Complex Numbers

Complex Numbers

The ability to manipulate complex numbers is very handy in circuit analysis and in electrical engineering in general. Complex numbers are particularly useful in the analysis of ac circuits. Again, although cal- culators and computer software packages are now available to manip- ulate complex numbers, it is still advisable for a student to be familiar with how to handle them by hand.

B.1 Representations of Complex Numbers

A complex number z may be written in rectangular form as

z ⫽ x ⫹ jy

(B.1)

where j ⫽ 1⫺ 1; x is the real part of z while y is the imaginary part of z; that is,

The complex plane looks like the The complex number z is shown plotted in the complex plane in

x⫽ Re(z), y ⫽ Im(z)

(B.2)

two-dimensional curvilinear coordinate

Fig. B.1. Since j ⫽ 1⫺ 1 ,

space, but it is not.

⫽jⴢj ⫽ ⫺j

Figure B.1

Graphical representation of a

complex number.

j n⫹ 4 ⫽j n

A second way of representing the complex number z is by speci- fying its magnitude r and the angle it makes with the real axis, as u Fig. B.1 shows. This is known as the polar form. It is given by

z⫽ 0z 0 l u⫽r l u

(B.4)

where

2 2 r ⫽ 2x ⫺ 1 ⫹y y , u ⫽ tan (B.5a)

or

x⫽r cos u , y ⫽ r sin u (B.5b)

that is,

z ⫽ x ⫹ jy ⫽ r l u⫽r cos u ⫹ jr sin u (B.6)

A-10

Appendix B

Complex Numbers

In converting from rectangular to polar form using Eq. (B.5), we must exercise care in determining the correct value of u . These are the four possibilities:

z ⫽ x ⫹ jy ,

u ⫽ tan 1 (1st Quadrant)

x z ⫽ ⫺x ⫹ jy ⫺ , u ⫽ 180⬚ ⫺ tan 1 y

(2nd Quadrant)

(B.7)

z ⫽ ⫺x ⫺ jy , u ⫽ 180⬚ ⫹ tan 1 (3rd Quadrant)

z ⫽ x ⫺ jy ,

u ⫽ 360⬚ ⫺ tan 1 (4th Quadrant)

x assuming that x and y are positive.

In the exponential form, z ⫽ re j u so The third way of representing the complex z is the exponential that dz 兾d u⫽ jre j u ⫽ jz.

form :

z ⫽ re j u (B.8)

This is almost the same as the polar form, because we use the same magnitude r and the angle u .

The three forms of representing a complex number are summa- rized as follows.

z ⫽ x ⫹ jy, (x ⫽ r cos u , y ⫽ r sin u )

Rectangular form ⫺ y

z⫽r

2 2 l 1 u , ar ⫽ 2x ⫹y , u ⫽ tan Polar form

xb

2 ⫺ z ⫽ re y , ar ⫽ 2x ⫹y 2 , u ⫽ tan 1 Exponential form xb

(B.9)

The first two forms are related by Eqs. (B.5) and (B.6). In Section B.3 we will derive Euler’s formula, which proves that the third form is also equivalent to the first two.

Example B.1

Express the following complex numbers in polar and exponential form: (a) (b) (c) (d) z 1 ⫽ 6 ⫹ j8, z 2 ⫽ 6 ⫺ j8, z 3 ⫽⫺ 6 ⫹ j8, z 4 ⫽⫺ 6 ⫺ j8.

Solution:

Notice that we have deliberately chosen these complex numbers to fall in the four quadrants, as shown in Fig. B.2.

(a) For z 1 ⫽ 6 ⫹ j8 (1st quadrant),

Hence, the polar form is 10 53.13⬚ and the exponential form is 10e j l 53.13⬚ .

(b) For z 2 ⫽ 6 ⫺ j8 (4th quadrant),

⫺ 8 360⬚ ⫺ tan 1 ⫽ 306.87⬚

10, u 2 ⫽

Appendix B

Complex Numbers

A-11

so that the polar form is 10 l 306.87⬚ and the exponential form is

Im

10e z . The angle u

j 306.87⬚

2 may also be taken as ⫺ 53.13⬚ , as shown in

Fig. B.2, so that the polar form becomes 10 l ⫺ 53.13⬚ and the

exponential form becomes 10e ⫺j 53.13⬚ .

