How can I do a frequency sweep when the frequency modulation index is dependent on the amplitude?

I have a question regarding frequency modulation and the modulation index. I know that the modulation index can be given by

$$\beta = \frac{\Delta \omega}{\omega_m}$$

So the value of the modulation index is highly dependent on the value of $\omega_m$.

When we calculate the coefficients for Bessel functions, we need to get $J_n(\beta)$, which is a function of $\beta$. So that means, whatever $\omega_m$ we choose, will affect the value of $\beta$.

So then my question is how can I do a frequency sweep then with $s = j\omega$? The $\beta$ value is always changing and thus so is the amplitude of my signals. Can I just choose the lowest value of $\omega_m$, and therefore, the largest $\beta$ as my worst case, and thus have "constant" J values? Hope this is making sense.

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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.

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So the value of the modulation index is highly dependent on the value of ωm

The usual assumption is that ωm is the carrier frequency, which is constant, whereas the time-varying frequency ω(t) is not constant, and Δω is the peak-to-peak variation of ω(t).

If you are varying ωm very slowly (e.g. modulation timescale much longer than the carrier period), then you should be able to do your analysis as a function of ωm.

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