To go further with radar, we need a mathematical model for radio signals. The standard model of the sinusoid (sine wave), which dates back to the 19th century, is still valid. In a sinusoid, the signal rises and falls in a repeating pattern, which we identify as a wave.
For one repeating cycle, which the text identifies as a single revolution of a wheel, we say that phase has gone full circle, from 0° to 360°. The graph above shows two full cycles.
We can represent any point on a sinusoid by a combination of two numbers. The first of these is the peak amplitude, generally called the amplitude In the graph above, the amplitude is 1. The second number required is the phase. From the graph above, we can see the change in the signal as the phase varies.
When we wish to add radar signals, we are adding sinusoids. These sinusoids may be of different amplitudes, phases, and frequencies. When we consider signals at a particular frequency, we can just use amplitude and phase lead. The graph below shows several different phase lags. Note that a phase lead or lag is just a shift on the phase axis.
Mathematically, this is just
Amplitude · sin(Phase + Phase lead)
Question 1 - For phase leads of 0°, 60°, 120°, and 180°, draw curves of the resulting sine waves. Draw all the sine waves on a single graph.
Since everything is based on sine waves, we will represent each sinusoid by amplitude and phase lead. Although the terminology is mildly confusing, from now on we will say phase although phase lead is the technically correct term. Remember that when we say phase in the vector addition that is to follow, we really mean the phase shift of the sine wave relative to an unshifted ideal sinusoid.
As an example of this representation, consider amplitude = 1 and phase = 180°. Using the graph above, we see that the sine wave has now been shifted through half a cycle.
Question 2 - If we take two equal amplitude sine waves, one 180° out of phase with the other, and add them, what is the result?
We can simplify this operation by using what is known as a phasor. The phasor is just a vector, that is, it has amplitude and phase. The amplitude of the phasor will be the peak amplitude of the sine wave and the phase will be the phase lead of the sine wave relative to an unshifted sine wave. Some phasors of equal amplitude are shown below.
Question 3 --
What is the result of adding a 0° and a 180° phasor of equal amplitude?
To get a sum for other phasor additions, draw the first phasor. When you draw phasors, make sure you have the magnitude and direction drawn accurately so that the addition will work out right. Now take the second phasor and move it so that the start of the second phasor is at the end of the first phasor. Finally, draw a "resultant" phasor from the start of the first phasor to the end of the second phasor. This is shown in the drawing below:
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+ |  |
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Question 4 --
What is the result of adding a 0° and a 45° phasor of equal amplitude?
What is the result of adding a 0° and a 90° phasor of equal amplitude?
What is the result of adding a 0° and a 135° phasor of equal amplitude?
What is the result of adding a 0° and a 225° phasor of equal amplitude?
What is the result of adding a 0° and a 270° phasor of equal amplitude?