RF and Microwave Engineering: Fundamentals of Wireless Communications

A.2 Logarithmic Representation

A.2.1 Dimensionless Quantities

Dimensionless real and positive quantities (like antenna gain G) and absolute values of complex-valued quantities (like absolute values of scattering parameter, i.e. |s_ij|) are often given in logarithmic scale. Logarithmic representations are advantageous if quantities vary over several orders of magnitude. Logarithmic values maintain a good resolution for both small and large values.

For power-based quantities (like antenna gain) a factor of 10 is used, whereas for voltage, current or field strength-based values (like scattering parameters) a factor of 20 is used. So, the logarithmic values are given as

A.44

where is the common logarithm. We do not use different symbols for linear and logarithmic representation. The pseudo-unit ‘dB’ (decibel) indicates the logarithmic scale and avoids confusing linear and logarithmic values: for example a gain of G = 1 in linear scale equals a gain of G = 0 dB in logarithmic scale. Table A.1 correlates commonly used linear and logarithmic values.

Table A.1 Conversion between linear and logarithmic scale

Logarithmic scale dB	Linear scale (Voltage ratio)	Linear scale (Power ratio)
+40	100	10 000 = 104
+30	≈31.6	1 000 = 103
+20	10	100 = 102
+10	≈3.16	10 = 101
+6	≈2	≈4