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Министерство путей сообщения
Российской Федерации
Дальневосточный государственный
университет путей сообщения
Кафедра “теоретические
основы электротехники”
В.С. Матющенко
ТЕОРЕТИЧЕСКИЕ ОСНОВЫ ЭЛЕКТРОТЕ=
ХНИКИ
ЛИНЕЙНЫЕ ЭЛЕКТРИЧЕСКИЕ ЦЕПИ ПОСТОЯННОГО
И ОДНОФАЗНОГО СИНУСОИДАЛЬНОГО =
ТОКОВ
Учебное пособие
Рекомендовано
редакционно-издательским совет=
ом ДВГУПС
в качестве учебного пособия
Хабаровск =
p>
2002
&nb=
sp;
&nb=
sp;
Рецензенты:
&nb=
sp;
Кафедра “Электросвязь, микроэлектроника и метрология” Хабаровского филиала =
Сибирского
государственного университета телекоммуникации и информатики,
(за=
ведующий
кафедрой, профессор В.А. Константинов)
Начальник отдела в представительстве РАО ЕЭС Рос=
сии Ю.Г.
Грибов
=
М 353
|
Матющенко
В.С. Теоретические основы электротехники. Линейные электрические цепи
постоянного и однофазного синусоидального токов: Учебное пособие. –
Хабаровск: Изд-во ДВГУПС, 2002. – 111 с.: ил.=
|
В
пособии рассмотрены цепи постоянного и однофазного синусоидального токов,
включая цепи с взаимной индуктивностью, излагаются теоретический материал и
примеры решения задач.
Уч=
ебное
пособие предназначено для студентов ИИФО ДВГУПС, изучающих теоретические основы электротехники=
.
ОГЛАВЛЕНИЕ
ВВЕДЕНИЕ
1.
Линейные электрические цепи постоянного тока
&nb=
sp;
1.1.
Электрическая цепь и ее элементы
&nb=
sp;
1.2.
Закон Ома для участка цепи с ЭДС
&nb=
sp;
1.3
Расчет сложных электрических цепей постоянного тока <=
/p>
&nb=
sp;
1.3.1.
Метод уравнений Кирхгофа
&nb=
sp;
1.3.2.
Метод узловых потенциалов
&nb=
sp;
1.3.3.
Метод контурных токов
&nb=
sp;
1.3.4.
Метод наложения
&nb=
sp;
1.3.5.
Эквивалентное преобразование треугольника и звезды сопротивлений <=
/o:p>
&nb=
sp;
1.4.
Пассивный и активный двухполюсники. Теорема об активном двухполюснике <=
o:p>
&nb=
sp;
1.5.
Метод эквивалентного генератора
&nb=
sp;
1.6.
Линия электропередачи постоянного тока
2.
Электрические цепи однофазного синусоидального тока <=
/p>
&nb=
sp;
2.1.
Закон электромагнитной индукции
&nb=
sp;
2.2.
Получение синусоидальной ЭД=
С.
Характеристики синусоидальных величин. Обозначения в цепях переменного тока=
&nb=
sp;
2.3.
Действующее значение переменного тока
&nb=
sp;
2.4.
Представление синусоидальной функции времени вращающимся вектором. Векторные
диаграммы
&nb=
sp;
2.5.
Основные сведения о комплексных числах
&nb=
sp;
=
2.6.
Представление синусоидальных функций времени комплексными числами =
&nb=
sp;
2.7.
Способы задания синусоидального тока
&nb=
sp;
2.8.
Законы Кирхгофа в цепях синусоидального тока. Методы расчета цепей
синусоидального тока
&nb=
sp;
2.9.
Понятие об активном сопротивлении. Синусоидальный ток в активном сопротивле=
нии
&nb=
sp;
2.10.
Самоиндукция. Индуктивность. Синусоидальный ток в индуктивности =
o:p>
&nb=
sp;
2.11.
Синусоидальный ток в емкости
&nb=
sp;
2.12.
Последовательное соединение активного сопротивления, индуктивности и емкост=
и
&nb=
sp;
2.13.
Параллельное соединение активного сопротивления, индуктивности и емкости
&nb=
sp;
2.14.
Пассивный двухполюсник в цепи синусоидального тока. Эквивалентные сопротивл=
ения
и проводимости
&nb=
sp;
2.15.
Закон Ома в символической форме для произвольной цепи
&nb=
sp;
2.16.
О расчете цепей синусоидального тока
&nb=
sp;
2.17.
Резонансы в электрических цепях
&nb=
sp;
2.18.
Энергия и мощность в цепи синусоидального тока
3.
