Question

If the number of zeros is less than the number of poles, i.e. \( Z<P \), then the value of the transfer function becomes zero for \( s \to \infty \). Hence, some zeros are located at infinity and the order of such zeros will be:

• A. Z
• B. P + Z
• C. Z - P
• D. P - Z

Hint:

The number of zeros at infinity for a rational transfer function is simply the difference between the degree of the denominator (number of poles, P) and the degree of the numerator (number of zeros, Z). This only applies when \(P>Z\).

Answer 1

The Correct Option is D

Step 1: Define the general form of a transfer function.A rational transfer function \(H(s)\) is expressed as the ratio of two polynomials:\[ H(s) = K \frac{N(s)}{D(s)} = K \frac{s^Z + b_{Z-1}s^{Z-1} + \dots + b_0}{s^P + a_{P-1}s^{P-1} + \dots + a_0} \]Here, \(Z\) represents the number of finite zeros (degree of the numerator \(N(s)\)), and \(P\) represents the number of finite poles (degree of the denominator \(D(s)\)).
Step 2: Examine the behavior as \( s \) approaches infinity.To determine the limit of \(H(s)\) as \( s \to \infty \), focus on the highest-power terms in the numerator and denominator:\[ \lim_{s \to \infty} H(s) = \lim_{s \to \infty} K \frac{s^Z}{s^P} = \lim_{s \to \infty} K s^{Z-P} \]
Step 3: Apply the condition \( Z<P \).Given that \(Z<P\), the exponent \(Z - P\) is negative. Let \(Z - P = -N\), where \(N = P - Z\) is a positive integer.The limit is then:\[ \lim_{s \to \infty} H(s) = \lim_{s \to \infty} K s^{-N} = \lim_{s \to \infty} \frac{K}{s^N} \]Since \(s^N\) approaches infinity as \(s\) approaches infinity (because \(N>0\)), the term \(K/s^N\) approaches 0.
Step 4: Determine zeros at infinity.A transfer function is said to have zeros at infinity if it approaches zero as \(s \to \infty\). The order of these zeros is the value of \(N\) such that \(H(s)\) behaves as \(1/s^N\).Based on our analysis, \(N = P - Z\).Therefore, the order of the zeros at infinity is \(P - Z\).