Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:The elements of the electrical circuit theory. Tutorial
.pdf
E
R
E
Fig. 14.6
First, let us obtain the circuit that corresponds to the no-load mode. Let
us remove from the circuit the branch with the sought current and replace it
with two output terminals. Between the output terminals, let us denote the
no-load voltage
U
that is directed in the same direction as the sought cur-
nl
rent (Fig. 14.7).
Fig. 14.7
For the left loop, the equation of Kirchhoff’s voltage law is
IRU E. (Ex. 14.1.1)
11 1nl nl
The current can be found using Ohm’s law:
E
II
12
nl nl
12
12
. (Ex. 14.1.2)
R
The no-load voltage (to which the EMF of the equivalent generator is
equal) can be found from formula (Ex. 14.1.1):
UEIR . (Ex. 14.1.3)
111equ nl nl
71
Next, let us find the resistance of the equivalent generator relative to the
R
R
R
output terminals. It can be done using the passive circuit, from which all the
sources are removed and replaced by their internal resistances (Fig. 14.8).
Fig. 14.8
From the analysis of Fig. 14.8, the formula is obvious for determining
the resistance of the equivalent generator:
R
R
equ
12
. (Ex. 14.1.4)
R
12
Finally, let us find the current
I
using Ohm’s law:
3
E
I
3
equ
equ
. (Ex. 14.1.5)
R
3
15. THE SINUSOIDAL ELECTRIC CIRCUITS
The sinusoidal current is the current that varies in the time according to
the harmonic law (Fig. 15.1):
2
it I t I t
() sin sin
The maximum value of the harmonic function of the current is called the
amplitude of the current and denoted as
as T) is the time during which the harmonic function makes the single oscil-
⎛⎞
mm
⎜⎟
T
⎝⎠
72
I . The period (that is denoted
m
. (15.1)
lation. The quantity that is inverse to the period is called the frequency (its
dimensionality is Hertz, or
1s
):
1
fT .
Fig. 15.1
The frequency multiplied by the constant 2 is called the angular fre-
quency (its dimensionality is radian per second, or
2
2
f
.
1s
):
T
In the expression (15.1), the argument of the harmonic function (i.e. the
sum
) is called the phase and the term Ψ is called the initial phase.
t
Thus, any harmonic function is determined by three quantities: the amplitude, the angular frequency and the initial phase.
The value of the harmonic function (15.1) at any arbitrary point of time
is called the instantaneous value.
The low-frequency sinusoidal currents and electromotive forces are obtained using the synchronous generators. The high-frequency sinusoidal currents and electromotive forces are obtained using the electron-tube generators or the semiconductor oscillators.
The energy characteristics of the alternating current are not determined
by its instantaneous value. Therefore, integral characteristics (the average
73
value of the alternating current and the root-mean-square value of the alter-
E
E
nating current) are used to estimating the alternating current.
The average value of the harmonic function is its average value during
half the period. Thus, the average value of the sinusoidal current is
T
2
22 2
IItdtII
av
mmm
∫
TT
0
⎛⎞
sin 0.638
⎜⎟
⎝⎠
. (15.2)
The average values of the sinusoidal electromotive force and the sinusoidal voltage are determined in the same way:
T
2
UUtdtUU
22 2
av
av
EtdtEE
mmm
∫
TT
0
T
2
22 2
mmm
∫
TT
0
⎛⎞
sin 0.638
⎜⎟
⎝⎠
⎛⎞
sin 0.638
⎜⎟
⎝⎠
, (15.3)
. (15.4)
The root-mean-square value of the sinusoidal current (that is also called
the effective value) is determined by the following formula:
TT
112
222
IitdtI tdt I
() sin 0.707
∫∫
TTT
00
mm
⎛⎞
⎜⎟
⎝⎠
I
m
. (15.5)
2
The root-mean-square values of the sinusoidal electromotive force and
the sinusoidal voltage are determined in the same way:
TT
112
222
() sin 0.707
UutdtU tdt U
e t dt E t dt E
∫∫
TTT
00
TT
112
222
() sin 0.707
∫∫
TTT
00
mm
mm
⎛⎞
⎜⎟
⎝⎠
⎛⎞
⎜⎟
⎝⎠
E
m
, (15.6)
2
U
m
. (15.7)
2
The root-mean-square value is the extremely important characteristic of
the sinusoidal current, since it is the root-mean-square value that determines
the correspondence between the sinusoidal current and the direct current.
