Electronic – frequency modulation

If we go back to basic theory, we have a carrier signal of the form :-

\$E_c\cos\phi_c\$

... and a sinusoidal modulation signal of the form ...

\$E_m\cos(\omega_mt)\$

and if we let the frequency deviation be proportional to the modulation amplitude, so

\$\Delta\omega\propto E_m\$

the instantaneous frequency is given by ->

\$\dot{\phi_c}=\omega_c+\Delta\omega.\cos(\omega_mt)\$

Integrating this to get the instantaneous phase ->

\$\phi_c=\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\$

So the modulated output is ->

\$E_c\cos\Big[\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\Big]\$

As you say, the modulation index is dependent upon \$\omega_m\$ so the relative amplitudes of the spectral components will vary with \$\omega_m\$, but the modulation index is also a measure of the peak phase deviation, so if you want the spectral amplitudes to be independent of \$\omega_m\$ you must have \$\omega_m\propto \Delta_\omega \propto E_m\$, ie phase modulation.

One technique of producing phase modulation is to use a frequency modulator with pre-emphasis of the modulating signal to get the amplitude proportional to the frequency.

You've got the choke on the wrong side of the varactor. It needs to go between the audio source and the varactor, in order to keep the RF (oscillator) signal out of the audio circuit. You want the RF to get to the varactor.

Also, 10µF for the DC-blocking capacitor between the varactor and the oscillator tank is way too large (although it'll probably work in simulation). You really only need a few hundred pF here; say, 330 pF or 470 pF.