## 1 Introduction  
Flexible printed electronic circuits have recently attracted attention as an alternative promising methodology due the additive process being more environmentally friendly and using less raw material compared to conventional lithography and chemical vapor deposition [1; 2; 3; 4; 5; 6]. As printed circuits become more complex and thinner, preventing circuit defects becomes a significant issue. Circuits are often scratched during the roll-to-roll (R2R) conveyance process, which can result in breaks in the conductive tracks, cracks, or pinholes [7; 8; 9; 10; 11]. This has prompted several suggestions of using flotation of the conveying substrate with a pneumatic cylinder [12; 13; 14; 15; 16]. An equivalent model of the flotation height, angle of air holes, and effective pressure has been proposed [12]. Also, the aerodynamic forces have been studied by considering the effect of flotation height and out-of-plane stability characteristics [13]. A governing partial differential equation was developed for the lateral deflection and stability of both straight and cambered substrates. Also, dimensionless parameters were proposed, including for the tension in and deflection of the substrate [14]. The flotation height has also been derived using an extensional resiliency model [15].  
We have previously suggested a method for preventing scratching of the printed pattern during R2R conveyance [17]. In the R2R conveyance process for multilayered printing, individual printing roll was rotated in precisely the same phase for the register control. Thus, the tension fluctuation of substrate was not able to be controlled, which resulted in the correlation between tension in substrate and flotation height. The correlation between the tension in the substrate and the flotation height was derived using the mass conservation law. The strain variation according to the aerodynamic lifting force was determined based on the floating height. In this approach, the substrate was assumed to be a spring with an empirical coefficient, which varied with the design of the air bar.  
One limitation of our previous study is that it investigated a fixed structure of the air bar and operating conditions. Even though the proposed mathematical model of the air bar system was validated in simulations and experiments, no previous study has investigated the design of the air bar in terms of the size, density, and distribution of air holes under various tension and velocity conditions. To prevent scratching the printed circuit due to roll contact, the distance between the roll surface and floating substrate needs to be investigated between the locations where it enters and exits the air bar.  
In the present study, we analyzed the optimal design for an air bar by considering the height of the substrate throughout the area where the substrate faces the air bar. FSI (fluid-structure interaction) numerical simulations were carried out to estimate the substrate behavior caused by pneumatic flotation. The distribution of air pressure in the space between the substrate and the surface of the air bar was first determined, and then the deformation of the substrate induced by this air pressure was calculated. The optimal hole density was determined from among several design candidates based on numerical simulations and experiments.