74
That is why the root-mean-square values of the electromotive force, the cur-
p
rent and the voltage are denoted by the same symbols that are used for the
electromotive force, the current and the voltage in the direct current circuits
(by the capital letters without the subscripts). Instantaneous values of electromotive force, current and voltage are denoted by the lowercase letters.
Let us compare the thermal action of the sinusoidal current with the
thermal action of the direct current, which flows during the same time in the
same resistor
R.
The quantity of heat that the resistor dissipates during the period of the
sinusoidal current is
T
2
()
QitRdt RT
sin
∫
0
2
IRT I
mm
2
2
⎛⎞
⎜⎟
2
⎝⎠
. (15.8)
The quantity of heat that the resistor dissipates during the same time in
the direct current circuit is
The coefficient
2
I
⎛⎞
m
⎜⎟
⎝⎠
in the expression (15.8) is the square of the
2
QIRT . (15.9)
dc
2
root-mean-square value of the sinusoidal current. If we compare the expression (15.8) with the expression (15.9), we conclude that the root-meansquare value of the sinusoidal current is equal to such direct current, which
dissipates the same quantity of heat during the period of sinusoidal current.
When the sinusoidal current flows in the branches of the electrical circuit, these branches consume the power of the sources. The instantaneous
power is the product of the instantaneous value of the branch voltage by the
instantaneous value of the branch current:
() ()()
tutit . (15.10)
Let us consider the relationships between currents and voltages in the
simplest electric circuits. The elements of the real existing sinusoidal circuits are the resistors, the inductance coils, and the capacitors. The resistors
dissipate the energy of the sources in the form of heat. The reactive elements (the inductance coils and the capacitors) at one moment accumulate
the energy and at another moment return it back to the circuit.
75
As we found out above, the resistor can be characterized by the depend-
p
ence of the voltage across the resistor on the current that flows in the resistor (this dependence is called the volt-ampere characteristic), or can also be
characterized by the resistance. The resistance of the resistor is constant in
the linear sinusoidal circuits:
where
u is the instantaneous value of the voltage across the resistor; and i is
u
Ri
const
(15.11)
the instantaneous value of the current that flows in it.
Let there is the known current in the branch:
() sin
it I t
m
. (15.12)
At any arbitrary point of time, according to Ohm’s law, the voltage
across the resistor is equal to the product of the current and the resistance. If
the circuit is linear, the voltage across the resistor can be determined by
multiplying the expression (15.11) and the expression (15.12):
() () sin sin
ut Rit RI t U t
mm
. (15.13)
As we can see from the obtained expression (15.13), the voltage across
the resistor and the current in the resistor coincide in the phase.
Let us insert the expression (15.12) and the expression (15.13) into the
expression (15.10) in order to find the instantaneous power:
UI UI
() sin cos2
tUI t t
mm
2
mm mm
22
. (15.14)
The curves of the current, the voltage and the instantaneous power in the
resistor are shown in Fig. 15.2. The instantaneous power is always positive
or equal to zero. This means that the resistor only consumes energy, but
does not returns it back to the circuit.
The power in the resistor, as a rule, is estimated as the average value of
the instantaneous power during the period:
TT
1
Pptdt tdt UI
() sin
∫∫
TT
00
UI UI
mm mm
2
2
. (15.15)
The power in the resistor, which is determined by the expression (15.15),
is called active (its dimensionality is Watt).
76
Fig. 15.2
The inductive element (i.e. the inductance coil) allows us to take into
account the electromotive force that induced by means of changing the
magnetic flux. The inductance coil can be characterized by the dependence
of the flux linkage on the current that flows in the coil (this dependence is
called the weber-ampere characteristic), or can also be characterized by the
inductance (its dimensionality is Henry). The inductance of the coil is constant in the linear sinusoidal circuits:
constL
(15.16)
i
where
L is the inductance of the inductance coil; ψ is the instantaneous val-
ue of the flux linkage of the inductance coil; and
i is the instantaneous value
of the current that flows in the inductance coil.
