and their proper signs may be done in accordance with the following scheme. For convenience, the first and second columns are repeated at the right of the array: Appendix A 695

(A. 16)

Thus, detA = alla22a33 + a12a23a31 +a13a21a32 - a13a22a31 - alla23a32 - a12a21a33. (A. 17) For example, -1 -2 3 detA = 2 3 4 = 3 + 24 + 12 - (-27 - 8 + 4) = 70. -3 2-1 A more general procedure for evaluating the determinants of any order is by expanding determinant in terms of a row or column, which is called Laplaces' expansion. If such an expansion is made along the ith row of an array, it has the form n n det A = IAI = L aik Aik = L aik8.ik> (A. 18) 1=1 k=1 where all aik are the elements of A and all Aik (8.ik) are cofactors. These cofactors are formed by deleting the ith row and kth column of the array (so that the remaining elements form a determinant, called , M, which is of order one less than IAI) and prefixing the result by the multiplier (-1 )i+k, which predetermines the of the minor. Thus, for example, if i = 2, k = 3 and n = 4, we have a11 al2 ~13 al4 I a11 al2 al4 1- "tin - ~- -az4 A23 = (_1)5 I =- a31 a32 a34 = -·M23, (A. 19) a31 a32 4133 a34 I a41 a42 a44 a41 ~2 043 a44 and expanding the above fourth-order array determinant IAI by its second row yields (A. 20) By the use of equation A.18 the determinant of an nth order array can be expressed as a of (n - l)th order determinants. These, in turn, 696 Appendix A

may be expressed as a function of (n - 2)th order determinants, and the process may be continued until the value of the determinant is obtained.

A.S.2 We shall next introduce the adjugate matrix of square matrix A. This is formed by replacing each element aij of A by its cofactor Aij and transposing the result:

(A.21)

Finally, we define the inverse matrix of A as its adjugate matrix divided by the determinant of A. (JAJ i= 0):

-1 adjA 1 A = /AI = JAJ [Aji]. (A.22)

As an example, we illustrate the evaluation of the inverse of a given matrix

A = [ : -4 ~l' -1 2 1

The cofactors of JAJ are

-4 1 1 All = 2 1 = -6, A31 = 1 1 21 = 9, .... 1 -4 1

Continuing this process yields

-6 3 [ adjA = -: 5 -7 17~l' Then, the determinant JAJ may be obtained by expansion along the first row: JAJ = 3(-6) + 1(-6) + 2 x 6 = -12.

Therefore,

0.5 -0.25 75 [ -0. A -1 = a~t = 0.5 -0.42 -0.58 l. -0.5 0.58 1.42 Appendix A 697

A.9 METHODS OF SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS In applications to circuit analysis, we are concerned with a of simulta• neous equations formulated on the of one of Kirchhoff's laws (nodal analysis or mesh analysis). These equations have the matrix form AX=B, (A.23) where A may represent the conductance or resistance matrix, X may repre• sent a vector of unknown node voltages or mesh currents, and B is a column matrix of constants usually formulated from voltage or current sources.

A.9.1 Solution using the inverse matrix To solve the matrix equation A.23, we pre-multiply both sides of the equa• tion by the inverse matrix A-I:

A-lAX = A-lB.

With equation A.13b we have

or (A. 24) which is the solution of equation A.23. This may also be written in the form

X = adjA B. (A.25) det A For example, consider the following set of equations, which could arise from a simple circuit of two nodes and two meshes: il + i2 + i3 =0 3il +i2 +2i3 =2 2iJ + 2i2 + i3 = 6. We may write these as a single matrix equation

or AI =B. 698 Appendix A

The inverse of A is evaluated as ~]. -2

Therefore, in accordance with equation A.24, the solution is

A.9.2 Solution by Cramer's rule Let the column matrices of unknowns and the right-hand member constants be

XI

x= B= (A.26)

The solution of equation A.23 by using Cramer's rule results in

(A.27) where AA is the determinant of matrix A (equation A.14) and Ai is the determinant formed by replacing the ith column of a coefficients in IAI by the column of b-constants, i.e.

bl a12 aln all bl aln

b2 a22 a2n a21 b2 a2n AI= A2 =

bn an2 Gnn ani bn ann (A.28)

This method is sometimes referred to as the determinant solution of the simultaneous equations. Appendix A 699

With Cramer's rule and the method of expansion of a determinant by cofactors, the solution of equation A.27 can be obtained in the following form:

(A.29)

As an example of the use of Cramer's rule, consider the following equa- tions which arise from a specific three-loop network: 5il -2i2 - 3i3 = 10 -2il +4i2 - i3 = 4 -3il - i2 +6i3 =-2

In accordance with equation A.27 we write the solution i1 as 10 -2 -3 4 4-1

-2 -1 6 = 262 = 6.09A 5 -2 -3 43 -2 4-1 -3 -1 6

5 10 -3 -2 4-1

6 = 212 = 4.93A 43

5 -2 10 -2 4 4 -3 -1 -2 = 153 = 3.53A 43 43

A.9.3 Solution by the Gauss-Jordan method This is a slight variation of the procedure which allows the values of Xl> X2, •• _, Xn to be read off directly_ The elimination 700 Appendix A procedure is extended to reduce the array of coefficients A = [ajj] to the form o o

o o (A. 30)

o o o which can be extended even further to give a unit matrix U. (By the Gaussian elimination, the array of coefficients is reduced to the form of a 'triangular' matrix:

al1 al2 aln o aZ2 aZII (A.31)

o o

This gives the value Xn = b~n/a~ .. , and by back-substitution, the rest of the unknowns can be calculated.) We illustrate this procedure in the following example. Consider the equations

2Xl - 4X2 + X3 = -9

4Xl + 2X2 - 3X3 = 17 (A.32)

Xl - X2 + 5X3 = -11 The steps in the Gauss-Jordan procedure can be put into compact form by forming a joint matrix of equation coefficients and right member constants: 2 -4 r4 2 -3 ~~] (A.33) 1 -1 5 -11

As a first step we eliminate a21 and a31 from equation A.33 by the two following operations:

(1) multiplying the first row by 2 and subtracting the result from the second row; (2) multiplying the first row by 1/2 and subtracting the result from the third row. Appendix A 701

This gives (note that the above operations are shown in the symbolic form on the right-hand side near to the matrix) 2 -4 1 -9] [010 -5 35 Row2-2xRowl (A. 34) o 1 9/2 -13/2 Row 3 - (1/2) x Row :I

We now eliminate an by: (3) multiplying the second row of equation A.34 by 1110 and subtracting the result from the third row, 2 -4 1 -9] [o 10 -5 35 o 0 5 -10 Row 3 - (1/10) x Row 2 (At this point the Gaussian elimination is accomplished, so that (1) the third equation results in X3 = (-10)/5 = -2; (2) the second equation results in X2 = (1/10)(35 - 5(-2» = 2.5; (3) the third equation results in Xl (1/2)( -9 + 4 x 2.5 - 1(-2» = 1.5.) Following the Gauss-Jordan method, we continue the above operations to eliminate al3 and a23 as follows:

Row 1 - (1/5) x Row 3 [2 4 0 o 10 0 -7]25 Row 2 + 1 x Row 3 o 0 5 -10 and also a12, which results in

0 Row 1 + (4/10) x Row 2 o 10 o 253] [: 0 5 -10 and finally

0 0 1.5] Row 1 x (1/2) 0 2.5 Row 2 x (1/10) [~ 0 1 -2 Row 3 x (1/5) giving directly

Xl = 1.5, X2 = 2.5, X3 = -2. 702 Appendix A

A.9.4 Efficiency of methods for the solution of linear equations It is worth comparing the methods considered above for solving simulta• neous linear equations. For n simultaneous equations, using the method of solution involving determinants, i.e. according to Cramer's rule, about n2 x n! multiplications and/or divisions have to be performed (/ being regarded as taking negligible time compared with multiplications/ divisions). For two equations the number of multiplications/divisions is exactly eight, for three equations it is 51, and for 100 equations it is '" 10162 , whose solution would take a fast computer (which performes 5 x 106 operations per second) about -0.6 x 10148 years to perform. To solve the equations using an inverse matrix requires slightly fewer ('" (n2 - 2n) x n!) multiplica• tions/ divisions and will take almost the same time as with Cramer's rule. Thus, using these two methods for solving large sets of linear equations is not practicable. The Gauss elimination method requires [(n2/3) +n2 - n/3] multiplica• tions/ divisions. For two equations, this means six operations, for three equation 17 operations and for 100 equations 343 300 operations, which means a considerable saving compared with the determinant methods. Thus, for large sets of simultaneous equations it is best to use the Gauss elimination or Gauss-J ordan method. Moreover, using Cramer's rule or an inverse matrix it may happen that the determinant of the A matrix (det A) is small compared with the other determinants involved, so that as a result of dividing by small numbers we are likely to get problems of ill-conditioning.

