To determine the range of K for which a system is stable, we need to analyze its characteristic equation. The stability of a linear time-invariant (LTI) system is directly related to the location of the roots of its characteristic equation in the s-plane.

System Stability Fundamentals

For a continuous-time linear system to be stable, all the roots of its characteristic equation must have negative real parts. This means all roots must lie strictly in the left half of the complex s-plane. If any root has a positive real part, the system is unstable. If any root has a zero real part (i.e., lies on the imaginary axis) and is not a simple root, or if there are repeated roots on the imaginary axis, the system is considered unstable or marginally stable in specific cases. For simple roots on the imaginary axis, the system is marginally stable.

The given characteristic equation for the system is:

$$6s + K = 0$$

Determining Stability Range for K

To find the range of K for system stability, we first need to determine the roots of the characteristic equation. In this specific case, it's a first-order system, which means there is only one root.

• Step 1: Isolate the root 's'
  From the characteristic equation \(6s + K = 0\), we can solve for \(s\) by rearranging the terms:
  $$6s = -K$$
  $$s = -\frac{K}{6}$$
• Step 2: Apply the stability condition
  For the system to be stable, the real part of the root \(s\) must be negative. Since \(s = -\frac{K}{6}\) is a real root, its value must be strictly less than zero.
  $$-\frac{K}{6} < 0$$
• Step 3: Solve for the parameter K
  To solve the inequality \(-\frac{K}{6} < 0\), we need to eliminate the negative sign and the denominator. We can multiply both sides of the inequality by \(-6\). It is crucial to remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.
  $$(-\frac{K}{6}) \times (-6) > 0 \times (-6)$$
  $$K > 0$$

Therefore, for the system described by the characteristic equation \(6s + K = 0\) to be stable, the value of the parameter K must be strictly greater than zero.

K Range for Stability

The determined range of K for which the system exhibits stability is \(K > 0\).