Hmmm.
"Does nothing".....
First question in my mind is "how do you know?"
The carrier oscillator frequency trimmers can change the way VR4 behaves in a big way.
Have you set the 7.8015 on USB, 7.7985 on LSB and 7.800 on AM? If those are too far off, there won't be enough residual carrier for VR4 to make a difference you can easily measure.
What are you using to monitor the unsuppressed carrier output from the radio while transmitting in sideband?
I prefer to use a nearby receiver, with the clarifier turned away from the center of the channel. That way I get a heterodyne tone from the SSB transmitter's residual carrier output. We set VR4 for minimum S-meter reading on the monitor receiver. You will also 'hear' that null setting, since the noise you hear along with the weak carrier will be loudest when the carrier is nulled.
The driver and final bias settings will have a BIG influence on the radio's residual carrier output level in sideband transmit. If the bias is set too low on either the driver, final or both you may not see any response from adjusting VR4. Missing bias can completely remove your residual SSB carrier no matter where VR4 is set. Makes the SSB audio raspy, as well. Easiest clue is to turn down the mike gain so that the radio only shows a quarter of a Watt of SSB power. If the audio quality in your monitor receiver is raspy this way, and sounds cleaner when turned back up to normal power level, you have a bias trimmer or two that is set too low. Or a bias-circuit fault that prevents you from turning it up to a proper level.
There will be some interaction here. Too much SSB carrier can fool the meter's current reading when setting the bias trimmer, resulting in a bias setting that is too low for proper SSB audio. If a quarter Watt of residual carrier drives up the meter's current reading, you'll end up setting the bias-current trimpot too low for clean SSB audio.
I ALWAYS leave the test jumper for the driver UNHOOKED when setting the final bias. This shuts down the driver transistor and eliminates any carrier leakage that can falsely drive up the final transistor's current reading.
I would suspect a flaw in your setup before getting too excited about having some sort of mysterious failure in the radio's balanced modulator circuit.
The residual carrier leakage in SSB transmit from a Cobra 2000 should be VERY low when everything is aligned properly, but there should be enough of it to hear on a monitor radio a couple of feet away. Enough to set VR4 for a minimum S-meter reading, at least.
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