A.I0 PARTITIONING OF MATRIXES If, in any matrix Z of any order, some rows and some columns are deleted, the remaining partial rows and columns of Z form a matrix of a lower order than Z. Such a matrix within a matrix Z is said to be a sub-matrix of Z. Thus, if

Zll Z12 Z13 Z14] Z21 Z22 Z23 Z24 Z= [ Z31 Z32 Z33 Z34

Z41 Z42 Z43 Z44 is a given matrix, then by deleting row 3 and columns 2 and 4, we have the sub-matrix

Zll Z13] [ Z21 Z23

Z41 Z43 Appendix A 703 or, by deleting rows 1 and 4 and columns I and 4, we have the sub-matrix

It is sometimes useful to partitition a given matrix into a number of sub• matrices in such a way that these sub-matrices are formed of adjacent partial rows and columns of the original matrix:

Zll ZI2 ZI3 I Z14 j Z = t_ZI1_ !2I _ Z13_:~2.1 _ Z31 Z32 Z33 I Z34

Here, the given matrix is partitioned into four sub-matrices, as indicated by the broken lines. We can then write

where it is obvious that

Zll ZI2 Z13 ] ZI= [ • Z3 = [Z31 Z32 Z33]. Z21 Z22 Z23

Of course a given matrix may be partitioned into sub-matrices in a number of ways.

A.IO.I Multiplication of partitioned matrixes Two matrices A and B are partitioned as

i.e. matrix A is partitioned according to the rows only and matrix B is parti• tioned according to the columns only. Then we can write

Thus, the process of multiplication, in terms of sub-matrices, may be carried out following the usual rule of row into column multiplication. (The verification of this rule is left to the reader as an exersise.) 704 Appendix A

A.tO.2 Use of partitioning Partitioning is sometimes used in carrying out the analysis of numerical matrices as it breaks down the formation of the row into column products into smaller, more manageable portions. More important are the cases in which the use of partitioned matrices allows theoretical analysis to be done in a compact form. At this stage let us consider an example of partitioning used in the solu• tion of equations when only some of the unknowns are of interest. Suppose that in the set of equations

ZI1 ZI2 1 Z13 ZI4 ZI5 il VI 1 Z22 1 Z23 ----Z21 _1- ______Z24 Z25 • i2 V2 1 Z31 z32 1 Z33 Z34 Z35 i3 = V3 1 Z41 Z42 1 Z43 Z44 Z45 i4 V4 1 ZSI Z52 1 Z53 Z54 Z55 i5 V5 only the input (i1) and output (i2) currents are of interest. For this reason we partition the column vector i into two and three rows, and accordingly we must also partition the matrixes Z and V in the same way, as show by the broken lines. Thus, we can now write the given matrix equation in terms of sub-matrices as

or (A.35a)

(A.35b) where

II=[il i2f, 12 = [ i3 i4 i5 f

VI = [VI v2f. V2 = [V3 V4 Vs f and also note that Za and Zd are square matrixes. Now we solve two simultaneous matrix equations (equation A.35) for II as follows: (1) Multiplying equation A.35b by Zd"1 yields

-IV -1 I I 2 = Zd 2 - Zd Zc 1 (A. 36) (note that all these multiplications are possible). Appendix A 705

(2) Substituing equation A.36 into equation A.35a and collecting terms, we have or, in short,

Zlll = V;.

(3) Finally, pre-multiplying these equations by Zjl we get the desired current II (i.e. i l and i2):

A.ll SOME BASIC RULES OF MATRIX ALGEBRA In this section we shall summarize some general rules of matrix algebra. 1. The ru1e of matrix algebra is that the operations with matrices, such as of equality, addition, subtraction and multiplication, can be performed only if the matrices are conformable for these operations, I.e. a. conformable for equality, addition and subtraction are matrices of the same order; b. conformable for multiplication are two matric€:s in which the number of columns of the multiplicand matrix is equal to the number of rows of the multiplier matrix. 2. Matrix addition is associative: A + (B + C) = (A + B) + c. 3. Matrix addition is commutative: A+B=B+A. 4. Multiplication by a scalar:

IXR = (IXrij]. 5. Matrix mu1tiplication is associative: (AB)C = A(BC). 6. Matrix multiplication is not (in general) commutative: AB =F BA. 7. Matrix multiplication is distributive: A(B+C) =AB+AC. 706 Appendix A

8. Any square matrix of any order, A, being multiplied by unit matrix, 1, of the same order equals itself: lA = Al = A. 9. If A and B are square matrices of the same order (say n x n) then AB and BA both exist and are of the order n x n, but will in general be different. 10. If A is a square matrix, then the product of n As exist and are given the nth power of A (which is of the same order as the order of A): AAA···A (n-factors) = An. It also follows that

and (A + B)2= (A + B)(A + B) = A2 + AB +BA + B2. 11. The transpose of a matrix

and The transpose of a product (AB)T = BTAT, and in a more general case (ABCD)T= DTCTBTAT. Note also that (ppT) T= ppT.

12. Any square matrix is associated with a numerical value (an algebraic function of its elements) which is called the determinant and denoted as detA = IAI =-= .:1A. Note that

13. The adjugate matrix of A is

adjA = [Aijf= [Aji] ,

where Aij are co-factors of aij (equation A.19). 14. The inverse matrix of A is

A-I =jAfa1 d'A J , A-1A = AA-i = 1, Appendix A 707 and

15. The inverse of the product (AB)-I= B-1A-I, 16. A solution of the matrix equation AX = B for X is given with help of the inverse matrix: X = A-lB. 17. A transformation from one set of variables (I,V) to another (I' ,V' ) in accordance with the transform matrix C,

I=CI', V=CV', yields a matrix transformation of ZI = Vas Z(CI)' = CV' or C-IZCI' = V', where C-IZC is referred to as a transform of Z. Then I' = (C-1ZC)-IV' = C-IZ-ICV'. Appendix B Complex numbers

In circuit studies involving sinusoidal steady-state analysis, we use numbers and variables which are complex . Using these numbers/variables we can greatly simplify the analysis. The rules for manipulating such complex quantities are quite different from the corresponding rules for manipulating real numbers. In this appendix we present a summary of such rules, which are defined as the algebra of complex numbers.

