Basic Laws 3

7/28/2019 Basic Laws 3

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Electric Circuits Basic Laws

2. Basic laws

The fundamental laws are presented in this chapter

which will be applied to determine the variables as

current, voltage or power at each element of a resistive

circuit.


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2.1. Ohms Law

The Ohms law states that the voltage across a resistor is directly

proportional to the current flowing through the resistor. This represented by

(2.1)

Ohm defined the constant of proportionality for a resistor as the resistance

. Thus, the Ohms Law is written as

(2.2)

Then, the resistance is defined as the ability of an element to resist the flow of

electric current which is measured in Ohms designed by . One must be careful

about the current direction or voltage polarity whose must be conform with the

passive sign convention shown in the next figure

Figure 2.1. Passive sign convention.


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Since can take values from zero to infinite; then, there are two extreme

cases. The case where 0 is called Short Circuit (sc) and the case where

is called Open Circuit (op). For the first case, it has zero voltage and

maximum current and for the second case the current is zero and maximum

voltage.

A useful concept is the Conductance denoted by ( ) which is defined as the

reciprocal of the resistance. Thus

= 1 = (2.3)

The unit of the conductance is the mho () or also siemens ( ).

The power dissipated by a resistor can be expressed in terms of using the

previous equations as

= = =

= =

(2.4)


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from these equations, it can be conclude that The dissipated power by the resistor is not linear function of either current or

voltage.

Since and are positive values, the resistor is a passive element that

always consume energy and it is incapable of generate or store energy.


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2.2. Series and parallel elements

It is better to understand the concepts of branch, node and loop from the

electrical point of view before to define the series or parallel connection.

Branch is a single element of the circuit.

Node is the point of connection between two or more branches. In otherwords, a node is all that can be enclosed without include any branch.

Loop is any closed path in a circuit.

Mesh is any loop that does not include any other loop.

In this way it is possible to say that

i. Series connection. Two elements are connected in series if both share a

node and there is not any other element connected at shared node.

ii. Parallel connection. Two or more elements are connected in parallel if they

are connected to the same pair of nodes.

Examples:


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2.3. Kirchhoffs laws

i. Kirchhoffss Voltage Law (KVL) states that the algebraic summation

of all voltages around a closed path (or loop) is zero. Mathematically

this can be expressed like

= 0

(2.5)

where N is the number of elements in the loop.

ii. Kirchhoffs Currrent Law (KCL) states that the algebraic summation ofcurrents entering a node (or a closed boundary) is zero. This KCL can

be written as

= 0

(2.6)

where is the number of branches connected to the node or closed

boundary.


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2.3.1. Circuits with a pair of nodes.

The procedure to solve this kind of circuits, which is limited to circuits that

contain only current sources, is:

i. Define a voltage variable.

ii. Apply KCL to the positive node and solve.

iii. With the previous result, find the asked values.

Examples:

2.3.2. Circuits with only one loop

The procedure, which is limited to circuits that contains only voltage sources, is

i. Define the current variable

ii. Apply KVL to the loop and solve.

iii. With the previous result, find the asked values.

Examples:


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2.4. Equivalent resistance of arrangements of resistors

At this point many students believe that the resistors should be connected in

series or parallel. However, it is good to clarify that a very common case implies

that the resistors are not series neither parallel.

2.4.1. Equivalent resistance for series resistors and

conductances

Considers series resistors as it is shown in the next figure

Figure 2.2. A single-loop of series resistors.

By applying KVL, it is possible to say


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= + ++ = + ++ (2.7)

or also

= + ++ = (2.8)

in consequence, it is possible to say that the equivalent resistance is

determined by

= + ++ (2.9)

and the respective equation for parallel conductances is

1

=1

+1

++1

(2.10)

It is useful to deduce the formula for the case of two series conductances,

which is

=

+ (2.11)


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2.4.2. Equivalent resistance for parallel resistors and

conductances

Considers series resistors as it is shown in the next figure

Figure 2.3. A circuit with parallel resistors.

By applying KCL, it is possible to say

(2.12)


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or also

= 1

+1

++1

= 1

(2.13)

in consequence, it is possible to say that the equivalent resistance is

determined by1

=1

+1

++1

(2.14)

and the respective equation for parallel conductances is

= + ++ (2.15)

It is useful to deduce the formula for the case of two parallel resistors, which is

=

+

(2.16)

which can be enunciated as: The equivalent resistance of two parallel

resistors is equal to the product of their resistances divided by their sum.


