Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:The elements of the electrical circuit theory. Tutorial
.pdf
Fig. 15.4
16. THE VECTOR DIAGRAMS
Kirchhoff’s laws are true in the linear sinusoidal circuits for the instantaneous values of the currents and the voltages. However, it is very difficult
to apply Kirchhoff’s laws directly to harmonic functions. The vector diagram is just that very tool, which facilitates the operations on the harmonic
functions.
As we found out above, any harmonic function (for example, the function that describes the current in the circuit) is determined by three quantities: the amplitude, the angular frequency and the initial phase. Let the current in the circuit is described by the harmonic function (15.1). In the certain
coordinate system, let us draw the vector OA
of which is equal to the amplitude of the current
the vector and the positive half-axis ox is equal to the initial phase of the
current Ψ.
Let us rotate the vector OA
counterclockwise around the origin of the
coordinates with the angular frequency that is equal to
number of seconds that is equal to t, the vector will move by the angle that
is equal to t (Fig. 16.1 b).
(Fig. 16.1 a), the magnitude
I
, and the angle between
m
. After the certain
81
a) b)
Fig. 16.1
There is no doubt that, when the vector
axis y is identical with the harmonic function (15.1). This means that there
is the univocal correspondence between the rotating vector OA
rent
it .
As a rule, there are only the sources with the same frequency in the circuits. Therefore, if the vectors of all the currents and voltages will rotate
with the same angular frequency, then their mutual position will not change.
Consequently, we can use the addition of the vectors instead of the addition
of the harmonic functions.
Since in the addition of the vectors only their mutual position matters,
there is no need to rotate the vectors. For the same reason, the choice of the
coordinate system and the reference point is absolutely arbitrary.
Nota bene! Since the root-mean-square value is the main characteristic
of the sinusoidal currents and voltages, often it is the root-mean-square value of the current or the voltage (not the amplitude value) that is used as the
vector magnitude.
Example 16.1 (using the addition of the vectors)
rotates, its projection on the
OA
and the cur-
There are two sinusoidal currents:
find the unknown current
it by means of the addition of the vectors.
3
it and
1
82
it. It is necessary to
2
The given data are
( ) 11.31sin(314 15 )
it t A
1
( ) 5.66 sin(314 60 )
it t A
2
() () ()
it it it
312
For the addition of the vectors, let us use the vector diagram that will be
drawn for the root-mean-square values of the currents. Therefore, first, let
us find the root-mean-square values of the currents:
11.31
IA
1
8
2
5.66
IA
2
4
2
Vectors must be drawn from the origin of the coordinates, the magnitudes of the vectors must be equal to the currents (on the certain scale), and
the initial phases of the vectors should be counted off from the positive halfaxis
ox (Fig. 16.2 a). The vectors must be added according to the law of par-
allelogram (Fig. 16.2 b).
a) b)
Fig. 16.2
Both the root-mean-square value and the initial phase of the unknown
current
it can be found from the vector diagram:
3
11.2
IA
3
3
83
30
The amplitude of the current
mean-square value of the current by
( ) 11.2 2 sin(314 30 ) 16 sin(314 30 )it t t A
3
it can be found by multiplying the root-
3
. As a result, the final solution is
2
17. THE SYMBOLIC METHOD
The symbolic method allows us, for the analysis of the sinusoidal circuits, to use the analytical calculations instead of the graphic-analytical calculations realized by means of the vector diagrams. On the one hand, this
increases the accuracy of the calculations; on the other hand, it greatly facilitates the mathematical manipulations with the harmonic functions. In addition, the symbolic method allows us, without limitation, to use for the analysis of the sinusoidal circuits all the methods that are used for the analysis
of the direct current circuits.
The main idea of the symbolic method is carrying over to the complex
plane the very vector diagram that we know. Let the vector, which represents to the harmonic function of the current, rotates counterclockwise on
the complex plane with angular frequency that is equal to (Fig. 17.1).
Then the certain complex function can correspond to the vector. It is this
complex function that will represent the real existing sinusoidal current.
Fig. 17.1
84
It is necessary to note that, in the technical sciences, it is customary to
j
denote by the letter j the imaginary unit, in order to not to mix up it with the
current. Also, let us will denote the complex values of the electromotive
force, the current, the voltage and the power by the capital letter with the dot
on top, and the modulus of the complex number we will denote by the capital letter without the dot. The complex values of the resistances and the conductances we will denote by the capital letter with the line below.
