https://r2.erweima.ai/mathbot/image/1721527729868dmhi2hhu.png

To solve for the magnitude response of the given transfer function \( H(s) \), we need to convert it to the frequency domain by substituting \( s = j\omega \), where \( j \) is the imaginary unit and \( \omega \) is the angular frequency.

Given transfer function:

\[ H(s) = \frac{s}{s + \frac{1}{RC}} \]

Substitute \( s = j\omega \):

\[ H(j\omega) = \frac{j\omega}{j\omega + \frac{1}{RC}} \]

To find the magnitude response \( |H(j\omega)| \), we compute the magnitude of the complex function:

\[ |H(j\omega)| = \left| \frac{j\omega}{j\omega + \frac{1}{RC}} \right| \]

The magnitude of a complex number \( \frac{A}{B} \) is \( \frac{|A|}{|B|} \), so we need to find the magnitudes of the numerator and the denominator separately.

Magnitude of the numerator \( j\omega \):

\[ |j\omega| = \omega \]

Magnitude of the denominator \( j\omega + \frac{1}{RC} \):

\[ |j\omega + \frac{1}{RC}| = \sqrt{(\omega)^2 + \left(\frac{1}{RC}\right)^2} \]

Therefore, the magnitude response \( |H(j\omega)| \) is:

\[ |H(j\omega)| = \frac{\omega}{\sqrt{\omega^2 + \left(\frac{1}{RC}\right)^2}} \]

This expression gives the magnitude response of the system in terms of the angular frequency \( \omega \), the resistance \( R \), and the capacitance \( C \).

Would you like more details or have any questions about this process? Here are 5 related questions you might consider:

1. How do you derive the phase response of the given transfer function?
2. What is the significance of the cut-off frequency in this magnitude response?
3. How do you determine the bandwidth of the system from the magnitude response?
4. Can you explain the behavior of the magnitude response at low and high frequencies?
5. How would the magnitude response change if the resistance \( R \) or capacitance \( C \) is altered?

**Tip:** Understanding the frequency response of a system helps in analyzing how the system behaves to different input frequencies, which is crucial in filter design and signal processing.

Learn how to solve for the magnitude response of the given transfer function H(s) = s / (s + 1/(RC)) using frequency domain analysis and complex number theory. This problem covers key concepts in transfer functions, including the substitution of s with jω, and the computation of magnitude responses. Suitable for advanced college students and professionals in electrical engineering and signal processing.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation