## 2 Materials and Methods
### Performance Evaluation of the Proposed IAC Actuator  
In this section, we will evaluate the operability of the designed SMA-based IAC actuator through MATLAB (R2023b) simulation using mathematical algorithms and Simulink. We do not conduct experiments for the proposed actuator at this phase. Still, the same examinations will be performed in this section by integrating the designed actuator into the intake manifold and comparing the engine outputs with the real practical operated engines and literature experimental results. The proposed IAC actuator is assembled into the structure of the throttle body to control the flow of bypass air to the intake manifold through the displacement of its pintle, as shown in Figure 8.  
The SMA-based actuator closes the bypass, and fresh air from the environment enters the intake manifold through the throttle valve, as shown in Figure 8a. The actuator is in a closed state, and there is no supply of electric current to the SMA spring. This state of the actuator always occurs whenever the engine is turned off, or the driver presses the  
Figure 8: Simplified structural representation of engine intake manifold with throttle body holding an SMA-based IAC actuator. (**a**) Air flowing in the main path with the opening throttle valve and bypass closed by the SMA-based IAC actuator; (**b**) air passes through the bypass path opened by the SMA-based IAC actuator and full closed throttle valve. (1. throttle body, 2. Bypass, 3. throttle valve butterfly, 4. SMA-based IAC actuator). Figure 7: Numerical modelling process of the IAC actuator. (**a**) Generated meshing of IAC actuator model, (**b**) induced electric potential from the input current flowing into the actuator, and (**c**) temperature in the actuator to activate shape memory effect. accelerator pedal. Figure 8b shows the fresh air flowing through the bypass to enter the intake manifold with an opening actuator. The throttle valve butterfly is in full close with an angle of 90\({}^{\circ}\). This state of the actuator always occurs during engine idling conditions. The dynamic characteristics of air through the bypass and their corresponding engine idling characteristics are expressed in Equations (10)-(13). The main inputs of the equations forming the algorithms in this section include the actual parameters for the spark ignition engine from previous research with experiments [44; 45; 46] and practical engines [47; 48]. The triggering signal from the SMA-based IAC actuator is the opening and closing strokes that vary with time (\(S\)). \[\dot{m}_{\rm A}(t)=\rm D_{p}sin(\varphi_{c})S(t)\frac{P_{atm}(t)}{(RT_{atm}(t) )^{\frac{1}{2}}}\bigg{(}\frac{2P_{in}(t)}{P_{atm}(t)}\bigg{(}1-\frac{P_{in}(t) }{P_{atm}(t)}\bigg{)}\bigg{)}^{\frac{1}{2}} \tag{10}\]  
where \(\dot{m}_{A}\), \(T_{atm}\), \(P_{atm}\), \(P_{in}\), \(R\) are the air mass flow rate through the pintle of the SMA-based IAC actuator, ambient temperature, atmospheric pressure, air pressure in the intake manifold, and air constant with a value of 287.05 (\(\rm{J/kg^{\circ}C}\)). The opening or closing stroke (\(S(t)\)) of the actuator describes the air mass flow rate through the bypass (\(\dot{m}_{\rm A}\)) before reaching the intake manifold. The longer the opening stroke of the actuator, the more flowing air enters the intake manifold. This air mass flow rate was used to define the other characteristics of the engine in idle mode. \[\frac{d}{dt}P_{in}(t)=\frac{RT_{in}(t)}{V_{in}}\Bigg{(}\dot{m}_{\rm A}(t)- \frac{\dot{m}_{en}(t)}{1+\frac{1}{\lambda(t)s_{t}}}\Bigg{)} \tag{11}\]  
In the above, \(T_{in}\), \(V_{in}\), \(\dot{m}_{en}\), \(\lambda\) represent the temperature in the intake manifold, the volume of the intake manifold, the air mass flow rate that passes through and induced by the engine, and the air-fuel ratio. \(s_{t}\) is the stochiometric air-fuel ratio, with a value of 14.7 for the spark ignition engine.