(c) For 3 z

3 ⫽⫺ 6 ⫹ j8 (2nd quadrant),

Hence, the polar form is 10 l 126.87⬚ and the exponential form

is 10e 126.87⬚ .

−j 4 r

(d) For −j z

4 ⫽⫺ 6 ⫺ j8 (3rd quadrant),

6 Figure B.2

For Example B.1.

l 233.13⬚

so that the polar form is 10 and the exponential form

is 10e j 233.13⬚

Convert the following complex numbers to polar and exponential forms:

Practice Problem B.1

(a) (b) (c) (d) z 1 ⫽ 3 ⫺ j4, z 2 ⫽ 5 ⫹ j12, z 3 ⫽⫺ 3 ⫺ j9, z 4 ⫽⫺ 7 ⫹ j.

Answer: 67.38⬚ (a) , 5 l 306.9⬚, 5e j (b) 13 67.38⬚, 13e l j ,

(c) 9.487 251.6⬚, 9.487e j 251.6⬚ , (d) 7.071 171.9⬚, 7.071e j l 171.9⬚ l .

Convert the following complex numbers into rectangular form:

Example B.2

(a) 12 l ⫺ 60⬚ , (b) ⫺ 50 l 285⬚ , (c) 8e j 10⬚ , (d) 20e ⫺jp 兾3 .

Solution:

(a) Using Eq. (B.6),

12 l ⫺ 60⬚ ⫽ 12 cos(⫺60⬚) ⫹ j12 sin(⫺60⬚) ⫽ 6 ⫺ j10.39 Note that u⫽⫺ 60⬚ is the same as u⫽ 360⬚ ⫺ 60⬚ ⫽ 300⬚.

(b) We can write

⫺ 50 l 285⬚ ⫽ ⫺50 cos 285⬚ ⫺ j50 sin 285⬚ ⫽ ⫺12.94 ⫹ j48.3 (c) Similarly,

8e j 10⬚ ⫽ 8 cos 10⬚ ⫹ j8 sin 10⬚ ⫽ 7.878 ⫹ j1.389 (d) Finally, 20e ⫺jp 兾3 ⫽

20 cos(⫺p 兾3) ⫹ j20 sin(⫺p兾3) ⫽ 10 ⫺ j17.32

Find the rectangular form of the following complex numbers:

Practice Problem B.2

(a) ⫺ 8 l 210⬚ , (b) 40 l 305⬚ , (c) 10e ⫺j 30⬚ , (d) 50e j p 兾2 .

Answer: (a) 6.928 ⫹ j4 , (b)

22.94 ⫺ j32.77 , (c)

8.66 ⫺ j5 , (d) j 50 .

A-12

Appendix B

Complex Numbers

B.2 Mathematical Operations

We have used lightface notation for complex numbers—since they are

Two complex numbers z 1 ⫽x 1 ⫹ jy 1 and z 2 ⫽x 2 ⫹ jy 2 are equal if not time- or frequency-dependent—

and only if their real parts are equal and their imaginary parts are equal, whereas we use boldface notation for phasors.

x 1 ⫽x 2 , y 1 ⫽y 2 (B.10)

The complex conjugate of the complex number z ⫽ x ⫹ jy is

z ⫺ju * ⫽ x ⫺ jy ⫽ r l ⫺u ⫽ re

(B.11)

Thus, the complex conjugate of a complex number is found by replac- ing every j by . ⫺j

Given two complex numbers z 1 ⫽x 1 ⫹ jy 1 ⫽r 1 l u 1 and z 2 ⫽x 2 ⫹

jy 2 ⫽r 2 l u 2 , their sum is

z 1 ⫹z 2 ⫽ (x 1 ⫹x 2 ) ⫹ j( y 1 ⫹y 2 )

(B.12)

and their difference is

z 1 ⫺z 2 ⫽ (x 1 ⫺x 2 ) ⫹ j( y 1 ⫺y 2 )

(B.13)

While it is more convenient to perform addition and subtraction of complex numbers in rectangular form, the product and quotient of the two complex numbers are best done in polar or exponential form. For their product,

z 1 z 2 ⫽r 1 r 2 l u 1 ⫹u 2 (B.14)

