Transfer Function (Solved Problem 2)

Introduction

This tutorial guides you through solving a problem related to the transfer function of an RC circuit acting as a high pass filter. Understanding transfer functions is crucial in control systems as they describe the relationship between the input and output of a system. This step-by-step guide will help you grasp the concepts and calculations involved.

Step 1: Understand the RC High Pass Filter Configuration

• An RC high pass filter consists of a resistor (R) and a capacitor (C) connected in series.

• The output is taken across the resistor.

• The formula for the transfer function (H(s)) in the s-domain is given by:

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

• Here, s represents the complex frequency variable in Laplace transform.

Step 2: Identify Circuit Parameters

• Determine the values of R and C in your specific circuit.
• For example, if R = 1 kΩ and C = 1 µF, these values will be substituted into the transfer function.

Step 3: Substitute Values into the Transfer Function

• Substitute the identified values into the transfer function formula.

• Using the earlier example:

  [ H(s) = \frac{s \cdot (1000) \cdot (1 \times 10^{-6})}{1 + s \cdot (1000) \cdot (1 \times 10^{-6})} ]

• This simplifies to:

  [ H(s) = \frac{s \cdot 0.001}{1 + s \cdot 0.001} ]

Step 4: Analyze the Transfer Function

• To analyze the behavior of the high pass filter, consider the following:
  • Magnitude: Calculate the magnitude of H(s) to see how the filter responds to different frequencies.
  • Phase: Determine the phase shift introduced by the filter.

Step 5: Plot the Frequency Response

• Use software tools such as MATLAB or Python to plot the frequency response:
  • Generate a range of frequencies.
  • Calculate the magnitude and phase for each frequency.
  • Plot these values to visualize how the filter behaves over the frequency spectrum.

Conclusion

This guide provided you with a structured approach to solving a transfer function problem for an RC high pass filter. Key steps included understanding the filter configuration, identifying parameters, substituting values, analyzing the transfer function, and plotting the frequency response. Familiarizing yourself with these concepts will enhance your understanding of control systems and their applications. For further study, consider exploring more complex filters or additional control system components.