FOR BOTH CIRCUITS Derive the transfer functions. Find the frequency where the ou
ID: 2990591 • Letter: F
Question
FOR BOTH CIRCUITS
Derive the transfer functions.
Find the frequency where the output voltage is 90 degrees out of phase with the input
voltage. Find the amplitude of the output voltage at this frequency.
Find the frequency where the amplitude of the output voltage is 1/3 that of the input
voltage. Find the phase difference between the input and output voltages.
Are the frequencies found in step 2 and 3 the same?Please explain.
FOR BOTH CIRCUITS Derive the transfer functions. Find the frequency where the output voltage is 90 degrees out of phase with the input voltage. Find the amplitude of the output voltage at this frequency. Find the frequency where the amplitude of the output voltage is 1/3 that of the input voltage. Find the phase difference between the input and output voltages. Are the frequencies found in step 2 and 3 the same?Please explain.Explanation / Answer
circuit 1 :
call the middle node V , then applying nodal analysis to it u get :
(Vin - V) / R = V (Cs) + (V - Vout)/R
multiplying by R u get :
Vin - V = (RCs) V + (V - Vout)
Vin = (2 + RCs) V - Vout ... (1)
and nodal at right node gives :
(V - Vout)/R = (Cs) Vout
so V - Vout = (RCs) Vout
hence V = (RCs + 1) Vout ... (2)
substituting for V in eqn (1) gives :
Vin = (2 + RCs) (RCs + 1) Vout - Vout
=> Vout [ (2 + RCs) (RCs + 1) - 1] = Vin
so H1(s) = Vout / Vin = 1 / [ (RC)2 s2 + 3RCs + 1]
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circuit 2 :
call the middle node V , then applying nodal analysis to it u get :
Cs (Vin - V) = V /R + Cs (V - Vout)
multiplying by R u get :
(RCs)(Vin - V) = V + (RCs)(V - Vout)
Vin - V = V / (RCs) + V - Vout
=> Vin = V (2 + 1/RCs) - Vout ... (1)
and nodal at right node gives :
(Cs)(V - Vout) = Vout /R
so (RCs) (V - Vout) = Vout
hence V = Vout (1 + 1 / RCs) ... (2)
substituting for V in eqn (1) gives :
Vin = Vout (1 + 1 / RCs) (2 + 1/RCs) - Vout
=> Vin = Vout [ (1 + 1 / RCs) (2 + 1/RCs) - 1]
=> Vout / Vin = 1 / [ (1 + 1 / RCs) (2 + 1/RCs) - 1]
mult by (RCs / RCs) u get :
Vout / Vin = RCs / [ (RCs + 1) (2 + 1/RCs) - RCs]
=> Vout/Vin = RCs / [ 2RCs + 1 + 2 + 1/RCs - RCs ]
= RCs / [ RCs + 3 + 1/RCs ]
=> H2(s) = (RC)2s2 / [ (RC)2s2 + 3RCs + 1]
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