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2.5. Voltage division

Consider the circuit shown in Figure 2.2. From Eq. (2.8), the current is

determined by

= + ++

(2.17)

and the voltage in the resistor is

=

+ ++ (2.18)

which is known as the voltage division relation. The most practical relation

for voltage division is the case of in two resistors, which is written as follow

=

+ or =

+ (2.19)


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2.6. Current division

Consider the circuit shown in Figure 2.3. From Eq. (2.13), the voltage is

determined by

=1

1

+

1

++

1

= (2.20)

and the current in the resistor is

=

=1

1+

1

++

1

(2.21)

which known as the current division relation. The most practical relation for

current division is the case of current division in two resistors, which is

written as follow

=

+ or =

+ (2.22)


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2.7. Series voltage sources

Consider the n voltage sources connected in series as it is presented in the

next figure

Figure 2.4. Series sources.

By KVL the total voltage is given as the summation of the sources as follow

(2.23)

It is convenient to note that for the parallel voltage sources case it cannot be

established any relation because there is a violation of Kirchhoffs Laws. Also it

should be noted that the polarity of the source must have the same. If there is the

case where the any source has opposite polarity, it must subtract in place of sum.


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2.8. Parallel current sources

Consider the current sources connected in parallel as it is presented in the

next figure

Figure 2.5. Parallel sources.

By KCL the total current is given as the summation of the sources as follow

(2.24)

Noted that for the series current sources case it cannot be established anyrelation because there is a violation of Kirchhoffs Laws. Also if there is any source

that has different direction of the current, the value must be subtracted in place of

added.


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2.9. Wye-Delta transformations

The wye (Y) or tee (T) and delta () or pi () connections shown in the next

figure are arrangements of resistors which occurs as part of larger networks.

Typical applications are the three-phase networks, electrical filters, matching

networks, etc. Sometimes it is possible to simplify the circuit analysis if these, wye

or delta connection, is converted to delta or wye respectively.

a) b)


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c)d)

Figure 2.6. Connections a) wye (Y), b) tee (T), c) delta (), d) pi ().

To get the equivalent delta circuit of a wye connection, first it should be

considered the next figure to establish the location of each resistor. For instance, in

the figure, in delta connection is the opposite of in wye connection.

Figure 2.7. Superposition of Y or networks.


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Thus, it is possible to say that the resistance between the node 1 and 2 for the

wye connection is

Y = + (2.25)

and for the delta connection is

= ( + ) (2.26)

because Y = (), then

= + =( + )

+ + (2.27)

similarly for and


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= + =( + )

+ + (2.28)

= + =( + )

+ + (2.29)

By subtracting Eq. (2.29) from Eq. (2.27) leads to

=( )

+ + (2.30)

By adding Eqs. (2.28) and (2.30) yields

=

+ + (2.31)

By subtracting Eq. (2.30) from Eq. (2.29) gives

=

+ + (2.32)


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By subtracting Eq. (2.31) from Eq. (2.27) yields

=

+ + (2.33)

The previous three equations can be summarized as Each resistor in the Y

network is the product of the resistors in the two adjacent branches, divided by

the sum of the three resistors.

To convert from wye to delta, the Eqs. (2.31) to (2.33) are multiplied pair by pairthen summed yielding

+ + = ( + + )

(+

+

)

(2.34)

or also

+ + =

+ + (2.35)


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By dividing the previous equation by each Eqs. (2.31) to (2.33) gives the final

relations employed to convert from wye to delta networks. These are

= + +

(2.36)

=

+ +

(2.37)

= + +

(2.38)

The previous equations can summarized in the next rule Each resistor in the

network is the sum of all possible products of Y resistors taken two at a time,

divides by the opposite Y resistor.

It is called that the and Y networks are balanced when

= = = or = = = (2.39)

when this happen the Eqs. (2.31) to (2.33) and (2.36) to (2.38) become

=

3 (2.40)


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2.10. Applications of Basic Laws

Now that the basic laws are known and that it has established the equivalent

resistance for basic arrangements of resistors, it is possible to explain how the

voltage controller, voltmeter, ammeter or ohmmeter for DC can be constructed.

2.10.1. Controller of potential levels

Figure 2.8. Potential level controller.


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2.10.2. Volmeter

The construction of the DC voltmeter, ammeter or ohmmeter implies the

application of the dArsonval meter movement. The scheme of this meter

movement is shown in the next figure.

Figure 2.9. dArsonval meter movement.


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Thus the voltmeter can be designed as it shown in the next figure

Figure 2.10. Voltmeters.

Figure 2.11. Ammeters.


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a) b)

Figure 2.12. Two way to measure resitance: a)using avoltmeter and a ammeter, b) using a ohmeter.

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