From our previous discussion of Nodal Analysis we have seen, how voltage sources affect nodal analysis. We have also seen how a voltage source makes it easier for us to calculate the node voltages when connected with a reference node. But things get complicated when a voltage source cannot be referenced i.e. it comes in between two non-reference node. This voltage source along with two non-reference nodes forms a supernode. 

In summary, When a voltage source comes in between two non-reference node then these two non-reference nodes and the voltage source form a supernode and we take this supernode as a single node and apply KCL and KVL to solve the circuit.

To solve a problem of supernode follow these steps:

Let us explain this procedure with an example:

First of all we have marked a reference node V0. Then we marked all other nodes as we do normally for nodal analysis.

Now we have given a dotted circle to remind us that this is a supernode along with V1 and V2.

Now we apply KCL at the circuit:

2 = (V1 – 0) / 2 + (V2 – 0) / 4 +7

8 = 2 V1 + V2 +28 (multiplying both sides by 4)

2 V1 + V2 = -20 ………………………………………………. (a)

Now we apply KVL at the supernode loop:

-V1 - 2 +V2 = 0

V2 = V1 + 2 ……………………………………………………. (b)

Putting this value of V2 in equation (a):

2 V1 + (V1 + 2) = -20

3 V1 = -22

V1 = -22/3 V

Now from (b): V2 = -22/3 +2 = -16/3V

Note that the 10 Ohm resistor connected across the supernode does not make any difference in the calculations as it is connected across the supernode.

We have already gain understandings of nodes, branches and loops. German physicist Gustav Robert Kirchhoff gave two laws which alongside with Ohm’s laws give us a powerful tool of circuit analysis.

Kirchhoff’s Current Law (KCL):

Kirchhoff stated that, amount of current entering or leaving a node should be equal. That means the algebraic sum of currents with respect to a node should always be zero.

A water pipe analogy would make it easier for us to understand this law. Suppose in a network of water pipes, water comes and leaves through different branches. At a single point of this network the amount of water coming to that node or point from different branch is always equal to the amount of water leaving from that point or node. If we think water as current then the amount of current coming through different branches at a node in a circuit must be equal to the amount of current leaving that node. This is Kirchoff’s Current Law (KCL).

To illustrate KCL look at the figure below:

 

At figure A, current I is entering node X and I1 and I2 is leaving node X. And at node Y, I1 and I2 is entering node Y and I is leaving node Y.

Hence, according to KCL:   

I=I1+I2=I-I1-I2 = 0

Here current I1 and I2 has opposite polarity of I. So we can express it as: I+I1+I2=0

At figure B, same thing happens. Current I1 and I2 are entering node Z and I3, I4 and I5 is leaving node Z.

Hence, according to KCL:

I1+I2+I3+I4+I5=0

Kirchoff’s Voltage Law (KVL):

The statement of KVL is: In a closed path or loop the algebraic sum all voltages must be zero. That means simply, the voltage sources will generate voltages and other elements will consume it. 

We already know the resistors have always a voltage drop, i.e they consume voltage. Other elements like capacitors and inductor also have voltage drops as they consume real or reactive power from the source. 

To illustrate KVL, let us see the figure below,

The voltage source E generates voltage or potential. Whereas Resistors have drops of E1, E2, E3 and E4. According to KVL, 

E = E1 + E2 + E3 +E4

As we can see from the figure, the signs of potential of resistors is opposite of voltage source. So,

E= -E1-E2-E3-E4

E+E1+E2+E3+E4=0

Hence, the algebraic sum of voltages in a loop is zero. 

Nodal analysis is a procedure to analyze circuits. It uses node voltages as circuit variables. It is a very easy procedure for calculation as it reduces equations and makes it convenient to solve large networks. In nodal analysis we don’t take element voltages rather we take node voltages which actually reduces number of equations to be solved simultaneously. Nodal analysis is basically the implementation of KCL (Kirchoff’s Current Law).

We are interested in node voltages in nodal analysis, and after finding out node voltages we form independent circuit equations to solve the circuit. We should follow the steps below to solve a circuit using nodal analysis:

Now let’s explain these three steps using an example. Look at the following circuit.

