### 6.10. NICHOLS CHART

The Nichols chart [21] is a very useful technique for determining stability and the closed-loop frequency response of a feedback system. Stability is determined from a plot of the open-loop gain versus phase characteristics. At the same time, the closed-loop frequency response of the system is determined by utilizing contours of constant closed-loop amplitude and phase shift which are overlaid on the gain-phase plot.

In order to derive the basic Nichols chart relationships, let us consider the unity-feedback system illustrated in Figure 6.45. The closed-loop transfer function is given by

**Figure 6.45** Block diagram of a simple feedback control system.

or

where *M*(*ω*) represents the amplitude component of the transfer function and *α*(*ω*) the phase component of the transfer function. The radian frequency at which the maximum value of *C*(*jω*)/*R*(*jω*) occurs is called the resonant frequency of the system, *ω*_{p}, and the maximum value of *C*(*jω*)/*R*(*jω*) is denoted by *M*_{p}. For the system illustrated in Figure 6.45, we would expect a typical closed-loop frequency response to have the general form shown in Figure 6.46.

From Section 4.2 we know that a small margin of stability would mean ...