Alternatively, using the rectangular form, z 1 z 2 ⫽ (x 1 ⫹ jy 1 )(x 2 ⫹ jy 2 )

(B.15)

⫽ (x 1 x 2 ⫺y 1 y 2 ) ⫹ j(x 1 y 2 ⫹x 2 y 1 )

For their quotient,

l r u 1 ⫺u 2 (B.16)

Alternatively, using the rectangular form, z 1 ⫽ x 1 ⫹ jy 1

(B.17)

z 2 x 2 ⫹ jy 2

We rationalize the denominator by multiplying both the numerator and

denominator by . z 2 *

z 1 ⫽ (x 1 ⫹ jy 1 )(x 2 ⫺ jy 2 ) ⫽ x 1 x 2 ⫹y 1 y 2 x 2 y 1 ⫺x 1 y 2

⫹j 2 (B.18)

Example B.3

If A⫽ 2 ⫹ j5, B ⫽ 4 ⫺ j6 , find: (a) A *(A ⫹ B) , (b) (A ⫹ B) 兾(A ⫺ B) .

Solution:

(a) If A⫽ 2 ⫹ j5 , then A * ⫽ 2 ⫺ j5 and A⫹B⫽ (2 ⫹ 4) ⫹ j(5 ⫺ 6) ⫽ 6 ⫺ j

so that

Appendix B

Complex Numbers

A-13

(b) Similarly, A⫺B⫽ (2 ⫺ 4) ⫹ j(5 ⫺ ⫺6) ⫽ ⫺2 ⫹ j11 Hence, A⫹B

6⫺j

(6 ⫺ j)(⫺2 ⫺ j11)

⫽ ⫽ A⫺B ⫺ 2 ⫹ j11 (⫺2 ⫹ j11)(⫺2 ⫺ j11)

Given that C⫽⫺ 3⫹j 7 and D⫽ 8 ⫹ j, calculate:

Practice Problem B.3

(a) (C ⫺ D*)(C ⫹ D*) , (b) D 2 兾C* , (c) 2CD 兾(C ⫹ D) . Answer: (a) , ⫺ 103 ⫺ j26 (b) ⫺ 5.19 ⫹ j 6.776, (c) 6.045 ⫹ j11.53.

Evaluate:

Example B.4

(2 ⫹ j5)(8e j 10⬚ )

j (3 ⫺ j4)*

(a) (b)

2 ⫹ j4 ⫹ 2 ⫺ 40⬚

(⫺1 ⫹ j6)(2 ⫹ j) l 2

Solution:

(a) Since there are terms in polar and exponential forms, it may be best to express all terms in polar form:

2 ⫹ j5 ⫽ 22 2 ⫹ 5 2 tan ⫺ l 1 5 兾2 ⫽ 5.385 l 68.2⬚

(2 ⫹ j5)(8e j 10⬚ ) ⫽ (5.385 l 68.2⬚)(8 l 10⬚) ⫽ 43.08 l 78.2⬚

2 ⫹ j4 ⫹ 2 l ⫺ 40⬚ ⫽ 2 ⫹ j4 ⫹ 2 cos(⫺40⬚) ⫹ j2 sin(⫺40⬚) ⫽ 3.532 ⫹ j2.714 ⫽ 4.454 l 37.54⬚

Thus,

(2 ⫹ j5)(8e j 10⬚ )

43.08 l 78.2⬚

⫽ 9.672 l 40.66⬚

2 ⫹ j4 ⫹ 2 l ⫺ 40⬚ 4.454 l 37.54⬚ (b) We can evaluate this in rectangular form, since all terms are in that

form. But j(3 ⫺ j4)* ⫽ j(3 ⫹ j4) ⫽ ⫺4 ⫹ j3

(2 ⫹ j) 2 ⫽ 4 ⫹ j4 ⫺ 1 ⫽ 3 ⫹ j4 (⫺1 ⫹ j6)(2 ⫹ j) 2 ⫽ (⫺1 ⫹ j6)(3 ⫹ j4) ⫽ ⫺3 ⫺ 4j ⫹ j18 ⫺ 24

⫽ ⫺27 ⫹ j14 Hence, j (3 ⫺ j4)*

⫺ 4 ⫹ j3

(⫺4 ⫹ j3)(⫺27 ⫺ j14)

(⫺1 ⫹ j6)(2 ⫹ j) 2

⫺ 27 ⫹ j14

108 ⫹ j56 ⫺ j81 ⫹ 42

⫽ 0.1622 ⫺ j0.027

A-14

Appendix B

Complex Numbers

Practice Problem B.4

Evaluate these complex fractions:

(15 ⫺ j7)(3 ⫹ j2)* * (a) (b)

6 l 30⬚ ⫹ j5 ⫺ 3

(4 ⫹ j6)*(3 l 70⬚) Answer: (a) (b) 3.387 l ⫺ 5.615⬚, 2.759 l ⫺ 287.6⬚.

