The critical point (-1, j0) is mapped to ________ on the Nichols chart.
(0, dB, -180°)
The Nichols chart is a graphical tool used in control systems engineering for frequency response analysis and stability assessment. It plots the logarithm of the magnitude of the open-loop transfer function in decibels (dB) against its phase angle in degrees. Understanding how specific points, like the critical point, map onto this chart is crucial for system analysis.
Critical Point Definition
In control systems, the critical point is typically referred to as the point (-1, j0) in the complex plane, often seen on a Nyquist plot or polar plot. This point is significant because it represents the condition where the magnitude of the open-loop transfer function is 1 and the phase angle is -180° (or 180°). When the Nyquist plot encircles this critical point, it indicates instability in the closed-loop system, according to the Nyquist stability criterion.
Mapping to Nichols Chart
To map any point from the complex plane (like (-1, j0)) to the Nichols chart, we need to convert its magnitude into decibels (dB) and determine its phase angle in degrees.
1. Magnitude Conversion
First, let's find the magnitude of the critical point (-1, j0):
Let the complex number be $$G(j\omega)H(j\omega) = -1 + j0$$
The magnitude is calculated as:
$$|G(j\omega)H(j\omega)| = \sqrt{(-1)^2 + (0)^2} = \sqrt{1 + 0} = \sqrt{1} = 1$$
Now, convert this magnitude into decibels (dB) using the formula: $$ \text{Magnitude (dB)} = 20 \log_{10} (\text{Magnitude})$$
$$ \text{Magnitude (dB)} = 20 \log_{10} (1) = 20 \times 0 = 0 \text{ dB} $$
So, the magnitude on the Nichols chart corresponding to the critical point (-1, j0) is 0 dB.
2. Phase Angle Determination
Next, let's find the phase angle of the critical point (-1, j0):
The complex number -1 + j0 lies on the negative real axis in the complex plane. The angle measured counter-clockwise from the positive real axis to the negative real axis is 180°. Alternatively, when considering phase lag in control systems, it is often represented as -180°.
$$ \text{Phase Angle } (\phi) = \arctan\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) $$
For (-1, j0):
$$ \phi = \arctan\left(\frac{0}{-1}\right) $$
Since the real part is negative and the imaginary part is zero, the angle is 180° or -180°. In the context of control systems and the typical range of a Nichols chart (which often extends to negative angles), -180° is the standard representation for the critical phase.
Thus, the phase angle on the Nichols chart corresponding to the critical point (-1, j0) is -180°.
Nichols Chart Point
Combining the magnitude and phase, the critical point (-1, j0) maps to the coordinates (0 dB, -180°) on the Nichols chart. This specific point on the Nichols chart is known as the "critical point" or "stability margin reference point," as it corresponds to the -1 point on the Nyquist plot.
Therefore, the critical point (-1, j0) is mapped to (0 dB, -180°) on the Nichols chart.
Important Questions from Nyquist Plot
______indicates not only whether a system is stable, but also its degree of stability and how stability may be imposed if necessary.
In Nyquist plot of a system on adding a pole at s = 0, then plot will -
The Nyquist plot of the transfer function \(G\left( s \right) = \frac{K}{{\left( {{s^2} + 2s + 2} \right)\left( {s + 2} \right)}}\)
Does not encircle the point (–1 + j0) for K = 10 but does encircle the point (-1 + j0) for K = 100 . Then the closed-loop system (having unity gain feedback) is
A closed-loop control system is stable if the Nyquist plot of the corresponding open-loop transfer function
The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE:
