Array Solutions AIM4170 - Page 39

Antenna Analyzer AIM4170 page 39
Let the equation be: x^2 – x + 2 = 0
a = 1, b = -1, c = +2
Changing “c” from –2 to +2 gives us:
x = [1 + SQRT(1 – 8)]/2
and
x = [1 – SQRT(1-8)]/2
Now we have to deal the problem of evaluating the square root of –7.
We write this as: -7 = (-1) * (+7)
Note the SQRT(A*B) = SQRT(A)*SQRT(B), so SQRT(-7) = SQRT(-1)*SQRT(+7).
The SQRT(+7) is 2.646 and SQRT(-1) we define as “j”, so SQRT(-7)=j*2.646.
One solution to the equation is:
x = [ 1 + j2.646 ]/2 = 0.5 + j1.323
To confirm that the value x=0.5+j1.323 actually does cause the equation to equal zero,
we have to do some arithmetic with complex numbers.
Addition is straightforward:
The real part of one number is added to the real part of the second number.
Similarly, the imaginary part of one number is added to the imaginary part of the second
number.
(a + jb) + (c + jd) = (a+c) +j(b+d)
For example: ( 1 + j4 ) + ( 5 + j8 ) = (5+1) +j(4+8) = 6 + j12
Multiplication is a little tricky:
The two complex numbers have to be multiplied term by term:
(a+jb)*(c+jd) = a*c + jb*c + a*jd + jd*jb
We get 4 terms. Note that j*j = -1, so the last term = -d*b (this is a real number)
The first and fourth terms are real, so we can add them directly to get: (a*c – d*b)