ЦЕПИ С ВЗАИМНОЙ ИНДУКТИВНОСТЬЮ
&nb=
sp;
3.1.
Явление взаимной индукции
&nb=
sp;
3.2.
Последовательное соединение индуктивно связанных элементов <=
/span>
&nb=
sp;
3.3.
Параллельное соединение индуктивно связанных элементов
&nb=
sp;
3.4.
Разметка зажимов индуктивно связанных катушек
&nb=
sp;
3.5.
Сложная цепь с взаимной индуктивностью
&nb=
sp;
3.6.
Эквивалентная замена индуктивных связей
&nb=
sp;
3.7.
Трансформатор без стального сердечника
СПИСОК
ЛИТЕРАТУРЫ
=
p>
ВВЕДЕНИЕ
Ро=
ль и
значение электрической энергии в жизни и экономической деятельности человека
общеизвестны. Благодаря удобству ее получения, передачи на расстояние и
распределения между потребителями она получила широкое распространение и ст=
ала
основой современной техники, базой для развития промышленности, транспорта,
сельского хозяйства, связи, нашла свое место в медицине и образовании, проч=
но
вошла в быт.
Вся
область практического применения электрических и магнитных явлений получила
название электротехники. Все ее отрасли тесно связаны между собой, техничес=
кие
средства построены на общих принципах, подчиняются одним и тем же законам.
Базой для исследования этих принципов и законов стал курс “Теоретические ос=
новы
электротехники” (ТОЭ), задачами которого являют=
ся
анализ электромагнитных процессов и явлений, изучение механизма действия
электрических машин и аппаратов, разработка и обобщение методов расчета
различных электротехнических устройств. Курс ТОЭ
имеет большое значение в формировании научного мировоззрения
инженера-электрика. На нем базируются все специальные электротехнические
дисциплины.
Предлагаемое учебное пособие ставит целью помочь студенту разобраться в сло=
жных
теоретических вопросах, понять суть изучаемых явлений, научиться решать
конкретные практические задачи. Пособие не является в полном смысле учебник=
ом
по ТОЭ. Оно включает в себя только отдельные ра=
зделы
курса. Ряд вопросов излагается без доказательств и подробных выводов. Для
лучшего усвоения материала в пособие включено большое количество численных
примеров с решениями.
По=
собие
предназначено для студентов, обучающихся без отрыва от производства и изуча=
ющих
ТОЭ самостоятельно.
<=
/span>
1. Линейные электрические цепи постоянного тока=
b>
1.1. Электрическая цепь и ее элементы
Электрическая цепь представляет собой совокупность устройств, предназначенн=
ых
для производства, передачи и потребления электрической энергии. Пример
простейшей электрической цепи показан на рис. 1.1. Кружок со стрелкой внутр=
и и
стоящей рядом буквой Е=
(рис. 1.1, а) обознача=
ет так
называемый источник ЭДС (его еще называют источ=
ником
напряжения). Это идеализированный источник энергии, внутреннее сопротивление
которого равно нулю, а напряжение постоянно по величине, равно ЭДС реального источника и не зависит от протекающего =
по
нему тока. Стрелка показывает направление возрастания потенциала внутри
источника. Плюс находится у острия, минус – у хвоста стрелки. Ток во внешней
цепи протекает по направлению стрелки ЭДС – от =
плюса
источника к минусу. Внутреннее сопротивление реального источника R0<=
/i> соединяется последовательно с ЭДС Е, и в совокупности они образуют схему замещения реального
источника (на рис. 1.1, а обведена пункти=
ром).
=
&nb=
sp;
а) &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
б)
Рис. 1.1. Простейшая электрическая цепь
Др=
угое
представление схемы генератора осуществляется в виде параллельного соединен=
ия
источника тока и сопротивления R0 (рис. 1.1, б). Под источником тока понимают также
идеализированный источник энергии, внутреннее сопротивление которого бескон=
ечно
велико, и который вырабатывает ток J, не зависящий от величины нагр=
узки R и равный частному от деления =
ЭДС реального источника на его внутреннее сопротивлен=
ие J =3D E/R0=
. На схеме он изображается кру=
жком с
двойной стрелкой, рядом с которым ставится буква J (рис. 1.1, б).
В =
схеме
рис. 1.1, а ЭДС равн=
а
сумме напряжений на нагрузке и внутреннем сопротивлении источника:
Е =3D U + IR0.
Отсюда
U =3D E – IR0<=
/sub>.
&nb=
sp; =
&nb=
sp;
(1.1)<=
/span>
Рис.
