How do I figure out the equivalent resistance of a network of resistors on a microcontroller?
Quick Answer

The equivalent impedance at P.O.I. is simply the two resistors in parallel \$R_1 \parallel R_3\$.

Detailed Answer

We make the following assumptions in this analysis:


The node IN has "infinite" input impedance -- its impedance is high enough that near-zero current flows through \$R_2\$ and so \$R_2\$ has no influence on the equivalent impedance seen.
The node OUT has "zero" output impedance -- when it is high we can treat OUT as a dead short to 3.3V, and when it is low we can treat OUT as a dead short to ground.


The equivalent impedance for the other resistors is found using Thévenin's theorem, which basically says that we can model this point by a voltage \$V_{th}\$ and a resistance \$R_{th}\$. To find these values, we find the open-circuit voltage \$V_{oc}\$ at P.O.I. if it is left as an open circuit, and the short-circuit current \$I_{sc}\$ through P.O.I. if we tie it to ground. Then \$V_{th} = V_{oc}\$, and \$R_{th} = {V_{oc} \over I_{sc}}\$. So the analysis becomes pretty straightforward:

If OUT is high (3.3V):

$$V_{th} = V_{oc} = 3.3V$$
$$I_{sc}= {3.3V \over R_1 \parallel R_3}$$
$$R_{th} = {V_{oc} \over I_{sc}} = {3.3V \over {3.3V \over R_1 \parallel R_3}} = R_1 \parallel R_3$$

If OUT is low (0V):
$$V_{th} = V_{oc} = 3.3V \cdot {R_3 \over R_1 + R_3}$$
$$I_{sc}= {3.3V \over R_1}$$
$$R_{th} = {V_{oc} \over I_{sc}} = {3.3V \cdot {R_3 \over R_1 + R_3} \over {3.3V \over R_1}} = {R_1R_3 \over R_1 + R_3} = R_1 \parallel R_3$$

So in both cases the impedance at P.O.I. looks like \$R_1 \parallel R_3\$, but the impedance is connected to a different effective voltage at the other end, depending on the voltage at OUT.