B.1 DEFINITIONS When encounter equations whose solutions cannot be expressed in terms of known numbers, they often define the solution with respect to new numbers and set about determining their main features. In the past, the failure of the existing rational numbers to provide a solution to x2 = 2 led to the extension of the number system to the irrational numbers. A similar problem happened when we tried to solve the equation x2 = -1, since no satisfies it (in other words, the square of no number, positive or negative, can give a ). To be able to solve the above equation, the sixteenth-century Italian Girolamo Cardano introduced the number R, which being squared gives -1, i.e. obeys the equation x2 = -1. This, actually not real number, R, is custo• marily and unfortunately called an imaginary number, or imaginary unit or imaginary . In electrical , the j is used to define the imaginary operator*:

j=vCt". (B.I)

*In mathematics and other disciplines, the imaginary operator is designated by the symbol i, but in electrical engineering that might be confused with current. 710 Appendix B

We now define the complex number to be of the form

A = a+jb, (B.2) where a and b are real numbers. Thus, the complex number has two compo• nents or two parts: a real part, a, and an imaginary part, b, which are usually expressed as a=ReA (B.3) b=ImA

(Note that it is wrong to say that jb is the imaginary component of A). A complex number (equation B.2) may be represented on a rectangular coordinate , or a , by interpreting it as a point (a,b), where the horizontal coordinate is a and the vertical coordinate is b, as, for example, the number 3 + j2 is shown in Fig. B.1(b). (This is an extension of the idea of the graphical interpretation of the real numbers by a horizontal , where a number is associated with a unique point on the line, Fig. B.I(a).) It is worth noting that as soon as complex numbers are introduced, the real numbers may be regarded as a special, or particular, case of complex numbers having imaginary parts equal to zero, say, for example, 2 = 2+jO. Complex numbers may be represented in different forms. The form in equation B.2 is called the rectangular form of the complex number A, since it corresponds to its representation on a rectangular co• ordinate system. The complex number A may also be uniquely located in the complex plane by specifying its along a straight line from the origin and the

1m (imaginary axis)

j3

j2 ------, A = 3 +j2

jl

Re -2 -I o 2 3 4 -I o 2 3 4 (real axis) jt

j2 -

(a) (b) Figure B.l (a) The real . (b) The complex plane and rectangular form of a complex number. Appendix B 711

1m

jb - -- A= a +jb

-¥~ __L- ______~~ Re a Figure B.2 Polar form of a complex number. e which this line makes with the real axis, as shown in Fig. B.2. It is obvious that r = IAI = J a2 + b2 (B.4a)

e = tan-I~, (BAb) a where r or IAI is called the amplitude or , and e is called the argument or angle ofa complex number. Then, by using an angle sign L, we may represent the complex number as A = rLe = IAILe, (B.5) which is called the polar form, since it corresponds to polar coordinates. It is also obvious from Fig. B.2 that a = r cos e = IAI cos e (B.6) b = r sin e = IAI sin e. Then A = r cos e + jr sin e, or A = IAI cos e + jlAI sin e, (B.7) which may be denoted as a trigonometrical form of the complex number. Finally, using 's identity (also called Euler's formula),

ejO = cos e + j sin e, (B.8) we may easily obtain the exponential form of the complex number: A = rejIJ = IAle jO (B.9) (For readers who are interested in deriving Euler's formula we now show one of the ways of doing it. We start with the trigonomc!tric form of a complex number of unit magnitude f = cos e + j sin e. (B.IO) To eliminate the trigonometric quantities in equation B.IO, we differentiate it, and then noting that l/j = jl/ = -1 yields

~ = j(cos e + j sin e) = jf. 712 Appendix B

By separating the variables, dill = j dB, and integrating we have lnl =jB+K, (B.ll) where K is a constant of integration, which must, however, be zero, since for B = 0 in equation B.lO, I = 1 and In 1 = O. Therefore, we have lnl = jB, or

1= ejO. (B.12) Finally, comparing this and equation B.lO we get ejli = cos e + j sin e, which proves Eu1er's formula.) Note that the exponential form of a complex number is essentially the same as the polar form since both of them are represented by the same two quantities (the magnitude IAI and the angle B), and the only slight difference is in the symbolism, i.e. the form of writing. It is apparent, therefore, that transformation from rectangular to expo• nential form or from exponential form to rectangular form is basically the same as transformation between rectangular and polar form, i.e. by using equations B.4 and B.6. For example, the number A, shown in Fig. B.l(b) is given in polar / exponential form,

A = 3.6lL33.4° = 3.6lej33.4°, since

IAI = J3 2 + 22 = 3.61 e = tan-1 ~ = 33.4°.

A complex number having a negative imaginary part is given in polar form with a negative angle, A = 3 - j2 = 3.6lL - 33.4°, which is also evident from Fig. B.3. Note that, as can be seen from this figure, the result 360° - 33.4° = 326.6° can also be used for the angle of a complex number in this example. However, using the smaller angle when• ever possible for the representation of complex numbers in polar/exponen• tial forms is usually preferable. If the rectangu1ar form of the complex number has a negative real part, we may consider it as a negative number, thus avoiding greater than 90° in magnitude. For example, given A = -4 ±j3, we can write A = -(4±j3), Appendix B 713

1m j2 -

-j2 ------

Figure B.3 Polar form of a complex number having a negative, imaginary part. and then transform it to polar/exponential form:

A = -SL ± 36.9° = _Se±j36.9° The negative sign can then be removed from the complex number, if required, by adding +180° when the angle is negative and --180° when the angle is postive. Thus, in the above example we have A = Sej(-36.9°+1800) = Sej143.1o or A = Sej(36.9°-1800) = Se-j143.1o. (Note that using an electronic calculator to determine the angles by calcu• lating the inverse tangent always gives a wrong answer when the complex number lies in the second or third quadrant. Thus, tan-1[3/(-4)] gives -36.9°, and tan-1[-3/(-4)] gives +36.9°, which of course is wrong in both cases. However, using the above procedure avoids this kind of mistake. Calculators that provide rectangular-to-polar conversion give the correct angle in this mode of calculation.) Some useful relations for complex numbers are summarised in Table B.I (see at the end of Appendix B).

B.2 OPERATIONS WITH COMPLEX NUMBERS

(aJ Equality Two complex numbers, both represented in rectangular form, are equal if and only if (ill) their real parts are equal and their imaginary parts are equal. Thus, if A =a+jb and B = c+jd, then A = Biff a=c and b=d. (B. 13) 714 Appendix B

Indeed, if a + jb = c + jd then a - c = jed - b), which requires a = c and b = d. Otherwise we would have a real number equal to an imaginary number, which, of course, is impossible. In polar / exponential form, if

A = IAlej/lA and B = IBlej/lB then A = Biff IAI=IBI and k = 0, 1,2, 3, ... (B.l4) According to this rule, an equation with complex numbers may be divided into two equations with real numbers. As an example, if 2 + x + j(7 - 2x) = 4 + jy, then 2+x=4 7 - 2x = y, which results in x = 2 and y = 3.

(b) Addition and subtraction These two operations (along with multiplication and division) apply to complex numbers exactly as they do to real numbers. That is, to add or subtract two complex numbers we simply add or subtract their real and their imaginary parts, i.e. A + B = (a + c) + j(b + d) (B.1Sa) A - B = (a - c) + j(b - d). (B.1Sb) As an example, let A = 4 + j2 and B = 2 - j6. Then A + B = (4 + j2) + (2 - j6) = 6 - j4 A - B = (4 + j2) - (2 - j6) = 2 + j8 Addition and subtraction of complex numbers may also be done graphi• cally on the complex plane, where each complex number may be treated as a vector. The sum is obtained by completing the parallelogram, or by connecting the vectors in a head-to-tail manner, as shown in Fig. B.4. Subtraction is done by adding the subtracted vector turned ± 180°, Fig. B.4(a). Of course, the graphical calculation is less accurate than the numer• ical one, but the graphical sketch of the solution is often useful to check the numerical result. Note that addition and subtraction should be performed in the rectan• gular forms. Appendix B 715