When the magnetic flux in the inductance coil changes, the selfinduction electromotive force occurs that opposes this change:
()
where
()
et L
L
()
et
is the self-induction electromotive force in the inductance coil.
L
77
di t
(15.17)
dt
Let us insert the expression of the sinusoidal current (15.12) into the ex-
L
L
X
p
pression of the self-induction electromotive force (15.17):
et L I t I L t
() sin cos
Lmm
d
dt
. (15.18)
The voltage across the inductance coil is directed opposite the selfinduction electromotive force:
ut e t I L t
() () cos
IL t U t
mm
⎛⎞ ⎛⎞
sin sin
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
m
. (15.19)
22
If we analyze the expression (15.19), we conclude that the current
curve lags behind the curve of the voltage across the inductance coil by
the angle that is equal to
. Thus, the inductance coil is the phase-shifting
2
element.
The product
tance and is denoted as
L , in the expression (15.19), is called the inductive reac-
(it is obvious, from the expression (15.19), that
the dimensionality of the inductive reactance is Ohm).
The instantaneous power in the inductance coil is
UI
() ()() sin cos sin2
tutitUI t t t . (15.20)
mm
mm
2
The curves of the current, the voltage and the instantaneous power in the
inductance coil are shown in Fig. 15.3. The instantaneous power in the inductance coil, passing through zero, changes sign with double frequency.
This means that, in the first quarter of the period, the inductance coil consumes the energy of the source (i.e. accumulates it in the magnetic field), in
the second quarter of the period, the inductance coil returns it back to the
circuit.
Since all the accumulated energy, somehow or other, is returned back to
the source, this energy does not do the work. Therefore, the power of the
inductance coil is called reactive (its dimensionality is reactive VoltAmpere) and the inductance coil is called the reactive element.
78
Fig. 15.3
The capacitive element (i.e. the capacitor) allows us to take into account
the energy accumulation in the electrostatic field. The capacitor can be
characterized by the dependence of the electric charge on the voltage across
the capacitor (this dependence is called the coulomb-volt characteristic), or
can also be characterized by the capacitance (its dimensionality is Farad).
The capacitance of the capacitor is constant in the linear sinusoidal circuits:
q
where
C is the capacitance of the capacitor; q is the instantaneous value
Cu
of the electric charge of the capacitor; and
const
(15.21)
u is the instantaneous value of the
voltage across the capacitor.
If the voltage across the capacitor does not change over time, then its
charge also does not change over time, and the current does not flow in the
capacitor. However, if the voltage across the capacitor changes over time,
for example, according to the harmonic law
() sin
ut U t, (15.22)
m
then the electric charge of the capacitor will also change according to the
harmonic law:
() () sin
qt Cut СUt
m
79
. (15.23)
Thus, the capacitor will be recharged periodically. The periodic recharge
X
p
of the capacitor is accompanied by the charging current that flows in it:
()
dq t d
() sin
it CU t
dt dt
m
U
UC t t
cos sin
m
m
1
C
⎛⎞
⎜⎟
⎝⎠
. (15.24)
2
If we analyze the expression (15.24), we conclude that the curve of the
voltage across the capacitor lags behind the current curve by the angle that
is equal to
. Thus, the capacitor is also the phase-shifting element.
2
The fraction
1
, in the expression (15.24), is called the capacitive reac-
C
tance and is denoted as
(it is obvious, from the expression (15.24), that
C
the dimensionality of the capacitive reactance is Ohm).
The instantaneous power in the capacitor is
22
UU
() ()() sin cos sin2
tutit t t t
mm
1
C
2
X
C
. (15.25)
The curves of the current, the voltage and the instantaneous power in
the capacitor are shown in Fig. 15.4. In the first quarter of the period, the
capacitor consumes the source energy and this energy creates the electric
field in it. In the second quarter of the period, the capacitor returns the energy back to the circuit. The power of the capacitor is also reactive (its
dimensionality is also reactive Volt-Ampere) and the capacitor is also the
reactive element.
The current in the capacitor can be expressed in the following form:
() ()
dq t du t
()
it C
. (15.26)
dt dt
Let us integrate the expression (15.26) in order to obtain the formula
that expresses the voltage across the capacitor in terms of the current:
1
() ()ut itdt
∫
C
. (15.27)
80
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]