The roots of a system having a transfer function G(s) = 6(s + 2) / [(s + 3)(s + 4)]

12) The roots of a system having a transfer function

G(s) = 6(s + 2) / [(s + 3)(s + 4)]

will be: 

1. either -3 or -4 [Option ID = 2877]
2. either -2 or -4 [Option ID = 2878]
3. either -3 or -2 [Option ID = 2879]
4. either 3 or 4 [Option ID = 2880]

Answer: A

Solution

The Fundamental Concept:
In Control Systems, the "Roots of a System" refer to the roots of the Characteristic Equation, which is the denominator of the Transfer Function. These are also known as Poles.

• Zeros: Roots of the Numerator (top).
• Poles (Roots): Roots of the Denominator (bottom).

1. Identify the Transfer Function

G(s) =

6(s + 2)

(s + 3)(s + 4)

2. Set the Denominator to Zero

To find the system roots (poles), we solve for s in the characteristic equation:

(s + 3)(s + 4) = 0

3. Solve for 's'

Using the zero-product property:

• s + 3 = 0   ⇒   s = -3
• s + 4 = 0   ⇒   s = -4

Why the other options are incorrect:

The value s = -2 (found in options 2 and 3) is a Zero of the system, not a root. Options featuring positive values (3 or 4) are mathematically incorrect based on the factors given.

The roots of the system are -3 and -4, which is Option 1.

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