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Файл:The elements of the electrical circuit theory. Tutorial
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a)
b) c)
Fig. 2.3
The electromotive force of the equivalent EMF source is equal to the
no-load voltage of the real source. The current of the equivalent current
source is equal to the short-circuit current of the real source.
The electromotive force of the equivalent EMF source and the current of
the equivalent current source submit to the following relation:
11
E
E
J
k
JR
kin
R
;
(2.3)
.
in
The relation (2.3) indicates that the EMF source with the seriesconnected resistance can always be replaced by the current source with the
parallel-connected same resistance and vice versa. Which of the two equivalents to use is completely indifferent, it is determined only by the convenience of analysis in each case.
Nota bene! It is necessary to note that the electromotive force of the
ideal EMF source is always directed from the lower potential to the higher
potential, and the current of the ideal current source is always directed in the
same direction as the current of the real current source.
3. THE SERIES-CONNECTED
AND PARALLEL-CONNECTED ELEMENTS
OF THE ELECTRIC CIRCUIT
The electrical circuit can be converted to simplify the analysis of the
circuit. This conversion consists of reducing the number of branches and
nodes. When performing the conversion, we must remember that after analyzing the converted circuit, it is necessary to do the reconversion to return
to the original circuit.
All the conversions of the circuit must be equivalent. That is, the conversion of any part of the circuit should not change the distribution of currents in the unconverted part of the circuit. This is only possible if, during
conversing, the potentials of the nodes in the unconverted part of the circuit
and the currents flowing from the outside to the converted part of the circuit
remain unchanged.
The simplest conversion of the electric circuit is to reduce the number of
series-parallel circuit elements.
In the series connection of circuit elements, the end terminal of the previous element is connected to the initial terminal of the next element
(Fig. 3.1). The main feature of the series connection is the same current in
each of the elements.
12
Fig. 3.1
If we apply Ohm’s law (1.1) to the series connection of elements, we
conclude that the element voltages are distributed in direct proportion to
their resistances, and the total resistance of the series connection is equal to
the sum of the resistances of the elements. For the circuit shown in Fig. 3.1,
the following is true:
UIR
ab
UIR
bc
UUUUIR IRRR
ad ab bc cd equ
UIR
cd
RRRR
equ
1
2
3
123
(3.1)
123
Thus, if the elements are series-connected, they can be replaced by one
equivalent element whose resistance is equal to the sum of the resistances of
all the elements.
Nota bene! Resistances add up when elements are series-connected.
In the parallel connection of circuit elements, the initial terminals of all
elements are one node, and the end terminals of all elements are another
node (Fig. 3.2). The main feature of the parallel connection is the same
voltage across each of the elements.
If the elements are parallel-connected, they can be replaced by one
equivalent element whose conductance is equal to the sum of the conductances of all the elements.
Nota bene! Conductances add up when elements are parallel-connected.
13
R
R
R
Fig. 3.2
For the circuit shown in Fig. 3.2, the following is true:
UIRIRIR
11 2 2 3 3
ab
UIR I
ab equ
gggg
equ
1111
RRRR
equ
123
132
1
g
equ
(3.2)
During the conversion of parallel elements, the equivalent resistance is
always less than the smallest of the converted resistances.
If n equal resistances are parallel-connected (Fig. 3.3), the equivalent resistance is n times less than the resistance of any of the elements.
Rn . (3.3)
equ
If only two resistances are parallel-connected (Fig. 3.4), formula (3.2)
becomes simpler. Equivalent resistance can be defined as the product of
these two resistances divided by their sum:
R
R
equ
12
R
12
14
. (3.4)
Fig
Fig
R
R
. 3.3
. 3.4
4. THE BASIC LAWS OF THE ELECTRIC CIRCUITS
The basic laws of the electric circuits are Ohm’s law and Kirchhoff’s
laws.
Ohm’s law
Let us consider the branch that has the resistance
and has not the
1
EMF sources (Fig. 4.1). Ohm’s law for the passive branch (1.1) is true for
this branch. At the same time, Ohm’s law for the passive branch can also
be formulated as follows: the current in the branch without the EMF
sources is equal to the voltage drop across the branch divided by the
branch resistance:
U
ab
. (4.1)
I
1
Ohm’s law for the branch that has the EMF
sources allows finding the branch current by the
known potential difference at the ends of the
branch. The current in the branch with the EMF
sources is the fraction whose denominator is the
branch resistance. The numerator of this fraction is
15
Fig. 4.1
the potential difference at the ends of the branch plus the algebraic sum of
the EMF sources enclosed between the ends of the branch. The voltages and
the EMF sources whose direction coincides with the current direction are
considered positive. The voltages and the EMF sources opposite to the current direction are considered negative.
