# Significance of 1/s in Root Loci & Nyquist Stability

• maverick280857
In summary, the conversation discusses the significance of transforming s into 1/s and its impact on root loci and Nyquist stability. The transformation allows mapping points at infinity to the origin, giving a better understanding of high frequency behavior. However, it is debated whether this transformation is necessary as we already have an understanding of points at infinity. Additionally, modifying the laplace or Fourier transform may be necessary for accurately representing the high frequency region.
maverick280857
Hi

Suppose we have a closed feedback system with loop gain = L(s) = G(s)H(s). The characteristic equation is

$$1 + L(s) = 0$$

What is the significance of the transformation $s \rightarrow 1/s$ and what bearing does it have on root loci and Nyquist stability?

I can see that the points $s = \pm \infty$ will be mapped to $s = 0$.

Thanks,
Vivek.

Anyone?

I don't get why you transform s to 1/s. Can you elaborate more on the question? Do you mean the invertibility of the char. eq i.e. $(1+L(s))^{-1}$ ?

Last edited:
Well, the question really is: what happens to the Root Locus and the Nyquist Stability criterion when I replace s by 1/s? Are they valid? Also, what is the physical significance of such a transformation. Intuitively, I think that such a substitution allows us to map points at infinity to the origin (and conversely)...so, it allows us to get a better "idea" of the high frequency behavior. But I am not fully convinced.

But we have already an understanding of the points at infinity, it is completely meaningful when we take $s|_{j\omega}\to\infty$.

Besides that, though I am not sure, I don't think that it will map the high frequency region as such because you have to also modify the laplace or Fourier transform accordingly.

## 1. What is the significance of 1/s in Root Loci and Nyquist Stability?

The term 1/s in Root Loci and Nyquist Stability refers to the transfer function of a system in the Laplace domain. It is used to represent the inverse of time, and is essential in analyzing and designing control systems.

## 2. How does 1/s affect the Root Loci plot?

The presence of 1/s in the transfer function affects the Root Loci plot by creating a pole at the origin. This pole can influence the behavior and stability of the system, and its location on the plot can indicate the system's response to different inputs.

## 3. Why is 1/s considered important in Nyquist Stability analysis?

In Nyquist Stability analysis, the 1/s term is crucial as it helps determine the number of encirclements around the critical point on the Nyquist plot. These encirclements are used to determine the stability of a system, and 1/s plays a significant role in this calculation.

## 4. Can the location of 1/s on the Root Loci plot change the stability of a system?

Yes, the location of 1/s on the Root Loci plot can affect the stability of a system. If the pole at the origin is on the right-hand side of the plot, it can make the system unstable. However, if it is on the left-hand side, it can contribute to the stability of the system.

## 5. How does the presence of 1/s impact the overall stability of a system?

The presence of 1/s in a transfer function can affect the overall stability of a system. It can introduce a pole at the origin, which can change the stability margin and impact the system's response to external disturbances. Therefore, it is essential to consider 1/s when analyzing the stability of a system.

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