A-C only, thank you! ### Design and Analysis of a Fourth Order Lowpass Filter#### Transfer FunctionThe transfer function of the fourth-order lowpass filter is given by:\[H(s) = \frac{1.4587 \times 10^7}{s^2 + 4868.2138s + 1.4587 \times 10^7} \times \frac{5.1506 \times 10^7}{s^2 + 2016.4802s + 5.1506 \times 10^7}\]#### Filter DesignTo design this filter, we will cascade two Sallen-Key second-order lowpass filters using the equal R, equal C method. The Sallen-Key second-order lowpass filter schematic is shown below. The resistors \( R_a \) and \( R_b \) are chosen to provide a gain of 1 when \( \omega = 0 \). The design constant \( k_m \) is set at 1000.![Schematic Diagram](attachment://image1.png)#### Problem Statements(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) **Determine the Normalized Component Values** for the first second-order section based on the transfer function \( H_1(s) \).#### Schematic Details- **Input**: \( V_{in} \)- **Output**: \( V_o \)- **Components**: - **Capacitors**: \( C1 \), \( C2 \) - **Resistors**: \( R_a \), \( R_b \), \( R2 \) - **Operational Amplifier**: \( U1 \) (configured in a Sallen-Key topology)This setup is crucial for achieving the desired frequency response, allowing flexibility in tuning \(\omega_0\) and \(Q\). ### Steps and CalculationsThe text does not cover specific calculations but prompts the reader to solve for the natural frequency \(\omega_{01}\) and quality factor \(Q_1\), then normalize component values for practical implementation in circuit design. (c) Find the scaled (both magnitude and frequency) component values for the first second-order section with transfer function \( H_1(s) \).(d) Find \(\omega_{02}\) 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 \( H_2(s) \).

Solution (a) - For the first section, The transfer function is given…

(a) Find wo1 and Q1 for the first second-order section with transfer function 1.4587 × 107 H,(s) : s2 %3D + 4868.2138s + 1.4587 × 107 (b) Find the normalized component values for the first second-order section with transfer function H1(s).

(a) Find wo1 and Q1 for the first second-order section with transfer function 1.4587 × 107 H,(s) : s2 %3D + 4868.2138s + 1.4587 × 107 (b) Find the normalized component values for the first second-order section with transfer function H1(s).

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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Thermistors

A thermistor is a special type of resistor whose resistance is more temperature-dependent than standard resistors. The resistance of the thermistor decreases with increasing temperature. This is the basic principle of thermistors. This allows the thermistor circuit to detect very small temperature changes that cannot be observed with the Resistance temperature detectors (RTDs) or the thermocouples. Since the resistance of a thermistor is temperature dependent, it can be connected to an electrical circuit to measure body temperature. This high sensitivity to temperature changes makes thermistors very useful for accurately controlling and calibrating temperature measurements. In general, resistance increases with increasing temperature in most metals, but thermistors react negatively.

Question

A-C only, thank you!

### Design and Analysis of a Fourth Order Lowpass Filter

#### Transfer Function

The transfer function of the fourth-order lowpass filter is given by:

\[
H(s) = \frac{1.4587 \times 10^7}{s^2 + 4868.2138s + 1.4587 \times 10^7} \times \frac{5.1506 \times 10^7}{s^2 + 2016.4802s + 5.1506 \times 10^7}
\]

#### Filter Design

To design this filter, we will cascade two Sallen-Key second-order lowpass filters using the equal R, equal C method. The Sallen-Key second-order lowpass filter schematic is shown below. The resistors \( R_a \) and \( R_b \) are chosen to provide a gain of 1 when \( \omega = 0 \). The design constant \( k_m \) is set at 1000.

![Schematic Diagram](attachment://image1.png)

#### Problem Statements

(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) **Determine the Normalized Component Values** for the first second-order section based on the transfer function \( H_1(s) \).

#### Schematic Details

- **Input**: \( V_{in} \)
- **Output**: \( V_o \)
- **Components**:
- **Capacitors**: \( C1 \), \( C2 \)
- **Resistors**: \( R_a \), \( R_b \), \( R2 \)
- **Operational Amplifier**: \( U1 \) (configured in a Sallen-Key topology)

This setup is crucial for achieving the desired frequency response, allowing flexibility in tuning \(\omega_0\) and \(Q\).

### Steps and Calculations

The text does not cover specific calculations but prompts the reader to solve for the natural frequency \(\omega_{01}\) and quality factor \(Q_1\), then normalize component values for practical implementation in circuit design.

(c) Find the scaled (both magnitude and frequency) component values for the first second-order section with transfer function \( H_1(s) \).

(d) Find \(\omega_{02}\) 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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