Phasor diagram of Phase modulated signal

Question

While studying the phasor diagrams of Amplitude modulation and Frequency modulation, I am stuck at a problem involving the phasor diagram of an angle modulated signal whose equation is given by $$s(t) = A_c \cos(2\pi f_ct)-\alpha A \sin(2\pi (f_c+f_m)t)-\alpha A \sin(2\pi (f_c-f_m)t)$$ I have studied the procedure of how we write the signal as a real part of complex phasors and then plot the phasor by taking the carrier frequency as a reference but am unable to proceed.

Answer 1

The phasor diagram is a notion adapted from AC circuit analysis methods in which each sinusoid (at a fixed frequency) is represented as a rotating vector (rotating counterclockwise at the fixed frequency) but is shown on paper as a nonmoving vector (usually as a straight line with arrowhead) corresponding to the position of the vector at time $t=0$. (The whole diagram on the piece of paper as well as the observer_ is assumed to be rotating at the fixed frequency and so, as Albert Einstein observed, the phasor in the phasor diagram on the piece of paper appears to be in a fixed position as far as the observer is concerned). Thus, the phasor for $\cos(2\pi f_c t)$ is an arrow along the $x$ axis while the phasor for $-\sin(2\pi f_c t)$ (note the $-$ sign) is an arrow along the $y$ axis. Now that you have multiple frequencies involved, your phasor diagram should look something like a long right arrow with two uparrows at the arrowhead, something like $${\Large\longrightarrow\!\!}^{\large\textstyle\uparrow}$$ You also need to indicate that there are two phasors rotating in opposite directions (at relative frequencies $\pm f_m$) on that vertical phasor.

If the modulating signal had been $\sin(2\pi f_mt + \theta)$, the AM signal would have worked out to be $$A_c \cos(2\pi f_ct)-\alpha A \sin(2\pi (f_c+f_m)t+\theta)-\alpha A \sin(2\pi (f_c-f_m)t-\theta)$$ and those two vertical phasors would have separated into two different (non-vertical) phasors and been easier to see, but not as easy to render in MathJax and so I won't attempt to include a diagram.