Answer to Question #246878 in Mechanical Engineering for MDS

Question

QUESTION 3 Figure 2 depicts the schematic diagram of a phase shift oscillator . Figure 2 Phase shift oscillator for Question 3 The circuit will oscillate if it is designed to have poles on the jv-axis. 3.1Show that the transfer function for the passive network in the circuit is given by 𝑉2(𝑠) 𝑉1(𝑠) = −1 (1 + 1 𝑠𝑅𝐶) (2 + 1 𝑠𝑅𝐶) 2 − 3 − 2 𝑠𝑅𝐶 (10) 3.2Show that the oscillator’s characteristic equation is given by 1 − 𝐾 1 (1 + 1 𝑠𝑅𝐶) (2 + 1 𝑠𝑅𝐶) 2 − 3 − 2 𝑠𝑅𝐶 = 0 (10) [20]

Answer 1

3.1)

We know that;

$\frac{V_f(s)}{V_2(s)}=\frac{1}{1+\frac{6}{sCR}+\frac{5}{s^{2}C^{2}R^{2}}+\frac{1}{s^{3}C^{3}R^{3}}}$

$V_1(s)=\frac{R}{R+\frac{1}{Cs}}V_f(s)$

$\frac{RCs+1}{RCs}V_1(s)=V_f(s)$

$\frac{V_2(s)}{V_1(s)}=\frac{-1}{(1+1sRC)(2+1sRC)^{2}}-3-2sRC$

3.2)

1+AB=0

$1+(\frac{R_1}{R_2})\frac{1}{1+\frac{6}{sRC}+\frac{5}{s^{2}R^{2}C^{2}}+\frac{1}{s^{3}R^{3}C^{3}}}$