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**TOPOLOGY IN CIRCUIT ANALYSIS**

**Nodal Analysis using Topology**

The **nodal analysis method **is based on
the fundamental cut-set**
technique**. Any forming **tree **(see*
figure C*), **should** **exclude the current sources, and include the
voltage sources of the circuit **(see *figure A*), because all the
equations derived from the node voltage method involve only the voltages of the
circuit. It is worth mentioning that an example tree of the circuit appears on
*figure C*. It joins together the 4 nodes, passing through the (i3-i4-i6)
branches. Another example tree could have been the (i2-i4-i5-i7) tree and so on.
On this web-page, all the terminologies and the mathematical equations would be
based on the tree showed on *figure C ***(i3-i4-i6)***,* because
only **one** tree is needed in order to analyse a circuit.

*figure A : the circuit*

*figure B
figure C*

By observing *figure C *above, the** **__
fundamental cut-set (i)__:** i7- i3-i2-i1** is
formed (*see figure D*).

*figure D
*

Another **fundamental** **cut-set (ii)**
can be: i**7- i5-i4-i2-i1 **(see *figure E*)

**
**

*figure E*

A **cut-set **can be: i**6-i4-i2-i1**
(see *figure F*). This cut-set **is not** a fundamental cut-set because
it contains more than one tree branches.

*figure F*

(__note that:__ if a node joins **more
than one tree branches**, **it cannot be isolated**. That is why node 2 is
never isolated and that's why *figure F* does not represent a fundamental
cut-set)

As mentioned above, in order for the circuit of
*figure A* to be analysed, **3** equations are needed. **Each
fundamental cut-set would provide a single equation**. *Figure D* and *
figure E* above represent 2 fundamental cut-sets, so they represent 2
equations. The last equation is given from *figure G* below, representing
the** **__fundamental cut-set (iii):__ i5-i6-i7.

*figure G*

By **observing ****figure C** and
**summing the currents in the fundamental cut-sets**:

...(1)

**fundamental cut-set (ii):**

currents i_{4 }and i_{5}
have** negative sign** because they **flow out of the node**

......(2)

**fundamental cut-set (iii):
**

.....................(3)

but

........................(4)

Using equations (1), (2) and (4) (neglecting
equation (3) because of equation (4)).

Substituting equation (4) into equation (2) =>

....................(5)

Substituting equation (4) into equation (1) =>

....................(6)

**V**_{1 }= 3.79V

**V**_{2} = 6.047V

**V**_{3} = 26.04V