bruce mayer, pe registered electrical & mechanical engineer bmayer@chabotcollege

[email protected] • ENGR-25_Functions-1.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Chp3 MATLABFunctions:

Part1



Learning Goals Understand the difference between

Built-In and User-Defined Functions Write User Defined Functions Describe Global and Local Variables When to use SUBfunctions as

opposed to NESTED-Functions Import Data from an External Data-File

• As generated, for example, by an Electronic Data-Acquisition System



Functions MATLAB Has Two Types of Functions1. Built-In Functions Provided by the

Softeware• e.g.; sqrt, exp, cos, sinh, etc.

2. User-Defined Functions are .m-files that can accept InPut Arguments and Return OutPut Values



Getting Help for Functions Use the lookfor command to find

functions that are relevant to your application

For example typing lookfor complex returns a list of functions that operate on complex numbers (more to come):

>> lookfor complexctranspose.m: %' Complex conjugate transpose. COMPLEX Construct complex result from real and imaginary parts.CONJ Complex conjugate.CPLXPAIR Sort numbers into complex conjugate pairs.IMAG Complex imaginary part.REAL Complex real part.CPLXMAP Plot a function of a complex variable.



Built-In Exponential FunctionsCommand Conventional Math Function

exp(x) Exponential; ex

sqrt(x) Square root; xlog(x) Natural logarithm; lnx

log10(x) Common (base 10) logarithm; logx = log10x

Note the use of log for NATURAL Logarithms and log10 for “normal” Logarithms• This a historical Artifact from the

FORTRAN Language – FORTRAN designers were concerned with confusing ln with “one-n”



Built-In Complex-No. Functions

Useful for Analyzing Periodic Systems• e.g., Sinusoidal Steady-

State Electrical Ckts

Command Conventional Math Functionabs(x) Absolute value (Magnitude or Modulus)

angle(x) Angle of a complex number (Argument)

conj(x) Complex Conjugate

imag(x) Imaginary part of a complex number

real(x) Real part of a complex number



Built-In Rounding FunctionsCommand Conventional Math Function

ceil(x) Round to nearest integer toward +fix(x) Round to nearest integer toward zero

floor(x) Round to nearest integer toward −round(x) Round toward nearest integer.

sign(x) Signum function:+1 if x > 0; 0 if x = 0; −1 if x < 0.

Graph



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/MTH/Phys 25

Complex Numbers

1- j 1- i



Complex Numbers – Math What do We Do

with?

1- i

7x Factoring

7171 x Let’s Make-Up or

IMAGINE

Def.)(Engr 1

Def.)(Math 1

j

i

jx 646.2



Complex No.s – Basic Concept

World of REAL numbersa x2 + b x + c = 0 x =

± 2ab2– 4ac b

Discriminant D

Solution(s) of a quadratic equation exist only for non-negative values of D !

World of COMPLEX numbersSolution(s) of a quadratic equation exist also for negative values of D !

x =

± j 2a

|b2– 4ac| b 1jIn Engineering √(-1) = j in Math √(-1) = i



Complex Number, z, Defined1 ji

z = x + j yReal partx = Re(z)

Imaginary party = Im(z)

Complex numbersIm(z) 0

Real numbersIm(z) = 0 Re(z)

Im(z)

Complex numbersReal

numbers



Complex No.s – Basic Rules

Powers of j j2 = –1 j3 = –j j4 = +1 j –1 = –j

j4n = +1; j4n+1 = +j ; j4n+2 = –1; j4n+3 = –j for n = 0, ±1, ± 2, …

Equality z1 = x1 + j y1 z2 = x2 + j y2

z1 = z2, x1 = x2 AND y1 = y2

Addition

z1 + z2, = (x1 + x2) + j ( y1 + y2)

1 ji



Complex No.s – Basic Rules cont

z1 = x1 + j y1 z2 = x2 + j y2

Multiplication z1 z2, = (x1 x2 – y1 y2) + j (x1 y2 + x2 y1)

The complex conjugate

z = x + j y z* = x – j y

(z + z*) = x Re(z)21 (z – z*) = j y j Im(z)2

1z z* = x2 + y2

Division z2

z1 = (x22 + y2

2)z2 z2*z1z2* =

x1 x2 + y1 y2 + j (x22 + y2

2)y1 x2 – x1 y2

1 ji



Complex No.s – Graphically

x

yr

z = x + i y

Im(z)

