MAT290 Matlab Module1

8/4/2019 MAT290 Matlab Module1

1/12

-5-20000

-15000

-10000

-5000

0

5000

UNI

FAC

Dep

ADVA

W

T

-4

ERSITY O

LTY OF A

rtment of

CED

MA

rki

o-

-3

TORONT

PPLIED S

Electrical

MA

NGIN

TLA

g

men

-2

O

IENCE AN

and Comp

290H

ERIN

Mo

ith

sion

-1x

ENGINE

uter Engi

1F

MAT

ule

rra

al P

0

RING

eering

HEMA

1

s a

otti

1

ICS

d

g

2 3


2/12

Course

Universit

The ficomman

however

distribute(Matrices

Preparat

1.2.

3.

Workin

Input via

Get y

the fo

Comdispla

>> 3

This

ans10.

The ufor mDown

comm

Previo

The reof a

>> 1ans

>> aans7.00

AT290H1F

of Toronto

rst step in ls. There a

we will pri

s MATLAB).and Vector

on (RequirRead throu

Watch the

http://www.or if you ar

Start button

In the exerc

you try thes

With Arra

the comman

urself setu

lowing exa

ands in Med on the sc

+ 7.5

ill give a re

5000

and down

king small ckeys allow

and: 3 + 7

us comman

sult of the lATLAB built

/43.5000

s *2

00

EXEearning hoe many onl

arily focus

This first) and two-di

d to do befo

h the posted

etting Start

mathworks.e running

, .

ise tables gi

e commands

s (Matrice

d-line

at one of t

ples, enter

TLAB arereen immedi

ult in the co

arrow keys

hanges to coou to navig

.5.

s are also ac

st performein variable.

MAT

CISto use M

ne sources

n those res

module wimensional p

e you start

Introductio

d with MATL

om/demos/

ATLAB, it c

en below, p

in MATLAB.

and Vecto

he facultys

these in M

xecuted byately. As an

mand win

llow you to

mmands yote through t

cessible thro

computatiot can be use

AB Module

2

ES -TLAB is to

hich can g

urces availa

ll cover thlotting.

hese exerci

to MATLAB

AB video de

ATLAB/gettin be access

redict the ou

s)

computers

TLAB and

pressing Eexample, tr

ow as follo

recall previ

have alreade command

ugh the Co

n is ascribed in the next

1

RR become famive you a b

ble through

general a

es)document.

onstration

ng-started-ed by select

tput that wil

and run M

ork along

nter or Re

s

us comman

y entered.window. P

mand Histo

to the variacommand.

YSiliar with itasic introdu

Mathworks

eas of wor

hich is ava

ith-matlab-ing Demos

l be given b

TLAB. As

ith the mo

turn, with

ds, which c

s well, theess the up a

ry window.

ble ans, whor instance,

Fall

ECE Depar

s basic setution to MAthe compan

ing with

lable at:

ideo-tutoriarom the M

MATLAB,b

ou read thr

ule.

the output

n be very h

ageUp andrrow to rec

ich is an ex

2011

tment

and

LAB,

that

rrays

.html

TLAB

efore

ough

being

elpful

Pagell the

mple


3/12

Course MAT290H1F MATLAB Module #1 Fall 2011

University of Toronto 3 ECE Department

You can also define your own variables. Type in the following commands to create the variables a and b:

>> a = 14/4a =3.5000

>> b = a+1b =4.5000

The command whos displays a list of variables, their sizes, and their data class. Try it now.

The entry of very large and very small numbers can be simplified using scientific notation. For example,

enter the numbers0 = 0.000000000008854 = 8.854 x 10-12, 0 = 4 x 10

-7, and 6.022 x 10

23 by typing:

>> epsilon = 8.854e-12

and

>> mu = 4*pi*1e-7

and

>> Avogadros_constant = 6.022e23

Do the following exercises:

1.Evaluatethe following expressions using MATLAB. Fill in the table as you go.

Expression Prediction Result from MATLAB

8 * 5 + 3

8 * (5 + 3)

10 / 2 / 5 3 + 2 * 4

3 ^ 2 ^ 3

exp(j*pi/4)

c = a^39 where a = 3

1/sqrt(epsilon*mu)where epsilon and mu are defined

as above.

2. When the command is followed by a semicolon ;, the output is suppressed. Check thedifference between the following expressions:

>> 3 + 7.5 % without semicolon>> 3 + 7.5; % with semicolon

Note that the last stored answer can always be viewed by typing ans in the command window.

Also note that the text following the % sign is ignored by MATLAB. It is useful for insertingcomments, explanations or reminders.


