Design of hybrid circuits and systems based on CMOS and nano-device requires rethinking of fundamental circuit analysis to aid design exploration. Conventional circuit analysis with modified nodal analysis (MNA) cannot consider new nano-devices such as memristor together with the traditional CMOS devices. This paper has introduced a new MNA method with magnetic flux (Φ) as new state variable. New SPICE-like circuit simulator is thereby developed for the design of hybrid CMOS and memristor circuits. A number of CMOS and memristor-based designs are explored, such as oscillator, chaotic circuit, programmable logic, analog-learning circuit, and crossbar memory, where their functionality, performance, reliability and power can be efficiently verified by the newly developed simulator. Specifically, one new 3-D-crossbar architecture with diode-added memristor is also proposed to improve integration density and to avoid sneak path during read-write operation.