Re(z)

The Argand diagram Modulus (magnitude) of z

arctan = arg z =

xy + p, if x < 0

arctan xy , if x > 0

r = mod z = |z| = x2 + y2

x = r cos

y = r sin

Argument (angle) of z

Polar form of a complex number z

z = r (cos + j sin

1 ji



Complex No.s – Polar Formz1 = r1 (cos 1 + j sin 1 z2 = r2 (cos 2 + j sin 2

x

yr

z = x + j y

Im(z)

Re(z)

|z1z2| = |z1| |z2| ; arg(z1z2) = arg(z1) + arg(z2)

z1 z2 = r1r2 (cos (1 + 2) + j sin(1 + 2))

= (cos (1 – 2) + j sin(1 – 2))z2

z1

r2

r1

= ; arg( ) = arg(z1) – arg(z2) |z2||z1|

z2

z1

z2

z1

Multiplication

Division

1 ji



Polar Multiplication Proof Consider:

22221111 sincossincos jdwjdw

Then 21

221212121

22112121

sinsincossinsincoscoscos

sincossincos

jjjdd

jjddww

But 12 j Then factoring out j, & Grouping 212121212121 sincoscossinsinsincoscos jddww

Recall Trig IDs

sinsincoscossincossinsincoscos

1 ji



Polar Multiplication Proof cont

Using Trig ID in the Loooong Expression

So Finally

21212121 sincos jddww

21cos

212121212121 sincoscossinsinsincoscos jddww

21sin

1 ji



De Moivre’s Formulaz1 z2 = r1r2 (cos (1 + 2) + j sin(1 + 2))

z1 z2…zn = r1r2 …rn [cos (1 + 2 …+ n) + j sin(1 + 2 …+ n)]

zn = rn (cos (n) + j sin(n))z1 = z2=…= zn

r = 1 (cos + j sin )n = cos (n) + j sin(n)

French Mathematician Abraham de Moivre (1667-1754)

1 ji



Complex Functions

f(x) = g(x) + j h(x)

A complex function

Real function

Real function

A complex conjugate functionf*(x) = g(x) – j h(x) f(x) f*(x) = g2(x) + h2(x)

Example: f(z) = z2 + 2z + 1; z = x + j y

f(z) = g(x,y) + j h(x,y)g(x,y) = (x2 – y2 + 2x + 1)h(x,y) = 2y (x + 1)

1 ji



Verify by MuPad From the last Line

(collect comand) collect real and imaginary parts

12,

22,12, 22

xyyxhyxyyxhxyxyxg



Euler’s Formula

A complex conjugate is also inverse

A power series for an exponential

!

u!

uueu

321

32

5!3!4!2!1

3!)(j

2!)(1

5342

32

θθθθθ

θjθjθe jθ

j

θ sin θez jθ j cos

jθu ei

Im(z)

Re(z)

1

1

–1

–1–

e–i

jθ-jθ

jθ-jθ

eej

θ

eeθ

21 sin

21 cos

θ sin θezz* -jθ j cos-1

1 ji

cos(θ) sin(θ)



Complex Numbers – Engineering



Complex Number Calcs Consider a General

Complex Number

This Can Be thought of as a VECTOR in the Complex Plane

This Vector Can be Expressed in Polar (exponential) Form Thru the Euler Identity

Where

)sin(cos

jrrejyxz j

jyxz

11 jjj Then from the Vector Plot

xy

yxr

1

22

tan

x

yr

z = x + j y

Im(z)

Re(z)

1 ji



Complex Number Calcs cont

Consider Two Complex Numbers

The SUM, Σ, and DIFFERENCE, , for these numbers

The PRODUCT n•m

j

j

Fejdcm

Aejban

Complex DIVISION is Painfully Tedious• See Next Slide

dbjcamndbjcamn

jjj AFeFeAemn

adbcjbdacbdjadbcjac

jdcjbamn2

1 ji



Complex Number Division For the Quotient n/m in

Rectangular Form

The Generally accepted Form of a Complex Quotient Does NOT contain Complex or Imaginary DENOMINATORS

Use the Complex CONJUGATE to Clear the Complex Denominator

jdcjba

mn

The Exponential Form is Cleaner• See Next Slide

22

22

2

dcadbcjbdac

mn

dcddcjcbdjadbcjac

mn

jdcjdc

jdcjba

mn

1 ji



Complex Number Division cont.

