Q: How to calculate voltage of these nodes How exactly does one calculate the voltage of the marked nodes and arrive at the numbers in the picture (1.75 and 1.628)? I fail when trying. :(

simulate this circuit – Schematic created using CircuitLab
Background:
I'm trying to get a better basic understanding of electronics and as an exercise for myself I'm trying to grasp the maths in a 7-bit resistor ladder DAC. I've managed to calculate the various voltages of all the nodes for each case where one digital pin is HIGH and all other LOW. Things get more complicated for me when two or more pins are HIGH at the same time and now I'm confused on how to arrive at the numbers in the picture. The picture is the circuit when two pins are HIGH. When trying to calculate the voltage of the marked node rightmost in the picture (with the actual value of 1.628V) I concluded that R1 and R3 are two resistors in series, so I treated them as one resistor of value 330 ohm. Then when I bring R2 into the calculation I arrive at 330*350/(330+350)=169.85. So using this value together with R4 and calculating voltage I come up with this: 5*150/(150+169.85)=2.34. Now, obviously I'm failing big somewhere, the question is where?

A: You can use mesh equations for this (the most general and fundamental way), but the simpler way is to successively work your way from the bottom left up to the top right combining resistors in parallel and series.



*

*The rule for series it that the same current must flow through both
components.

*Similarly, the rule for parallel is that the voltage across both
components must be the same.


Therefore, R1 and R3 are NOT in series. There is some current flowing out through the middle into R5. You can't just ignore that. That means the current running in R1 is not the same as the current running in R3 which means they are not in series. The only resistors in series with each other in this entire schematic are R13 and R14.
That means, for example, R12 is NOT in parallel with R10, nor is it in parallel with R13 or R12. However, the series combination of R13+R14 is in parallel with R12.

You also cannot use the resistor divider equation here due to its limitations. That limitation is as follows:
The resistor divider equation assumes that the output node of the divider has zero current flowing out of it. That means that all the current stays in the divider. As soon as some current flows out the divider's central output node, the current flowing in the bottom resistor is no longer the same as the current flowing in the top resistor which then causes your voltage stepdown ratio to be dependent on the load.
In reality, this means you can only use the resistor divider equation to approximate the voltage at the divider's output when the load impedance is much much greater than the resistor values in the divider itself.
