# Nodal analysis of a basic DC circuit

I am trying to find the v1, v2, v3 and i5 using nodal analysis.

I gave the name to the current over the respective resistors i3 and i4 and put it as i0 - i3 - i4 = 0 also write the equations as i3 = (V2-V1)/R3 and i4 = (V2-V3)/R4 but couldn't progress further to find i5 and others.

The only info about the direction of currents flow was, i5 is flowing clockwise (to left).

This is my first question on the exchange so please let me know if I'm doing something wrong.

Thanks

simulate this circuit – Schematic created using CircuitLab

• There's something not exactly right about how you label current in this schematic. How do we know which way the current is flowing with $i_1 , i_2, ....$ etc.? I do believe CircuitLab has arrows you can use to indicate that. Or can we assume that all current is traveling clockwise/counterclockwise? (Please edit your question to clarify this...) – KingDuken Apr 24 at 16:54
• I couldn't find the arrows in CircuitLab but only info about current direction was i5 was flowing clockwise (to left). I will add this to question as well – Dora Apr 24 at 16:58
• Well it's actually up to you to decide whether or not you want your current to travel clockwise or counterclockwise (mathematically speaking). If you want all current to travel to clockwise like $i_5$, that's fine :) – KingDuken Apr 24 at 17:04
• Thanks, in that case yes I think it's okay to have all of them flow clockwise if that's not a problem. Can you help me with solving the equations please? All the ways I could see is getting stuck at the point I mentioned in the question – Dora Apr 24 at 17:09

Sometimes, it helps to redraw the schematic:

simulate this circuit – Schematic created using CircuitLab

I've removed your indicators of current, since they are associated in an obvious fashion to the resistors and with $$\I_0\$$ associated with the voltage source's current. In re-drawing the schematic, I decided to make $$\V_2\$$ the ground reference. I've annotated that fact in the schematic, just to be pedantic about it.

As you can see, there are only two unknown voltages. So this means just two equations and two unknowns. In nodal form, this is just:

\begin{align*} \frac{V_1}{R_1}+\frac{V_1}{R_3}+\frac{V_1}{R_5}&=\frac{9\:\text{V}}{R_1}+\frac{0\:\text{V}}{R_3}+\frac{V_3}{R_5}\\\\ \frac{V_3}{R_2}+\frac{V_3}{R_4}+\frac{V_3}{R_5}&=\frac{9\:\text{V}}{R_2}+\frac{0\:\text{V}}{R_4}+\frac{V_1}{R_5} \end{align*}

You should be able to solve the above for $$\V_1\$$ and $$\V_3\$$. With those in hand, all of the currents are easily developed. (To read how I layout nodal, see the section called nodal, done my way.)

P.S. There is absolutely NO NEED for super-nodes. Certainly not in your circuit. But in general, not ever. I've no idea why anyone insists otherwise. (There never is a need, though of course you can use them if you like. No requirement, though. A topic for another day.) To me, supernodes inspire "tricky" thinking that can too readily lead to unintended errors in analysis. Different people will have different opinions. But no one can show that they are required. There is always an equivalent set of equations, more prosaic in nature and more easily developed, without using them.

You can't do a straightforward nodal analysis in this case because you can't directly write a term for the current through the voltage source. Instead, you need to create a supernode that encloses the voltage source. Write one equation that is the sum of the currents for the supernode (the currents through R1 through R4). As mentioned before, you need to mind the directions of those currents.

You need one more equation. That equation comes by inspection and simply specifies the voltage difference between the nodes at either end of the voltage source. Really, the first thing you should do is number all of the nodes in the circuit.

• Could you please elaborate what you mean by supernode with currents through R1 through R4 – Dora Apr 24 at 18:12
• You draw a big circle that encloses the voltage source...that is your supernode. Just like any other node, the sum of the currents into the supernode must be zero, so write an equation for all of the currents flowing into the big circle. Those currents would be the currents through R1 through R4. – Elliot Alderson Apr 24 at 18:14
• Is it possible you can add a drawing or schematic? – Dora Apr 24 at 18:26
• Supernodes refer to basically combining two nodes into a single (bigger) node. As Elliot said, it will require an additional equation for that single supernode. – KingDuken Apr 24 at 19:34
• @DenizArıyan Just do an internet search for "nodal" and "supernode". There is a lot of instructional material out there, I just wanted to give you a nudge in the right direction. – Elliot Alderson Apr 24 at 21:42