According to the Fourier transform, a complicated signal can be represented by the sum of sinusoidal signals with different frequencies, so we select sinusoidal signals with some certain frequencies to simulate the blade signals of the two-stage compressor in order to investigate how different perturbations influence the result of correlation measure and explain the mechanism of this stall warning approach.

As mentioned above, the rotation frequency of the shaft (*f*_{s} ) is 50 Hz, and the blade number of the first rotor is 47. Thus, the blade passing frequency (BPF) is 2350 Hz (47*f*_{s} ). Ignoring the blade-to-blade difference, we describe the steady blade signal as:

(2)

In axial flow compressors, two typical low frequency perturbations appear before the stall occurs: “modal wave” and “spike”. The “modal wave” rotates at a relatively low speed below 50% of rotor speed (*f*_{s} ), and “spike” propagates more quickly at speeds between 70% and 90% of the rotor speed (*f*_{s} ). According to the rotating speed of the “modal wave” and “spike”, 0.5*f*_{s} (25 Hz) and 0.9*f*_{s} (45 Hz) signals are chosen to represent the low frequency perturbations in the compressor. Due to the interaction effect between two adjacent blade rows, there are some perturbations whose frequencies are close to multiples of the BPF, so the signals with frequencies of 94.5*f*_{s} (4725 Hz) and 94.9*f*_{s} (4745 Hz) are selected to stand for the high frequency perturbations in the compressor. Here, four signals whose frequencies are 0.5*f*_{s} (25 Hz), 0.9*f*_{s} (45 Hz), 94.5*f*_{s} (4725 Hz) and 94.9*f*_{s} (4745 Hz) are added into the steady blade signal to represent four different perturbations. The expressions of these four signals are

(3)

Compared with the steady blade-passing signal, the power of perturbations in a compressor is rather small, so the amplitude *a*_{i} is set equal to 0.1*a*_{0} and 0.3*a*_{0} for each perturbation with different frequency.

Fig. 5 depicts the *Rc* value of the steady blade-passing signal. In Fig. 5, the value of *Rc* keeps at 1 from begin- ning to the end, which indicates a perfect periodicity of the blade-passing signal at the design point of the com- pressor. Figs. 6-9 depict the correlation measure results of the signal *P*_{0} with different perturbations *P*_{1} ,* P*_{2} ,* P*_{3} and *P*_{4} . The mean value of *Rc* for signals with each perturb- bation with different amplitude is detailed in Table 2.

It can be identified in Table 2 that, when the amplitude is 0.1*a*_{0} , the mean value of *Rc* for signals with different

**Fig. 5***Rc* of blade-passing signal

**Table 2** Mean value of *Rc* for signals with different perturbations

signals | *a*_{i}*=* 0.1*a*_{0} | *a*_{i}*=* 0.3*a*_{0} |

*P*_{0}*+P*_{1} | 0.980 | 0.841 |

*P*_{0}*+P*_{2} | 0.998 | 0.984 |

*P*_{0}*+P*_{3} | 0.980 | 0.835 |

*P*_{0}*+P*_{4} | 0.998 | 0.984 |

perturbation is 0.980, 0.998, 0.980 and 0.998, and if the amplitude increases to 0.3*a*_{0} , it decreases to 0.841, 0.984, 0.835 and 0.984. This result indicates that for signals with a certain frequency perturbation if the amplitude is amplified, the mean value of *Rc* will decrease with verifying degrees.

As evident from Table 2, the correlation measure results of signal *P*_{0}*+P*_{2} and *P*_{0}*+P*_{4} are the same, and the mean values of *Rc* for signal *P*_{0}*+P*_{1} and *P*_{0}*+P*_{3} are also highly identical. It should be noticed that the frequency differences between *P*_{1} , *P*_{3} and *P*_{2} , *P*_{4} are both 4700 Hz. This suggests that two perturbations between which the

**Fig. 6***Rc* of blade-passing signal with 25 Hz perturbation

**Fig. 7***Rc* of blade-passing signal with 45 Hz perturbation

**Fig. 8***Rc* of blade-passing signal with 4725 Hz perturbation

**Fig. 9***Rc* of blade-passing signal with 4745 Hz perturbation

difference of frequency is a certain value may have a similar influence on the mean value of *Rc*. For a certain perturbation *P*_{i} , its frequency *f*_{i} can be expressed as *f*_{i}*=* (*x*_{i}*+y*_{i} )*f*_{s} , *x*_{i} ∈N, *y*_{i} ∈[0,1). The correlation measure compares the blade-passing signals of two adjacent revolutions. The signal *P*_{0}*+P*_{i} at present and one revolution before can be written as

(4)

(5)

Comparing two equations shows that *x*_{i} has no influence to the difference between two signals, which indicates if the difference of frequency between two perturbations is equal to the integral multiple of shaft frequency this two perturbations will have an identical effect on the mean value of *Rc*. According to the periodicity property of sinusoidal signals, perturbation whose *y*_{i} is close to 0 or 1 brings fewer differences to signals of two adjacent revolutions than perturbation of which the value of *y*_{i} is near 0.5 does. This explains why the mean value of *Rc* for *P*_{0}*+P*_{2} and *P*_{0}*+P*_{4} is obviously higher than *P*_{0}*+P*_{1} and *P*_{0}*+P*_{3} .

Although the mean value of *Rc* for *P*_{0}*+P*_{1} and *P*_{0}*+P*_{3} is almost equal. The range of *Rc* in Fig. 6 is significantly larger than the *Rc* in Fig. 8. This difference in *Rc* range can be attributed to the short length of the calculating window. As can be seen in the Fig. 10, when the scale of calculating window is rather smaller than the perturb- bation wavelength, the influence that perturbation brings to the signal changes distinctly with time. In this figure, the *Rc* at *j* is lower than the *Rc* at time *k*. If the calculating window is extended from 3 blade pitches to 47 blade pitches (one revolution), the range of *Rc* for *P*_{0}*+P*_{1} and *P*_{0}*+P*_{3} is almost identical (Fig. 11). According to reference [15], in a compressor working at near stall point, the perturbations mainly consist of flow separation and vortex shedding on the blade. The scale of these flow phenomenon usually covers two to three blade pitches, so a calculating window covering three blade pitches is an appropriate choice. Hence, the distribution of *Rc* can be used to distinguish the low and high frequency perturb- bations for the range of *Rc* for signals with low frequency perturbations is large whereas for signals with high frequency perturbations it is small.

**Fig. 10** Influence of perturbation at different time

**Fig. 11***Rc* of signals with 25 Hz perturbation and 4725 Hz perturbation (*a*_{i}*=* 0.3*a*_{0} )

Thus, the mechanism of this stall warning approach can be explained as below. Considering that the unstea- diness of flow separation and vortex shedding, the frequ- ency components of the perturbations are complicated and when the operating point of a compressor is slowly driven to stall boundary, the power of the perturbations will grow and the value of *Rc* will decrease.