In particular, for the circuit shown in Fig. 4.2, the following is true:
UEE
12
ae
I
RR
12
. (4.2)
Fig. 4.2
Kirchhoff’s current law
Kirchhoff’s current law is formulated as follows: at the node of the electric circuit, the algebraic sum of the currents is zero.
Usually, the currents directed towards the node
are considered positive, and the currents directed from
the node are considered negative.
For the node shown in Fig. 4.3, the following is
true:
0II I I
. (4.3)
Fig. 4.3
circuit contains
As many equations of Kirchhoff’s current law can
1234
be obtained as many nodes the circuit contains. However, not all of these equations are independent. If the
n nodes, then the number of independent equations is 1n .
The equation that corresponds to the last remaining node would be the consequence of all the other equations.
16
Kirchhoff’s voltage law
Kirchhoff’s voltage law is formulated as follows: in the closed loop of
the electric circuit, the algebraic sum of the voltages is equal to the algebraic
sum of the EMF sources that are in the closed loop.
The voltages and the EMF sources whose direction coincides with the
way around the loop are considered positive. The voltages and the EMF
sources whose direction is opposite to the way around the loop are considered negative. The way around the loop can be chosen arbitrarily.
Nota bene! The following algorithm can be proposed for the practical
use of Kirchhoff’s voltage law:
1.
Choose the current directions in the branches that form the closed
loop.
2.
Choose the way around the loop.
3.
Write the equation, the left-hand member of which is the algebraic
sum of the voltage drops across the branch resistances; the right-hand member of the equation is the algebraic sum of the EMF sources that are in the
closed loop; the voltage drop across the resistance must be written in accordance with Ohm’s law for the passive branch (1.1).
For the loop shown in Fig. 4.4, the following is true:
IR IR IR E E E. (4.4)
11 2 2 3 3 1 2 4
Fig. 4.4
17
Kirchhoff’s voltage law is true only for the closed loop. At the same
time, any open part of the circuit can be complemented to the closed loop by
the voltage drop across the breakage of the open part.
Example 4.1 (using Kirchhoff’s voltage law for the open part of the
circuit)
The open part of the circuit denoted as
complement it to the closed loop, add the voltage between the point
the point
d (Fig. 4.5 b). Then Kirchhoff’s voltage law can be written for the
abcd is shown in Fig. 4.5 a. To
c and
abcd loop:
IR IR U E.
11 1 2 1cd
a)
b)
Fig. 4.5
If there is the current source in the circuit, the important circumstance
should take into account. The resistance of the current source is infinity,
therefore the current source does not form the closed loop and cannot be the
term of the equations of Kirchhoff’s voltage law. However, the current
source must certainly be in the equations of Kirchhoff’s current law.
18
If it is necessary to write the equation of Kirchhoff’s voltage law for the
loop that contains the current source then the current source should be replaced by the voltage at its terminals.
Example 4.2 (using Kirchhoff’s laws if there is the current source in the
circuit)
It is necessary to write the equation of Kirchhoff’s current law for the
a, and the equation of Kirchhoff’s voltage law for the loop that is de-
node
noted as
abcd (Fig. 4.6 a).
There is the current source in the equation of Kirchhoff’s current law,
and this equation is
II J .
13
0
k
a)
b)
Fig. 4.6
19
In order to write the equation of Kirchhoff’s voltage law for the abcd
loop, replace the current source by the voltage at its terminals. If the way
around the loop is counterclockwise (Fig. 4.6 b), then the equation is
IR U E.
33 CS 2
To simplify the analysis, the current source with the parallel resistance
can be replaced by the equivalent EMF source (see Fig. 2.3 b, c and the
formulas (2.3)). After the analysis, it is necessary to return to the original
circuit.
The independent loop of the circuit
In principle, as many equations of Kirchhoff’s voltage law can be obtained as many closed loops the circuit contains. However, not all of these
equations are independent. To determine the independence of the equations
of Kirchhoff’s voltage law, let us introduce the concept of the independent
loop of the circuit.
The independent loop is one that contains at least one new branch that is
not included in the other loops of the circuit. The independent loops are
chosen arbitrarily, but the easiest way is to choose independent loops so that
they coincide with the cells of the circuit (Fig. 4.7).
Fig. 4.7
20
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