Question

The system with the open loop transfer function \(\frac{1}{s(1+s)\) is:

• A. Type 2 and order 1
• B. Type 1 and order 1
• C. Type 0 and order 0
• D. Type 1 and order 2

Hint:

Given a general transfer function layout: \[ G(s)H(s) = \frac{K \cdot \prod (s + z_i)}{s^N \cdot (s^1 + a_1s^0)\cdots} \] - The value of the exponent $N$ immediately tells you the System Type. - The total count of all poles combined across the denominator tells you the System Order.

Answer 1

The Correct Option is D

Concept: The performance and behavior of a feedback control system depend heavily on two basic properties of its Open-Loop Transfer Function $G(s)H(s)$:
• System Type: Defined as the total number of independent poles located exactly at the origin ($s = 0$) of the complex $s$-plane.
• System Order: Defined as the highest power of $s$ present in the denominator polynomial of the transfer function when it is fully expanded. This represents the total number of open-loop poles in the entire system.

Step 1: Finding the System Type
The given open-loop transfer function is: \[ G(s)H(s) = \frac{1}{s(1+s)} \] Let us look at the independent term $s^N$ in the denominator that stands on its own outside of any polynomial brackets. Here, we can write the denominator as: \[ s^1 \cdot (s+1) \] The exponent of this standalone $s$ term at the origin is exactly $1$ ($N=1$). This tells us there is one pole located at $s = 0$. Therefore, the system is a Type 1 system.

Step 2: Finding the System Order
To find the system order, expand the full denominator polynomial in the transfer function: \[ D(s) = s(1+s) = s^1 + s^2 = s^2 + s \] The highest power of the variable $s$ in this expanded denominator expression is $2$. This means the system has two total poles ($s = 0$ and $s = -1$). Therefore, it is a 2nd Order (or order 2) system.

Step 3: Conclusion
Combining these two findings, the system is classified as Type 1 and order 2, which matches option (D).