## 1 Introduction  
Plasma actuators have attracted a lot of research attention in recent years because of the associated advantages, such as the absence of moving components, fast response, and wide bandwidth. So far promising control outcomes have been revealed in suppressing flow separation [1,2], alleviating turbomachinery stall [3,4], and shock wave/boundary layer interaction [5,6]. A plasma synthetic jet actuator (PSJA) is one of the typical plasma actuators. The PSJA makes use of the heat released from the spark discharge process between electrodes within a small cavity where the electrodes are housed. As the velocity of the resulting jet is high (larger than 100 m/s), the PSJA is suitable for high-speed applications. It should be noted that the dielectric barrier discharge (DBD) plasma actuator with annular electrodes can also generate a vertical jet [7,8], and has the name of plasma synthetic jet actuator too [9]. However, the jet velocity from DBD-PSJA is much less. The present work has focused on the PSJA with the principle of spark discharge. As the diameter of the PSJA orifice is in the order of millimeters, to enlarge the affected area, an array containing distributed actuators is necessary for practical applications [10]. In 2009, [PERSON] et al. [11] attempted to reduce the jet exhaust noise using six PSJAs. Unfortunately, little effect was achieved, and it was concluded that this was as a result of an insufficient number of actuators. In 2012, an array of three PSJAs was used to control the unsteadiness of a shock wave/turbulent boundary layer by [PERSON] et al. [6]. In 2013, an attempt of using 20 PSJAs is made to reduce the trailing edge separation on a NACA-0015 profile [12,13]. Therefore, it can be found that there is a strong desire to implement more PSJAs in various flow controls to deliver a strong actuation. The arc discharge channel was revealed to feature negative resistant characteristics [14,15], namely, an increase in the current leads to a decrease in the voltage. As a result, the multichannel discharge circuit is difficult to power when using a single power supply. In 2005, [PERSON] et al. [16] tested the discharge in eight channels connected in parallel by using a 30 k\(\Omega\) current-limiting resistor. This large current-limiting resistor results in a large energy waste. As introduced above, 20 PSJAs were driven simultaneously, but 20 high power supply systems are used by Caruana [12], which is apparently not suitable for on-board applications. Recently, a multichannel discharge technique based on a voltage relay concept is proposed by the authors' group [17,18,19]. It allows multi-channel discharge without increasing the input voltage and without adding an additional current-limiting resistor. The new voltage relay circuit is a breakthrough. Despite its initial success in experimental testing, the manner in which the channel number affects the jet flow is still not clear, which is of great importance in optimizing the PSJA array's aerodynamic performance. One efficient way to study the effect of actuator number is to use an analytical model. In 2003, [PERSON] et al. [20] developed a first-order PSJA model. However, it can only predict the basic performance parameter, such as the jet velocity. In 2010, [PERSON] et al. [21] made a further development where the jet generation process is included, but without modeling the recovery process. In 2015, [PERSON] et al. [22] developed a model that can predict a PSJA's entire working cycle. However, the models introduced above only simulate the operation of a single PSJA. No model is available to predict the performance of a PSJA array, namely a number of PSJAs. The plasma synthetic jet actuator array (PSJA array) developed by the authors can enlarge the area affected by the actuator greatly, which is more suitable for practical application. A new analytic model is developed in the present work. It consists of two parts: The multichannel discharge model and the PSJA aerodynamic model.