A compact metal halide lamps have showed high luminous efficacy and long lamp life time, when they are ignited by an antenna-excited microwave discharge (hereafter abbreviated by AEMD). The impedance of the lamp in the steady state ignition has been derived to discuss properties and structures of the lamp, using the equivalent electric circuits for the microwave circuit and the lamp. The expressions for the resistance and reactance of the lamp impedance are applied to deduce the matching and resonance conditions to efficiently feed the microwave power into the lamp. It is found that these expressions not only explain measured lamp properties but also contribute to design the HID lamp structures.