No, shutting off your water supply does not directly affect your air conditioner. Residential HVAC systems are completely separate from plumbing systems. They don't share any connections.
Understanding the Separation
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Independent Systems: Your air conditioner and your plumbing are entirely independent systems. Turning off the water won't impact the operation of your AC unit. [Reference: Does Shutting Off Water Affect AC? "Fortunately, residential HVAC systems are separate from home plumbing systems."]
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No Shared Components: There are no pipes or components shared between your AC unit and your water lines. [Reference: Does Shutting Off Water Affect AC? "Central air systems aren't connected to your home's drain pipes at all."]
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Water Heater Exception: While turning off the main water supply won't affect your AC directly, it will affect your water heater. If you shut off the water supply, your water heater needs to be turned off to prevent damage. [Reference: Vacation, turn off main water? Affect on water heater & A/C - "As others mentioned your water heater will need to be turned off if you turn the water off to prevent problems and your AC unit is not at all…"]
Indirect Effects
While there's no direct connection, turning off the water might indirectly impact your home's comfort:
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Reduced Humidity: If you're using a whole-home humidifier that relies on your water supply, turning the water off will disable it. This could potentially lead to slightly drier air, though this won't affect your AC's cooling ability itself.
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Water Heater Issues (Indirect): As mentioned, issues with the water heater due to lack of water supply are unrelated to the AC unit itself, but a malfunctioning water heater could impact overall home comfort.
Conclusion
In summary, turning off your home's water supply will not affect your air conditioner's operation. The two systems are completely separate.