For the Quotient n/m in Exponential Form

However Must Still Calculate the Magnitudes and Angles

j

j

j

eFAFeAe

mn

Look for lotsof this inENGR43

1 ji



Root of a Complex Number How to Find?

3 197 j

jrejyxz Use Euler

xy

yxr

1

22

tan

In This Case

Note that θ is in the 2nd Quadrant

Thus

2.1107

19tan

25.20197

1

22

r

3 2.110

3

25.20

197

je

j



Root of A Complex Number Now use

Properites of Exponents

Use Euler in Reverse

74.36

32.110

31

2.11031

31

2.110

3 2.110

726.2

726.2

25.20

25.20

25.20

j

j

j

j

j

e

e

e

e

e sincos jrz

In this Case

631.11841.274.36sin74.36cos726.2

jjz

By MATLAB(-7+19j)^(1/3)ans = 2.1841 + 1.6305i



ln of a NEGATIVE Number (1) What is Im(z)

Re(z)

r

–r

–r = πz=(-19,0) 19ln

180

190arctan19019 22

p

r

State −19 as a complex no.

019 jz

Find Euler Reln Quantities r, & θ



ln of a NEGATIVE Number (2) Note that θ is

180º, NOT Zero Thus the Polar

form of −19

Im(z)

Re(z)

r

–r

–r = πz=(-19,0)

pjez

jz

19

019

Taking the ln

eje

ezj

j

ln9444.2ln19ln

19lnln

p

p

p

1416.39444.2ln jz



Log of a NEGATIVE Number Recall complex

forms for −19

Im(z)

Re(z)

r

–r

–r = πz=(-19,0)

pjez

jz

19

019

Taking the common (Base-10) log

ej

e

ezj

j

log2788.1log19log

19loglog

p

p

p

3644.12788.1

4343.02788.119loglog

jjez j

p

p



sin or cos of Complex number Recall from Euler Development

By Sum & Difference Formulas

sinh22

sin2

1sin

cosh22

cos2

cos

jeejeejjeej

eeeejee

jjjjjj

jjjjjj

jy

x

sinsincoscossincossinsincoscos



sin or cos of Complex number Thus

From PreviousSlide

So Finally

jyxjyxyx

jyxjyxjyxsincoscossinsin

sinsincoscoscos

yjjy

yjysinhsin

coshcos

yxjyxyx

yxjyxjyxsinhcoscoshsinsin

sinhsincoshcoscos



MATLAB Complex Operations>> a = 3+2j;>> b = -4+5i;>> c = -5-j*4;>> d = i;

>> ac = a*cac = -7.0000 -22.0000i

>> Mag_b = abs(b)Mag_b = 6.4031

>> c_star = conj(c)c_star = -5.0000 + 4.0000i

1 ji



Complex Ops

>> b_d = b/db_d = 5.0000 + 4.0000i

>> c_b = c/bc_b = 0 + 1.0000i

>> c_a = c/ac_a = -1.7692 - 0.1538i

>> Re_d = real(d)Re_d = 0

>> Im_b = imag(b)Im_b = 5



Complex Ops

>> b_sq = b^2b_sq = -9.0000 -40.0000i

>> b_cu = b^3b_cu = 2.3600e+002 +1.1500e+002i

>> cos_a = cos(a)cos_a = -3.7245 - 0.5118i

>> exp_c = exp(c)exp_c = -0.0044 + 0.0051i

>> log_b = log10(b)log_b = 0.8064 + 0.9752i



Complex Ops>> r = 73;>> theta = 2*pi/11;>> theta_deg = 180*theta/pitheta_deg = 32.7273

>> z = r*exp(j*theta)z = 61.4115 +39.4668i

>> abs(z)ans = 73

>> 180*angle(z)/pians = 32.7273

>> x = -23; y = 19;>> z2 = complex(x,y)z2 = -23.0000 +19.0000i

1 ji



Caveat MATLAB Accepts>> z2 = 3+7j;>> z3 = 5 + i*11;>> z4 = 7 + 13*j;>> z2z2 = 3.0000 + 7.0000i>> z3z3 = 5.0000 +11.0000i>> z4z4 = 7.0000 +13.0000i

But NOT>> z5 = 7 + j5;??? Undefined function or variable 'j5'.

