Modelling of Rotational Speed of a Digital Spin Coater Using Multi-Level Periodic Perturbation Signals

Thin-film deposition technology has an essential role in the development of an integrated circuit (IC). Building integration of circuits requires thin-film preparation in semiconductor production. Since demands for smaller electronic components with faster processing time are inevitable because of the industry’s automation revolution, thin-film deposition techniques have also been advancing. Several methods in a thin-film deposition have been developed, such as chemical vapor deposition (CVD) [1], physical vapor deposition (PVD) [2], and spin coating [3].

Spin coating is the most preferred method with regards to its superiorities compared to the others [4]. It is a method for depositing thin films by spreading the solution on the substrate first, and then the substrate is rotated at a constant speed to obtain evenly distributed thin film deposits on the substrate [5]. One of the advantages is the cheap manufacturing cost. It is possible to build a spin coater with a low budget and minimum power consumption but having acceptable performances [6]. Besides, the spin coating method tends to have excellent performance in terms of uniformity [7]. It is crucial to reach uniformity of thin-film production to minimize the manufactured semiconductor's error tolerances [8].

It is necessary to pay attention to the factors that affect thin films' uniformity level in terms of making a suitable spin coater. One of them is the optimal rotational disc speed [9]. The spin coater should have an excellent rotational speed control with a stable and fast response.

Several works have succeeded in getting the desired response of spinning speed through implementing the concept of a closed-loop control system. In Hossain et al. [10], the implementation utilized ATmega 32 microcontroller together with a DC motor triggered by PWM signal. The desired speed and spinning time could be set using matrix keyboard and shown through LCD display. The digital spin coater produced in that research could vary the speed up to 3000 rpm and reach the desired speed within 10 seconds. Another trial to build a digital spin coater using the closed-loop control scheme was conducted by Manikandan et al. [11]. The hardware structure was quite similar to the previous work, except for the ARM-based microcontroller used, which is well-known for its fast processing time. The system could reach the speed reference up to 3000 rpm, despite the unevaluated timing response.

However, both works did not consider involving modeling steps for designing the controller. Based on the study in Triwiyatno et al. [12], building a mathematical model of a system that can incorporates its physical dynamics encourages a more robust and better controller design [12]. It can make the spin coater become more flexible for preliminary testing of the designed controller through simulation to get the best performance.

This paper presents modeling steps for a digital spin coater. It uses the concept of a single-input and single-output identification system to generate a representative mathematical model. Since commonly a DC motor is used as the actuator of a spin coater, a PWM (Pulse Width Modulation) signal is used to drive the speed [13]. This work considers the PWM signal as an input of the spin coater system and rotational spinning speed as an output. This approach simplifies the modeling steps because it is not compulsory to get detailed information about an actuator's internal physical parameter, which can be challenging to obtain. The system's generated input applies a multi-level periodic perturbation signal (MLPPS) to produce a model with considerable fitness value.

The article is organized as follows. In section 2, the hardware and electronic design of the digital spin coater are described. The following section elaborates on how the steps conducted to model the digital spin coater. The next part aims to present results and discussions about the obtained mathematical model. Section 5 concludes the results along with suggestions for possible future research about digital spin coater.

This study employs a system modelling method based on the concept of input and output identification of a system. In general, engineers can conduct system identification by directly applying a step signal as an input of the system, and its response in the scheme of an open-loop arrangement [14]. However, dynamic identification that involves random signals when generating input data is preferable than direct identification. The created signals enable the identification process to consider noise because its value varies with a specific frequency [15]. Dynamic identification in this study uses multi-level periodic perturbation signals as the input signals for the system.

3.1 Input-output data sets generation

The first stage in modelling of a spin coater's rotational speed is determining the input and output data sets. Several multi-level periodic perturbation signals (MLPPS) are prepared as input in the form of a PWM signal before it is used by the Arduino microcontroller to drive the DC motor, as shown in Figure 3. The motor's rotational speed detected by the rotary encoder sensor is utilized to obtain the real-time system response. Real-time system response for each input signal is managed as the corresponding output signal pair. Thus, there are four data sets, each containing input and output signal pairs, as depicted in Figure 4.