4/12

Course

Universit

Workin

A mat

columor spa

rows.

by typ

>> AA =1 24 57 8

Try th

>> AA2 =1 2-1 1

Notice

from 1

Trans

rows

>> AA2 =1 2-1 1

>> Aans1 -12 13 34 5

Stand

would

>> [ans410 1

AT290H1F

of Toronto

with Ma

rix of size n

n vectors arces are used

or example

ng

= [1 2

3

9

is example o

= [1:43 43 5

how the c

to 4 in step

osing a ma

ith the corr

3 43 5

% tr

rd matrix m

be executed

2;3 4]

rices

kis a rec

special typto separate

, define the

; 4 5 6;

f how to spe

-1:2:5]

lon ':' is us

of 1; -1:2

rix can be

sponding co

nspose o

ltiplication

in MATLAB

[2 0;1 3

MAT

angular arra

s of matriceelements in

atrix

7 8 9]

cify a matri

d to specify

:5 means fr

chieved wit

lumns, as sh

f A2

is done with

by typing:

]

AB Module

4

y of number

s, with sizea row, and

% row b

in MATLAB

a range of

om -1 to 5 i

a simple c

own in the f

the * comm24

2 01 3

1

s having nrofn 1, anemicolons

row in

:

numbers as

increments

ommand, w

ollowing ex

and, so the c

ows and kc1 k, resp

re used to s

ut

elements. 1

of 2.

here transpo

mple:

alculation gi

Fall

ECE Depar

olumns. Ro

ectively. Coeparate indi

:4 simply

sing interch

ven by:

2011

tment

and

masidual

eans

anges


5/12



While the dot product, or element by element multiplication, is done by using the .* command.For example, the result of

>> [1 2;3 4].*[2 0;1 3]ans =2 0

3 12

Trytheseexamples:

Expression Prediction Result from MATLAB

A2*A2

A2.*A2

A2.*A2

Building Matrices and Extracting Parts of Matrices

It is often necessary to build a larger matrix from a smaller one; for example

>> x = [4; -1], y = [-1 3]x =4-1y =-1 3

>> z = [x y] % z consists of the columns x and yz =4 -1-1 3

Right after the above example, try

>> z^2

>> z.^2

What is the difference between the two results?

Next type in

>> z = x + y

>> z^2 % does this work?

>> z.^2 % what about this?


6/12

Course

Universit

What

may h

Specif

first r>> Aans

and en>> Aans

>> Aans

Partial

>> Aans

We ca>> AA =

Revie

the M

SS

E

Many

that th

AT290H1F

of Toronto

do you get

elp you ans

ic elements

w of the ma

(1,2)

2

tire rows, or

(1,:)

1

(:,2)

258

rows, or col

(1:2,2)

25

n also make

(1:2,1)

1 21 57 8

the summ

TLAB Prim

anaging Coanaging Va

atrix and A

ecial Charecial Varia

lementary

rigonometriof these co

ey are there.

s a result n

er these que

f a matrix c

rixA (as de

columns, ar

2 3

umns, can b

use of the co

[1;1]

369

ries of the

er:

mmands andiables and t

ray Operato

cters,les and Co

ath Functio

Functions.mands are

You can al

MAT

w? Why do

stions.

an also be a

ined above),

e displayed t

e displayed

lon operator

ommands li

Functions,he Workspa

rs,

stants,

s, and

not immedi

ays use he

AB Module

6

es one work

cessed. For

you would

hrough the u

y limiting t

to assign va

sted in the f

e,

tely useful

lp to get mo

1

and not th

example, to

ype:

se of the col

e range:

lues to a par

llowing tab

o our purpo

re informati

other? Th

display the

on operator:

t of a matrix

es of the Re

ses, but you

on about the

Fall

ECE Depar

size com

2nd element

:

ference secti

should be

.

2011

tment

mand

of the

on of

ware


7/12



Do the following exercise:

1. Given the vectors (or row matrices) x = [1 3 7], y = [2 4 2] and the matrices A = [3 1 6; 5 2 7],B = [1 4; 7 8; 2 2], determine which of the following statements will correctly execute (and if

not, try to understand why) and provide the result or explanation why it doesnt work:

Expression Prediction Result from MATLAB/Explanation

x + y

x + A

[x; y]

B [x y]

A 3

B * A

A * B

A* B

2 * B

B(:,2)

A.^2

exp(B(1,1))

cos(pi*A)

sin((pi/2)*A(1,2))

exp(j*pi/4*B)


8/12



Applications Arrays

1. Solving sets of simultaneous equationsIn ECE212 (Circuit Analysis) you often need to solve sets of simultaneous equations which

result from the basic techniques of circuit analysis (nodal and mesh analysis). Doing this byhand can be a time consuming process, but in MATLAB the solutions to these equations can be

easily found.

