Frequency Modulation is encode a message signal on a carrier signal by changing the frequency of it.

the carrier signal has a basis frequency and the form is like: $S_C(t) = A_C cos( \omega_C t )$

the message signal can take any form. after the Modulation, the output signal is: $S_O (t) = A_C cos( \omega_C t + \Delta f \int{S_M dt } )$

the integrated message signal should be normalized to 1. and the Δf is the range of frequency change.

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The frequency produced by the Gunn Oscillator can be modulated, in order to matching the resonance frequency of the microwave chamber.

there is a adjusting knob on the Gunn Oscillator, which determine the basic frequency of the microwave. and a modulation signal is from the power supply. the frequency of the Gunn Oscillator is ranging from 8.6 GHz to 9.6 GHz.

the modulation signal is a simple linear function with frequency 23ms.

since our modulation frequency is a linear function. thus, the change of  the amplitude will change the output frequency of the microwave.

but the actually frequency modulation is by a Varicap or Varactor, which the capacitance can be changed by applied voltage. by changing the capacitor of the Gunn Oscillator, the output frequency changed.

However, the principle of frequency modulation unchanged.

In microwave engineering,  the Gunn Oscillator with the modulator will be called Voltage-Controlled Oscillator or VCO.

I still not fully understand the mathematic of the Gunn Oscialltor, and how it reacts with modulation signal. since i don’t have the internal structure of the oscillator.