## 1 Introduction  
The blade is an important part of blade rotating machinery. Ensuring the normal operation of the blade is the key to ensuring the normal working of impeller machinery. When turbomachinery is running, the blade often vibrates violently in harsh environments. In such an environment and working condition, the performance and fatigue life of the blade are greatly reduced, which may eventually lead to the destruction of the turbomachinery and even lead to major accidents. Therefore, in order to ensure that the abnormal operation of the blade can be detected in time and to determine the remaining service life of the turbine engine, it is necessary to monitor the operating status of the blade. Of aeroengine failures, 60% can be attributed to vibration, of which 70% can be attributed to blade failure [1]. Therefore, online monitoring of blade vibration is needed to avoid accidents caused by vibration. The BTC is the distance between the tip of the engine blade and the engine casing. With the continuous development of turbine technology, the accurate measurement of the BTC also puts forward higher requirements. In order to reduce the weight of the turbine,turbine blades are mostly small and thin. When the distance between the blade tip and the engine casing is too large, resulting in an increase in the leakage of the blade tip, the engine efficiency decreases, and in severe cases, it also causes engine wheezing. Relevant data indicate that more than one-third of aerodynamic loss is due to the blade clearance flow [2]. An increase in BTC by an amount equal to 1% of the blade length reduces the efficiency by 0.8-1.2% [3]. For every 0.0254 mm reduction in the BTC, the fuel consumption rate decreases by 0.1% and the exhaust gas temperature decreases by 1 \({}^{\circ}\)C [4]. The fuel consumption rate loss caused by the tip clearance accounts for about 67% of the total loss of the blade profile and clearance seal [5]. When the BTC is too small, friction occurs between the casing and the blade tip, which even causes damage to the engine. Therefore, it is necessary to monitor the gas turbine's BTC online to detect abnormal operation in time and prevent accidents. Existing BTC monitoring methods mainly include five kinds of methods: the discharge probe method [6; 7], the optical fiber method [8; 9; 10; 11; 12], the capacitance method [13; 14; 15; 16; 17], the eddy current method [18; 19; 20; 21] and the microwave method [22; 23; 24]. The first method, the discharge probe method, also known as the spark discharge method, is a gap-monitoring method based on the principle of spark discharge. The BTC value is measured with a discharge probe device combined with a mechanical displacement mechanism, and the maximum accuracy is 25 \(\upmu\)m.  
The second method, the optical fiber method, uses optical fiber sensors for the non-contact measurement of BTC values. The optical fiber method has the characteristics of high measurement accuracy, high sensitivity and a fast frequency response. However, the measurement system has a complex structure, is susceptible to optical surface contamination, achieves long-term measurement with difficulty, has poor heat resistance and has a high cost. According to different measurement principles, this method can be divided into optical fiber methods such as the reflection intensity method, the laser triangulation method and the Doppler frequency shift method, with a maximum accuracy of 10 \(\upmu\)m.  
The third method, the capacitance method, is based on the working principle of bipolar capacitance, in which the end face of the conductive blade to be measured and the core of the sensor probe form two plates of the capacitance, and the distance between the plates is the BTC value. The measurement of the BTC is realized by detecting the change in the capacitance value.