Question #246878

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) = βˆ’1 (1 + 1 𝑠𝑅𝐢) (2 + 1 𝑠𝑅𝐢) 2 βˆ’ 3 βˆ’ 2 𝑠𝑅𝐢 (10) 3.2Show that the oscillator’s characteristic equation is given by 1 βˆ’ 𝐾 1 (1 + 1 𝑠𝑅𝐢) (2 + 1 𝑠𝑅𝐢) 2 βˆ’ 3 βˆ’ 2 𝑠𝑅𝐢 = 0 (10) [20]


1
Expert's answer
2021-10-06T00:30:51-0400

3.1)

We know that;

Vf(s)V2(s)=11+6sCR+5s2C2R2+1s3C3R3\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}}}

V1(s)=RR+1CsVf(s)V_1(s)=\frac{R}{R+\frac{1}{Cs}}V_f(s)

RCs+1RCsV1(s)=Vf(s)\frac{RCs+1}{RCs}V_1(s)=V_f(s)

V2(s)V1(s)=βˆ’1(1+1sRC)(2+1sRC)2βˆ’3βˆ’2sRC\frac{V_2(s)}{V_1(s)}=\frac{-1}{(1+1sRC)(2+1sRC)^{2}}-3-2sRC


3.2)

1+AB=0

1+(R1R2)11+6sRC+5s2R2C2+1s3R3C31+(\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}}}


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