Question

Which of the following statements is true?

• A. More zeros always improve transient response
• B. Zero in right hand plane affects steady-state error
• C. Improper systems are physically unrealizable
• D. Stability depends on zeros

Hint:

A system is physically realizable only if it is proper (\(n \ge m\)). Improper systems (\(m > n\)) act as high-order differentiators that would amplify high-frequency noise to infinite power, which cannot happen in real-world circuits.

Answer 1

The Correct Option is C

Concept: The structural configuration of a dynamic system is defined by its s-domain transfer function rational polynomial: \[ H(s) = \frac{N(s)}{D(s)} = \frac{b_m s^m + b_{m-1}s^{m-1} + \cdots + b_0}{a_n s^n + a_{n-1}s^{n-1} + \cdots + a_0} \] Where \(m\) represents the degree of the numerator polynomial (number of system zeros), and \(n\) represents the degree of the denominator polynomial (number of system poles).
• Proper System: A system is proper if the denominator degree is greater than or equal to the numerator degree (\(n \ge m\)).
• Improper System: A system is improper if the numerator degree exceeds the denominator degree (\(m > n\)).

Step 1: Explaining why improper systems are physically unrealizable.
Let us look at a simple improper system where \(m = 1\) and \(n = 0\): \[ H(s) = s \] This transfer function represents a pure ideal differentiator. If we evaluate its frequency response by substituting \(s = j\omega\): \[ |H(j\omega)| = |j\omega| = \omega \] As the input signal frequency (\(\omega\)) approaches infinity, the system's gain also approaches infinity: \[ \lim_{\omega \to \infty} |H(j\omega)| = \infty \] In a real-world physical environment, high-frequency thermal noise is always present. An improper system would amplify this high-frequency noise to infinite power. Because physical components cannot supply infinite power, instantaneous differentiation is impossible. Therefore, improper systems are physically unrealizable, making Statement (C) completely true.

Step 2: Evaluating why the alternate choices are false.

• Statement (A) is false: Adding zeros does not always improve transient performance. For example, adding a zero in the right-half of the s-plane introduces "undershoot" (non-minimum phase behavior), which degrades the transient response.
• Statement (B) is false: Steady-state error depends entirely on the system type (the number of pure integrators at the origin, \(s=0\)) and the low-frequency loop gain, not on right-half plane zeros.
• Statement (D) is false: Absolute system stability depends entirely on the locations of the system's poles (which must sit in the left-half of the s-plane). Zeros affect the shape and peak amplitudes of transient responses, but they do not alter absolute stability. Thus, statement (C) is the only true option.