This is of course only useful if we are dealing with cases where the coefficients of theequations are numbers. In cases where the coefficients contain one or more variables we must

know how to solve this by hand. This might occur in a design-type problem where the value ofa resistor is not known and the purpose of the problem is to determine the correct resistor valuefor a desired circuit operation.

To start, consider the set of equations given by:

5 6 = 0

10 + 14 = 2

This can be converted to matrix form as:

5 610 14 =

02

With the solution given by:

= 5 6

10 14

02 = 1.21

In MATLAB this can be solving using the inv command (see help inv):>> inv([5 -6;-10 14])*[0;2]

ans =

1.20001.0000

Note, that the matrix multiplier, *, is used rather than the dot multipler, .*, since we wantreal matrix multiplication rather than element-by-element multiplication.

Use MATLAB to solve the following sets of equations:

a)

16 2 4 = 62 + 6 2 = 0

4 2 + 4 + 6 = 0 + = 3

b)

9 2 2 = 42 + 10 4 = 6

2 4 + 9 = 6 + 8 3 = 0 3 + 4 = 6


9/12

Course

Universit

2.Tc

(l

p

I

hb

b

o

F

r

AT290H1F

of Toronto

c) Ufoth

oots of Pol

he fundamemplex coef

herezcanter in the ter

olynomials.

our study o

gher-orderilt-in functi

owser in th

how this fu

se this funct

a)

b) c)

d) 2or the first t

sult.

e circuit an

r the mesh cese mesh cur

nomials

tal theoremicients has a

e real or com, but for n

f ordinary di

olynomials.ons, roots.

command

nction can b

ion to deter

+ 3 2 + 1

+ 2 + 2 + 9 +

ree polyno

MAT

lysis techni

rrents givenrents.

of algebra stt least one r

plex). Wew we will f

fferential eq

MATLAB caFind the de

indow, ,

e used in the

ine the root

= 0

= 0

= 02 +

ials, verify

AB Module

9

ues to deter

in the circu

ates that anyot, meaning

will see a sicus on usin

ations we

n be used toscription of

or by typing

reference p

s of the follo

2 + 2

y hand calc

1

ine the fou

t below. Th

non-constathatf(z) = 0

ple proof oMATLAB t

ill find that

find these rthis function

help ro

ge for this f

wing polyn

+ 2 +

lation that

r equations n

en use MAT

t polynomiafor at least

f this using cfind the roo

we need to f

ots throughthrough eit

ts. Note, t

nction (do

mials.

= 0

ATLAB is g

Fall

ECE Depar

eeded to sol

AB to solve

l with real one number,

omplex analts of

ind the roots

one of its mer the functi

ere are exa

roots).

ving the cor

2011

tment

ve

for

z

ysis

of

nyon

ples

rect


10/12

Course

Universit

Plottin

Prepara

a.b.Thes

MAT

Two-Di

The plo

matrix)

with strai

[x(n), y(n

same nu

Type the

>> x =

>> y =

>> plo

>> plo

More tha

comman

>> clf>> plo

Or, the h

>> fig

>> plo

>> hol

>> plo

Multiple

comman>> fig

>> sub

>> plo

>> sub

>> plo

AT290H1F

of Toronto

In MAT

ion: To beg

isualizingsing Basic

e are availa

AB, it can

mensional

t command

, the comma

ght lines. Th

)]. Note tha

ber of elem

following c

linspac

2.^x;

t(x)

t(x,y)

n one plot c

. Try:

% Tht(x,2*x.

old comma

ure % T

t(x,2*x.

d

t(x,4*x.

plots can a

:

ure

plot(2,1

t(x,2*x.

plot(2,1

t(x,4*x.

XEAB

in, first wat

ata (4:07)lotting Fun

le at http://

e accessed

Plots

can be used

nd plot(y

e command

t x and y m

ents).

mmands aft

(0,10,10

to get

an be made

is clear2,x,4*x.

d can be us

is opens

2)