1m 1m _ A+B ~ B

Re

r L -B (a) (b) Figure B.4 Graphical addition and subtraction of complex numbers: (a) by constructing a parallelogram; (b) in head-to-tail manner. (e) Multiplication The product of two complex numbers is also a complex number, and this may be performed in either the rectangular or the polar form. In rectangular form, two complex numbers can be multiplied as algebraic binominals: AB = (a + jb)(e + jd) = ae + jad + jbe + Pbd or AB = (ae - bd) + j(ad + be), (B.16) since l = (.;=T)2 = -1. For example, (4 + j2)(2 - j6) = (8 + 12) + j(4 - 24) = 20 - 520. Using the trigonometrical form of the complex numbers (equation B.7), we have

AB = (rl cos 01 + jrl sin 01)(r2 cos O2 + jr2 sin 02)

= r)r2[(cos 0) cos O2 - sin 01 sin O2) + j(sin 01 cos O2 + cos 01 sin 02)] = rlr2[cos(01 + O2) + j sin (0) + O2)]. This result may be interpreted as follows: (B.17) which gives the product of two complex numbers in polar form, i.e. we may multiply the numbers by multiplying their magnitudes and adding their angles. This result may also be obtained by using the exponential form of complex numbers 716 Appendix B

(according to the rule of multiplying powers having the same base). For example transforming A and B of the previous example to the polar form, we have A = 4.472L26.57° and B = 6.325L - 71.57°. Therefore, AB = 28.285L - 45° = 20 - j20. The product of two complex numbers may also be sketched in the complex plane, as show in Fig. B.5. Before defining the operation of division, we must define the conjugate of complex numbers.

(d) Conjugations The conjugate of a complex number A = a + jb, written A*, is found by replacing eachj by -j, i.e in rectangular form

A* = a - jb, (B.18a) and in the exponential and polar form A* = IAle-j8 = IAIL - 8. (B.18b) It is evident that the congugate of A* is the complex number by itself, in other words,

(A*)* = A. (B.19) A complex number and its conjugate are said to form a conjugate complex pair (Fig. B.6). The following results of operations with conjugate pairs may easily be obtained:

A + A* = (a + jb) + (a - jb) = 2a (B.20a)

A - A* = (a + jb) - (a - jb) = j2b (B.20b)

AA* = (a + jb)(a - jb) = if + b2 = IAI2. (B.20c)

1m AB

A -joI==--P---J...... j'------Re Ss SA Figure B.5 The product of two complex numbers on the complex plane. Appendix B 717

1m A

----JoE'---t----- Re

A* Figure B.6 A conjugate complex pair.

In other words, the sum of a complex number and its conjugate is a real number, which is equal to the double real part; the difference of a complex number and its conjugate is an imaginary number, which is equal to the doubled imaginary part; the product of a complex number and its conjugate is a real number, which is equal to the square of the magnitude of the complex number. Now we consider the division of two complex numbers.

(e) Division The quotient of two complex numbers in rectangular form results in a complex denominator. c=~= a+jb. B e+jd We may get rid of the complex number in the denominator and express the result as a simple complex number having one real and one imaginary part. To do that, we multiply the nominator and denominator by the conju• gate of the denominator C _ (a + jb)(e - jd) _ (ae + bd) + j(be - ad) - (e + jd)(e - jd) - e2 + d2 ' or (B.21) where the real part is p = (ae + bd)/(e2 + tf) and the imaginary part is q = (ae - ad)/(c2 + tf). The quotient of two complex numbers in exponential form is found as

jO, C - IAle _ ~ j(OI-02) (B.22) - IBlej02 - IBI e , or in polar form

(B.23) 718 Appendix B

For example, considering the complex numbers A = 4 + j2 and B = 2 - j6, we shall find their quotient in rectangular form as 4 + j2 8 - 12 . 4 + 24 . C = 2 _ j6 = 22 + 62 + J 22 + 62 = -0.1 + JO.7, and in polar form as

4.472L26.57° o· C = 6.325L _ 71.570 = 0.7070L98.14 = -0.1 + JO.7. Evidently, it is easier to multiply and divide complex numbers given in polar / exponential form. If the complex numbers are given in rectangular form they should first be transformed into polar form (which can be done quickly with the help of a calculator). However, for addition and subtrac• tion the reverse situation applies, i.e. complex numbers given in polar form must be transformed into rectangular form.

B.3 OF COMPLEX NUMBERS The logarithm of a complex number is found by expressing this number in exponential form, and also observing that actually e = e + k2n, where k is any . Then the natural logarithm of A is In A = In IAle j(9+k2>r) = In IAI + j(e + k2n), (B.24) i.e. the logarithm of a complex number is also a complex number. The value when k = 0 is known as the principal value of the logarithm. If the complex number is given in rectangular form, it should first be transformed into exponential form, and the angle must be expressed in radians. As an example, let us find the logarithm of the complex number A = 3 + j4. Thus, In A = In 5ejO.927 = In 5 + j(0.927 + k2n), with the principal value In A = 1.61 + jO.927. It is worth mentioning that by extending the system of numbers to complex numbers, we can also find the logarithm of negative numbers. Thus, In(-I) = In lej7t = In 1 + jn =jn. In a similar manner we can find the common logarithm of complex numbers. Thus, log A = log IAI + log ej9 = log IAI + j()(log e), or log A = log IAI + j0.4343 e, where log e = 0.4343. Appendix B 719

B.4 POWERS AND ROOTS OF COMPLEX NUMBERS Powers and roots of complex numbers are found by using the laws of expo• nents. Thus,

(B.25) i.e. to raise a complex number to any integral power we must raise its magnitude to that power and multiply the argument (angle) by the exponent of the power (n). For example, let us find the sixth power of the complex number A = 1.970 + jO.3473. Transforming it to the exponential form yields

As a second example, raising (1/2 - j.J3/2) to the sixth power we get 1, since the magnitude of this number is 1 and the argument is -60°, i.e. A6 = le-j60' x 6 = 1. Extracting the root of the number may be interpreted as raising it to a fraction l/n, i.e.

(B.26)

In other words, extracting the root of a complex number, being the inverse of raising a number to a power, is done by extracting that root of the magni• tude and by dividing the argument by the index of the root.. Note that in this procedure we also have to indicate the argument by its full value e+ k2n. For example, let us determine the of -j9, whose magnitude is 9 and the argument is L - 90° ± k360°. Thus,

(_j9)1/2 = .Jgej(-90'+k360')/2, which gives two different values of the root

-j45' 3 . 3 (1) 3 e =--J- for k = 0, 2, 4, ... ./2 ./2 (2) 3 -jl35' 3. 3 e =-./2+J./2 for k = 1, 3, 5, ...

It is useful to check these results. Multiplying the number (1) or (2) by itself, we have 720 Appendix B

Taking, in the same manner, the square root of 16 (which may be treated as a complex number 16 + jO), we have

jO vl6ei(o+k3600 )/2 = [4e = 4 for even k 4eil80 = -4 for odd k. As a next example, let us extract the cube root of 1.

j24o' le Ik=2 or

rl = cos 00 + j sin 00 = 1

120 0 •• 120 0 1 . .j3 r2 = cos + J sm = - 2 + J 2

0 •• 240 0 1. .j3 r3 cos 240 + J sm J -. = = - -2 - 2 Note that for k greater than 2, the root values repeat themselves. In Fig. B.7 these values are represented by the points AI, A2, A3• The AIA2A3 is an equilateral triangle inscribed in a circle of radius 1. For the next example, find the sixth root of -1. It is left to the reader to convince himself that the result, obtained graphically, is as shown in Fig. B.8. The points AI, A2 , ••• , A6 , which represent the root values, are vertices of a regular hexagon inscribed in a circle of radius 1. In accordance with this, the root values are .j3 . 1 .j3 . 1 rl =2+J 2; r3 = -2+J 2;

.j3 . I .j3 . 1 r4= -2-J 2; r5 = -j; r6 =2-J 2'

In each of the above examples the number of distinct root values is equal to the index of the root. In general, the nth root of any complex

1m

-+--~~h-----~lRe

Figure B.7 Graphical representation of the cube root of 1. Appendix B 721

-:t-J-----7>IE------r+_ Re

As Figure B.8 Graphical representation of the sixth root of -1. number has n distinct values. From equation B.26 it also follows that the n roots of some complex number and the n roots of the conjugate of that number are a pairwise conjugate.

