Practical_Antenna_Handbook_0071639586
A p p e n d i x A : U s e f u l M a t h 725 serve that the current and voltage through either of these families of components are in quadrature—that is, they are 90 degrees out of phase with each other. We say that capacitors and inductors are reactive elements, and it turns out to be very helpful to represent the reactive component of a signal or a circuit parameter such as impedance as imaginary, and to plot it on the y, or imaginary, axis. The real part of a signal or circuit parameter is plotted along the x axis. Throughout this book we will find many antenna and transmission line impedances expressed in the form Z = IN R + jX (A.6.4) Sometimes, however, we need the input admittance, Y IN , corresponding to that impedance. Specifically, Y IN 1 1 = = Z R + jX IN (A.6.5) This often turns out to be very inconvenient to work with because we can no longer easily see how to break the complex admittance into its real part (conductance) and a separate imaginary part (susceptance). However, a little math sleight of hand will help us. If we multiply an equation by 1, we don’t change the equation. Furthermore, 1 can be any messy number form divided by itself; for instance: A − jB 1 = A − jB So let’s multiply both the numerator and denominator of Eq. (A.6.5) by the term (R – jX): 1 1 R jX × 1 = × − R + jX R + jX R − jX R − jX = ( R + jX)( R − jX) R − jX = ( R 2 + jXR − jXR − j 2 X 2 ) R − jX = 2 2 R + X In other words, for an impedance of the form Z IN = R + jX, Y R − jX = R + X R X = (real part) − j (imaginary part) 2 2 2 2 R + X R + X = G (conductance) + jB (susceptance) IN 2 2 (A.6.6)
726 A p p e n d i x A : U s e f u l M a t h (R – jX) is called the complex conjugate of (R + jX). Multiplication of both the numerator and the denominator of a complex number such as that in Eq. (A.6.5) by the complex conjugate of the denominator (wherein all “j” terms are replaced with “–j” terms and vice versa) always results in a pure real denominator, allowing separation of the original complex number into separate real and imaginary components. Sample Base-10 Logarithms N Log 1 0.000 2 0.301 3 0.477 4 0.602 5 0.699 6 0.778 7 0.845 8 0.903 9 0.954 10 1.0 20 1.3 100 2.0 1000 3.0 5000 3.699 10,000 4.000 37,500 4.574 600,000 5.778 1,000,000 6.0
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726 A p p e n d i x A : U s e f u l M a t h<br />
(R – jX) is called the complex conjugate of (R + jX). Multiplication of both the numerator<br />
and the denominator of a complex number such as that in Eq. (A.6.5) by the complex<br />
conjugate of the denominator (wherein all “j” terms are replaced with “–j” terms and<br />
vice versa) always results in a pure real denominator, allowing separation of the original<br />
complex number into separate real and imaginary components.<br />
Sample Base-10 Logarithms<br />
N<br />
Log<br />
1 0.000<br />
2 0.301<br />
3 0.477<br />
4 0.602<br />
5 0.699<br />
6 0.778<br />
7 0.845<br />
8 0.903<br />
9 0.954<br />
10 1.0<br />
20 1.3<br />
100 2.0<br />
1000 3.0<br />
5000 3.699<br />
10,000 4.000<br />
37,500 4.574<br />
600,000 5.778<br />
1,000,000 6.0
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