Question

Consider the transfer function \[ H_c(s) = \frac{1}{(s + 1)(s + 3)} \] Bilinear transformation with a sampling period of 0.1 s is employed to obtain the discrete-time transfer function \( H_d(z) \). Then \( H_d(z) \) is _________

• A. ( \frac{(1 + z^{-1})^2}{(19 - 21z^{-1})(23 - 17z^{-1})} \)
• B. ( \frac{(1 - z^{-1})^2}{(21 - 19z^{-1})(17 - 23z^{-1})} \)
• C. ( \frac{(1 + z^{-1})^2}{(21 - 19z^{-1})(23 - 17z^{-1})} \)
• D. ( \frac{(1 + z^{-1})^2}{(21 - 19z^{-1})(17 - 23z^{-1})} \)

Hint:

The bilinear transformation is commonly used to convert continuous-time transfer functions to discrete-time transfer functions while preserving stability.

Answer 1

The Correct Option is C

Step 1: Bilinear Transformation.
The bilinear transformation is used to map a continuous-time transfer function \( H_c(s) \) to a discrete-time transfer function \( H_d(z) \). The transformation is given by: \[ s = \frac{2}{T} \cdot \frac{1 - z^{-1}}{1 + z^{-1}} \] where \( T \) is the sampling period. Substituting \( T = 0.1 \) s, we apply this transformation to the given transfer function \( H_c(s) = \frac{1}{(s+1)(s+3)} \). Step 2: Analyzing the options.
- (A) Incorrect, the poles do not match after applying the transformation.
- (B) Incorrect, this is not the correct form after applying the bilinear transformation.
- (C) Correct, this matches the correct form after the bilinear transformation is applied to \( H_c(s) \).
- (D) Incorrect, the poles are not correctly transformed in this case. Step 3: Conclusion.
Thus, the correct answer is (C).