Question

A unity feedback system has the open loop transfer function shown below. Find the imaginary axis crossing of the root locus s=±jω Enter ω to two decimal places. Do not enter ±j

HG(s)= K/s(s² +4s+4.5)

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Answer 1

The imaginary axis crossing of the root locus for the given unity feedback system occurs at ω = 1.41.

Step-by-step explanation:

In a unity feedback system, the root locus analysis is crucial for understanding the system's stability and behavior. The open-loop transfer function (OLTF) is provided as \(HG(s) = \frac{K}{s(s^2 + 4s + 4.5)}\). To find the imaginary axis crossing of the root locus, we set the real part of the denominator to zero since the imaginary axis crossing occurs when the real part of the characteristic equation is zero.

The characteristic equation is obtained by setting the denominator of the OLTF to zero:

`\[s^2 + 4s + 4.5 = 0\]`

Now, we can apply the quadratic formula to find the roots of this equation. The discriminant
`(\(\Delta\))` is calculated as
`\(\Delta = b^2 - 4ac\),` where
`\(a = 1\), \(b = 4\), and \(c = 4.5\). If \(\Delta > 0\),` the roots are real; if
`\(\Delta = 0\),` there is a repeated real root; and if
`\(\Delta < 0\),` there are complex conjugate roots.

After calculating
`\(\Delta\),` we find that
`\(\Delta < 0\)`, indicating complex conjugate roots. The roots can be expressed as
`\(s = -2 + j\omega\)` and
`\(s = -2 - j\omega\)`, where
`\(\omega\)` is the angular frequency. Setting the real part to zero, we get
`\(\omega = 1.41\)`, which is the imaginary axis crossing of the root locus. This result signifies the frequency at which the system transitions from stable to unstable behavior along the imaginary axis.