Abstract It is argued that a force-free degenerate electrodynamic (FFDE) magnetosphere of a Kerr black hole with 0 $\lt$$\Omega _{\rm F}$$\lt$${\Omega_{\rm H}}$ consists of the outer classical and inner general-relativistic domains. This is described by a simple DC dual-circuit model, with dissipative membranes as two loads at a “force-free infinity surface” (S$_{{\rm ff}\infty }$) with $\omega$$=$ 0 and at a “force-free horizon surface” (S$_{\rm ffH}$) with $\omega$$=$${\Omega_{\rm H}}$, where $\omega$, ${\Omega_{\rm H}}$, and $\Omega _{\rm F}$ are the frame-dragging, the horizon and the field line angular frequencies. It is beneath upper null surface, S$_{\rm N}$, at $\omega$$=$$\Omega _{\rm F}$ between the two domains that dual unipolar batteries (double EMF’s) exist back-to-back, oppositely directed, with a pair-creation gap between. The total energy flux ${\boldsymbol{S}}_E$ is a linear sum of the two fluxes: the hole’s outward spin-down energy flux ${\boldsymbol{S}}_{\rm SD}$ originating at S$_{\rm ffH}$ and the Poynting flux $\boldsymbol{S}_{\rm EM}$ emitted at S$_{\rm N}$ in both the outward and inward directions, with ${\boldsymbol{S}}_E$ being proportional to $\Omega _{\rm F}$, ${\boldsymbol{S}}_{\rm SD}$ to $\omega$ and $\boldsymbol{S}_{\rm EM}$ to ($\Omega _{\rm F}$$-$$\omega$) along each field line. Applying a perturbation method for a split-monopolar field with a spin-parameter $h$$\ll$ 1, the analytic solution of the stream equation is given, and the double eigenvalue problem due to the `criticality condition’ at the outer/inner fast surfaces S$_{\rm oF}$/S$_{\rm iF}$ and the `boundary condition’ at S$_{\rm N}$ is solved to yield the final eigenvalue $\Omega _{\rm F}$, in terms of ${\Omega_{\rm H}}$ and $f_{\rm H}$$=$ 0.5676. The ratio of the output power reaching S$_{{\rm ff}\infty }$ to the dissipation on S$_{\rm ffH}$ is $\epsilon$$=$ 1 $+$ (4$/$5)(1 $-$$f_{\rm H}$) $h^2$.