4.a is based on the Barkhausen criterion
which states that the looped system starts oscillating at frequency [F.sub.0] so that between the input of the LNA and the output of the resonator it has a voltage gain higher than 0 dB and a phase shift multiple of 360 [degrees].
Several novel delay cells have been demonstrated to compose the two-stage ring-VCO, but extra power is inevitably needed to provide an excess phase shift for oscillation satisfying Barkhausen criterion
. On the other hand, implementation of 4 stages of RO consumes considerable amount of power due to additional stages.
Though several 2-stage ring VCO can be composed by different delay stage, extra power is certainly needed to provide an excess phase shift for oscillation fulfilling well-known Barkhausen criterion
. Conversely, implementation of 4-stage ring VCO can be possible by differential topology, however, this oscillator consumes significant amount of power.
and applying the steady-state Barkhausen criterion
([I.sup.c] + j[I.sup.s])[H.sub.0] = V(1 + j2Q([omega]-[[omega].sub.0])/[[omega].sub.0]), the balance of the imaginary components leads to the following:
This relation is commonly known as the Barkhausen criterion, which states that the loop gain must be 1 and the loop phase shill multiples of 360[degrees] to obtain oscillation.
Since the oscillator has a group delay, the Barkhausen criterion changes to
This is known as the Barkhausen criterion
for the oscillation.
The Barkhausen criterion
for oscillation implies that the phaseshift in the loop must be zero and the gain equal to one.