2)

lso be creat

1) % Bre

2) % Plo

2) % Swi

2) % Plo

MAT

CISE

h the follo

ctions (5:3

ww.mathw

y selecting

o produce a

) draws the

plot(x,y

st be both ei

r predicting

0);

a graphi

in the same

the cu^2)

ed. Try:

up a ne

ed on diffe

aks the

ts the f

tches th

ts the f

AB Module

10

S - P

ing video d

)

orks.com/p

Demos fro

graphical di

oints [1, y(

does the s

ther row or

the results:

c repres

figure by s

rent fig

w figure

ent axes in

figure w

unction

e curren

unction

1

OT

emonstratio

oducts/matl

the MATL

splay of a se

)], [2, y(2)],

me for the p

olumn vect

ntation

ecifying m

ure wind

window

the same

ndow in

n the t

axes t

n the b

ING

ns:

ab/demos.h

B Start but

t of data poi

. . ., [n, y(n)

oints [x(1),

rs of the sa

ltiple argum

ow

indow, usi

o two ax

p set of

the bot

ttom set

Fall

ECE Depar

ml or if yo

on, .

ts. For a ve

] and conne

(1)], [x(2),

e length (i.

ents in the

g the sub

es (top

axes

tom set

of axes

2011

tment

are runnin

tor (or row

ts these poi

(2)], . . . ,

. must have

lot

lot

and bott


11/12

Course

Universit

Often itfollowin

>> fig

>> sub

>> plo>> sub

>> sem

>> sub

>> log

Review t

MATLAB Two

Grap

Figur

Do the f

1. Create

by us

comm

>> x

Now

and s

repres

The p

by selof intthe pl

2. Create

over t

and y

Add a>> t

>> x

>> y

Note t

comm

AT290H1F

of Toronto

s useful tocommands:

ure

plot(3,1

t(x,x.^2)plot(3,1

ilogx(x,

plot(3,1

log(x,x.

e summarie

Primer:

imensionalAnnotation

Window/A

llowing ex

a new plot

(0, 0), (2,

ing the co

and:

= [0,2,

ype help

earch for

enting them

operties of t

ecting the mrest, then yt to have a

a new plot

e range 0 u will need

n appropriatitle(A

abel(T

abel(V

hat the tit

and.

plot a funct

1)

2) % Swi

.^2) % P

3) % Swi

2) % Plo

s of the com

Graphs

es Creation

rcises:

f the 8 poin

4), (3.5, 7),

mand plo

.5,5,7.2

plot follo

lot. Afte

with blue sq

he plot can a

enu Viewu can adjustreen dotted

f the expon

10. Yto use the do

title andxecaying

me (s))

ltage (V

e, xlabel

MAT

ion on a se

tches th

lots the

tches th

ts the f

mands liste

and Control

s whose (x

(5, 9.3),

(x,y,'r

,9.34,11

ing the MA

reading a

ares instea

lso be chang

Property Eits colour, line with gre

ntially deca

u can use th

t operator, .

ndy label toSinusoid

))

, and ylab

AB Module

11

ilog or a l

e curren

functio

e curren

unction

in the follo

,y) coordin

7.2, 6.12),

'). You c

,12.44];

TLAB promp

out the pl

of red circl

ed through t

itor. Onceine thicknesen stars.

ing sinusoi

= . sine linspac

*, to properl

your plot.)

l comman

1

og-log set o

axes t

with t

axes t

ith bot

ing tables

ates are

(9.34, 2), (

an create t

=[0,4,7,

t, or access

ot comma

s.

he figure pr

he editor iss, line type,

voltage fu

4e command

y multiply t

or example,

s must follo

f axes. For

the mid

e x-axis

the bot

axes se

f the Refere

11, 4.2), (1

e required

9.3,6.12

the main M

d, plot the

perties tool

open, if youetc. Use thi

ction, given

to create the

e exp and

, you can use

w rather tha

Fall

ECE Depar

example, tr

dle set

set to

tom set

t to a l

nce section

.44, 0)

vectors wit

,2,4.2,0

ATLAB help

same poin

ar button,

click on thes editor to c

by

required t

in functio

the comma

n precede th

2011

tment

y the

a log sc

og scale

of the

h the

];

menu

ts by

, or

traceange

ector

s.

ds:

e plot


12/12



You can adjust the range of the axes of this plot by using the command axis. For example try thecommand:>> axis([0 5 -1 1])

What are the new ranges of the

and

axes?

Note, the command axis equal sets the ranges of the and axes to be the same.

Applications Two-Dimensional Plots

1. Graphical Determination of the Roots of a PolynomialOne of the great uses of programs such as MATLAB is that it can be used to easily graph

complicated functions. One use for this utility is that the real-valuedroots to a polynomial canbe determined graphically by plotting the function and identifying where the curve crosses the

independent variable axis (i.e., thex-axis).

As an example, determine the real-valued roots of the polynomials given by:

a) + 2 + 2 = 0b) 2 + 9 + 2 + + 2 + 2 + 2 + 1 = 0

You plot for part b) should look similar to the figure on the first page of this module.

Documents

MAT290 Matlab Module1