Introductory Circuit Analysis (13th Edition)
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ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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![### Fourth Order Lowpass Filter Design
#### Transfer Function
The transfer function of the fourth order lowpass filter is given by:
\[
H(s) = \frac{1.4587 \times 10^7}{s^2 + 4868.2138s + 1.4587 \times 10^7} \times \frac{5.1506 \times 10^7}{s^2 + 2016.4802s + 5.1506 \times 10^7}
\]
#### Sallen-Key Lowpass Filter Design
To design this filter, cascade two Sallen-Key second-order lowpass filters using the equal R, equal C method. The Sallen-Key second-order lowpass filter circuit is shown with components \( R_a \), \( R_b \), \( R_2 \), \( C_1 \), and \( C_2 \). \( R_a \) and \( R_b \) ensure the circuit has a gain of 1 at \( \omega = 0 \). The parameter \( k_m = 1000 \).
##### Circuit Diagram Explained
- **Input Source**: \( V_{in} \)
- **Output**: \( V_o \)
- **Operational Amplifier**: Indicated as "OPAMP" internally denoted by U1
- **Components**:
- Resistors: \( R_a \), \( R_b \), \( R_2 \), \( R_A \), \( R_B \)
- Capacitors: \( C_1 \), \( C_2 \)
#### Tasks
(a) **Find \(\omega_{01}\) and \(Q_1\)** for the first second-order section with transfer function:
\[
H_1(s) = \frac{1.4587 \times 10^7}{s^2 + 4868.2138s + 1.4587 \times 10^7}
\]
(b) **Normalized Component Values**: Determine for the first second-order section as per \( H_1(s) \).
This setup helps in creating a fourth-order lowpass filter with desired characteristics by efficiently pairing Sallen-Key filters using straightforward calculations and known configurations.](https://content.bartleby.com/qna-images/question/d230794e-e523-4ec2-92f4-d9c51a06a3bf/9e9fcb33-2efd-4a80-97a2-c02747f4f4c0/s0aoxaq_processed.png)
Transcribed Image Text:### Fourth Order Lowpass Filter Design
#### Transfer Function
The transfer function of the fourth order lowpass filter is given by:
\[
H(s) = \frac{1.4587 \times 10^7}{s^2 + 4868.2138s + 1.4587 \times 10^7} \times \frac{5.1506 \times 10^7}{s^2 + 2016.4802s + 5.1506 \times 10^7}
\]
#### Sallen-Key Lowpass Filter Design
To design this filter, cascade two Sallen-Key second-order lowpass filters using the equal R, equal C method. The Sallen-Key second-order lowpass filter circuit is shown with components \( R_a \), \( R_b \), \( R_2 \), \( C_1 \), and \( C_2 \). \( R_a \) and \( R_b \) ensure the circuit has a gain of 1 at \( \omega = 0 \). The parameter \( k_m = 1000 \).
##### Circuit Diagram Explained
- **Input Source**: \( V_{in} \)
- **Output**: \( V_o \)
- **Operational Amplifier**: Indicated as "OPAMP" internally denoted by U1
- **Components**:
- Resistors: \( R_a \), \( R_b \), \( R_2 \), \( R_A \), \( R_B \)
- Capacitors: \( C_1 \), \( C_2 \)
#### Tasks
(a) **Find \(\omega_{01}\) and \(Q_1\)** for the first second-order section with transfer function:
\[
H_1(s) = \frac{1.4587 \times 10^7}{s^2 + 4868.2138s + 1.4587 \times 10^7}
\]
(b) **Normalized Component Values**: Determine for the first second-order section as per \( H_1(s) \).
This setup helps in creating a fourth-order lowpass filter with desired characteristics by efficiently pairing Sallen-Key filters using straightforward calculations and known configurations.
![(c) Find the scaled (both magnitude and frequency) component values for the first second-order section with transfer function \( H_1(s) \).
(d) Find \(\omega_2\) and \(Q_2\) for the second second-order section with transfer function
\[
H_2(s) = \frac{5.1506 \times 10^7}{s^2 + 2016.4802s + 5.1506 \times 10^7}
\]
(e) Find the normalized component values for the second second-order section with transfer function \( H_2(s) \).
(f) Find the scaled (both magnitude and frequency) component values for the second second-order section with transfer function \( H_2(s) \).](https://content.bartleby.com/qna-images/question/d230794e-e523-4ec2-92f4-d9c51a06a3bf/9e9fcb33-2efd-4a80-97a2-c02747f4f4c0/eis6i28_processed.png)
Transcribed Image Text:(c) Find the scaled (both magnitude and frequency) component values for the first second-order section with transfer function \( H_1(s) \).
(d) Find \(\omega_2\) and \(Q_2\) for the second second-order section with transfer function
\[
H_2(s) = \frac{5.1506 \times 10^7}{s^2 + 2016.4802s + 5.1506 \times 10^7}
\]
(e) Find the normalized component values for the second second-order section with transfer function \( H_2(s) \).
(f) Find the scaled (both magnitude and frequency) component values for the second second-order section with transfer function \( H_2(s) \).
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