 

It’s a simple circuit with a voltage source of 10V and a current source of 1 mA and three different resistors. We have to find the voltage across the resistor R3. Now let us assign a reference node first. Here we can see three nodes are there. We choose the lowermost one as the reference node and assign it Vo = 0V. We also define other nodes having voltages as V1 and V2.

 

 

Now as we can see, node V1 has a voltage source of 10V and the other terminal of the source is connected to the reference node. Hence we can say that the whole 10V should appear at V1.

Therefore V1=10V.

Well, here is an important thing to mention: most of the time we assign the node which connected to a voltage source’s negative terminal as a reference node which in turn simplifies our calculation by giving the other node voltage value directly from the voltage source’s value.

 Now, let us apply KCL at node V2:

(V2 – V1) / 20 k  + V2 / 10 k – 1 m = 0

 (V2 – V1) / 20 k  + V2 / 10 k = 1 m

V2 –V1 + 2 V2 = 20  (multiplying both sides by 20k)

3 V2 – V1 = 20

3 V2 – 10 = 20 (putting the value of V1)

3 V2 = 20 + 10

3 V2 = 30

V2 = 30 / 3

V2 = 10V

Now, we have all the node voltages. We can find the voltage Vr3 easily.

Vr3 = V1 – V2 = 10 V – 10 V = 0V

We have taken arbitrary values of resistors and sources which lead the voltage across R3 to be zero.

 Most of the time we can solve the circuit using our idea of series and parallel connection and ohm’s law or KCL or KVL. But sometimes we encounter some circuit where the formations of elements are neither in series nor in parallel. For example look at the figure of bridge network below:

 

Now look at the following two formations which look like Wye and Delta. These formations usually found in three phase connections, filters, or in matching networks.

  

 (a) Y netwrk (b) Delta Network

 (a) Delta Network

 

 (b) Pie Network

When we analyze a circuit we may find a certain formation to be helpful. And we might want to change one formation into other. Suppose it might helpful for us to work with Wye formation in three phase rather in Delta formation. So we have to know how to transform a Delta network into Wye network.

Delta to Wye conversion:

Look at the figure below:

 

Now for Delta to Wye conversion:

R1 = (Rb . Rc) / (Ra + Rb + Rc)

R2 = (Rc . Ra) / (Ra + Rb + Rc)

R3 = (Ra . Rb) / (Ra + Rb + Rc)

We don’t have to memorize these formulas rather we can easily remember these. Each resistor of the Wye network is the product of the two adjacent resistors of Delta network, divided by the sum of all three resistors of the Delta network.

Wye to Delta Conversion:

Look at the figure again:

Now for Wye to Delta conversion:

Ra = (R1.R2 + R2.R3 + R3.R1) / R1

Rb = (R1.R2 + R2.R3 + R3.R1) / R2

Rc = (R1.R2 + R2.R3 + R3.R1) / R3

We don’t have to memorize this also. The easy way to remember is, each resistor in the Delta Network is the sum of all possible product formation of Wye network’s resistors taken two at a time, divided by the opposite Wye resistor.

When we discuss network topology we have to understand certain terms. Among these terms node, branch or loops are used most of the time. Let’s see what they mean.

Branch: A single element with its terminals is usually called a branch. For example a voltage source or a resistor is a branch.

Node: When two or more branches are connected at a point then that point is called a node.

Loop: A closed path in a circuit is called a loop.

Mesh: Mesh is a kind of loop which has no loop inside it. But you have to remember that all meshes are also loops. But all loops are not meshes. 

For example let’s have look at the following figure:

In a network or circuit, number of loop, nodes and branches has to satisfy the following fundamental relationship:

                b=l+n-1

where, b = number of branches,

l = number of loops and

n = number of nodes.

There is another important thing to remember in circuit analysis. 

We see lightning occurring at a  stormy day or night. Lightning might occur between clouds, or between clouds and ground in the atmosphere. This lightning is nothing but a magnified version of static electric discharge. Before the lightning happens, there is a static charge in the clouds, between different clouds of parts of clouds. In the following figure cloud to cloud (A) and cloud to ground (B) static charge build ups are shown. At figure (A) when two different clouds come sufficiently close to each other then the static discharge happens and lightning occurs. In case of (B) the positive charge in the earth attracts the negative charges of cloud and then discharge happens.

