Practical_Antenna_Handbook_0071639586

710 A p p e n d i x A : U s e f u l M a t h
Occasionally you will run across the word neper. The neper is to natural logarithms (ln)
what the decibel is to base-10 logarithms. Thus, they are related by a fixed ratio:
1 Np ≈ 8.69 dB or 1 dB ≈ 0.115 Np
(A.1.2)
At this point you may be thinking, “If we have easy access to calculators these days,
why do we care about logarithms at all?” Again, the primary reasons are these:
• Working with logarithms of numbers instead of the numbers themselves allows
us to deal with huge spans in the magnitudes of the numbers more easily. The
logarithm of 15,000,000,000 is 10.3. Which number would you rather work
with?
• Many electronic and auditory phenomena exhibit a logarithmic response to
applied signals. In these cases, displaying input/output relationships on graph
paper that has a logarithmic scale on one axis (called semi-log paper) or on both
axes (called log-log paper) often enhances our understanding of the physical
principles involved.
• When we compare the relative performance of two devices (two different
antenna designs, say), or we test a device under two different sets of conditions
(one antenna at two different heights above ground, say) we usually express the
result as a ratio of two numbers. As we shall see in Section A.2, the use of decibels
(which are logarithms multiplied by certain standard scale factors) is ideally
suited to our purposes.
A.2 Decibels
Throughout this book we shall use decibels to help us quantify and compare the performance
of antennas and feedlines. The beauty of decibels is that we can multiply and
divide what may be very large ratios with simple addition and subtraction of their
logarithms, appropriately scaled. This will be especially important whenever we are
determining the overall performance of an antenna or an entire RF transmission and
reception system consisting of multiple components.
Definition “A decibel is the base-10 logarithm of a number (or a ratio of two numbers),
multiplied by a scale factor.” For electronic circuits (including antennas) the formula is:
P
Decibels (dB) = 10log 2 P1
(A.2.1)
where P 1 and P 2 are power levels at two different points in the circuit or at one specific
point under two different sets of test conditions. (If you are not familiar with logarithms,
check out Sec. A.1 of this appendix.)
Example A.2.1 An amplifier delivers 1000 W of power (P 2 ) to a load when driven with
an input level of 25 W (P 1 ). What is the power gain of the amplifier, expressed in decibels?