⫺ 1 ⫹ j ⫹ 2e j

B.3 Euler’s Formula

Euler’s formula is an important result in complex variables. We derive it from the series expansion of e x , cos u , and sin u . We know that

e 1⫹x⫹

Replacing x by gives j u

3 5 u (B.21) 7

sin u⫽u⫺ ⫹

so that

u 2 u 3 u 4 u 5 cos u⫹j sin u⫽ 1 ⫹ ju ⫺ ⫺j ⫹

⫹j ⫺ p (B.22)

4! 5! Comparing Eqs. (B.20) and (B.22), we conclude that

e j u ⫽ cos u⫹j sin u (B.23)

This is known as Euler’s formula. The exponential form of represent- ing a complex number as in Eq. (B.8) is based on Euler’s formula. From Eq. (B.23), notice that

cos u⫽ Re(e j u ), sin u⫽ Im(e j u )

(B.24)

and that

j u cos 0e 2 0⫽2 u⫹ sin 2 u⫽ 1

Replacing by u ⫺u in Eq. (B.23) gives

e ⫺ju ⫽ cos u⫺j sin u (B.25)

Adding Eqs. (B.23) and (B.25) yields

cos u⫽ (e j u ⫹e ⫺ju )

(B.26)

Appendix B

Complex Numbers

A-15

Subtracting Eq. (B.25) from Eq. (B.23) yields

Useful Identities

The following identities are useful in dealing with complex numbers. If then z ⫽ x ⫹ jy ⫽ r l u ,

2 2 zz 2 *⫽x ⫹y ⫽r

(B.28)

j 2z ⫽ 2x ⫹ jy ⫽ 2re 兾2 u ⫽ 2r l u 兾2

(B.29)

nn

z n ⫽ (x ⫹ jy) ⫽r l n n u ⫽r jn u e n ⫽r (cos nu ⫹ j sin nu) (B.30)

1 1 z 1 兾n 兾n 兾n ⫽ (x ⫹ jy) ⫽r l u 兾n ⫹ 2 p k 兾n

(B.31)

k ⫽ 0, 1, 2, p,n⫺1

ln(re j u j ) ⫽ ln r ⫹ ln e u ⫽ ln r ⫹ ju ⫹ j2k p

(B.32)

(k ⫽ integer)

e ⫺jp 兾2 ⫽ ⫺j Re(e (a⫹j␻)t ) ⫽ Re(e at j␻t e )⫽e at cos ␻ t

(B.34)

Im(e

(a⫹j␻)t

j␻t

at

) ⫽ Im(e at e )⫽e sin ␻ t

If ,

A 4 ⫽ 6 ⫹ j8 find: (a) 1A, (b) A .

Example B.5

Solution:

(a) First, convert A to polar form:

r ⫽ 26 ⫹8 ⫽ 10, u ⫽ tan

6 ⫽ 53.13⬚, A ⫽ 10 l 53.13⬚

Then

1A ⫽ 110 l 53.13⬚兾2 ⫽ 3.162 l 26.56⬚ (b) Since A ⫽ 10 l 53.13⬚,

A 4 4 ⫽r 4 l 4u ⫽ 10 l 4 ⫻ 53.13⬚ ⫽ 10,000 l 212.52⬚

If A⫽ 3 ⫺ j4 , find: (a) A 1 兾3 (3 roots), and (b) ln A.

Practice Problem B.5

Answer: (a) 1.71 l 102.3⬚, 1.71 l 222.3⬚, 1.71 l 342.3⬚,

(b) 1.609 ⫹ j5.356 ⫹ j2n p (n ⫽ 0, 1, 2, . . . ).