B.5 ROTATING VECTOR When a complex number reiIJ is multiplied by the imaginary unit j (the magnitude of which is 1 and the argument 90°, i.e. 1e j90), the magnitude of the product remains equal to r. The argument, however, is increased by 90°, which means that the given vector is rotated through 90° without changing its length, as shown in Fig. B.9. The vector OA is rotated to OB, while mUltiplying this vector once again by j yields the of OB through another 90° to the position ~C. Moreover, multiplication of a given vector r by a factor ejrol results in the rotation of this factor around a circle of radius r in the counterclockwise direction, with an angular of co rad/s. In t seconds it has moved through an angle rot, as shown in Fig.B. 10. Its real part is the projection on the horizontal/real axis, given by r cos cot, and its imaginary part is the projection on the vertical/imaginary axis, given by r sin cot. Thus, the projections the cosine and waves as the vector rotates (Fig. B.l 0). Euler's formula in this case is written (for the , r = 1) as

re jrot = cos cot + j sin cot.

1m B

C A --~~--~~~--Re

Figure B.9 Vector rotation. 722 Appendix B

1m

OJ! 211

Figure B.IO Graphical illustration of rotating vector and consine and sine waves.

Some useful relationships for complex numbers are given in Table B.I.

Table B.l Some useful relationships for complex numbers Given form Result l -I ( -j)U) 1 I/j -j 1 ejO" or ILO° j e j90' or lL90° -1 e±jI80' or IL ± 1800 -j e-j9O' or lL - 900 Index

ABCD parameters of transmission Attenuation constant, 532 line, 540 Autotransformer, 263-267 Ac (alternating current), 57 step-down, 264 Ac steady-state analysis, 57 step-up, 264 circuit theorems, 113 Average power, 162 compensation (substitution) Additive connection, 207 theorem, 121 linearity principle, 123 Balanced polyphase generator, 299 mesh analysis, 108 Balanced three-phase circuits, 319 nodal analysis, 104 Balanced three-phase generator, substitution theorem, 121 309 superposition principle, 114 Balanced two phase system, 297 Thevevin and Norton theorems, 118 Bandwidth, 140-141 Active power, 162 Bell (decibel), 563 Admittance, 76, 80 Bergeron, L., 663 angle of, 80 Bergeron diagram, 663-666 complex, 76, 80, 89 Bewley, L. V., 661 imaginary part (susceptance) of, 80 Bilateral Laplace transform, 574 in , 81, 86 Branch of graph, 2 in series, 87 magnitude of, 80 Capacitance/capacitor, 76 matrix, 105 current-voltage relation, 76-77 Norton admittance of transformer energy storage in, 136 250 equivalent, 89 real part (conductance), 80 impedance, 77 Amplitude spectrum, 452 initial voltage, 614 Angle of complex number, 711 of transmission line, 529 Angular frequency, 391 parallel connection of, 87 Apparent power, 167 phasor relationship for, 76 Argument of complex number, 711 series connection of, 88 Argument of sinusoidal, 60 Characteristic equation of transmission Armature winding, 64 line, 532 Artificial neutral point, 350 Characteristic impedance of Associated reference direction, 14 transmission line, 532, 543 Asynchronous machines, 384 Characteristic parameters of see also induction machines transmission line, 543 724 Index

Circuit(s) Complex (current) response, 73 capacitive, 81 Complex exponential forcing function, coupled magnetically (mutually), 201 68 dual, 49 Complex forcing (voltage) function, 73 equivalent of coupled coils, 222-226 Complex frequency plane, 574 equivalent of transmission line, Complex numbers, 709 566-569 addition/subtraction of, 714 inductive, 81 argument! angle of, 711 ladder, 90 conjugate of, 715 mesh analysis of, 34, 108 definitions, 709 mutually (magnetically) coupled, 201 division of, 717 nodal-analysis of, 19, 104 exponential form of, 711 non-planar, 2 imaginary component, 710 phasor model of, 85 imaginary number, 709 planar, 2 imaginary operator, 709 polyphase, 296 imaginary part, 710 reactive, 161 logarithm of, 718 resonant, 132 magnitude of, 711 RLC-circuit, 70, 81 multiplication of, 715 three-phase unbalanced, 336 polar form of, 711 three-phase, 316 powers of, 719 three-phase balanced, 319 real part of, 710 Wheatstone bridge of, 91 rectangular form of, 710 with more than two coupled representation of sinusoids, 70 elements, 283-284 roots of, 720 Circuit analysis trigonometrical form of, 711 computer-aided, 49-55 Complex plane, 710 mesh analysis, 34, 108 Complex power, 168-170 nodal-analysis, 19, 104 Computer-aided circuit analysis, 49-55 with Fourier series, 477-479 Conductance, 22, 80 with Laplace transform, 608 common, 22 Circuit elements in parallel, 87 capacitor, 76 input, 124 dual,44-45 matrix, 21 inductor, 74 mutual,22 mutually coupled, 201 selfconductance, 29 passive, 17 transconductance, 29, 32, 124 resistor, 731 Conjugate of complex number, 715 Closed path, 2 Conjugate pair, 607, 716 (s) Conservation of complex power, coupling, 208 171-172 exponential damping, 143 Conservation of energy (Tellegen's Fourier, 448 theorem),41-42 sampling, 141, 143 Controlled (dependent) source, 26 transmission, 641, 644 Convergence region, 575 Co-factors, 694 Convolution theorem, 595-597 Coll(s),202 integral, 595 mutually coupled, 202-209 Copper losses, 257 Column matrix, 687 Core losses, 255 Index 725

Cosinusoidal function, 66 Laplace's expansion of, 695 Co-tree (of graphs), 5 using for equation solution, 699 Coupled elements Diagram(s) in parallel, 216-219 circle, 189-199 in series, 216-231 linear, 188-190 Coupling coefficient, 208 phasor,96 Cramer's rule of solution equations, potential, 99 698 Difference, potential, 101 Current , 609 alternating (ac), 57 Direct current, 57 direct (dc), 57 Displacement neutral voltage, 337 effective value of, 163 Dissipation factor, 141 frequency- representation of, Distortion factor of non-sinusoidal 72 waves, 514 lag, 76 Distortionless transmission line, lead, 77 548-550 leaking, 527 Distributed-parameter system mesh, 34, 108 (network), 527 phase, 327 Domain resonance, 151 time, 72, 573 r.m.s., 163 frequency, 72, 573 sinusoidal, 58-61 Dot convention, 204, 207 time-domain representation of, 72 Doubled-tuned circuit, 271 Current ratio of transformer, 246, 248 Double subscript notation, 298-299 Current source Dual circuits/networks, 44-48 current-controlled (CCCS), 26 exact, 46 dependent (controlled), 26 graphs, 46-49 'killing', 124 Dual elements, 45 transfer ratio, 124 Duality, 44 voltage-controlled (VCCS), 26 Duhamel's integral, 599-600 Current transformer, 269 Cycle, 60 Eddy-current losses, 256n Cycle per second, 60 Edison, Thomas A., 57 Cut set of graphs, 6 Effective (rms) value, 163-164 periodic function of, 497 Damping coefficient, 143 Electromotive (EMF), 65 Damping factor, 141 Element (see: Circuit element) Datum node, 9 EMF (electromotive force), 65 Dc (direct current), 57 Energy Decibel, 563 in capacitors, 136 .1-connection, 313, 331 conservation of (principle), 42 Delta connection, 313, 331 in inductors, 136. Delta (unit-impulse) function, 577 in mutual inductance, 235 .1-Y transformation, 93 Equation(s) Dependent (controlled) sources, 26 branch, 18 Determinants, 694, characteristic (of transmission line), co-factors of, 694 532 evaluation of, 695 differential, solution by Laplace expanding of, 695 transform, 609 726 Index

