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QUESTION 4 (10 marks) The operational amplifiers in the circuit in Figure 3 are ideal. (a) (7 marks) Using the state-variable method, show that the transfer function of the circuit in Figure 3 is that of a bandstop filter $H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{s^2 + R_4/(R_2R_3R_5C_1C_2)}{s^2 + s/(R_1C_1) + R_4/(R_2R_3R_5C_1C_2)}$ (b) (3 marks) Assume $C_1 = C_2 = 1\mu F$. Select resistor values to ensure that the circuit is overdamped and has the centre frequency of $10^6 rad/s.$

          QUESTION 4 (10 marks)
The operational amplifiers in the circuit in Figure 3 are ideal.
(a) (7 marks) Using the state-variable method, show that the transfer function of the circuit in Figure 3
is that of a bandstop filter
$H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{s^2 + R_4/(R_2R_3R_5C_1C_2)}{s^2 + s/(R_1C_1) + R_4/(R_2R_3R_5C_1C_2)}$
(b) (3 marks) Assume $C_1 = C_2 = 1\mu F$. Select resistor values to ensure that the circuit is overdamped
and has the centre frequency of $10^6 rad/s.$

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QUESTION 4 (10 marks)
The operational amplifiers in the circuit in Figure 3 are ideal.
(a) (7 marks) Using the state-variable method, show that the transfer function of the circuit in Figure 3
is that of a bandstop filter
H(s) = (Vout(s))/(Vin(s)) = (s^2 + R4/(R2R3R5C1C2))/(s^2 + s/(R1C1) + R4/(R2R3R5C1C2))
(b) (3 marks) Assume C1 = C2 = 1μ F. Select resistor values to ensure that the circuit is overdamped
and has the centre frequency of 10^6 rad/s.

Added by David P.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert David Morabito

Step 1

First, let's define the state variables: x1 = V1 (voltage across C1) x2 = V2 (voltage across C2) x3 = V3 (voltage at the output of the op-amp) Show more…

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QUESTION 4 (10 marks) The operational amplifiers in the circuit in Figure 3 are ideal. (a) (7 marks) Using the state-variable method, show that the transfer function of the circuit in Figure 3 is that of a bandstop filter: s^2 + R4/(R*Rg*R*C*C2) -------------------------------- Vin(s) (s^2 + s/(R*C1)) + R4/(R*R*R*C*C) (b) (3 marks) Assume C1 = C2 = 1F. Select resistor values to ensure that the circuit is overdamped and has the center frequency of 103 rad/s. R R3 W W Rs Figure 3: The electrical circuit for Question 4.

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