1817_05Basic Math for process control

7/28/2019 1817_05Basic Math for process control

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5ComplexQuantities

BackgroundA quadrat ic equat ion involving the var iable "x" can be wr i t ten in i t s general form and can then be solved using an a lgebraic procedure . The general form of the quad rat ic e qu at io n in x, i s

a x2 + b x + c = 0.Because of the quadrat ic ( second power) nature of the equat ion, two val ues of x wil l sat isfy i t . These values of x are cal led, in mathematical parlance, the "roots of the equat ion." They can be designated m 1 and m 2 , andthei r values are

The solut ion for any quadrat ic equat ion can consequent ly be found byapplying th is formula for m 1 a n d m 2 . For example , g iven that

x2 + x - 6 = 0 (i.e., a = 1, b = 1, c = -6 ) ,

Therefore, x = 2 and x = -3 are the solutions for x 2 + x - 6 = 0.

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62 Basic Math for Process Control

The determinat ion of the roots of a quadrat ic equat ion i s s t ra ight forward,p rov ided tha t the quan t i ty under the square roo t s ign (b 2 - 4ac) does nottur n ou t to be negat iv e , as i t w ou ld if the equ at ion to be solved w erex2 - 6 x + 13 = 0.

In this case,

The term ? can be s impl i f ied one s tep fur ther by takin g the factor 4outs ide of the squa re root s ign, leaving only the factor ( -1 ) . Tha t is ,? = 4 ?. The roots of the equat ion then become

This in t roduces the concep t of a "nu m be r" w hos e va lue is ? -1 T h i s n u m ber wil l be identif ied by the let ter j , th at is, j = ?-1 . Since ?-1 c an no t beevaluated, or f rom another v iewpoint , i t i s not possib le to draw a l ine thati s ?-1 un i ts long, the qu an t i ty j must be imag inary .I t i s impor tant to understand, however , that as far as mathemat ics i s concerned , s t a t ing tha t a nu m be r i s imaginary is entirely different from stat in gthat there i s no such number. The quan t i ty j i s ima g inary beca use i t canno tbe o bse rve d in na tu re , b ut i t defini tely exists be ca use x = (3 + 2 j) an d x =(3 - 2 j) are valu es th at sat isfy th e eq ua tion (x2 - 6 x + 13) = 0. Th is can bepro ve n by s ubs t i tu t ing the valu e x = (3 + 2 j) in to the or ig inal equ at ion ,bear ing in mind tha t j 2 = - 1 .

x2 = 9 + 12j + 4 j 2 = 9 + 12 j - 4 = 5 + 12 j- 6 x = - 1 8 - 1 2 j+ 13 = 13

x2 - 6 x + 13 = 5 + 12 j - 18 - 12 j + 13 = 0The nu m be rs (3 + 2 j) an d ( 3 - 2 j ) , w hic h in th is case are the roots of(x 2 - 6 x + 13) = 0, are ca lled complex numbers. The character ist ic of acom plex nu m be r i s that i t conta in s an imag ina ry par t , that i s , a pa r t tha tc o n t a in s t h e n u m b e r j . A complex nu m be r usua l ly con ta ins a r ea l pa r t a swel l , as it do es in the case of the com plex n um be r (3 + 2 j) . In th is instanc e,the real pa r t i s 3 an d the im agin ary p ar t i s 2 j .


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Chapter 5 - Com plex Quantit ies 63

Figure 5 -1 . Graphical Representation of Com plex Num bers.

The real par t of a complex number may be zero , leaving only the imagina ry par t . If the ima gina ry pa r t were zero , how eve r , only the real pa r twould r emain , and the number would be a r ea l number r a the r than a comp l e x n u m b e r .

Graphical RepresentationA com plex num be r can be r epresen ted g raph ica l ly by p lo t t ing it on a complex plan e, in w hich the axis for the real pa r t i s hor iz onta l , whi le the axisfor the im agin ary p ar t i s ver t ica l . In Figure 5-1, the point P1 represents thecom plex n um be r (3 + 2 j ) , w hi le P2 rep resen ts (3 - 2 j ).W he n the roots of a qu ad rat ic eq uat io n w i th real coeff ic ients are com plexnu m be rs , they wi l l a lways occur in pa i r s , ca l led conjugate pairs. T h is m e a n sth at th e roots are of the form (a + jb) an d (a - jb). In any conjugate pair , thereal pa r ts of bo th num be rs are the sam e, wh i le the im agin ary pa r ts d if ferin sign only.


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The Complex VariableVar iab les, a s we l l a s nu m be rs , can be complex , wh ich m ean s they can hav ean im agi nar y pa r t . If z is a comp lex var iable , the n i t may have a real par t ,but i t will have an imag inary par t . The r ea l and imag inary par t s can bede sig na ted by x an d y, respectively, so tha t z = (x + j y) .F igure 5-2 is the graph ical rep resen ta t ion of a comp lex var iable in both rect i l inear and polar coordinates . In rect i l inear coordinates , the x componenti s sh ow n m eas ured a long the hor i zon ta l ( rea l) ax i s , wh i l e the y com pon en tis m ea su red a lon g the ver t ical ( ima ginary or j) axis . The su m of x hor iz on tal ly and jy vert ical ly, wh ich is z, creates a vector , w hi ch star ts at the orig inO and ends a t P .Since z ha s not only the qu al i ty of leng th but a lso the qua l i ty of d i rect ion,de pe nd ing on the va lues o f the r ea l and im ag inary par t s x and y, the vector z i s des igna ted OP. The ba r over the le t ters is the sho r th an d sym bolthat indicates that OP is a vector .

