Tutorial: Part I
Describing Signal Voltage and Power
In most radio systems, we are interested in periodic currents and voltages since unchanging currents or voltages don't radiate as discussed above. Thus, a time-dependent signal voltage is usually written as the product of a magnitude (here v0) and a periodic function like the sine or cosine:
with an analogous expression for periodic currents. The instantaneous power dissipated into a load is the product of the voltage across the load and the current flowing through it. For a resistive load with a DC current flowing, we find:
To get the average power for a periodic signal, we add up the total power over a cycle and divide by the cycle time. This gives us a factor of (1/2), the average value of cos2:
Sometimes people introduce the root-mean-square (RMS) voltage to eliminate the extra factor of (1/2) from the expression for power:
It isn't always obvious which definition of voltage is being used; confusion on this score leads to erroneous factors of 2 floating around. In this book, we will always use peak voltages and currents rather than RMS quantities, and thus, explicitly include the factor of (1/2) in calculating average power. It is often of interest to display the amount of power associated with a sinusoidal signal of a given frequency as a power spectrum; a simple example of such a display for the single-frequency signal of equation is shown in Figure 4.
Figure 4. A Single-frequency Sinusoidal Signal and Corresponding Power Spectrum
Signal power can vary over a huge range in typical radio practice: power dissipated into a
typical 50-ohm load can range from tens of watts to 0.000 000 000 000 001 (10-15) watt. Related quantities, such as voltage, current, and gains and losses, span similar ranges. It is inconvenient to write out and manipulate such quantities as decimal numbers; instead, we use logarithmic notation. Recall that the base-10 logarithm is defined as:
For example, log(10) = 1, and log(1000) = 3. Negative logarithms denote numbers less
than 1: log(0.001) = -3.
It is traditional to use not raw logarithms, but deciBels (dB) in communications engineering. The ratio of two powers--for example, the ratio of the output power from an amplifier to the power that went in, which is the power gain of the amplifier--can be written in dB as: