Nodal Analysis with Voltage source

Greetings our beloved instructor! At last the final topic for the midterm has come. Yet we know it is too early to celebrate because we still have a long way to go. But nevertheless we are quite relieved knowing our coverage for the exams.

Doing nodal analysis can be achieve in two ways, Kirchhoff's current law and Kirchhoff's voltage law. These laws has been discussed on our earlier topics so we assume that elaboration is of no significance anymore. Knowing such we can continue to nodal analysis with voltage source. Of course steps in doing nodal analysis are the following.

Steps in Nodal Analysis

1. Select a node as the reference node, Assign voltages v1, v2, . . . . . ,
vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.

2.Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express currents in terms of node voltages.

3. Solve the resulting simultaneous equations to obtain the unknown node voltages.

In nodal analysis with voltage source, we are presented with two cases;

Nodal Analysis with Voltage source Cases:

Case 1:

If the voltage source (dependent or independent) is connected between two non-reference nodes, the two non-reference nodes form a generalized node or super node, we apply both KCL and KVL to determine the node voltages.

Presented in this example.

Case 2:

if a voltage source is connected between the reference node and a non-reference node, we simply set the voltage at the non-reference node equal to the voltage of the voltage source

A Super-node is formed by enclosing a (dependent or independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it.

Steps in calculating voltage drops on supernodes:

Step 1. Take off all voltage sources in super-nodes and apply KCL to super-nodes.

Step 2. Put voltage sources back to the nodes and apply KVL to relative loops.

As shown in this illustration:

Learning:

This week we have learned another nodal analysis but this time with a voltage source. It is quite similar to nodal analysis with current source,but nodal analysis with voltage source will have a supernode, which is a combination of two non-reference nodes connected to the same voltage source. And will have another solution by introducing KVL to the computation together with the KCL on the supernodes. Aside from those, nodal analysis with voltage source is similar to our last week's topic, nodal analysis with current source.
Doing and solving for the values (i.e. voltage drop and current) across a circuit is a lot of fun! We know these topics will help us push through our ever dearly loved careers. Though it is hard at first, we know we can achieve such if we put our utmost effort unto it. It has been said that if there is a will, there is and will always have a way.

Nodal Analysis

Our fine greetings! Today we have again have an amazing week with an also amazing spectacular topic. And it is the nodal analysis, one of the techniques we will have to use in order to calculate the values of any elements we would like in a given more seemingly complicated circuit.

Without further ado, nodal analysis is one of the many methods in solving and finding a specific value of a parameter in electronic circuit analysis. The aim of using nodal analysis is to determine the voltage in each node that’s relative to the reference node, which is the ground GND where voltage is equal to 0. This means that all the other nodes present in the circuit are referred to as the non-reference nodes; the ones that has voltage you are trying to solve for. Depends, of course, if you do need the voltage present in them or not.

Let’s do a quick recap about the parts of an electronic circuit; A node is a point of connection between two or more branches. A branch represents a single element such as a voltage source, or a resistor, etc. And a loop is any closed path in a circuit.

Here are the steps on how to determine node voltages:

Determine the nodes of the circuit, and then select a node as the reference node (ground GND). Then assign the non-reference nodes to voltages V1, V2, Vx, or whatever you feel comfortable with.
Apply KCL (Kirchhoff’s Current Law) to each of the non-reference nodes. Use Ohm’s law (V=IR) to express the branch currents in terms of node voltages. I use the shortcut method, though the same principles are still applied.
Solve the resulting simultaneous equations formed from the non-reference nodes to obtain the unknown node voltages.
Always remember that the number of equations formed should be equal to the number of unknowns. So, taking this simple circuit as an example:

This sample circuit has an AC voltage source, a current source, a resistor, an inductor, and a capacitor. We can also observe that there are four nodes present in this circuit. Assuming that we need to find the voltage across the capacitor, what node will we choose as our reference node GND that will make the problem easier? Note that the lesser the unknowns, the easier the problem will be. It’s like the unknowns determine the difficulty of the problem.

So, there are four nodes. If we were to select the top-left node as the GND, V1 as the top-mid node, V2 as the top-right node, and V3 as the bottom node, it would mean that we can get the voltage across the capacitor with V1 – V3 where V3 = -Vs (voltage source, since the negative terminal of the voltage source is connected to node V3). This would be a fine option but it’s kind of — maybe, unethical — to have the GND connected to the positive terminal of the voltage source.

If we were to select the top-mid node as the GND, V1 as the top-right, and the same position for V2 and V3, then a supernode (formed by enclosing a voltage source, either dependent or independent, connected between two non-reference nodes and any elements connected in parallel with it) would be present which adds more difficulty in solving the circuit. Same situation goes if the top-right node is the GND.

But if we were to select the bottom node as the GND, and V1, V2, and V3 as the top-left, top-mid, and top-right nodes, respectively, then we can get the voltage across the capacitor with V2 – GND = V2 – 0 = V2 (since current flows from a higher potential to a lower potential in a resistor), while V1 = Vs (since the positive terminal of the voltage source is connected to node V1). That makes two unknowns (V2 and V3), though we only need to solve for V2. And it’s more ethical compared to having the GND on top of the circuit.

Now to label the nodes and elements, assuming that all the elements’ respective values are given and have been converted to their equal impedances already. If you want to know how to solve for the impedance in a resistor, inductor, and a capacitor, click here.

 

There we go! So, what we are trying to find is the voltage across the capacitor Z3 and we have discussed earlier that Vz3 = V2, since Vz3 = V2 – GND and that the voltage at the GND is equal to 0. Now to form the equation at node V2 using the shortcut method. Brace yourself for I am about to make up names that i’ll be using to better explain the shortcut method.

@node V2:

V2( ) = 0

To form the equation at node V2, you must locate V2 (imagine V2 as a person or an animal or any object) and look around its surroundings. And I mean the lines or pathways that are connected to it. In this case, there are three pathways connected to V2 (path to V1, to V3, and to GND). And each path is connected to an element (resistor, inductor, and capacitor), or what I will be naming as a bridge. And this is how you start the equation:

V2( 1/bridge1 + 1/bridge2 + 1/bridge3) = 0

..or..

V2( 1/z1 + 1/z2 + 1/z3 ) = 0

It’s like connecting impedances in parallel. Continuing to the path across one of the bridges, you’ll encounter another node (it could either be the GND or not), or what i’ll be naming as your neighbour. This is how neighbours are treated in the equation:

V2( 1/z1 + 1/z2 + 1/z3 ) – neighbour1/bridge1 – neighbour2/bridge2 – neighbour3/bridge3 = 0

..or..

V2( 1/z1 + 1/z2 + 1/z3 ) – V1/z1 – V3/z2 – GND/z3 = 0

..and since V1 = Vs, and GND = 0..

V2( 1/z1 + 1/z2 + 1/z3 ) – Vs/z1 – V3/z2 – 0 = 0

..and since Vs/z1 is a constant, we transpose it to the other side..

V2( 1/z1 + 1/z2 + 1/z3 ) – V3/z2 = Vs/z1

And that’s it for the first equation, with V2 and V3 as unknowns. As I mentioned earlier, the number of unknowns should be equal to the number of equations. So, we are going to need one more equation, and that’d be the equation at node V3.

@node V3:

V3( ) = 0

Following the same procedures with the bridges and neighbours, we get–

V3( 1/z2 ) – V2/z2 = 0

As you can see, node V3 is connected to a current source. A current source in nodal analysis is considered as a constant unless it’s a dependent source. When the current source’s direction is away from the node, you add the current to the equation. If its direction is towards the node, you subtract the current to the equation. And since the current source in the sample circuit is directed towards node V3, it goes like this:

V3( 1/z2 ) – V2/z2 – Is = 0

..and since Is is already a given constant, we transpose it to the other side..

Vs( 1/z2 ) – V2/z2 = Is

And there’s your second equation. Now you can solve for Vz3 by fusing the two equations using matrices, substitution, elimination, or whatever method you know. Though our professor requires us to use the matrix method.

