Question

Order of the system of transfer function \((2s + 1) / (5s^2 + 2s + 1)\) is _______

• A. first order
• B. \(2^{\text{nd}}\) order
• C. \(3^{\text{rd}}\) order
• D. no order

Hint:

To quickly find the order of any transfer function, look at the highest exponent of \(s\) in the denominator.
If the denominator is of the form \(a s^2 + b s + c\), it is a second-order system.
If it is of the form \(a s + b\), it is a first-order system.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks to determine the order of the system described by the given Laplace domain transfer function.
The order of a dynamic system is a fundamental property that dictates its response behavior (such as speed, oscillation, and stability) to various inputs.

Step 2: Key Formula or Approach:
A general transfer function of a continuous system is represented as a ratio of polynomials in the Laplace variable \(s\):
\[ G(s) = \frac{N(s)}{D(s)} = \frac{b_m s^m + b_{m-1} s^{m-1} + \dots + b_0}{a_n s^n + a_{n-1} s^{n-1} + \dots + a_0} \] where:
\(N(s)\) is the numerator polynomial.
\(D(s)\) is the denominator polynomial.
The order of the system is defined as the degree of the denominator polynomial \(D(s)\), which corresponds to the highest power of \(s\) present in \(D(s)\) (assuming no pole-zero cancellations occur).

Step 3: Detailed Explanation:
We are given the transfer function:
\[ G(s) = \frac{2s + 1}{5s^2 + 2s + 1} \] Let us identify the numerator and denominator polynomials:
- Numerator polynomial, \(N(s) = 2s + 1\). The highest power of \(s\) in the numerator is \(1\) (so there is one zero at \(s = -0.5\)).
- Denominator polynomial, \(D(s) = 5s^2 + 2s + 1\). This polynomial is also known as the characteristic equation of the system.
Let us examine the powers of \(s\) in the denominator:
- The terms are \(5s^2\), \(2s^1\), and \(1s^0\).
- The highest exponent of the variable \(s\) in the denominator is \(2\).
Since there are no common factors between the numerator and denominator to cause pole-zero cancellation, the degree of the denominator polynomial is exactly \(2\).
Therefore, the transfer function represents a second-order system.

Step 4: Final Answer
The order of the system is second-order, which corresponds to option (B).