Question 3.1: Example of VCO Excess Phase Assume two VCOs are identical. O...
Example of VCO Excess Phase
Assume two VCOs are identical. One has a constant V_{C- nom} = 1.5V bias applied to its control line, and the other has a 1.5-V bias applied to its control line, but this voltage experiences a transient step of 300 mV for a period of 100 ps, 200 ps after the VCOs have been started. Each VCO has a K_{VCO} of 100 MHz/V and an f_{nom} of 2 GHz. Determine the total accumulated phase of each VCO after a period of 1 ns. What is the excess phase of the VCO that experiences the transient?
Each VCO will operate nominally at a frequency of 2 GHz, which corresponds to a period of 500 ps. Thus, in 1 ns, the first VCO will accumulate a total phase of 4π rad. The second VCO will run 300 mV · 100 MHz/V = 30 MHz faster for a time of 100 ps. Thus, during this time, f_{VCO} = 30 MHz. As a result, it accumulates an excess phase of
Excess phase = (2π · 30 MHz) ·100 Ps =1.88 · 10^{-2} rad
or 1.1°, in addition to the 4π of phase that it accumulated due to the dc voltage. Thus, it has been disturbed from its nominal operating point by 1.1°. This excess phase is the only phase that we generally consider when doing small-signal analysis for PLLs.