## 2 Consideration of the Energy Losses for Converting Chemically Stored Energy into Electrical Energy
The values for \(T\) and \(c_{p}\) of the supplied and exhaust gas are averaged over the time during which the specific valve is open. The volume change energy \(E_{vc}\) is the balance of the thermal energy, the wall heat losses and the thermal energy losses through the exhaust. The losses can be described by the thermal efficiency \(\eta_{th}\). The volume change energy, thus, results in:  
\[E_{vc}=E_{th}-E_{val}-E_{exh}=\eta_{th}\cdot E_{th}. \tag{5}\]  
The volume change energy leads to a motion of the piston, which in turns is always subject to friction. The friction losses arise mainly in our prototype in the piston rings, bearings and the valve train. The mechanical energy \(E_{M}\) is the subtraction of the friction energy from the volume change energy and can be described by the friction efficiency loss \(\eta_{Fr}\) as follows:  
\[E_{M}=E_{vc}-E_{Fr}=\eta_{Fr}\cdot E_{vc} \tag{6}\]  
As described in Equation (6), friction occurs, which can also be described as:  
\[P_{Fr}=v^{2}\cdot u=F_{Fr}\cdot v\, \tag{7}\]  
where \(\mu\) the coefficient of friction and \(v\) is the speed of the translator. The mechanical power \(P_{M}\) for the electrical machine can be expressed as:  
\[P_{M}=F_{EM}\cdot v, \tag{8}\]  
where \(F_{EM}\) is the force of the electric machine. Due to the oscillation of the translator, the system requires acceleration power \(P_{acc}\) in many operating points. The acceleration power can be expressed with:  
\[P_{acc}=m\cdot\frac{dv}{dt}\cdot v\, \tag{9}\]  
where \(m\) is the accelerated mass of the system and \(\frac{dv}{dt}\) the time derivative of the speed. The power output from the internal combustion engine \(P_{ICE}\) can be described with:  
\[P_{ICE}=F_{p}\cdot v=A_{cyl}\cdot p\cdot v, \tag{10}\]where \(p\) is the cylinder pressure, \(A_{cyl}\) is the area of the piston and \(F_{p}\) is the gas force. In addition, in the electrical machine the reluctance power \(P_{Rel}\) arises, which can be described as a function of position:  
\[E_{th}=\left(m_{fuel}-m_{unb}\right)\cdot LHV=\eta_{comb}\cdot E_{fuel}. \tag{2}\]0  
The power equilibrium on the mechanical transmission element results as follows:  
\[E_{th}=\left(m_{fuel}-m_{unb}\right)\cdot LHV=\eta_{comb}\cdot E_{fuel}. \tag{2}\]1  
In the case of a directly driven free-piston engine [17], the trajectory can be varied highly dynamically with the electric machine using the field oriented control. This is possible with the electrical machine force \(F_{EM}\), which is proportional to the current \(i_{q}\) via the force constant \(k_{F}\). The force of the electric machine can be expressed with:  
\[E_{th}=\left(m_{fuel}-m_{unb}\right)\cdot LHV=\eta_{comb}\cdot E_{fuel}. \tag{2}\]2  
The mechanical energy can be converted into electrical energy with the help of the electrical machine. The special feature of the free-piston engine is that it does not have a crankshaft. The required oscillation movement of the piston can be realized with the help of a linear electric machine. The electrical machines can be constructed in different ways [27; 33; 34]. Depending on the machine type, different power losses can arise. The power loss occurs when the iron is re-magnetized, which is referred to as hysteresis power loss \(P_{Hys}\). Eddy current power dissipation \(P_{Fi}\) can also arise. The sum of the hysteresis power loss and the eddy current power loss is referred to as the iron power loss \(P_{Fe}\):  
\[E_{th}=\left(m_{fuel}-m_{unb}\right)\cdot LHV=\eta_{comb}\cdot E_{fuel}. \tag{2}\]3  
In addition to the iron power loss, there is also a power loss due to the current \(i_{q}\) in a coil winding with ohmic resistance \(R\). This power loss is referred to as the copper power loss. The copper power losses are calculated in the d/q-reference frame as follows:  
\[E_{th}=\left(m_{fuel}-m_{unb}\right)\cdot LHV=\eta_{comb}\cdot E_{fuel}. \tag{2}\]4  
Due to the iron and copper power losses, the entire mechanical power on the electrical machine cannot completely be converted into electrical power.