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

46 Chapter 1 Review of algebraic techniques
TechnicalComputingExercises1.7
1 Useatechnical computing language suchas
MATLAB ® to verifythe solutionsto the problemsin
Exercises1.7. InMATLAB ® ,the functionresidue
calculates the partialfraction expansion. Forexample,
exercise1(a)wouldbesolvedbytypingthefollowing:
b = [6 14];
a=[143];
[r,p,k] = residue(b, a)
Notice how the coefficients ofthe numerator are input
in the formb = [6 14];thisisknown asarow
vector. The concept ofavector willbe discussed in
later chapters. Fornow itisadequate to treat thisasa
horizontal list ofnumbers which are passed to
MATLAB ® in a specific order.
Similarly, the coefficients ofthe denominator are
input bya = [1 4 3].
Each vectoris arranged with the coefficient ofthe
highestpower ofxfirst.
Theresult is:
r =
p =
4.0000
2.0000
-1
-3
k =
[]
Examining the solution we notethat the output for
bothrand pisarranged asavertical list.Thisway of
representing the outputis known asacolumn vector.
We note that the numbersreturnedin columnvector p
have anegative sign.Thisis because the result
calculated containsthepoles ofthe partial fraction
expansion. These are values ofthe variable which
make the denominator ofthe fractionzero. The
significance ofthiswillbecome clear later in the text
but fornow itisadequate to note the difference in
sign from what might have been expected.
1.8 SUMMATIONNOTATION
Inengineeringweoftenwanttomeasurethevalueofavariable,suchascurrent,voltage
or pressure.
Suppose we make three measurements of a variablex. We can label these measurementsx
1
,x 2
andx 3
. Inthis context, the numbers 1,2,3arecalledsubscripts.
Inmathematics,theGreeklettersigma,written ∑ ,standsfora‘sum’.Forexample,
the sumx 1
+x 2
+x 3
is written
3∑
k=1
x k
Notethatthesubscriptkrangesfrom1to3.Askrangesfrom1to3,x k
becomesx 1
then
x 2
and thenx 3
and the sigmasign tells us toadd up these quantities.
Ingeneral,
N∑
x k
=x 1
+x 2
+···+x N
k=1
This notation is often used to express some of the fundamental equations of electrical
circuitanalysis. Sometimes ‘Summation Notation’ isknown as ‘Sigma Notation’.