The conventional voltage source inverter is only a buck converter and conventional current source inverter is only a boost converter. By adding a Z-Source network , the buck-boost capability can be achieved. The recently proposed Quasi-Z-Source inverter (qZSI)  is an important improvement to Z-Source inverter (ZSI), which greatly reduces the passive component stress. The current-fed qZSI can buck-boost voltage and achieve bidirectional power flow without replacing the diode with an active switch . It has lower current stress on inductor compared to current-fed ZSI. However, the analysis and control methods proposed in  are based on the assumptions that the capacitor voltage is almost constant and equal to the input voltage. These assumptions become invalid when the capacitor is very small or the lower power factor is low in some applications that the volume is a very crucial factor. The capacitor voltage has high ripple or even becomes discontinuous. In these cases, the circuit has two new operations modes except for the normal three modes, which is called discontinuous operation modes. This paper analyzes the characteristics of the discontinuous operation modes, and derives the critical conditions for these new modes under different control strategies. Simulation and experiment results are given to verify the theoretical analysis.