Figure 5-2. z = x + j y in Recti l inear and P olar Co ord inate s.

The length , or m ag ni t ud e, of the vector O P is equ al to the d is tance O P a ndis custo ma r i ly ident i f ied by | z |. In m ath em at ics terminology, | z | i scal led the modulus of the complex number z . From the t r iangle OAP,| z |2 = x2 + y2 . Therefore,

The vector O P is vector th e su m of x + j y in a graphical repre senta t ionus ing r ec t il inear coord ina tes . Cer t a in p rob lem s m ay be dea l t wi th m oreconven ien t ly by r epresen t ing a comp lex num be r in a sys tem of po la r coor -


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dina tes . In the po lar sys tem, the m od ul us of z i s equ al to r. The di rect ion ofthe vector OP is sh ow n by the angle 0 , w hic h i s the angle forme d by OPand the horizontal or real axis. This angle is defined by the relat ion

tan =y/x,or = tan -1y/x (the angle whose tangent is y/x).The angle 0 is cal led the argument of the v ecto r z, or arg z for short .These two d iag ram s sho w th a t x = r cos , an d y = r s in .

Trigonometric and Exponential FunctionsIf the Mac Lau r in ser ies exp ans ion is us ed to dev elop an infin i te ser iesexpression for e j (bear ing in mind that j 2 = -1) , the result is

The expression in f irst set of brackets is the MacLaurin ser ies for cos 0,w hi le th e express ion in the sec ond set of brackets i s the ser ies for s in 0 .C o n s e q u e n t l ye j = cos 0 + j sin 0.

By using the same technique, i t can be shown that e -j = cos 0 - j sin 0.I f these expressions are added, e j + e -j = 2 cos , a n d

If the second expression is subtracted from the f irst ,

These re la t ionsh ips f requent ly c om e in ha n dy in the solut ion of d i fferent ia l equat ions.


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Sum of Two Com plex QuantitiesIf z1 = x1 + j y1, and z 2 = x2 + j y 2, then

z1 + z 2 = x1 + j y1 + x2 + j y 2 = (x 1 + x2) + j (y 1 + y 2).This indicates tha t to find the su m of two com plex qu ant i t ies , the real an dimaginary par ts should be added separate ly . F igure 5-3 i l lust ra tes th is .

Figure 5-3. Graphical Re presentat ion of the Sum of Two Comp lex Qu anti t ies.

Product of Two Complex QuantitiesFigure 5-4 is a graphical i l lust ra t ion of the pro du ct of tw o com plex qua nt i t ies.The p rod uc t of tw o complex quan t i t ie s z1 = x 1+ j y1 and z 2 = x2 + j y 2 ismost easi ly determined by conver t ing to the polar form.If z1 = r1e j 1 and z 2 = r 2 e j 2 , then

z1 z 2 = r1e j 1 r 2 e j 2=r1r2e j ( 1+ 2 )

= |z 1||z2|e


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Figure 5-4. Multipl ication (z 1 x z 2 ) .

This ver i f ies that the product of two complex quant i t ies i s obta ined bym u l t i p l y i n g t h e m a g n i t u d e s a n d a d d i n g t h e a r g u m e n t s .Separating the Real and Imaginary Parts

I t i s of ten ne cessary to rear ra nge th e term s of a com plex q uan t i ty to d eter min e the mag ni tu de a nd the a rg um ent . Th i s i s do ne by co ll ect ing theterms that are not mul t ip l i ed by j i n to one g rou p , and the t e rms tha t aremul t ip l i ed by j in to an other . For ex am ple , su pp os e z = (a + jb) (c + jd) . Mu lt ip ly ing th is y ie ldsz = ac + jad + jbc - bd .

The real pa r t i s (ac - bd ) , w hi le the im agin ary pa r t i s (ad + be) .


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Separat ion of the complex quant i ty in to i t s real and imaginary par ts i ss t ra ight forward, except in cases where the complex quant i ty i s a f ract ion.W he n th is hap pe ns , the a lgebraic re la t ion (a - b) (a + b) = (a2 - b2) i s used .Not ice that if b wer e an ima gin ary nu m be r , ( = jc) , the im agin ary pa r twould d i sappear when the p roduc t i s t aken . Tha t i s ,

(a + jc)(a - jc) = (a 2 - j2 c2) = (a 2 + c2).In th is way the denominator of the f ract ion can be c leared of imaginarynu m be rs , an d the fr ac t ion can be separa ted in r ea l and im ag ina ry par t sus ing the p rocedure

Example 1: Magnitude and Argument of a Complex ExpressionD e t e r m i n e t h e m o d u l u s a n d a r g u m e n t of t h e c o m p l e x q u a n t i t y

To c lea r the denom ina to r o f the imag ina ry num ber , mul t ip ly the nu m erato r and d eno m ina to r by (j t - 1 ).


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1817_05Basic Math for process control

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