Since the concept of spin superconductor was proposed, all the related studies concentrate on spin-polarized case. Here, we generalize the study to spin-non-polarized case. The free energy of non-polarized spin superconductor is obtained, and the Ginzburg-Landau-type equations are derived by using the variational method. These Ginzburg-Landau-type equations can be reduced to the spin-polarized case when the spin direction is fixed. Moreover, the expressions of super linear and angular spin currents inside the superconductor are derived. We demonstrate that the electric field induced by super spin current is equal to the one induced by equivalent charge obtained from the second Ginzburg-Landau-type equation, which shows self-consistency of our theory. By applying these Ginzburg-Landau-type equations, the effect of electric field on the superconductor is also studied. These results will help us get a better understanding of the spin superconductor and the related topics such as Bose-Einstein condensate of magnons and spin superfluidity.
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