Page 112 - Numerical Analysis Using MATLAB and Excel

Summary

• Two phasors A and B where A = a + jb and B = c + jd , are equal if and only if their real
parts are equal and also their imaginary parts are equal. Thus, A = B if and only if a = c and
b = . d
• The sum of two phasors has a real component equal to the sum of the real components, and an
imaginary component equal to the sum of the imaginary components. For subtraction, we
change the signs of the components of the subtrahend and we perform addition. Thus, if
(
)
(
)
A = a + jb and B = c + jd , then A + B = ( a + c + jb + d ) and A – B = ( a – c + jb – d )
• Phasors are multiplied using the rules of elementary algebra. If A = a + jb and B = c + jd ,
then AB⋅ = ac + jad + jbc bd = ( ac bd + jad + bc )
(
)
–
–
• The conjugate of a phasor, denoted as A∗ , is another phasor with the same real component,
and with an imaginary component of opposite sign. Thus, if A = a + jb , then A∗ = ajb .
–
• When performing division of phasors, it is desirable to obtain the quotient separated into a
real part and an imaginary part. This is achieved by multiplying the denominator by its conju-
gate. Thus, if A = a + jb and B = c + jd , then,

(
)
(
)
–
A a + jb ( a + jb cjd ) ( ac + bd + jbc – ad ) ( ac + bd ) ( bc – ad )
---- = -------------- = ------------------------------------- = ------------------------------------------------------ = ----------------------- + j----------------------
(
)
–
2
B c + jd ( c + jd cjd ) c + d 2 c + d 2 c + d 2
2
2
• The relations e jθ = cos θ + jsin θ and e – jθ = cos θ jsin θ are known as the Euler’s identi-
–
ties.
• To convert a phasor from rectangular to exponential form, we use the expression
1 b ⎞
⎛
–
2 2 jtan -- a ⎠ -
⎝
a + jb = a + b e
• To convert a phasor from exponential to rectangular form, we use the expressions
Ce jθ = Ccos θ + jCsin θ
– jθ
–
Ce = Ccos θ jCsin θ
• The polar form is essentially the same as the exponential form but the notation is different,
that is,
∠
Ce jθ = C θ
and it is important to remember that the phase angle is always measured with respect to the
θ
positive real axis, and rotates in the counterclockwise direction.

• The rectangular form is most useful when we add or subtract phasors; however, the exponen-
tial and polar forms are most convenient when we multiply or divide phasors.

Numerical Analysis Using MATLAB® and Excel®, Third Edition 3−25
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