# Mathematical expression of Single-tone Phase Modulated Wave

### Mathematical expression of Single-tone Phase Modulated Wave

As we already knew, phase modulation is another type of angle modulation. The PM wave is obtained by varying the phase angle Φ of a carrier in proportion with the amplitude of the modulating voltage.

If the carrier voltage is expressed as under:

Then, the PM wave can be expressed as under:

Here, Φm = Maximum phase change corresponding to the maximum amplitude of the modulating signal.

For the sake of uniformity, let us modify the above equation as under:

where, m_{p} = Φm = Modulation index of PM = phase deviation.

The FM and PM waves look identical when their modulation index are identical. However, if we change the modulating frequency f_{m}, then m_{f} will change but there is no change in the value of m_{p}.

### Deviation Sensitivity of PM

For the modulating signal :

the instantaneous phase deviation is given by,

Where,k_{p} is the deviation sensitivity of phase modulation.

Thus,