# Simulation of Parameters of Plasma Dynamics of a Magneto Plasma Compressor
## 3 Electrical Parameters and Power Regimes for Discharge  
The most important feature of the electrical efficiency dependence on the average power \(\mathrm{n}_{\text{el}}(\mathrm{P}_{\text{el1}})=\int\limits_{0}^{\mathrm{t}_{ \text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}_{ \text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}_{\text{t}}}}}}}}}}}}}}}}} \mathrm{n}_{\text{u}(\mathrm{t})\mathrm{J}^{2}(\mathrm{t})/\mathrm{W}_{0} \mathrm{d}t\) is a relatively weak influence of \(\mathrm{P}_{\text{el1}}\) (\(\mathrm{P}_{\text{el1}}\)--the average electrical power released in the MPC discharge plasma during the first half-cycle of the current) on \(\mathrm{n}_{\text{el}}\): an increase in \(\mathrm{P}_{\text{el1}}\) by two orders leads only to an insignificant decrease (\(10\div 15\%\)) \(\mathrm{n}_{\text{el1}}\), in contrast to discharges with the ohmic mechanism of plasma heating. At identical power parameters of the storage geometry magneto plasma compressor electric efficiency, \(\eta_{\rm el}\) increases (with simultaneous growth of \(\overline{\rm R}_{\rm pl}=\frac{1}{\rm f_{i}}\int\limits_{0}^{t_{1}}R_{\rm pl}(t)dt\) (here and further, the sign from above \"_\" means averaging over time \(t\)) and a reduction in the discharge current amplitude) with a decrease in the density \(\rho_{0}\) of the surrounding gas. Both mentioned facts testify to the manifestation and significant influence on \(R_{\rm pl}\) and \(\eta_{\rm el}\) of energy dissipation in the electric discharge systems under consideration. Ponderomotor forces are the forces acting on plasma in the magnetic and electric field. There is an electromagnetic (ponderomotor) force (for a thin conductor, this expression corresponds to the Bio-Savard law) for plasma generation in the MPC discharge (for the case of magnetic permeability). The question about the role of energy dissipation in magneto plasma compressor discharges is one of the key ones and needs to be considered independently. The calculated value of effective plasma load resistance can be represented as:  
\[R_{\rm pl}(t)=\frac{1}{\rm j^{2}(t)}\int\limits_{\sqrt[]{}}^{\rightarrow}\rm j \,EdV=R_{\rm om}(t)+R_{\rm pldyn}\]  
where \(R_{\rm om}(t)\) is the value of that part of the total effective resistance, which is associated with the process of ohmic heating of the plasma, \(R_{\rm pldyn}(t)\) determines the nature of energy dissipation, caused by processes of transformation of electromagnetic energy into the work of ponderomotive forces, i.e., it is responsible for the plasma dynamic mechanism of plasma heating. All systematic investigations reveal one dimensionless criterion, the value of the ratio \(\lambda_{\rm R}=\overline{R}_{\rm om}/\overline{R}_{\rm pl}\) will mainly depend on its value. This regime parameter is:  
\[A_{\rm m}=\left(\frac{\overline{\rm P}_{\rm m}(z=0,\ t_{\rm m})}{\rho_{0}\rm D ^{2}}\right)^{1/2}\]  
where \(\overline{\rm P}_{\rm m}(z=0,\ t_{\rm m})\) is the pressure of the magnetic field for \((z=0,\ t_{\rm m})\), and \(\rho_{0}\rm D^{2}\) is the full velocity of the gas flow coming up onto the shock wave. As numerical simulations have shown, the average (during the first half-cycle of the current) velocity \(\rm D\) of the head shock wave in the axial direction is satisfactorily approximated by the expression:  
\[\rm D=K_{D}\left(\frac{P_{\rm el1}}{\rho_{0}\pi\pi_{2}^{2}}\right)^{1/3}\]  
where \(K_{D}\) is a relatively weak (\(\sim\)1 for 10\({}^{7}\)-10\({}^{10}\) W, \(\rho_{0}=10^{-2}\) kg/m\({}^{3}\)) function of \(P_{\rm el1}\) and \(\rho_{0}\). The defining parameter \(A_{\rm m}\) can be written as:  
\[A_{\rm m}=\frac{\mu_{0}^{1/2}\Gamma_{\rm m}}{K_{D}}\left(\frac{\rm J_{m}^{3}}{ \rho_{0}^{1/2}P_{\rm el1}}\right)^{1/3}\]  
where \(\Gamma_{\rm m}=\left(\pi\pi_{2}\right)^{1/3}\)f(r\({}_{2}\)/r\({}_{1}\))/ \(\left(2\sqrt{2}\pi\tau_{1}\right)\) is the geometric factor.