The expansion of the magnetic field in a series of force-free constituents for the internal field and that of ‘semi’ force-free constituents for the external field results in the skew symmetry of the coupling tensor of a magnetohydrodynamic dynamo. The periodic solution which exists in this case is deduced by the method of eigenvalues. Based on the numerical results a dynamo model is proposed to draw up the characteristic features of the magnetographic observations carried along the central meridian and at the polar areas.