Given the following circuit, I would like to find the time constant \$\tau \$. I know that at \$t\rightarrow \infty\$, the capacitor acts as an open circuit, so no current flows in it. In an RC circuit, \$\tau=R\cdot C\$, so R in this case would be \$R_1 + R_2\$.


However, given the same circuit written with Norton equivalent, I get \$R=R_1 // R_2\$.

enter image description here

My question is: what is the right way to find \$R\$?


so R in this case would be R1+R2

No, R1 is not directly and uniquely in series with R2 hence that assumption is wrong.

The reasoning is simple; convert the voltage supply and series resistor (R1) to a current source equivalent. When you do that you'll find that the current source has R1 in parallel. And, if you look at your 2nd picture you can directly connect R1 and R2 in parallel.

So, when you reverse-convert the current source (in parallel with R1 and R2) to a voltage source, the new value for the voltage source series resistor is R1 || R2.

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