The interesting thing is that, the current flow in a lightning stroke is very high. Its about several tens of thousands or hundreds of thousands, of amperes. But this huge current can't do that much damage as it lasts for only a fraction of second.

Where V stands for voltage, I for current and R for resistance.


One have to remember only one and he/she can derive the others by simple mathematics.

An electrical device which can generate prescribed voltage irrespective of the current supplied to other elements is called an ideal voltage source. The ability to provide constant voltage will not be harmed by any other means like heavy load. Since load always destabilizes voltage sources and at full load no source can produce prescribed voltage so ideal voltage sources are imaginary. Ideal voltage sources are not available in real world. We use this idea of Ideal voltage source to make comparison among other voltage sources. 

An electrical device which can generate prescribed current irrespective of the circuit or other elements connected to it, is called an ideal current source.To maintain a constant current the voltage in the circuit will vary. Thus it will produce arbitrary voltage across its terminals. The figure below shows symbol of ideal current sources. 


We can say: An ideal current source maintains a prescribed or constant current irrespective of the circuit connected to it. The circuit connected to the ideal current source determines the voltage generated.

02. Charles A. Coulomb (1736–1806): He wan a French engineer and physicist. He published the laws of electrostatics in between 1785 and 1791. Unit of charge Coulomb was named after him.

03. JamesWatt (1736–1819): He was an English inventor. He developed the steam engine. Unit of power is named after him.

04. Alessandro Volta (1745–1827): He was an Italian physicist. He discovered the electric pile. His name is used to represent the unit of electric potential and the alternate name of this quantity (voltage).

05. Hans Christian Oersted (1777–1851): He was a Danish physicist. He discovered the connection between electricity and magnetism in 1820. His name is used to represent the unit of magnetic field strength.

06. Andr´e Marie Amp`ere (1775–1836): He was a French mathematician. He was also a chemist, and physicist. The relationship between electric current and the magnetic field was first electrically quantified by him. His name is used to represent the unit of electric current.

07. Georg Simon Ohm (1789–1854): He was a German mathematician. He investigated the relationship between voltage and current and quantified the phenomenon of resistance. First results of his work were published in 1827. Unit of resistance is named after him.

08. Michael Faraday (1791–1867): He was an English experimenter. He demonstrated electromagnetic induction in 1831. Beginning of the age of electric power was marked by his electrical transformer and electromagnetic generator. Unit of capacitance was named after him.

09. Joseph Henry (1797–1878): He was an American physicist. He discovered self-induction at around 1831. Unit of inductance was named after him. Essential structure of the telegraph was first recognized by him. Later on Samuel F. B. Morse perfected it.

10. Carl Friedrich Gauss (1777–1855): He was a German mathematician. He and Wilhelm Eduard Weber (1804–1891), a German physicist, first published the measurement of earth's magnetic field in 1833. Unit of magnetic field strength is Gauss, while unit of magnetic flux is Weber.

11. James Clerk Maxwell (1831–1879): He was a Scottish physicist. He discovered the electromagnetic theory of light. He also discovered the laws of electrodynamics. Maxwell's equations are the base of modern theory of electromagnetics.

12. ErnstWerner Siemens (1816–1892) andWilhelm Siemens (1823–1883), both were German inventors and engineers. They perfected selectrical science and contributed to the invention and development of electric machines. Their name is used to represent the modern unit of conductance.

13. Heinrich Rudolph Hertz (1857–1894): He was a German scientist and experimenter. He discovered the nature of electromagnetic waves. He then published his findings in 1888. The unit of frequency is named after him.

14. Nikola Tesla (1856–1943): He was a Croatian inventor. But in 1884 he immigrated to the United States. He invented poly-phase electric power systems. The modern AC electric power system and induction motor was his work. The unit of magnetic flux density is named after him.

The amount of current is always same at a single loop circuit. The current will not change anywhere in the circuit. But the voltage drop will be different. 

In a broken circuit, where there is a voltage source connected, then the full voltage of that source will appear across the points of the break.