FM Modulation – Achieving FM Modulation with a Single BJT

The instantaneous frequency of a signal \$A\cos(\phi(t))\$ where \$\phi(t)\$ is an arbitrary function of time is defined as the derivative of \$\phi(t)\$ if you want to measure frequency in radians per second and as \$\frac{1}{2\pi}\$ times the derivaive_ of \$\phi(t)\$ if you want to measure frequency in Hertz. Of course, in the common case of a fixed frequency this corresponds to the familiar \$\phi(t) = \omega_c t+\phi_0 = 2\pi f_c t + \phi_0\$.

The standard definition of a frequency-modulated signal is one in which the deviation of the instantaneous frequency (at time \$t_0\$, say), from the carrier frequency is proportional to the value \$x_m(t_0)\$ of the modulating signal \$x(t)\$ at time \$t_0\$. The constant of proportionality is denoted by \$f_{\Delta}\$ in your notation: a \$1\$ volt signal creates a deviation of \$f_{\Delta}\$ Hz. Thus, if \$A\cos(\phi(t))\$ is the FM signal, then we have that $$ \left.\frac{\mathrm d}{\mathrm dt}\phi(t)\right|_{t=t_0} = 2\pi f_c + 2\pi f_{\Delta} x_m(t_0) $$ so that the deviation of the instantaneous frequency \$f_c + f_{\Delta} x_m(t_0)\$ from the carrier frequency \$f_c\$ is \$f_{\Delta} x_m(t_0)\$, just as we want it to be. It then follows from the fundamental theorem of calculus that $$\phi(t_0) = \int_{0}^{t_0}2\pi f_c + 2\pi f_{\Delta} x_m(t_0)\, \mathrm dt = 2\pi f_c t_0 + \int_{0}^{t_0} 2\pi f_{\Delta} x_m(t_0)\, \mathrm dt$$ or, with a slight change in notation, the FM signal can be expressed as $$A\cos\left(2\pi f_c t + \int_{0}^{t} 2\pi f_{\Delta} x_m(\tau)\, \mathrm d\tau\right)$$ the way you have it. Note that \$A\$ is the amplitude of the FM signal and is fixed; it is the frequency that is varying. Surely we need to distinguish between the FM signal when it is created using a voltage-controlled oscillator with an amplitude of \$1\$ volt and when it comes out of the power amplifier and goes to the antenna with a power of 10 kW?

The frequency of this generator is determined by \$C_2||C_{bc}\$ of the transistor. \$C_5\$ connects the \$C_{bc}\$ in parallel with L and C2.

As long as the junction capacitance is voltage dependent, changing the base voltage changes \$C_{bc}\$ and thus the resonance frequency of the oscillator.

Of course, the quality of this modulation is not very high, but for such simple device it is acceptable.