Learning:

This week we have learned to use another method, called nodal analysis in solving some values of the elements present in the circuit. The first process we need to do is to locate a reference node, a node where most of the elements are connected, second is to assign voltages across each non-reference nodes. Third is to use KCL on each non-reference nodes. Lastly in order to solve for the voltages on each nodes, we are required to use the matrix and do the math. :D

Basic Laws: Wye-Delta Transformation II

CET acquaintance is coming! So we decided to make this blog a day earlier because we might be too tired to do stuffs after the event. Anyways. Greetings again our dear professor! We got a problem upon discussing what to write on this blog for because the topic that we discussed last week, is of the same topic we have discussed this week. But as we specified on our last blog, the wye-delta transformation that we have explained is only a kind of introduction. So now we decided to make wye-delta transformation a little broader.

PROPER USAGE OF WYE-DELTA TRANSFORMATION

How to apply this wye-delta transformation? Lets have this example.

 As said, we use wye-delta transformation if the resistors are neither connected in series or parallel. Thus getting the equivalent resistance may seem impossible if wye-delta transformation is not applied. On the given example, we first determine what is the best and easiest transformation in order to acquire the equivalent resistance. On this case, delta to wye transformation is the likely candidate, because in only one transformation we can solve directly the equivalent resistance. Also there are no wye (Y) circuit formation found on the circuit.  Never realized that one until few seconds ago.  Anyways applying these equation to R1, R2 and R3 we can arrive at a redrawn but equivalent circuit.

Thus the redrawn circuit will look like.

Our deepest apologies for a badly made redrawn circuit.  Now we can actually solve for the equivalent resistance which is simply. 

However, this equation is only applicable to our redrawn circuit.

Learning:

This week (although we had similar topic of the previous week) we have learned that doing wye-delta is not complicated as it first seems. The only weight is to analyze the circuit and understand what transformation should be used for us to acquire and be able to have the equivalent resistance of the circuit.

"Having to deal with technology can be painfully harsh, but every pain has its own hidden worth"

Basic Law: Wye-Delta Transformation

Good evening! Or good noon? morning? We don’t even know anymore. Should we greet with the time setting of when we made this blog, or when will you sir, our professor, will be reading our blogs? But of course we would not know when will you read so to be safe we would just rather say.


Greetings our beloved instructor! A week has passed again, and we are now on a new topic. Well it was kind of an introduction, we presume. Because Wednesday this week we had our first ever quiz, which by the way, we failed so hard. But maybe that is a challenge for us, to work harder, to be better! Or we did just failed the first quiz miserably. Anyways, our new topic is about:

WYE-DELTA TRANSFORMATION

Situations often arise in circuit analysis when the resistors are neither in parallel nor in series. This situations are where this wye-delta transformation technique is used, where it simplifies the analysis of an electric network or circuit. The Y-Δ transformation is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π. The concept is to transform a wye (y) electric network into a delta (Δ) electric network and vice versa in order to evaluate the circuit.

Basic Y-Δ transformation

The transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for complex as well as real impedances. Impedance actually is the measure of the opposition that a circuit presents to a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current (AC) circuit.

But in order to transform wye-delta or delta-wye. We must use a solution, transforming each would require that even if it is transformed, the elements would still be of equal value.

Equations for the transformation from Δ to Y

Equations for the transformation from Y to Δ

Learning:

This week we are introduced to a new way of reconstructing a circuit. It is called the wye-delta transformation. This transformation is used in order to reconstruct the resistance on the circuit to be able to calculate the equivalent resistance present on a seemingly complicated circuit. However just barely transforming wye-delta or delta-wye configurations of any given resistors may give or may not give you the desired configuration of the circuit to be able to calculate the equivalent resistance. A better understanding of how the resistors are connected and analyzing which way are the best is crucial in complicated circuits.

We are new to circuits and we still have a lot to learn. But with every new method, we are bound to unlock more understanding of what and how can we do better in this program.

“We are doing our best! So please grades, do catch up” – every engineering student ever.