Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:The elements of the electrical circuit theory. Tutorial
.pdf
complex plane), we will not need the coordinate axes, so they can be re-
L
X
L
X
L
X
moved. Since the current lags behind the voltage at the terminals of the
EMF source, the current in the circuit is inductive, and the angle φ must be
positive (that is, it must be counted off counterclockwise). Let us denote, in
the vector diagram, the angle φ considering this fact. In practice, this means
that we only have to replace the angle
with the angle φ (Fig. 19.3 a).
Thus, the vector diagram, which is shown in Fig. 19.3 a, is the final solution
to the problem in the first case, when
X . The current in the circuit is
C
inductive and the angle φ is positive, since the current lags behind the voltage at the terminals of the EMF source.
a) b) c)
Fig. 19.3
The second case, when
X . The current in the circuit is capaci-
C
tive and the angle φ is negative, since the voltage at the terminals of the
EMF source lags behind the current (Fig. 19.3 b).
The third case, when
X . The current in the circuit is purely re-
C
sistive and the angle φ is zero, since the current coincides in the phase with
111
the voltage at the terminals of the EMF source that is equal to the voltage
L
j
E
across the resistor (Fig. 19.3 c). This mode is called the voltage resonance.
Nota bene! In all the vector diagrams, which are shown in Fig. 19.2 and
in Fig. 19.3, for the sake of the greater clearness, the vectors are slightly
separated each from other. In the case when the vector diagram is used as
the tool of the calculations, this should not at any way be. In particular, in the
mode of the voltage resonance (see Fig. 19.3 c), the vector
jIX
tor
posite in the direction; the vector
coincide with each other, they are equal in the modulus and op-
C
and the vector IR are the same vector.
IX
and the vec-
20. THE PARALLEL-CONNECTED RLC-ELEMENTS
Let us consider the circuit with the parallel connection of the resistor,
the inductance coil, the capacitor and the EMF source (Fig. 20.1). In this
case, it is more opportune to use the conductances of the branches (but not
their resistances) therefore the resistor is denoted as g in the circuit diagram.
Fig. 20.1
Let us find the input current
I
in the circuit if the EMF source ensures,
at its terminals, the input voltage that varies according to the following harmonic law:
() sin
et E t. (20.1)
m
Let us write the equation of Kirchhoff’s current law for the instantane-
ous values of the currents:
() () () ()
it i t i t i t. (20.2)
gLC
112
Let us express the currents of the resistor, the inductance coil and the
g
capacitor in terms of the input voltage (which, according to Kirchhoff’s
voltage law, is equal to the electromotive force of the source). In order to do
this, let us choose, in the circuit, three independent loops and write for them
the equations of Kirchhoff’s voltage law:
()
it
g
()
et
, (20.3)
()
di t
()
et Ldt , (20.4)
L
Then, let us express the currents in the equations (20.3)-(20.5):
() ()
Now, if we insert the expressions (20.6)-(20.8) into the equation of
Kirchhoff’s current law (20.2), we will obtain the following integrodifferential equation:
() () ()
it etg etdt C
If the EMF source ensures, at its terminals, the harmonic voltage, then
the solution of the integro-differential equation (20.9) can also be obtained
in the form of the harmonic function with the same angular frequency:
The initial phase of the current in the expression (20.10) is the constant
that is not yet known to us. It is opportune to denote this constant as
() sin
it I t. (20.10)
1
et i t dt
() ()
it etg , (20.6)
g
it etdt
() ()
L
()
it C
C
C
∫
C
1
∫
L
. (20.8)
1()
∫
Ldt
m
. (20.5)
, (20.7)
()
de t
dt
de t
. (20.9)
,
113
because if φ is the angle between the input voltage and the input current,
then the initial phase of the input current will be
when the initial
phase of the input voltage is zero.
Let us insert the expression (20.1) and the expression (20.10) into the integro-differential equation (20.9):
E
ItEgt tECt
. (20.11)
sin sin cos cos
mm m
m
L
There is no doubt that the equation (20.11) must be true for any arbitrary
point of time. In particular, if 0t , then the equation (20.11) must take
the following form:
1
IE C
sin
mm
⎛⎞
⎜⎟
L
⎝⎠
. (20.12)
But, on the other hand, if
t , then the equation (20.11) must take
2
the following form:
cos
IEg . (20.13)
mm
Let us square both the equation (20.12) and the equation (20.13) and
then let us sum up these equations term by term. And we will must obtain
22
(since there is the well-known equality
sin cos 1 ) following ex-
pression:
222
IEg C
mm
⎡⎤
⎢⎥
⎢⎥
⎣⎦
1
⎛⎞
⎜⎟
L
⎝⎠
2
. (20.14)
It follows, from the expression (20.14), that
1
⎛⎞
IEg C Eg bb EY
mm m LC m
22
⎜⎟
L
⎝⎠
2
2
. (20.15)
If we divide the equation (20.12) by the equation (20.13), then we will
obtain the following expressions:
114
1
L
L
L
L
L
L
E
E
C
L
tan
, (20.16)
bb
C
gg
bb
arctan
C
. (20.17)
g
As a result, the solution of the integro-differential equation (20.9) is
it E Y t
where
bL
L
The conductances
1
() sin arctan
m
; CbC ;
Ygbb
b
and
but the equivalent conductance
⎛⎞
⎜⎟
⎝⎠
2
b
themselves are always greater than zero,
C
bb can be either positive, or nega-
C
bb
2
LC
C
(20.18)
g
.
tive, or zero. Let us analyze the vector diagrams that correspond to all these
cases.
