When I put my 1/2" lens on my 1/2" sensor, the image is perfect, but when I put the same lens on my 1/3" sensor, everything is very blur, even when I focus at maximum, so I need to put an extension tube, but the image seems of lower quality (the extension tube disturbs the image ?). I don't understand why it's blur without extension tube, in Lens for 1/2" sensors, installed on a 1/3" sensor the picture f/w = a/b does not explains it. The image is supposed to be formed at the same distance, this shouldn't depend on the sensor size right ?
Indeed I didn't see but one is C and the other is CS so that's why the extension tube of 5mm was working.
One of your cameras is C-mount with 17.526mm registration distance, the other is CS-mount at 12.526mm. There is a 5mm difference in the registration distance of those two mounts. You need a 5mm extension tube to use your C-mount lens having a registration distance of 17.526mm with a CS-mount camera having a flange only 12.526mm in front of the sensor.
But I've still a question : can the extension tubes lower the quality of the images? (for instance if it's an extension tube of poor quality?)
Extension tubes contain no optical elements, so they can not degrade the image projected by the lens.
What a poorly made extension tube might do is cause a misalignment between the lens and sensor, so that the optical axis of the lens is not perpendicular to the surface of the sensor or so that the exact registration distance is not exactly what it should be.
In the first case, you would see evidence of tilt in the image, as one side of the sensor would be too far from the lens and the other side would be too close when the center is the proper distance. The tilt could be from top to bottom, or side to side, or along any axis between horizontal and vertical.
In the second case, the entire sensor would be slightly too far or too close to the lens, because your 5mm tube isn't exactly 5.000mm in length, but maybe 4.92mm or 5.11mm long.
Of course, a poorly made extension tube could also demonstrate both tilt and improper length, so that the two types of misalignments are combined for even poorer results.