1 ji



All Done for Today

LeonhardEuler

(1707-1783)



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Appendix

6972 23 xxxxf



Complex Integration Example

I = xbxeax dcos I = Re xee ibxax d I = Re xe ib)x(a d

22

cos sinsin cos

sin cos

babx b bx aibx b bx ae

ibabx i bxe

ibaexe

ax

axib)x(a

ib)x(a d

22sin coscos

babx b bx aexbxe axax

d

22cos sinsin

babx b bx aexbxe axax

d

1 ji

Us EULER to Faciltitate (Nasty) AntiDerivation



Illuminate Previous Slide By EULER

xbxibxexeeI axibxaxC dsincosd

Using Term-by-Term Integration xbxeixbxexbxiexbxeI axaxaxax

C dsindcosdsindcos

• As “i” is just a CONSTANT Taking the REAL Part of the above

xbxeixbxeII axaxC dsindcosReRe

Cibxaxax IxeexbxeI RedRedcos



Rotation OperatorPolar form of a complex number

z = r (cos + j sin jθe r αθjjθjαjα eeezez' r r

z(x,y)

Im(z)

Re(z)

r

–r

–rz'(x',y')

α)(θ sin α)cos(θr

θ sin cosθr

j y'jx'z'j yjx z

x'= r cos( + = r(cos cos – sin sin y'= r sin( + = r(sin cos + cos sin

x'= x cos – y sin y'= x sin + y cos

The function ej can be regarded as a representation of the rotation operator R(x,y) which transforms the coordinates (x,y) of a point z into coordinates (x',y') of the rotated point z' : R(x,y) = (x',y') .

1 ji



Periodicity

z' = 2πjjθeez' = 2πθjez = jθe

zej e jθjθ sin2πcos2πz'

2, 1, 0,2π nee jθnθj ,

The function ej occurs in the natural sciences whenever periodic motion is described or when a system has periodic structure.

1 ji



Periodicity on a Circleor n nth roots of 1

z0

Im(z)

Re(z)1

z1

z2

2

z0 = ej0 ; z1 = ej 2p/3 ; z2 = ej 4p/3

zk = ej 2pk/3, where k = 0,1,2

zk3 = (ej 2pk/3)3 = ej 2pk = 1 1 = zk

3

z0 = ej0 = 1 z1 = ej 2p/3 = - + j

z2 = ej 4p/3 = - – j

21

21

23

23 zk = ej 2pk/3, where k = 0, ±1

3 roots of third degree of 1

1 ji



Periodicity on a Circle contor n nth roots of 1

z0 = ej0 = 1 z±1 = e±j p/3 = ± j

z±2 = e ±j 2p/3 = - ± j

21

21

23

23

6 sixth roots of 1

z0

Im(z)

Re(z)

z1z2

z3

z-2 z-1

Used for description of the properties of the benzene molecule.

n nth roots of 1

zk = ej 2pk/n for k =0, ±1, ±2, ... , ±(n –1)/2 if n is odd

0, ±1, ±2, ... , ±(n/2 –1), n/2 if n is even

Such functions are important for the description of systems with circular periodicity.

1 ji



Periodicity on a Line

Such functions are important for the description of periodic systems such as crystals.

... a a a a a

f(x) = f(x + a) f(x) = x/aje 2π

Periodic function

Generalization for three-dimensional periodic systems

f(x,y,z) = z/cjy/bjx/aj eee 2π2π2π

1 ji



Trig on complex numbers

bruce mayer, pe registered electrical & mechanical engineer bmayer@chabotcollege

Documents

complex numbers

complex numberrealxreal

pe engineeringmathphysics

conj complex conjugate

complex conjugate pairs

complex conjugate transpose

imag complex imaginary

labbruce mayer