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

Question

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.

Answer 1

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.

Answer 2

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

The usual assumption is that ω_m 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 ω_m 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 ω_m.