Electronic – How to find the equivalent resistance of a circuit with a node between each resistor

The first calculation you did is correct, R3 and R4 are in series, the combination is in parallel with R2 and when combined it is in series with R1.

The second method of trying to combine R3 and R2 in parallel results in an incorrect answer because they are not in parallel. Devices are in parallel if they have the same voltage across them. The voltage across R2 is split between R4 and R3, thus the voltage across R2 and R3 is not the same, which means they are not parallel.

1. You correctly zero'd out the current source while calculating the equivalent resistance. Remember we are trying to create a two terminal model for the network looking at it from the right side. Since the 5 ohm resistor is disconnected on 1 terminal, no current can flow through this resistor and we can discard it. The equivalent resistance is then the series resistance of the 10 ohm + 20 ohm resistors. In summary, if you have a resistor that is not connected on 1 terminal, you can ignore it.

2. Be careful of how you are combining resistors. Remember that when finding Thevenin resistance we are calculating an equivalent resistance with respect to the output terminal (the two open circles). When you combine the 6 ohm and 10 ohm resistors in your second step, you've destroyed the top output node and you are no longer calculating an equivalent resistance with respect to the output terminals. When combining resistors for Thevenin / Nortin equivalents, you can't touch the output terminals, those have to be present in both the original and equivalent models. Your first method is correct because it finds the equivalent resistance with respect to the the output terminal.

Note when I say "with respect to X" I mean that the final equivalent resistance / circuit has X has it's two terminals. This comes into play later in electronics, for example with amplifiers, when you might take the Thevenin equivalent with respect to the input and that will give you 1 model, and then take it with respect to the output which will give a different model.