082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

9.3 Operations with complex numbers 331
EXERCISES9.3
1 Ifz 1 =3+2jandz 2 =4−8jfind
(a) z 1 +z 2 (b) z 1 −z 2 (c) z 2 −z 1
2 Expressthe following in the forma +bj:
1
(a) (b) −2
1 +j j
1
(c)
j + 1 j
(d)
2 −j 1 +j
3
(e)
3+2j + 1
5 −j
3 Expressthe following in the forma +bj:
2
(a) (b) −2+3j
1 −j j
(c)3j(4 −2j)
(e) 5+3j
2+2j
(b) (7 −2j)(5 +6j)
4 Find aquadraticequationwhose rootsare 1 −3j and
1+3j.
5 If (x +jy) 2 =3+4j,findxandy,wherex,y ∈ R.
6 Find the real andimaginaryparts of
(a)
2
4 +j − 3
2 −j
(b) j 4 −j 5
(c) 1 j +j (d) 1
j 3 −3j
Solutions
1 (a)7−6j (b) −1+10j (c) 1−10j 4 x 2 −2x+10=0
1
2 (a)
2 − 1 2 j (b) 2j (c) 2 5 − 4 5 j 5 x=2,y=1;x=−2,y=−1
1
(d)
2 + 1 23 −11j
j (e) 6 (a) − 62
2 26
85 , −61 85
(b) 1, −1
3 (a) 1+j (b) 3+2j (c) 6+12j
(d) 47 +32j (e) 2 − 1 (c) 0, 0
2 j (d) 0, 1 4
TechnicalComputingExercises9.3
1 Many technicalcomputing languagesallow the user
toinputandmanipulatecomplex numbers.Investigate
howthe complex numbera +bjisinput to the
software to whichyou have access.
2 Usetechnicalcomputing software to simplifythe
followingcomplex numbers:
(a) (1 +j) 5 3j
(b)
(1 −2j) 7
4
(c)
(3 +j) 3 − 7
(2 −3j) 2
3 Solvethe polynomial equations
(a) x 3 +7x 2 +9x+63=0
(b) x 4 +15x 2 −16=0
YoumayrecallfromChapter1thatin MATLAB ® the
rootsfunctioncan be used to solvepolynomial
equationswith real roots.This functioncan also
calculatecomplex roots.
4 Use apackageto findthe real andimaginaryparts of
15+j
7−5j .
5 Acommonrequirementincontroltheoryistofindthe
poles ofarationalfunction. Useapackageto findthe
poles ofthe function
0.5
G(s) =
s 3 +2s 2 +0.2s +0.0556
To solvethisproblem automatically mayrequire the
use ofspecialist toolboxesin MATLAB ® orGNU
Octave which are not partofthe core program.For
example, the Signal Processingtoolboxin
MATLAB ® has the functiontf2zp which convertsa
transfer function expressed asaratio ofpolynomials
to pole/zero form.