Nodal Analysis

Nodal Analysis with Voltage Sources

Case 1

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

Case 2

            If a 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.

Note:

            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.

Principles of super node:

a.    The voltage source inside the super node provides a constraint equation needed to solve for the node voltages.

b.    A super node has no voltage of its own.

c.    A super node requires the application of both KCL and KVL.

Method of Analysis

With the use of the two laws which are the Kirchhoff’s Current and Voltage Laws, we can analyze any linear circuits by obtaining a set of simultaneous equations that are being solved to obtain the required values of the node voltages.

Nodal Analysis

            ---provides a general procedure for analyzing circuits using node voltages as the circuits variables assuming that circuits do not contain voltage sources.

Note:

            To determine node voltages, we should select first a node as your reference node.

Then assign voltage V1, V2, Vn-1 to the remaining n-1 nodes. You will apply KCL to each non-reference nodes. Now you can solve for the unknown node voltages.

Let’s consider the principle that current flow from a higher potential to a lower potential in a resistor. Express in:

                                                i = V(higher)- V(lower)

                                                                   R

In solving for the remaining unknown node voltages, there are some standard methods such as substitution method, elimination method, Cramer’s rule or matrix inversion.

                                                            

Current Division

Parallel Current Division 

Current division refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.

The formula describing a current divider is similar in form to that for the voltage divider. However, the ratio describing current division places the impedance of the unconsidered branches in the numerator, unlike voltage division where the considered impedance is in the numerator. This is because in current dividers, total energy expended is minimized, resulting in currents that go through paths of least impedance, therefore the inverse relationship with impedance. On the other hand, voltage divider is used to satisfy Kirchhoff's Voltage Law. The voltage around a loop must sum up to zero, so the voltage drops must be divided evenly in a direct relationship with the impedance.

Example 1.3: Circuits containing series and parallel connections -- Consider the circuit of figure 1.15. Assume that  and all values of resistors are known. The circuit can be solved on the basis of series and parallel connections. In particular we can find  as follows.

 
Figure 1.15: A circuit solvable by series and parallel concepts  

The total resistance seen from the source is

Therefore,

Using the current division formula, we can find

Finally, since , we have

Voltage Division

Series Voltage Division

             The voltage division rule (voltage divider) is a simple rule which can be used in solving circuits to simplify the solution. Applying the voltage division rule can also solve simple circuits thoroughly. The statement of the rule is simple:

Voltage Division Rule: The voltage is divided between two series resistors in direct proportion to their resistance.

Example:

KCL and KVL

KCL (Kirchhoff's Current Law) KVL (Kirchhoff's Voltage Law)

This fundamental law results from the conservation of charge. It applies to a junction or node in a circuit -- a point in the circuit where charge has several possible paths to travel.  

KVL - is one of two fundamental laws in electrical engineering, the other being Kirchhoff's Current Law (KCL).

KVL is a fundamental law, as fundamental as Conservation of Energy in mechanics, for example, because KVL is really conservation of electrical energy.

KVL and KCL are the starting point for analysis of any circuit.
KCL and KVL always hold and are usually the most useful piece of information you will have about a circuit after the circuit itself.

Here's a simple circuit.  It has three components - a battery and two other components.  Each of the three components will have a current going through it and a voltage across it.  Here we want to focus on the voltage across each element, and how those three voltages are related.

 

 

Lab experiment #1 Resistor Color Code and Measurement of Resistance

Objectives:

1. To measure the value of resistors from their color code.
2. To measure resistors of different values.
3. To measure a resistor using the various resistance rangers of an ohm-meter.
4. To measure resistance across each combination of two of the three terminals of a potentiometer and to observe the resistance change as a shaft of the potentiometer.

Each resistors has its own color codes and through these codes we can identify each resistors according to their colors. The ohm (symbol: Ω) is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm. Although several empirically derived standard units for expressing electrical resistance were developed in connection with early telegraphy practice, the British Association for the Advancement of Science proposed a unit derived from existing units of mass, length and time and of a convenient size for practical work as early as 1861. The definition of the "ohm" unit was revised several times. Today the value of the ohm is expressed in terms of the quantum Hall effect.

Reflection:

      

     I realized that it's so hard to identify each colors in a resistor because it's very small that you really to focus on it. That's why it takes time for us to recognize it because of that. But since that we are 5 in our group, we can catch up with the time and requirements gave by our professor .

As a conclusion, the following colors has its own identification like in the first band it indicates the significant digit, in the second band the multiplier, third band is the tolerance and the last as possible the failure rate.