Radio Frequency Integrated Circuit Design - Webs

I C_AVE = 1
Voltage-Controlled Oscillators
2� � 2�
0
I S e
307
Vtank sin (� )
2vT d� = IS IO�Vtank 2v � (8.107)
T
where IO (x) is a modified Bessel function of the first kind of order zero, and
IS is the saturation current. For fairly large V tank/2v T , there is an approximate
solution:
IC_AVE ≈ I Vtank S � e 2vT √
Vtank
2�
2vT (8.108)
Now we can write the gain of this part of the loop, which is a nonlinear
function of V tank (all other parts of the loop were described by linear functions):
A 3(s) = 2 ∂I C_AVE
∂V tank
= I Vtank S � e 2vT −
√�vTVtank
I Vtank S � e 2vT √ �
V
VT 3/2
tank
(8.109)
Here it is assumed that this part of the loop has poles at a significantly
higher frequency than the one in the VCO and the one in the mirror.
These equations can be used to design the loop and demonstrate the
stability of this circuit. The capacitor C 2 is placed in the circuit to create a
dominant and controllable pole P1 significantly below the other pole P2. Generally,
the frequency of P2 and gain of A 2(s) are set by the oscillator requirements
and are not adjustable. For more stability, the loop gain can be adjusted by
either changing the gain in A 1(s) (by adjusting the ratio of the current mirror)
or by adjusting the gain of A 3(s) (this can be done by changing the size of the
limiting transistors Q 3 and Q 4). Reducing the gain of the loop is a less desirable
alternative than adjusting P1, because as the loop gain is reduced, its ability to
settle to an exact final value is reduced.
Example 8.11 The Design of a VCO AAC Loop
Design an AAC loop for the VCO schematic shown in Figure 8.47. Use L =
5nH(Q = 10), VCC = 5V. The AAC loop should be set up so that the dc
gain around the loop is 40 dB to ensure that the final dc current through the
oscillator is set accurately. The loop is to have 0-dB gain at 1 GHz to ensure
that parasitic phase shift has minimal impact on the stability of the design.
Also, find the phase margin of the loop. Assume that Js = 1 × 10 −18 A/�m
and � = 100 in this technology. Use no more than a gain of 1:10 in the current
mirror.