Nodal Analysis - Supernode


Solve the circuit with nodal analysis and find  I_x and  V_y .
Nodal analysis with supernode

Solution
1) Identify all nodes in the circuit. Call the number of nodes  N .
There are four nodes in the circuit:

There are four nodes in total.
Therefore  N=4
2) Select a reference node
Because of symmetry in the circuit, any node can be chosen as the reference node.
Choosing the reference node

3) Assign a variable for each node whose voltage is unknown.
There are three nodes beside the reference node:
All nodes
Node I and Node III are connected to each other by a voltage source. Therefore, they form a supernode.
supernode
The voltage of Node III can be written in terms of the voltage of Node I. All we need to do is to apply KVL in the loop illustrated below.
Applying KVL to find the voltage of supernode
KVL:  -V_1 -Vs_2+V_3=0 \to V_3=V_1-4 .

A good practice is to avoid assigning a voltage label to Node III and use  V_1-4 as its voltage.
Voltages of supernode - nodal analysis
There is also another voltage source which connects Node II to the reference node. This voltage source must be used to obtain the node voltage. Since the negative terminal of the voltage source is connected to the node,  V_2=-Vs_1=-10V . You can verify this by applying KVL to the loop between the reference node and Node II.
4) Write down KCL equations.
We only need to write a KCL equation for the supernode:
 -Is_2 + \frac{V_1}{R_2} +\frac {V_3 -V_2}{R_1}-Is_1=0 . Substituting  V_3=V_1-4 and known variables,
  V_1 =-\frac{44}{9}V . Therefore,   V_3 =V_1-4=-\frac{80}{9}V

Now, we need to determine the required quantities:

KCL at node 1:

 -Is_2 +\frac{V_1}{R_2} +I_x=0 \to I_x=  \frac{20}{9} A

KVL around the loop shown in the figure below:
KVL to find Vy
 -V_3 -V_y +V_2=0\to V_y = - \frac{10}{9} v .

15 thoughts on “Nodal Analysis - Supernode

  1. Dear Dr. Li,

    I signed up already for your web site, but I still have problems in order to download the book named Nodal Analysis (I tried with IDM and other programs), please help me to solve this trouble.

    Awaiting your urgent reply.

    Best regards,

    Orhan TUNCOZ

  2. dear dr. yaz z. li
    i am about to start teaching network analysis to ug students of electrical and electronics engineering. thanks for explanation and your illustrations are too good that i want to follow. kindly let me know which software is used to draw the networks and highlighting relevant portions of the network. thanks and regards.

      1. Hi Dr. Yaz Z. Li,

        I am also teaching an electrical engineering class, and am playing around with circuitikz. Although I can do basic circuits, I have no idea how to do the colour highlighting that you have in this page. Would it be possible for you to post a sample code showing how to do this?

        Thank you!

        Alexis

        1. Hi Alexis,

          Actually, I used GIMP for highlighting here but for this post, I used fill as:

          \fill[magenta,opacity=0.5] (-.5,-.5) -- (-.5,2.5) --(.5,2.5) -- (.5,-.5) ;
          \fill[red,opacity=0.5] (-.5,4.5) -- (-.5,7.5) --(.5,7.5) -- (.5,4.5) ;
          \fill[blue,opacity=0.5] (2.5,4.5) -- (2.5,7.5) --(3.5,7.5) -- (3.5,5.5) -- (5.5,5.5) --(5.5,4.5);
          \fill[green,opacity=0.5] (2.5,-.5) -- (2.5,2.5) --(5.5,2.5) -- (5.5,1.5) --(3.5,1.5) --(3.5,-.5);
          .
          Let me know if you have any questions.

  3. hey, i have a text on circuit analysis on thrusday and im having difficulties in complex situations, for instints. can i select a node as a reference node if it is connected to a voltage source, and that mean the nodal voltage of the node of the other side of the voltage source will be that of the voltage source.

    thanks

    brendon

  4. hey, i have a test on circuit analysis on thrusday and im having difficulties in complex situations, for instints. can i select a node as a reference node if it is connected to a voltage source? If possible does that mean the nodal voltage of the node of the other side of the voltage source will be that of the voltage source.

    thanks

    brendon

    1. Hi Brendon,
      Yes, you can. We prefer to select such nodes as the reference node to have one of the node voltages known without need to solve. Please download my free ebook, these are explained there.

      Good luck :)

  5. hello...
    I have already signed in to your website but still i have problem loading the book NODAL ANALYSIS .can u plz help me regarding loading of this book.
    THANKS

    1. In that case, you consider all three nodes as a supernode. Label the voltage of one of the nodes and find the voltage of the other two nodes in terms of the voltage of the first node. (The same if there are even more than three nodes in a supernode.)

  6. I signed up already for your web site, but I still have problems in order to download the book named Nodal Analysis, please help me to solve this trouble.
    Randy

  7. Hi sir yaaz !
    Sir would you plz explain me how to solve a question in which a supernode contains. Another super node ...
    Thanks

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