Nyquist Plot Rotation due to a Pole at s=0

The Nyquist plot is a powerful graphical tool used in control systems to determine the stability of a feedback control system. It plots the frequency response of the open-loop transfer function \(G(s)H(s)\) in the complex plane as the frequency \(\omega\) varies from \(-\infty\) to \(+\infty\). The shape and encirclements of the Nyquist plot around the critical point \((-1, j0)\) provide crucial information about system stability.

Understanding the Effect of Adding a Pole at s=0

Adding a pole at \(s = 0\) to a system's open-loop transfer function means multiplying the original transfer function by \(\frac{1}{s}\). A pole at \(s=0\) represents an integrator in the system. Let's consider the original open-loop transfer function as \(G(s)H(s)\). If we add a pole at \(s=0\), the new open-loop transfer function, let's call it \(G'(s)H'(s)\), becomes:

\[G'(s)H'(s) = \frac{1}{s} G(s)H(s)\]

To understand the effect on the Nyquist plot, we need to analyze the change in the magnitude and phase of the frequency response when we substitute \(s = j\omega\).

The original frequency response is \(G(j\omega)H(j\omega)\). The new frequency response is:

\[G'(j\omega)H'(j\omega) = \frac{1}{j\omega} G(j\omega)H(j\omega)\]

Phase Shift due to the Pole at s=0

The phase angle of a complex number \(Z = |Z|e^{j\phi}\) is \(\phi\). When we multiply two complex numbers, their phases add up. So, the phase of the new transfer function will be the sum of the phase of \(\frac{1}{j\omega}\) and the phase of the original \(G(j\omega)H(j\omega)\).

• The complex number \(\frac{1}{j\omega}\) can be rewritten as \(\frac{-j}{\omega}\).
• The phase angle of \(\frac{1}{j\omega}\) is \(\angle\left(\frac{1}{j\omega}\right) = \angle\left(\frac{-j}{\omega}\right)\). Since \(-j\) lies on the negative imaginary axis, its angle is \(-90^\circ\) or \(-\frac{\pi}{2}\) radians.

Therefore, the new phase \(\phi'\) of the system with the added pole at \(s=0\) will be:

\[\phi' = \angle G'(j\omega)H'(j\omega) = \angle\left(\frac{1}{j\omega}\right) + \angle\left(G(j\omega)H(j\omega)\right) = -90^\circ + \phi\]

This means that for every point on the original Nyquist plot, its phase angle is decreased by \(90^\circ\).

Impact on Nyquist Plot Rotation

A decrease in the phase angle by \(90^\circ\) corresponds to a specific rotation of the Nyquist plot in the complex plane:

• A positive change in phase (e.g., \(+90^\circ\)) results in an anti-clockwise rotation.
• A negative change in phase (e.g., \(-90^\circ\)) results in a clockwise rotation.

Since adding a pole at \(s=0\) introduces a phase shift of \(-90^\circ\), the Nyquist plot of the system will rotate clockwise by \(90^\circ\).

Example Illustration

Consider a very simple open-loop transfer function: \(G(s)H(s) = K\) (a constant gain, where \(K\) is a positive real number).

Its Nyquist plot is simply a point \(K\) on the positive real axis at \(K \angle 0^\circ\).

Now, let's add a pole at \(s=0\): \(G'(s)H'(s) = \frac{K}{s}\).

The frequency response is \(G'(j\omega)H'(j\omega) = \frac{K}{j\omega} = -j\frac{K}{\omega}\).

• The magnitude is \(\left|\frac{K}{\omega}\right|\).
• The phase is \(-90^\circ\).

As \(\omega\) varies from \(0^+\) to \(\infty\):

• When \(\omega \to 0^+\), the magnitude \(\to \infty\), and the phase is \(-90^\circ\). So, the plot starts at infinity along the negative imaginary axis.
• When \(\omega \to \infty\), the magnitude \(\to 0\), and the phase is \(-90^\circ\). So, the plot approaches the origin along the negative imaginary axis.

This example clearly illustrates that the entire plot has rotated \(90^\circ\) clockwise from its original position (a point on the positive real axis) to a line along the negative imaginary axis.

Conclusion on Nyquist Plot Modification

In summary, the addition of a pole at \(s=0\) to the open-loop transfer function of a system introduces a constant phase lag of \(90^\circ\) (\(-\frac{\pi}{2}\) radians) at all frequencies. This consistent phase shift causes the entire Nyquist plot to rotate in a clockwise direction by \(90^\circ\). This is a fundamental concept in frequency response analysis and stability assessment in control systems, especially when dealing with system type and steady-state errors.