The input signal has a period Δt of 3 seconds, which is determined based on the settling time of the system defined from a given step input. With 15 times repetitions, the time spent for an input signal is 45 seconds. The PWM value that can be implemented into the spin coater system is limited from 0 to 255 because the signal used to drive a DC motor is an 8-bit PWM signal.

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Figure 3. Experimental setup for input-output data sets generation

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Figure 4. Input-output data set pairs of the spin coater system

3.2 Estimation of the mathematical model

Mathematical model estimation for the spin coater system was conducted using four generated data sets. Transfer function and state-space structures are utilized in this study since these structures have been considered by several studies for a system with DC motor as the actuator [16-18]. Four structures were built to express the system's model utilizing each data set; they are continuous transfer function, discrete transfer function, continuous state-space, and discrete state-space. Hence, there are sixteen candidates of models obtained from the estimation stage.

Model structures was arranged based on the differential equations of DC motor. It was started from physical equations for mechanical and electrical parts of DC motor, as listed in Eq. (1) and (2), respectively, with $\mathrm{K}$ is motor torque and electromotive force constant, $\mathrm{i}_{(\mathrm{t})}$ is armature current, b is motor viscous friction constant, $\omega_{(t)}$ is rotational speed of DC motor, $J$ is moment of inertia of the rotor, $V_{(t)}$ represents voltage source, $L$ is electric inductance, and $R$ is electric resistance.

$\mathrm{Ki}_{(\mathrm{t})}-\mathrm{b} \omega_{(\mathrm{t})}=\mathrm{J} \frac{\mathrm{d} \omega_{(\mathrm{t})}}{\mathrm{dt}}$      (1)

$\mathrm{V}_{(\mathrm{t})}-\mathrm{K} \omega_{(\mathrm{t})}=\mathrm{L} \frac{\mathrm{di}(\mathrm{t})}{\mathrm{dt}}+\mathrm{Ri}_{(\mathrm{t})}$      (2)

$\frac{\mathrm{d}}{\mathrm{dt}}\left[\begin{array}{c}\omega_{(\mathrm{t})} \\ \mathrm{i}_{(\mathrm{t})}\end{array}\right]=\left[\begin{array}{cc}\frac{-\mathrm{b}}{\mathrm{J}} & \frac{\mathrm{K}}{\mathrm{J}} \\ \frac{-\mathrm{K}}{\mathrm{L}} & \frac{-\mathrm{R}}{\mathrm{L}}\end{array}\right]\left[\begin{array}{c}\omega_{(\mathrm{t})} \\ \mathrm{i}_{(\mathrm{t})}\end{array}\right]+\left[\begin{array}{l}0 \\ \frac{1}{\mathrm{~L}}\end{array}\right] \mathrm{V}_{(\mathrm{t})}$;

$\mathrm{y}=\left[\begin{array}{ll}1 & 0\end{array}\right]\left[\begin{array}{c}\omega_{(\mathrm{t})} \\ \mathrm{i}_{(\mathrm{t})}\end{array}\right]$      (3)

From Eq. (1) and (2), we can obtain general state-space model of DC motor with $\omega_{(t)}$ as the output and $V_{(t)}$ as the input, as written in Eq. (3). Thus, it can be converted into discrete form with respect to Eq. (4), as shown in Eq. (5), with $\mathrm{T}$ represents sampling time and $\mathrm{k} \in \mathrm{Z}^{+}$. Eq. (1) and (2) can be transformed into mathematical model in the form of continuous transfer function as presented in Eq. (6). Utilizing forward rule method, Eq. (6) can be restructured into Eq. (7) that explains transfer function of DC motor in a discrete domain. Those four model structures of DC motor as shown in Eq. (3)-(7) are considered general equations for estimation process model conducted in this work.

