x\(\left(\theta\right)=R\left(1-\cos\theta\right)+L+x_{L}-\sqrt{L^{2}-R^{2}\operatorname{}\theta}\) (1)
where R is the length of the crank, L is the connecting rod, \(\theta\) is the angle of the crank measured from the top position (TP), and \(x_{L}\) is the distance between the top of the air chamber and the TP of the piston. The majority of the energy dissipation originates from the forced motion of the piston at the TP and the bottom position (BP) (see Fig. 3(a)). The piston head force (\(F_{\text{ph}})\) that is generated by an electric motor can be calculated using coordinate transformation in Equation (3). \(F_{\text{ph}}\left(\theta\right)=J(\theta)\bullet T\) (2) \(J\left(\theta\right)=\frac{\text{dθ}}{\text{dx}}=\frac{1}{-R\sin\theta\left(1+\frac{R\cos\theta}{\sqrt{L^{2}-R^{2}\operatorname{}\theta}}\right)}\) (3) where T is the nominal torque of the DC motor, and J is the Jacobian. From Equation (3), the Jacobian goes to infinity depending on the crank angle θ measured from the TP every multiple of π rad. (a) (b) Fig. 3 Modeling of a piston–crank mechanism for the compressor (a), compressor with double crank system (b).