On page 625 you state that a system with positive feedback can be stable if its closed-loop gain is less than one. How do I simulate the loop gain of
the BMR in Fig. 20.15 to see that itís less than one?
One way to simulate the loop gain, vf /vin or AOLβ, is seen below (click for a larger image). The inductor is a short for DC but blocks the AC input
signal. Note the use of a huge capacitor and inductor (we can do this in a simulation ;-). For DC purposes this schematic is exactly the same as the
one seen in Fig. 20.15. See additional comments at the bottom of the page.
Note that, as mentioned on page 625, itís easy to increase the loop gain by increasing the capacitance on M2ís source to ground (this is bad!, see below).
A few more comments (for the analog gurus ;-), there are many ways to look at this circuit. Here is one. The output is the drain voltage of M2, vd2. Since the
VSG of M4 is set by this output voltage we could also say that the output is the drain current of M4 or M2. The open circuit gain, AOL, is vd2/vin which is
(1/gm4)/(R1 + 1/gm2) noting that this is less than one. Also note that gm3 = gm4 and gm2 = Kgm1 where, above, K is 4. The feedback voltage, vf, is connected
directly to the input (no external source, that is, itís self-biased) and given by gm4vd2/gm1. So we can write β = gm4/gm1 which is often close to one. The loop
gain is βAOL which is simply AOL (the open-loop gain) when β = 1 (which is the case above). Since the circuit employs positive feedback we can write the
closed-loop gain as ACL = AOL/(1 Ė AOLβ). So, as can be seen here, if AOLβ is greater than or equal to one the circuit becomes unstable. A good design will
ensure that the loop gain, AOLβ, is well below one.
Again, repeating the above information, we can increase the loop gain by shunting R1 with a capacitor. This drives the impedance at the source of M2 towards
ground with increasing frequency, this is bad!
Note that we are using an AC analysis when discussing stability. A DC analysis is used to determine operating point.
note that itís more correct, in
the last paragraph on page 625, to use ďloop gainĒ instead of ď
closed loop gainĒ
since a loop gain, AOLβ,
of 0.6 will
result in a closed loop gain, ACL, of 1.5 (which obvious isnít less than 1 ;-). This typo is fixed in the third and later printings of the third edition.