Nodal Analysis – Super-node: a Non-grounded Voltage Source

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Electrical Circuits Problems, Resistive Circuits

Use nodal analysis to solve the circuit and determine the values of $I_x$ and $V_y$ .

I. Identify all nodes in the circuit.
There are four nodes in the circuit:

II. Select a reference node.
There is symmetry in this circuit and each node has three elements connected to. One of the elements connected to each node is a voltage source. Thus, any node has no advantage over others in being the reference node. We choose the bottom left node as the reference node.

III. Assign variables for unknown node voltages.

Three remaining nodes are labeled as shown in the figure above. With this labeling, the value of $V_2$ can be determined directly. Since there is a voltage source between the node of $V_2$ and the reference node, and the positive terminal of the voltage source is connected to the reference node, we have:

$V_2 = -V_{s_1} = -10 \, \text{V}.$

The label $V_2$ is assigned to simplify referencing this node in the analysis. Alternatively, one could avoid assigning a label to this node and instead directly substitute $-10 \, \text{V}$ wherever it is required in the equations.

IV. Incorporate dependent sources.
This circuit does not contain any dependent sources.

V. Apply Kirchhoff’s Current Law (KCL).

We write a single KCL equation for the super-node:

$-Is_2 + \frac{V_1}{R_2} + \frac{V_3 - V_2}{R_1} - Is_1 = 0.$

$-1 + \frac{V_1}{4} + \frac{V_3 - 10}{5} + 2 = 0.$

Multiplying by 20

$-20 + 5V_1 + 4V_3 - 40 + 40 = 0.$

(1) $5V_1 + 4V_3 = 20.$

VI. Handle super-nodes.
Nodes $V_1$ and $V_3$ are connected by a voltage source, forming a super-node. By applying Kirchhoff’s Voltage Law (KVL) to the loop shown below, $V_3$ can be expressed in terms of $V_1$ as follows:

(2) $-V_1 + V_{s_2} + V_3 = 0 \quad \Rightarrow \quad V_3 = V_1 + 4.$

VII. Solve the System of Equations.

Substituting (2) in (1):

$5V_1 + 4(V_1+4) = 20$

$9V_1 = 4$

$V_1 = \frac{4}{9} \, \text{V}$

Substituting $V_3 = V_1 + 4$ and the known values into the equation, we solve for $V_3$ :

$V_3 = \frac{40}{9} \, \text{V}.$

At this point, the circuit is fully solved, and all node voltages are determined.

VIII. Determine Additional Variables. Calculating $I_x$ and $V_y$

With the node voltages known, we can now calculate $I_x$ and $V_y$ .

KCL at Node 1:

$-Is_2 + \frac{V_1}{R_2} + I_x = 0 \quad \Rightarrow \quad I_x = \frac{10}{9} \, \text{A}.$

KVL around the loop shown below

$-V_3 - V_y + V_2 = 0 \quad \Rightarrow \quad V_y = \frac{50}{9} \, \text{V}.$

Download the Circuit File

To download the LTspice circuit file for your own simulations, click the link below.Please remember to unzip the file after downloading to access the .asc file for your simulations:

Nodal Analysis – Super-node Circuit.asc Download

Comments

18 responses to “Nodal Analysis – Super-node: a Non-grounded Voltage Source”

March 29, 2011

nicomedian

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

Reply
May 26, 2011

p c madhavarao

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.

Reply
1. May 26, 2011
  
  Dr. Yaz Z. Li
  
  Hi there,
  The latest one that I use is latex + circuitikz.
  
  Reply
  1. January 9, 2012
    
    Alexis M
    
    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
    
    Reply
    1. January 9, 2012
      
      Dr. Yaz Z. Li
      
      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.
      
      Reply
November 9, 2011

brendon

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

Reply
November 9, 2011

brendon

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

Reply
1. November 9, 2011
  
  Dr. Yaz Z. Li
  
  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 🙂
  
  Reply
  1. November 13, 2011
    
    brendon
    
    i had the test, it was not hard. hope i passed it. thanks for your asistance.
    
    Reply
December 26, 2011

mumu

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

Reply
February 25, 2012

Minor Rojas

Hi,

what about if you have a supernode that contains another supernode?

thanks

Reply
1. February 27, 2012
  
  Dr. Yaz Z. Li
  
  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.)
  
  Reply
March 29, 2012

randymarshall1

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

Reply
May 15, 2012

zeeshan ashraf

your analysis of super node realy realy give me a suppert to understand super node concept.please upload more then.

Reply
April 1, 2014

Yasir Amin

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

Reply
April 18, 2015

mandar kulkarni

sir i want to know that how to solve supernode problem when there is a resistance between two non reference nodes which makes a supernode i.e. the resistance is in series with the voltage source

Reply
1. April 18, 2015
  
  Yaz
  
  If there is a resistance in series with a voltage source, their connection is supernode from the voltage source side.
  
  Reply
August 27, 2018

Kawalya

Hello how can I solve a question that has a current source connected in parallel with two resistors

Reply