The first case, when
bb . Let us analyze drawing the first vector
C
diagram in detail. When the scales of the current and the voltage are chosen,
let us draw, first of all, the vector of the voltage across all the parallelconnected elements (this voltage, according to Kirchhoff’s voltage law, is
equal to the electromotive force of the source). Let us draw this vector hori-
zontally from the left to the right and denote it as
, but we will always
keep in our mind that this vector is the voltage across all the parallelconnected elements (Fig. 20.2 a). It is necessary to leave free space around
this vector for all the other vectors that will be oriented relative to the vector
of the voltage across the parallel-connected elements.
Next, let us draw the vector of the current in the resistor
IEg
g
(Fig. 20.2 b) that coincides in the phase with the vector of the voltage across
the resistor (i.e. with the vector
). Let us draw this vector so that it is
slightly shifted (so that it does not merge visually with the vector of the
voltage across the resistor). Let us do the same in the future, always when
vectors can merge visually.
115
g
L
L
E
L
a) b)
c) d)
e) f)
Fig. 20.2
Then let us draw (from the end of the vector of the current in the resis-
tor
I) the vector of the current in the inductance coil
IjEb
L
(Fig. 20.2 c). This vector is directed vertically downward, since the vector of
the current in the inductance coil
across the inductance coil (i.e. the vector
I
lags behind the vector of the voltage
) by the angle that is equal to
.
2
Then let us draw (from the end of the vector of the current in the induct-
ance coil
I) the vector of the current in the capacitor
IjEb
CC
116
(Fig. 20.2 d). This vector is directed vertically upward, since the vector of
E
L
the voltage across the capacitor (i.e. the vector
current in the capacitor
I
by the angle that is equal to
C
) lags behind the vector of
.
2
If we now connect the beginning of the first current vector to the end of the
last current vector, then we will obtain the vector that corresponds to the sum of
I
all three current vectors. It is the vector of the input current
At last, let us denote the initial phase of the input current
(Fig. 20.2 e).
I
(Fig. 20.2 f).
Since the input current lags behind the voltage at the terminals of the
EMF source, the input current is inductive, and the angle φ must be positive
(that is, it must be counted off counterclockwise). Let us denote, in the vector diagram, the angle φ considering this fact. In practice, this means that we
only have to replace the angle
with the angle φ (Fig. 20.3 a). Thus, the
vector diagram, which is shown in Fig. 20.3 a, is the final solution to the
problem in the first case, when
bb . The input current is inductive and
C
the angle φ is positive, since the input current lags behind the voltage at the
terminals of the EMF source.
a) b) c)
Fig. 20.3
117
The second case, when
L
L
L
g
bb . The input current is capacitive and the
C
angle φ is negative, since the voltage at the terminals of the EMF source
lags behind the input current (Fig. 20.3 b).
The third case, when
bb
. The input current is purely resistive and
C
the angle φ is zero, since voltage at the terminals of the EMF source coincides in the phase with the input current that is equal to the current in the
resistor (Fig. 20.3 c). This mode is called the current resonance.
Nota bene! In all the vector diagrams, which are shown in Fig. 20.2 and
in Fig. 20.3, for the sake of the greater clearness, the vectors are slightly
separated each from other. In the case when the vector diagram is used as
the tool of the calculations, this should not at any way be. In particular, in
the mode of the current resonance (see Fig. 20.3 c), the vector
vector
posite in the direction; the vector I
I coincide with each other, they are equal in the modulus and op-
C
and the vector
are the same vector.
I
I and the
118
THE LIST OF THE BOOKS THAT ARE RECOMMENDED FOR STUDY
1. Kuphaldt T. R. Lessons in electric circuits: Volume I – DC: Volume II –
DC / Tony R. Kuphaldt. – Koros press limited, 2006, 2007. – 560 pages, 568 pages.
2. Bird J. Electrical circuit theory and technology / John Bird. – Newnes,
2010. – 737 pages.
3. Mayergoyz I. D. Basic electric circuit theory: A one-semester / Isaak D. Ma-
yergoyz, W. Lawson. – Academic press, 2012. – 449 pages.
4. Sundararajan D. Introductory circuit theory / D. Sundararajan. – Springer in-
ternational publishing, 2019. – 297 pages.
5. Wing O. Classical circuit theory. / Omar Wing – Springer US, 1980. –
184 pages.
119
CONTENTS
Preface ....................................................................................................................... 3
1. The electric circuit and its elements ............................................................... 4
2. The EMF sources and the current sources ..................................................... 9
3. The series-connected and parallel-connected elements of the electric
circuit ............................................................................................................... 12
4. The basic laws of the electric circuits .......................................................... 15
5. The equations set of Kirchhoff’s laws for calculating the currents
of the circuit ................................................................................................. 21
6. The power balance of the electric circuit ..................................................... 23
7. The mesh-current method ............................................................................ 26
8. The superposition method ............................................................................ 37
9. The formulas for converting Y-Δ and Δ-Y .................................................. 41
10. The formulas for converting the parallel branches into one equivalent
branch .......................................................................................................... 48
11. The transference of the sources out of the branch ........................................ 52
12. The node potential method ........................................................................... 57
13. The theorem of the compensation ................................................................ 65
14. The equivalent generator method ................................................................. 67
15. The sinusoidal electric circuits ..................................................................... 72
16. The vector diagrams ..................................................................................... 81
17. The symbolic method ................................................................................... 84
18. The power of the sinusoidal mode................................................................ 98
19. The series-connected RLC-elements .......................................................... 106
20. The parallel-connected RLC-elements ....................................................... 112
The list of the books that are recommended for study ........................................... 119
120
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]