Spectrum of a VCO Modulated by White Noise

To illustrate the preceding theory,   we analyze the practical example of white voltage noise modulating a VCO.

We have a VCO with tuning sensitivity K_v (Hz/Volt).  The VCO control voltage has a DC level,  giving an output frequency f_c ,  and a noise voltage v_n  which has a white noise spectrum of e_n volts/sqrt(Hz).       

To determine the output spectrum we proceed as follows:

• consider the noise voltage to be made up of a large number of sinusoidal signals

• note that each sinusoidal signal results in very low level phase modulation

• so for each sinusoidal signal determine the power in the sidelobes

• add up the power in the sidelobes from each original sinusoidal component

We consider the noise to be made up of sine waves spaced at 1Hz intervals (to be rigorous we would have to consider the limit as this frequency interval went to zero).   As the noise is white,  there is equal energy in each 1Hz cell,  so we can represent each 1Hz cell at frequency f_m by

v(f_m) =   Ö2 e_n sin(2pf_mt + f_m)

.. (1.9)     

Note that the rms level in each 1Hz is e_n as required,  but with random phase f_m.   Each of these sinusoids results in narrowband frequency modulation,  with modulation index given by

b(f_m) =   Ö2 e_n  K_v / f_m 

.. (1.10)     

Note that K_v  is in Hz/volt and  f_m is in Hz.   As long as the total modulation index is small (as it typically is) then each 1Hz component of v_n contributes sidebands of amplitude 

Noise Sideband Amplitude =    e_n  K_v / (Ö2  f_m )

.. (1.11)     

relative to the carrier.   Thus the SSB Phase Noise is given by

L_f( f_m ) = e_n²  K_v² / (2  f_m² )

.. (1.12)      

Note that (e_nK_v)  is the rms deviation per Hz,  allowing the use of the following universal charts

Figure 2 - Universal VCO Noise Modulation Chart 1

Figure 2 shows how the phase noise at particular offset frequencies (100Hz, 1kHz, 10kHz, 100kHz and 1MHz) varies according to the RMS frequency deviation per Hz bandwidth - that is e_n. 

Example:  The noise from a 1kW resistor (4nV/sqrt(Hz) ) modulating a 10MHz/V VCO results in an rms deviation of 0.04Hz,  so the chart reveals that the  phase noise will be -111dBc/Hz at 10kHz offset.

Figure 3 - Universal VCO Noise Modulation Chart 2

Figure 3 shows the typical output spectrum chosen values of e_nK_v ,  the RMS deviation per Hz.   

Example: The noise from a 1kW resistor (4nV/sqrt(Hz) ) modulating a 10MHz/V VCO,  the resulting rms deviation of 0.04Hz will give a spectrum slightly worse than the third curve from the top.