Equation(s) (contc!) Fortescue, L. Charles, 401 Kirchhotrs voltage, 13 Fourier coefficients, 448 Kirchhotrs current, 9 Fourier, Jean-Baptiste Joseph, 447n matrix, 697 Fourier series, 448 mesh,34 amplitude spectrum, 452, 473-476 nodal,19 circuit analysis with, 477-480 simultaneous, solution to, 697 coefficients, 448 uncoupled, 425 complex (also exponential) form of, using Gaussian elimination, solution 471-472 to, 699-700 effective/r.m.s. value of, 496-497 using determinants, solution to, 699 even-function (also Equivalence of circuits, 93 symmetry about vertical axis), Equivalent circuit(s) 456-457 II of coupled coils, 225 graphical method of determining, II of transmission line, 566--567 466-469 T of coupled coils, 224 half-wave symmetry (also symmetry T of transmission line, 566-567 about horizontal axis), 458 based on Norton theorem, 119 harmonics, 448 based on Thevevin theorem, 120 odd-function symmetry (also of core transformer, 256--257 symmetry about origin), 456, 458 of coupled coils, 222-226 phase spectrum, 452, 473-476 of linear transformer, 241 symmetry properties, 455 Euler's identity /formula, 68, 712 trigonometric form, 448 Evans, R.D., 401 Frequency, 60 Even symmetry, 456-457 angular, 391 Exponential forcing function, 68 complex, 574 Exponential form of complex number, cyclic, 60 711 domain, 72, 573 Exponentially damped sinusoid, in fundamental, 448 transmission line, 533-535 half-power, 141 radian, 61 Farady's law, 202 resonance, 134 Filter(s) , 13 7, 616 high capacitance, 489 Frequency shift property (Laplace high-inductive, 489 transform), 558 L-C filter, 492 Full-wave rectified sine wave, 449 Final-value theorem (Laplace Function transform), 594 complex exponential, 68 Flux cosinusoidal, 66 leakage, 203 even symmetry, 456-457 magnetic, 202 exponential, 577 mutual,203 forcing exponential, 68 Forced response, 619 half-wave symmetry, 458 Forcing function odd symmetry, 456, 458 complex, 73 periodic, 59 exponential, 68 ramp, 577-578 sinusoidal, 66 rectangular wave, 464 Form factor of non-sinusoidal wave, sampling (sinc), 473 512 sinusoidal, 66 Index 727

transfer, 597 Harmonics, 451 unit impulse, 577 in three phase systems, 518-525 unit step, 576 resonance due to, 494-496 Fundamental frequency, 448 spectra, 452 Fundamental harmonic, 477 Heaviside's expansion theorem, 604 Histeresis losses, 256n Gaussian elimination, 699-700 Gauss-Jordan method of solution transformer, 247 equations, 701 current ratio, 248 Generalized Ohm's Law, 14 voltage ratio, 248 Generator (a.c.), 64 Imaginary component, 72, 709 air gap, 64 Imaginary number, 66-67, 709 distribution factor, 66 Imaginary operator, 72, 709 EMF (electromotive force), 65 Imaginary part, 68 magnetizing () winding, 64 Impedances one-phase, 64 angle of, 71 polyphase, 296 characteristic (of transmission line), rotor, 64 532,543 three phase, 307 complex, 73-76 two-pole, 64 in parallel, 82, 87 Graph, I input of transformer, 243 branch,2 in series, 86 chord,5 magnitude of, 71 circuit-graph, 1 primary (input) of transformer, 243, 249 closed path, 2 reactive component of, 79 connected graph, 3 reflected, 243 co-tree branches, 5 resistive component of, 79 cut set, 6 scaling, 137,616 datum or reference node, 9 Th6vevin of transformer, 249 dual,46 Impulse response, 597 fundamental, 11 Impulse (source) function, 577 hinged,3 Impulse voltage waveform, 657 link,5 matrix, 7 loop, 2 Incident wave, 535 mesh, 2,11 Independence of equations, 9, 12 network-graph, I Inductance(s)/ Inductor(s) node,2,9 current-voltage relation, 74 non-planar graph, 2 energy storage in , 136 oriented graph, 2 impedance, 74 path,2 in parallel, 88 planar graph, 2 in series , 87 subgraph,4 initial current, 613 tree, 5 magnetizing, 256 twig, 5 matrix, 217 unconnected, 3 mutual,202 of transmission lines, 529 Half-power frequency, 141 phasor relationship, 74-75 Half-wave rectified sine wave, 449, 462 reciprocal inductance matrix, 218 Half-wave symmetry, 458 self, 204 728 Index

Induction machines, 384-391 initial-value theorem, 593 basic relations, 389-391 integration in frequency domain, circuit model, 390 590-591 performance, 387 inverse transform, 600-602 principle of, 384 linearity theorem, 578 squirrel cage, 387 non-zero initial conditions, 613 torque production, 385 one-side transform, 574 Initial-condition generators, 614 ramp function, 577 Initial conditions, 613 scaling in frequency domain, 589 Initial value theorem (Laplace sinusoidal function response, transform), 593 621-623 Input impedance of transformer, 243 steady-state response, 583, 619 Input impedance of transmission line, Th€:venin and Norton equivalent, 556-559 623-628 Instantaneous power, 160-161 time differentiation theorem, 579 Insulation in transmission lines, 527 time scaling, 592t Inverse Laplace transform, 600-602 time integration theorem, 581 Iron-core transformer, 253 time-shift theorem, 583 transient response, 619 KCL (Kirchhoffs current law), 7 unit impulse function of, 577 definition, 7 unit-step function of, 576 equations (linearly independent), zero initial conditions, 611 7,9 zero-state response, 583 phasor notation, 84--85 diagram, 661-663 Kirchhoffs laws, 7, 84 Lead, current-voltages, 77 Kirchhoffs voltage law, 10,84 Leading phase angle, 62, 174 KVL (Kirchhoffs voltage law), 10 Leakage flux, 203 definition, 10 Leaking current, 527 equations, 11 velocity, 535 phasor notation, 84--85 Line currents, 327-328 Line terminals, 298 Ladder network, 569-570 Line voltages, 311, 314 Lag, current-voltage, 76 Linear transformer, 240-244 Lagging phase angle, 62, 174 Linearity properties, 123-128 Laplace transform, 573-574 Linearity theorem (Laplace transform), basic theorems of, 578-583 578 circuit analysis with, 608-611 Line-to-ground fault, 430-435 convolution theorem, 595-597 Line-to-line fault, , 436-439 definition of, 574-575 Links in graphs, 5 differentiation in frequency domain, Locus of phasors, 188 589-590 Long transmission line, 534 Duhamel's integral, 599-600 Loop in graphs, 5 equating coefficients method, 602 Loop matrix, 10 of, 577 Losses final-value theorem, 594 copper, 257 frequency-shift property, 588 core, 255 Heaviside's expansion theorem, eddy-current, 256n 604-607 histeresis, 256n initial-condition generators, 614 Lossless transmission line, 550 Index 729

