View source for Errata: In Exercise 4.12, some additional assumptions are required for oscillation

{{errata page
| chapter = Dynamic Behavior
| page = 129
| line = 6
| contributor = K. J. Astrom
| date = 8 Mar 2011
| rev = 2.10e
| version = Third printing
| text = 
As written, some additional assumptions of the values of the resistances and capacitances are needed in order to obtain an oscillation.  For a simple fix, change the last line of the exercise to read
<blockquote>
Show that, <font color=blue>under suitable conditions on parameter values,</font> the circuit gives an oscillation with a stable limit cycle
with amplitude <amsmath>v_0</amsmath>.
</blockquote>

A better solution is to modify the exercise to read as follows:
<blockquote>
An op amp circuit for an oscillator was shown in Exercise 3.5.  The oscillatory solution for that linear circuit was stable but not asymptotically stable. A schematic of a modified circuit that has nonlinear elements is shown in the figure below (omitted; see text).

The modification is obtained by making a feedback around each operational amplifier that has capacitors using multipliers. The
signal <amsmath>a_e=v_1^2+v_2^2/\alpha^2-v_0^2</amsmath> is the amplitude error. Show that the
system is modeled by
<center><amsmath>
  \begin{aligned}
  \frac{dv_1}{dt} &= \frac{R_4}{R_1 R_3 C_1} v_2 + 
    \frac{1}{R_{11} C_1} v_1 (v_0^2 - v_1^2 - \frac{v_2^2}{\alpha^2}), \\
 \frac{dv_2}{dt} &= -\frac{1}{R_2 C_2} v_1 + 
   \frac{1}{R_{22} C_2} v_2 (v_0^2 - v_1^2 - \frac{v_2^2}{\alpha^2}).
  \end{aligned}
</amsmath></center>
Determine <amsmath>\alpha</amsmath> so that the the circuit gives an oscillation with a stable limit cycle
with amplitude <amsmath>v_0</amsmath>.
</blockquote>
}}