General FM expression; FM using phase modulator

General expression for FM

Let the carrier be $c(t) = A_c\cos 2\pi f_c t$. A general angle-modulated wave is

$$s(t) = A_c\cos\theta_i(t)$$

The instantaneous frequency is $f_i(t) = \frac{1}{2\pi}\frac{d\theta_i}{dt}$.

In FM, the instantaneous frequency varies linearly with the message $m(t)$:

$$f_i(t) = f_c + k_f\, m(t)$$

where $k_f$ is the frequency sensitivity (Hz/V). Integrating,

$$\theta_i(t) = 2\pi\int_0^t f_i(\tau)\,d\tau = 2\pi f_c t + 2\pi k_f \int_0^t m(\tau)\,d\tau$$

Hence the general FM expression:

$$s_{FM}(t) = A_c\cos\left[2\pi f_c t + 2\pi k_f\int_0^t m(\tau)\,d\tau\right]$$

Single-tone case: for $m(t) = A_m\cos 2\pi f_m t$,

$$\theta_i(t) = 2\pi f_c t + \frac{k_f A_m}{f_m}\sin 2\pi f_m t$$ $$s(t) = A_c\cos\left[2\pi f_c t + \beta\sin 2\pi f_m t\right], \qquad \beta = \frac{k_f A_m}{f_m} = \frac{\Delta f}{f_m}$$

FM generation using a phase modulator

A PM wave is $A_c\cos[2\pi f_c t + k_p\, x(t)]$: its phase follows the input $x(t)$. Comparing with the FM expression, if the input to a phase modulator is the integral of the message, the output is FM.

 m(t)  +------------+ x(t)=int m  +-----------+  FM wave
 ----->| Integrator |------------>|  Phase    |---------->
       +------------+             | modulator |
                                  +-----------+
                                        ^
                                  carrier Ac cos wc t

With $x(t) = \int_0^t m(\tau)d\tau$:

$$s(t) = A_c\cos\left[2\pi f_c t + k_p\int_0^t m(\tau)\,d\tau\right]$$

This is FM with $k_p = 2\pi k_f$. This is the basis of the indirect (Armstrong) method: the phase modulator is built from a balanced modulator and a 90° phase shift of a crystal oscillator, giving NBFM with high carrier stability. Frequency multipliers and a mixer then convert it to wideband FM. (Conversely, a differentiator followed by a frequency modulator gives PM.)