# Difference between revisions of "Errata: In Exercise 4.12, some additional assumptions are required for oscillation"

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{{errata page | {{errata page | ||

− | | chapter = | + | | chapter = Dynamic Behavior |

− | | page = | + | | page = 129 |

− | | line = | + | | line = 6 |

− | | contributor = | + | | contributor = K. J. Astrom |

− | | date = | + | | date = 8 Mar 2011 |

| rev = 2.10e | | rev = 2.10e | ||

| version = Third printing | | version = Third printing | ||

| text = | | 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> | ||

}} | }} |

## Latest revision as of 23:44, 10 August 2012

Return to Errata page |

Location: page 129, line 6

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

Show that, under suitable conditions on parameter values, the circuit gives an oscillation with a stable limit cycle with amplitude .

A better solution is to modify the exercise to read as follows:

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 is the amplitude error. Show that the system is modeled by

Determine so that the the circuit gives an oscillation with a stable limit cycle with amplitude .

(Contributed by K. J. Astrom, 8 Mar 2011)