Is the DC-Link capacitor (C1,C2) necessary in cascaded H-bridge to achieve more balance and reduce stress on switches?
Suppose the load is inductive, like a motor.
T2 T3 are ON, and current flows according to the arrows.
Now T2 T3 turn off, but current in the inductive load does not drop to zero, it keeps flowing according to the blue arrows. So it dumps current into the positive rail.
If the power supply has too much inductance (L) or if it can only supply power (not absorb power) then the current that is dumped into the positive rail has nowhere to go, which causes a fast rise in voltage, which blows the semiconductors.
If there is enough capacitance, then this current is absorbed by the capacitors, charging them, which causes supply voltage to rise a little bit. And on the next cycle, this energy will be reused when T2 T3 turn back on.
Basically, current flowing in an inductive load stores energy in the magnetic field. If the inductor is big enough (say, a motor) it can be quite substantial, and when the bridge attempts to reduce or reverse the current then this energy will have to be transferred from the inductor to somewhere else. If the circuit is not designed to store this energy in a capacitor, or dissipate it safely in a power resistor, then it will be dissipated somewhere unsafe, for example by avalanching the power devices and blowing them up.