Magnetically /mutual coupled circuits, mesh impedance matrix, 109 201 mesh matrix equation, 36 transients in, 628-631 mesh matrix, 13 Magnetic flux, 202-206 mesh resistance matrix, 36 leakage, 203 mesh source matrix, 36 mutual, 203 mesh-equation, 36 Magnetizing winding, 64 mesh-resistance, 36 Magnetomotive force (MMF), 205, 235 of phasor circuits, 108-113 MMF (magnetomotive force), 205, 255 transform of mesh-currents, 35 Magnitude of complex number, 711 Mutual inductance, 202, 204 Magnitude scaling, 137, 616 dot convention, 204 Matrices reactance, 208 addition/subtraction of, 690, 705 Mutually coupled circuits, 201 adjugate, 696, 706 additive connection, 207, 213 branch-conductance, 21 subtractive connection, 207, 213 branch-resistance, 18 series connection of, 213 column matrix, 687 parallel connection of, 216 conformable, 692 equivalent of, 222-228 definitions, 687 II-equivalent of, 225 determinants of, 694, 706 T -equivalent of, 224 equality of, 688 resonant in, 271-282 equation of, 688 Gaussian elimination, 699 Natural response, 619 Gauss-Jordan method, 701 Negative phase , 311 incidence, 7 Neper, 562 inverse, 23, 36, 694, 707 Network(s) (see also circuits) loop, 10 II-equivalent, 225, 556 main diagonal of, 687 T-equivalent, 224, 556 mesh, 13 active, 624 multiplication of, 691, 705 dual,49 nodal-equation, 19 ladder, 569-570 partitioning of, 702 nonplanar,2 post-multiplied, 692 nonreciprocal pre-multiplied, 692 one-port, 160 row matrix, 687 passive, 624 scalar products, 693 planar, 2 source, 22 Network theorems square matrix, 687 for phasor circuits, 113-133 sub-matrix, 702 superposition principle, 114 symmetrical matrix, 23, 36, 688 Thevevin and Norton theorem, 118 transformation with, 707 Neutral conductor (wire), 320 transpose of, 693, 706 Neutral terminal, 298 unit matrix, 688 Nodal analysis zero matrix, 688 common conductance, 22 Maximum power transfer theorem, , 8 180-184 mutual conductance, 22 Mesh analysis node conductance matrix, 21 common resistance, 36 node source matrix, 20 mesh currents, 34 node voltage equations, 19 730 Index

Nodal analysis (contd) Passive networks, 624 of phas or circuits, 104-108 Passive sine convention, 14 transform of nodal-to branch- Path in graph, 2 voltages, 19 Peak factor of non-sinusoidal waves, Node reference, 9 516 Node voltage, 19 Per-phase method, 321 Nodes in graphs, 2 Period,59 Non-sinusoidal voltages and currents, Periodic function, 59 501 effective value, 497 average value, 500 Fourier series of, 448-452 deviation factor, 516 Laplace transform of, 587 distortion factor, 514 Permeance, 205 distortion power, 505 Phase equivalent reactive power, 511 angle, 60 equivalent sine waves of, 509 in phase, 62 form factor, 512 lagging, 62 high harmonics factor, 516 leading, 62 non-active power, 505 out of phase, 62 peak factor, 513 Phase constant (in transmission line), power due to, 501 532 power factor, 510 Phase current, 327 ripple factor, 516 Phase sequence, 311 Norton equivalent circuit, 119, 180, Phase spectrum, 452 623 Phase voltage, 296 Norton's theorem applied to phasor Phasor(s),67-68 circuit, 119 concept, 59, 66 currents, 71 Ohm's law, 14 diagrams, 96 in phasor notation, 84 model of circuit, 85 for generalized branch, 14 unit, 300, 309 One-port network, 160 voltages, 71 One-sided Laplace transform, 574 Phasor diagrams, 96 Open-circuit impedance of Planar circuit! network, 2 transmission line, 556 Plane, complex, 710 Open-conductor fault, 439-443 Polar form of complex number, 711 Operator a, 309-310 Polarity voltage reference, 10 Poles TI -circuit of transmission line, 566 complex, 607 TI-equivalent of magnetically coupled multiple, 605 coils, 224-225 simple, 604 Parallel connection Polyphase generator, 298-303 of capacitors, 87 line terminals, 298 of impedances, 82 neutral terminal, 298 of inductors, 88 power, 304 Parallel resonance, 149 delta connection, 299 Parameters of transmission line, Polyleg wye connection, 299 527-530 Polyphase systems, 295-295 Parseval's theorem, 502 Positive phase sequence, 311 Partial-fraction expansions, 600--601 Potential diagram, 99 Index 731

Potential difference, 111 Reactive power, 169-170 Potential zero node, 99 Real part (component) of complex Power number, 710 active, 162 Reciprocity principle, 131 apparent, 167 Rectangular form of complex number, average, 162 710 complex, 168-170 Rectangular pulses, 474 conservation of, 171 train of, 474 distortion, 505 Reference direction in terms of symmetrical components, associated, 14 431-432 current of, 7 instantaneous, 160-161 voltage of, 10 non-active, 505 Reference node, 9 polyphase, 304 Reflected impedance, 243 polyphase generator of, 304 Reflected waves in transmission line, reactive, 169-170 535 three-phase, 322-323 of waves in transmission total, 168 line, 660-668 transferred,234 Residues, 604 Power factor, 173 Resistance / resistor, 17, 73 correction of, 175-179 transresistance, 40 lagging, 174 common, 36 leading, 174 mutual, 36 Power measurements, 184-187 Resonance in three-phase systems, 349-359 bandwidth frequencies, 141 Power of forward travelling wave, 638 bandwidth, 141 Power of backward travelling wave, circuits, 274 638 condition, 134 Power transfer in coupled circuits, currents in parallel circuits, 149, 154 232-235 current in series RLC circuits, Power transfer, maximum, 180-184 133-134, 143 Primary winding of transformer, 240 current resonance, lSI Propagation constant, 532, 571 damping coefficient, 143 Proportional relation (principle), 126 damping factor, 141 Pulsating field, 374 dissipation factor, 141 Pulse(s) fractional deviation from, 272 rectangular, 474 frequency deviation, 147 train of rectangular, 474 frequency response, 139 triangular, 587-588 half-power frequencies, 141 in complex circuits, 151-158 Quadrature component, 164 in coupled circuits, 271-277 Quality factor, 135 magnitude curve, current/voltages, 143-145 Radian frequency, 61 parallel, 149-151 Ramp function, 577 partial resonance in coupled circuits, Reactance, 71 279 capacitive, 71 primary particular, 271 inductive, 71 quality factor, 135, 151, 153 mutual,208 series, 133 732 Index

Resonance (contd) Series connection scaled bandwidth, 141 of capacitors, 87 scaling, 13 7 of impedances, 86 secondary particular, 271 of inductors, 87 universal loci, 138 Series faults, 439 universal response curves of coupled Series resonance, 133 circuits, 274 Series RLC circuit, 70 voltage resonance, 135 Sequence Resonant circuits, 132 negative phase, 311 Resonant frequency, 134 positive phase, 311 Responses Short circuit impedance of complete, 608 transmission line, 556 complex, 73 Shunt faults, 436--439 forced,619 Sinc or sampling function, 473 imaginary, 72 Sine wave, 59--63 natural,619 full-wave rectified, 449 sinusoidal function to, 621-622 half-wave rectified, 449 sinusoidal steady-state, 583 Single phase circuit, 320 steady-state, 618-619 Sinusoidal excitation step-function to, 599-600 analyzing with complex numbers, time domain, 601 67-69 transient, 618-619 phasor representation, 66-67 zero-state, 583 Sinusoidal forcing functions, 58 Right-hand screw rule, 386 Sinusoidal function, 59-63 Ripple factor of non-sinusoidal waves, quadrature representation, 63 516 rectangular form, 63 R.m.s. value, 163 Sinusoid(s) characteristics, 59-63 Root-mean-square value (r.m.s.), Source(s) 163 additional, 30, 32, 40 Rotating magnetic field, 295, 371 controlled(dependent),26 in of three coils, 377-379 dependent(controlled),26 in group of two coils, 372-376 equivalent, current, 17 speed in radians per second, 379 equivalent ,voltage, 17 speed in rpm (revolutions per 'killing', 124 minute), 379 shifting technique, 14-17 Rotating unit radius-vector, 67 transformation, 15 Rotating vector, 721 Source-free (natural) response, 619 Row matrix/vector, 687 Spectrum Rpm (revolution per minute), 379 amplitude/magnitude, 452, 473 phase, 452, 473 s-plane, 618 Square wave, 464 Sampling, 466 Standing waves in transmission line, Sampling factor/coefficient, 141, 143 551-555 Scaling Steady-state response, 619 frequency, 137,616 sinusoidal, 583 magnitude, 137, 616 Steinmetz, Charles Proteus, 6611, 257n Secondary of transformer, 240 Strength of impulse function, 677 Secondary winding, 240 Subgraph,4 Self-inductance, 204 Substitution theorem, 121 Index 733

