Different time constants for charging and discharging of a circuit with a constant current source and a capacitor
Here's the circuit I think you're describing:







simulate this circuit &ndash; Schematic created using CircuitLab

There's no time constant in this situation. The current source makes the capacitor voltage rise linearly. Time constants only appear when there's exponential decay.

When the current source is disconnected from the capacitor, there's no way for the capacitor to discharge. You can't just connect both ends to ground since the voltage across a capacitor is not allowed to change instantaneously in circuit theory.

Sources don't have input resistance since they have no inputs. They do have output resistance. If you include an output resistance with the current source, that will get you an exponential decay with a time constant, and the resistance will affect the time constant.

In general, it is possible to have different time constants for charging and discharging if you switch other components in or out of the circuit.