Passive means a piece of hardware with no electronics, just cabling. Your DVI-HDMI cable is passive, as all it does is re-order the pins into a different plug. An active converter (which has a physical chip in it that decodes and re-encodes the signal) will make no difference.
Most likely your PC isn't set to the right resolution. Right-click on the desktop, go to Resolution, select the TV, and select the highest one it lets you. It might help.