Superposition of powers, 502 phasor diagram, 394 Superposition principle, 114-118 power angle, 395 Surge arrester, 667 power diagram, 395 Susceptance, 80 torque production, 393 Symmetrical components Synchronous speed, 386,391,393 analysis equations, 404 analysis of unbalanced systems, T -circuit of transmission line, 566 431-442 T -equivalent of magnetically coupled impedances of sequence networks, coils,224-225 417 Tesla, Nicola, 384 matrix transformation of, 439--443 Tellegen's Theorem, 41 measurements of, 410 Thevevin equivalent circuit, 120, 180, mutual impedances of rotating 623 maclUnes,427--429 Thevevin impedance, 121 negative-sequence impedance, 413 Thevevin theorem applied to phasor negative-sequence system, 401 circuit, 118 operator matrix a, 403 Tree branch voltages, 11 positive-sequence impedance, 413 Tree of graphs, 5 positive-sequence system, 401 Three-phase connections, 316-319 power in terms of, 431--442 ,1-,1 connection, 331--335 principal of, 401--409 Y-,1 connection, 326---330 sequence admittance matrix, 422 Y-Y connection, 319--325 sequence admittance, 414 Three-phase generator, 307 sequence currents, 417 angular velocity, 307 sequence impedance matrix, 421 ,1-connected,313 sequence impedances, 411 distribution factor, 307 sequence mutual impedances, induced voltages, 307 421--422 line terminals, 311 sequence networks, 418--419 line voltages, 312, 314 sequence rule, 413 neutral terminal, 311 sequence voltages, 403 phase voltages, 311, 313 synthesis equations, 404 Y-connected,312 zero-sequence impedance, 413 Three-phase systems (circuits), zero-sequence system, 401 316-319 Symmetry properties, 455 balanced, 319 even function, 457 per-phase circuit, 320 half-wave, 458 per-phase method, 321 odd-function, 458 unbalanced, 336 Synchronous condenser, 397 Three-phase transformer(s), 360 Synchronous inductor, 397 phase angle displacement, 362-365 Synchronous machines, 391--400 rated primary-to-secondary voltage angular velocity, 391 ratio, 367-369 basic relations, 393 tertiary windings, 362 brushless, 392 Y-,1 connection, 363 circuit model, 394 Y -Y connection, 362 damper winding, 397 Three-wire system, 268 falls out of step, 398 Time differentiation theorem (Laplace motor operation, 396 transform), 579 performance, 392 Time domain, 72, 573 734 Index

Time domain response, 601 non-linear resistive termination, Time-integration theorem (Laplace 667-667 transform), 581 open-circuit termination, 651 Time scaling theorem (Laplace short-circuit termination, 651 transform), 592t successive reflection of waves, Time-shift theorem (Laplace 660-668 transform), 583 sum of delayed waves, 672 Topological properties of networks, 2 termination resistance Transformer(s), 239-240 Thevevin equivalent, 646 see also Three-phase transformers transients in cables, 682-685 as coupling element, 368 transients in ground rod, 677-681 complete circuit model, 256 transmission coefficient, 641, 644 copper losses, 257 wave formation, 639 core losses, 255 wave reflections, 648 current, 269 Transients in magnetically coupled equivalent circuit of, 241 circuits, 628 ideal, 247-248 Transient response of transmission iron core, 253 line, 633 linear behaviour of, 240-244 Transmission line efficiency, 562 linear circuit model, 253-257 Transmission line(s) model referred to primary, 258 backward travelling (also reflected) model referred to secondary, 258 wave, 535, 636 model,240 characteristic impedance of, 532, 543 phasor diagram of, 258 characteristic parameters of, 543- primary losses, 246 decrement in voltage of, 532 primary, 240 distortionless, 548-550 reflected impedance, 243 efficiency of, 562 secondary losses, 246 equivalent circuits of, 566-569 secondary, 240 forward travelling (also incident) three-phase, 318, 361 wave, 535, 636 turns ratio, 248 incident wave, 535 voltage potential, 269 input impedance of, 556, 557 Transient behaviour of transmission ladder network of. 569-571 lines long, 534 capacitance connected to junction, loss-less, 550--551 655 open-circuit impedance of, 556 capacitance termination, 659, 668 parameters, 528-530 connecting to load, 643 propagation constant of, 532, 571 connecting to voltage source, 640 reflected wave, 535 differential equations, 633 reflection coefficient, 538, 539 electric energy, 638 short-circuit impedance of, 556 equivalent circuit for arriving wave, standing waves, 551-555 648 surge impedance loading (SIL) of, junction of two lines, 655 561 Laplace transform analysis, 668 transmission (ABCD) parameters of, line with LG/CR parameters, 540 675-677 velocity of wave propagation, 535, magnetic energy, 638 546, natural termination, 650 Transmission parameters, 540 Index 735

Triangular pulse, 587-588 polarity convention, 10 Turns ratio, 248 positive (abc) sequence of, 311 Two-phase systems, 297 quadrature component, 164 balanced, 297 resonance, 135 Two-port network (transmission line, r.m.s.! effective value, 163 as),566 sinusoidal, 58 Two-sided Laplace transform, 574 time-domain representation, 72 Two-wattmeter method, 353 Voltage-controlled current source, 26 Voltage (potential) transformer, 268 Unbalanced connection, 336 Voltage source(s) A-connection, 344-3248 current-controlled (CCVS), 26 four-wire Y-Y connection, 336-340 dependent (controlled), 26 three-wire Y-connection, 341-343 'killing', 124 Unbalanced three-phase circuits, 336 voltage-controlled (VCVS), 26 see also Symmetrical components Uncoupled equations, 425 Wagner, C. F., 401 Unilateral Laplace transform, 574 Watthourmeter, 186 Unit coupled transformer (ideal), Wattmeters, 184 247-248 measuring with, 185-187 Unit impulse function, 577 measuring with in three-phase Unit phasor, 300, 307, 309 system, 349-356 Unit step function, 576 Wave equations, 634 Wave(s) VA (voIt-amperes)/KVA (kilovolt backward, 535, 636 amperes), 167 forward, 535, 636 VAR (volt-ampere reactive), 164 full-wave rectified, 449 Velocity of wave (surge) propagation, half-wave rectified, 449, 462 533-535 incident, in transmission line, 535 Voltage-current relationships for in transmission lines, 535, 636 passive elements, 73-77 reflected, in transmission line, 535 Voltage ratio of ideal transformer, sawtooth, 463 248 sine, 59-63 Voltage resonance, 135 square, 464 Voltage(s) standing, in transmission line, of, 305 551-555 displacement neutral, 337 Wheatstone bridge circuit, 91 double-subscript notation, 229 Wye-connection, 312, 317 frequency-domain representation, 72 Wye-delta transformation, 94 generation of, 64 lag, 76 Y -connection, 312, 317 lead,77 Y-A transformation, 94 line voltage, 301 Y -Y connections, three phase, 319 line-to-line, 301 Y-A / A-Y systems, three phase, 126 negative (acb) sequence of, 311 nodal analysis, 19, 104-108 Zero potential, 99 node, 19 Zero initial conditions, 611 phase voltage, 296 Zero-state response, 583, 624