If the harmonic function of the current is
() sin
it I t, (17.1)
m
then the complex function, which represents this sinusoidal current, is
jt
()
I t Ie Ie e
mm m
cos sin
Itjt ⎡ ⎤
m
⎣⎦
jt j
. (17.2)
There is no doubt that the harmonic function (17.1) coincides with the
imaginary part of the complex function (17.2).
As a rule, there are only the sources with the same frequency in the circuits and therefore there is no need to rotate the vectors. This means that the
multiplier
te
in the expression (17.2), in which there is the time component, can be omitted; and we will manipulate only with the certain complex
number (but not the complex function).
Thus, all the real existing harmonic functions must be replaced by the
complex numbers and the analysis of the sinusoidal circuit must be carried
out in the complex form. The result must be represented in the harmonic
form.
It is necessary to note that, in the mathematical manipulations with the
complex numbers, both the root-mean-square values of the currents and the
voltages can be used and the amplitude values.
Now, let us consider some properties of the complex numbers.
It is known that the complex number can be represented as the point on
the complex plane. Depending on the way in which we denote the coordinates of this point, two forms of the complex number can be distinguished:
the algebraic form and the exponential form.
The algebraic form of the complex number corresponds to the Cartesian
coordinate system on the complex plane (Fig. 17.2):
85
A
j
A
A
A
where
is the complex number; a is the real part of the complex number
(it is the coordinate that is taken along the axis
ajb
(17.3)
Re); b is the imaginary part
of the complex number (it is the coordinate that is taken along the axis
Fig. 17.2
The exponential form of the complex number corresponds to the polar
coordinate system on the complex plane (Fig. 17.3):
Ae
(17.4)
Im).
where
vector that connects the origin of the coordinates to the point
A is the modulus of the complex number (it is the magnitude of the
); Ψ is the
exponent of the complex number (it is the angle that is counted off from the
positive half-axis
Re).
The algebraic form of the complex number and the exponential form of
the complex number are related to each other through the trigonometric
form of the complex number:
cos sinAA j
. (17.5)
The formulas can be obtained from the expression (17.5) for converting
the complex number from one form to another form.
86
j
Fig. 17.3
For converting the complex number from the exponential form to the
algebraic form, the following formulas must be used:
cos
aA
bA
(17.6)
sin
For converting the complex number from the algebraic form to the exponential form, the following formulas must be used:
Aab
Multiplying the complex number by
vector by the angle
22
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
b
arctan if 0
a
a
b
arctan if 0 and 0
ab
a
b
arctan if 0 and 0
ab
a
⎛⎞
, because
⎜⎟
2
⎝⎠
je
(Fig. 17.4).
87
(17.7)
j is equivalent to rotating the
2
Fig. 17.4
j
Multiplying the complex number by
⎛⎞
vector by the angle
, because
⎜⎟
2
⎝⎠
j is equivalent to rotating the
je
(Fig. 17.5).
Fig. 17.5
88
2
Multiplying the complex number by
j
j
j
j
vector by the angle
The differentiation of the harmonic function corresponds to multiplying
its complex representation by
, because
1
(Fig. 17.6).
Fig. 17.6
:
1 is equivalent to rotating the
ee
j
jI
(17.8)
() sin
it I t I Ie
m
()
di t
dt
The integration of the harmonic function corresponds to dividing its
complex representation by
j :
() sin
it I t I Ie
m
itdt
()
∫
89
I
j
(17.9)
If there is the imaginary number in the denominator of the fraction, then
the imaginary unit can be transferred to the numerator by multiplying both
the numerator and denominator by
equality:
If there is, in the denominator of the fraction, the number, in which the
real part is not zero, then both the numerator and denominator must be multiplied by the conjugate complex number. The conjugate complex number is
the number, which is symmetric to the original number relative to the real
axis (Fig. 17.7). The conjugate complex number is denote by the asterisk
on top.
1jj). For example:
1
jjj
j (there is the following well-known
j
j
. (17.10)
Fig. 17.7
Here the well-known equality is used, which is that the product of two
conjugate complex numbers is equal to the square of their modulus:
AA A
2
. (17.11)
90
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]