Antenna Analyzer AIM4170 page 40
The second and third terms are imaginary, so we can them to get: j*(b*c + a*d)
The final result is:
(a+jb)*(c+jd) = (ac – db) + j(bc + ad)
This is tedious. Fortunately, the computer is good at this sort of thing, so we usually don’t
have to worry about the details.
Now we’ll finish checking our equation by plugging in one of the answers that we found:
Let x = 0.5+j1.323
x*x = (0.5+j1.323)*(0.5+j1.323) = - 1.50 + j1.323
Then, the whole equation = (-1.50+j1.323) - (0.5+j1.323) –2 = 0 (good)
To relate complex numbers to electrical circuits, we make the following observations:
Resistance is a real number.
Inductive reactance is a positive imaginary number.
Capacitive reactance is a negative imaginary number.
The impedance of a circuit is:
Z = R + jX, X = reactance and it can be positive (inductor) or negative (capacitor)
For example, suppose we have a 100pf capacitor (100*10^-12 Farad) in series with a 500
ohm resistor and the frequency is 7 MHz.
At 7 MHz, the capacitive reactance X= -1/(2*pi*7000000*100*10^-12) = -227 ohms
Note: the minus sign is very important.
Z = 500 – j227 = impedance of the series R-C circuit.
Real_part_of_Z = Re(Z) = 500
Imaginary_part_of_Z = Im(Z) = -227
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