$A=\left[\begin{array}{cc}\frac{-b}{J} & \frac{K}{J} \\ -K & -R \\ \frac{-R}{L} & \frac{-R}{L}\end{array}\right] ; B=\left[\begin{array}{l}0 \\ \frac{1}{L}\end{array}\right]$      (4)

$\left[\begin{array}{c}\omega_{((\mathrm{k}+1) \mathrm{T})} \\ \mathrm{i}_{((\mathrm{k}+1) \mathrm{T})}\end{array}\right]=\mathrm{e}^{\mathrm{AT}}\left[\begin{array}{c}\omega_{(\mathrm{kT})} \\ \mathrm{i}_{(\mathrm{kT})}\end{array}\right]+\left[\left(\mathrm{e}^{\mathrm{AT}}-\mathrm{I}\right) \mathrm{BA}^{-1}\right] \mathrm{V}_{(\mathrm{kT})}$; $\mathrm{y}=\left[\begin{array}{ll}1 & 0\end{array}\right]\left[\begin{array}{c}\omega_{(\mathrm{kT})} \\ \mathrm{i}_{(\mathrm{kT})}\end{array}\right]$     (5)

$\frac{\omega_{(s)}}{V_{(s)}}=\frac{\mathrm{K} / \mathrm{LJ}}{\mathrm{s}^{2}+(\mathrm{Lb}+\mathrm{RJ} / / \mathrm{LJ}) s+\mathrm{Rb}+\mathrm{K}^{2} / \mathrm{LJ}}$      (6)

$\frac{\omega_{(s)}}{V_{(s)}}$$=\frac{\mathrm{K} / \mathrm{LJ}}{(\mathrm{z}-1 / \mathrm{T})^{2}+(\mathrm{Lb}+\mathrm{RJ} / \mathrm{LJ})(\mathrm{z}-1 / \mathrm{T})+\mathrm{Rb}+\mathrm{K}^{2} / \mathrm{LJ}}$      (7)

3.3 Evaluation of the obtained models

Each candidate obtained from the previous step was validated with all predefined input-output pairs derived from a real-time measurement. Each validation resulted in fitness percentage that indicates the closeness of performances between obtained models and the physical system. Eq. (8) shows how the percentage value was obtained, where $\hat{p}$ is the model output signal, $p$ is the physical output, and $\bar{p}$ is its mean. The higher the fitness value, the closer the candidate model to the real system.

$\%$ fitness $=100 \times\left[1-\frac{\operatorname{norm}(p-\hat{p})}{\operatorname{norm}(p-\bar{p})} \quad \right]$      (8)

$\operatorname{MSE}=\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{n}}(\mathrm{p}-\hat{\mathrm{p}})^{2}$      (9)

For every structure, a candidate with the best fitness percentage mean is considered as the most appropriate candidate. Thus, four best candidates from four model structures are then evaluated by comparing one to another through the value of mean squared error (MSE). Eq. (9) explains how the MSE value is obtained, with n represents the number of data for the output signal. The smaller the mean squared error, the better the candidate model resembles the system's dynamic behaviour. The chosen model is the candidate with the lowest MSE value.

This work contributes on the development of the mathematical model of a digital spin coater. The most appropriate model for the rotational speed of digital spin coater has been acquired. The concept of input-output identification and utilization of MLPPS were successfully employed in this work. Validation tests were conducted by evaluating the fitness percentage using given data sets and comparing the model candidates' step-response and the plant using MSE value. Model TFD3 has shown the best performance based on validation tests among all candidates. Step-response evaluation can ensure that the chosen model is the best in representing the digital spin coater's dynamics when utilized in thin-film preparation. This finding can help in simulating a designed controller for a digital spin coater using the achieved model. For future studies, investigation on artificial intelligent based model will be very interesting.