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Lesson 4

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Basic Laws of Electric Circuits

Nodes, Branches, Loops and

Current Division

Lesson 4

Basic Laws of Electric Circuits

Nodes, Branches, and Loops:

Before going further in circuit theory, we consider the

structure of electric circuits and the names given to various

member that make up the structure.

We define an electric circuit as a connection of electrical

devices that form one or more closed paths.

Electrical devices can include, but are not limited to,

resistors transistors transformers

capacitors logic devices light bulbs

inductors switches batteries

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Basic Laws of Electric Circuits

Nodes, Branches, and Loops:

A branch: A branch is a single electrical element or device.

Figure 4.1: A circuit with 5 branches.

A node: A node can be defined as a connection point between

two or more branches.

Figure 4.2: A circuit with 3 nodes.

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Basic Laws of Electric Circuits

Nodes, Branches, and Loops:

If we start at any point in a circuit (node), proceed through

connected electric devices back to the point (node) from

which we started, without crossing a node more than one time,

we form a closed-path.

A loop is a closed-path.

An independent loop is one that contains at least one element

not contained in another loop.

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Basic Laws of Electric Circuits

Nodes, Branches, and Loops:

The relationship between nodes, branches and loops

can be expressed as follows:

# branches = # loops + # nodes - 1

or

B = L + N - 1

Eq. 4.1

In using the above equation, the number of loops are

restricted to be those that are independent.

In solving most of the circuits in this course, we will not

need to resort to Eq. 4.1. However, there are times when it

is helpful to use this equation to check our analysis.

4

Basic Laws of Electric Circuits

Nodes, Branches, and Loops:

Consider the circuit shown in Figure 4.3.

Figure 4.3: A multi-loop circuit

give the number of nodes

give the number of independent loops

give the number of branches

verify Eq. 4.1

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Basic Laws of Electric Circuits

Single Node Pair Circuits:

Current division.

A single node pair circuit is shown in Figure 4.4

Figure 4.4: A circuit with a single node pair.

We would like to determine how the current divides (splits)

in the circuit.

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Basic Laws of Electric Circuits

Single Node Pair Circuits:

Current division.

Eq. 4.2

Eq. 4.3

Therefore;

Eq. 4.4

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Basic Laws of Electric Circuits

Single Node Pair Circuits:

Current division.

From Eq. 4.4 we can write,

Eq. 4.5

Equation 4.5 is a very important expression. In words it

says that the equivalent of two resistors in parallel equals to

the product of the two resistors divided by the sum.

The equivalent resistance of two resistors in parallel is always

less than the smallest resistor.

8

Basic Laws of Electric Circuits

Single Node Pair Circuits:

Current division.

In general, if we have N resistors in parallel as in Figure 4.5

Figure 4.5: Resistors in parallel.

Eq. 4.6

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Basic Laws of Electric Circuits

Single Node Pair Circuits:

Current division.

Back to current division: We can write from Figure 4.4;

In summary form;

Eq. 4.7

The above tells us how a current I divides when fed into

two resistors in parallel. Important

10

Basic Laws of Electric Circuits

Single Node Pair Circuits:

Current division.

In general, if we have N resistors in parallel and we want to

find the current in, say, the jth resistor, as shown in Figure 4.6,

Figure 4.6: General case for current division.

Eq. 4.8

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Basic Laws of Electric Circuits

Current Division:Example 4.1

Given the circuit of Figure 4.7. Find the currents I1 and I2

using the current division.

Fig 4.7: Circuit for Ex. 4.1.

By direct application of current division:

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Basic Laws of Electric Circuits

Current Division:Example 4.2

Given the circuit of Figure 4.8. Find the currents I1 and I2

using the current division.

Figure 4.8: Circuit for Ex. 4.2.

The 7 resistor does not change that the current

toward the 4 and 12 ohm resistors in parallel is 10 A.

Therefore the values of I1 and I2 are the same as in

Example 4.1.

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Basic Laws of Electric Circuits

Current Division:Example 4.3

Find the currents I1 and I2 in the circuit of Figure 4.9 using

current division. Also, find the voltage Vx

Figure 4.9: Circuit for Ex. 4.3.

We first find the equivalent resistance seen by the 20 V source.

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Basic Laws of Electric Circuits

Current Division:Example 4.3

We can now find current I by,

We now find I1 and I2 directly from the current division rule:

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Basic Laws of Electric Circuits

Current Division:Example 4.3

We can find Vx from I1x12 or I2x4. In either case we get Vx = 6 V.

We can also find Vx from the voltage division rule:

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Basic Laws of Electric Circuits

Current Division:Example 4.4

For the circuit of Figure 4.10, find the currents I1, I2, and I3

using the current division rule.

Figure 4.10: Circuit

for Example 4.4.

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Basic Laws of Electric Circuits

Current Division:Example 4.4

We notice that I1 + I2 + I3 = - 15 A

as expected.

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Basic Laws of Circuits

circuits

End of Lesson 4

Nodes, Branches, Loops, Current Division

Basic Laws of Electric Circuits

Current Division:Example 4.4

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Basic Laws of Electric Circuits

Current Division:Example 4.4

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