1.2. Внешняя характеристика генератора
|
Последнее выражение представляет так называемую внеш=
нюю
характеристику генератора. Оно говорит о том, что напряжение на его зажим=
ах
меньше ЭДС на величину падения напряжения на
внутреннем сопротивлении (рис. 1.2). Чем больше ток и внутреннее
сопротивление генератора, тем меньше выдаваемое им напряжение. При холост=
ом
ходе генератора (при I =
=3D 0) напряжение, измере=
нное
на его разомкнутых зажимах равно ЭДС: =
U =3D E.
|
<=
span
style=3D'font-size:13.5pt;font-family:Arial'>Рис. 1.3. Напряжение на зажи=
мах
источника
|
На
практике часто приходится сталкиваться с элементами схемы, показанными на
рис. 1.3. Разница между ними заключается во взаимном направлении стрелок =
ЭДС и напряжения. В первом случае (рис. 1.3, а=
i>),
когда эти стрелки направлены противоположно друг другу, напряжение
определяется как разность потенциалов положительного и отрицательного заж=
имов
источника и поэтому положительно. При одинаковых направлениях стрелок E=
и U (рис. 1.3, б) напряж=
ение
равно разности отрицательного и положительного потенциалов, а потому оно
отрицательно:
U =3D – E.
|
&nb=
sp; Пример
1.1. Напряжение холостого хода батареи равно 16,4 В. Чему равно ее
внутреннее сопротивление, если при токе во внешней цепи, равном 8 А, напряж=
ение
на ее зажимах равно 15,2 В?
Р е ш=
е н и е. В соответствии с уравнением (1.1) из схемы,
показанной на рис. 1.4, а, следует U<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D UX
=3D E =3D 16,4 В.
Схема 1.4, б д=
ает
Ом.
&nb=
sp; =
а
&nb=
sp; =
&nb=
sp; =
б
Рис. 1.4. Разомкнутая (а=
) и
замкнутая (б) цепи
При
решении задачи мы полагали, что измерение проводилось идеальным вольтметром,
имеющим бесконечно большое сопротивление. При конечной величине сопротивлен=
ия
вольтметра в измерение вносится погрешность.
=
Рис. 1.5. Измерение ЭДС<=
/span>
вольтметром
|
Пример 1.2. ЭДС б=
атареи
измеряется вольтметром, имеющим сопротивление =
RV. Чему равно
показание вольтметра при трех различных значениях его сопротивления, если=
Е =3D 80 В<=
/span>,
=3D 100 Ом?
Р
е ш е н и е. Пока=
зание
вольтметра U<=
sub>V
равно падению напряжения на его сопротивлении (рис. 1.5):
|
;
&nb=
sp; =
(1.2)
а) RV=
=3D 100 кОм:
|
В;
|
б) RV=
=3D 2,5 кОм:
|
В;
|
в) RV=
=3D 400 Ом:
|
В.
|
Чем
больше сопротивление вольтметра, тем меньше погрешность измерения. Как след=
ует
из формулы (1.2), только при RV=
span> ®
показание вольтметра равно ЭДС: UV =3D E.
Нагрузкой в схеме на рис. 1.1 служит сопротивление R. Напряжение на его зажимах св=
язано
с током законом Ома
I<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D GU=
span>,
&nb=
sp; =
&nb=
sp; =
(1.3)=
span>
где G=
span> – проводимость, величина, обра=
тная
сопротивлению R; единица
измерения – cименс (=
См).
При G =3D const<=
/i>
выражение (1.3) представляет собой уравнение прямой, проходящей через нач=
ало
координат. Его график (рис. 1.6) называется вольтамперной характеристикой.
Элементы электрической цепи, имеющие аналогичную (прямолинейную)
вольтамперную характеристику, называются линейными. Электрическая цепь,
состоящая только из линейных элементов, также называется линейной.
|
Рис.
1.6. Вольтамперная характеристика линейного сопротивления
|
Полагая в уравнении (=
1.3)
, получим U =3D IR. Последнее выражение справедл=
иво,
когда стрелки напряжения и тока у резистора направлены в одну сторону (рис.
1.7, а). При изменении на схеме направления любой из стрелок в правой
части закона Ома следует ставить минус (рис. 1.7, б). Здесь при
определении напряжения на элементе мы "идем по стрелке" напряжения
против стрелки тока.
Рядом с буквой U можно ставить два индекса, обозначающие точки, между
которыми определяется напряжение; например, <=
b>Uab – напряжение между точками<=
/span> а и b. При этом направление стрелки напряжения на схеме определяется
порядком следования индексов – от а к b (от первого индекса ко второму).
|
Рис.
1.7. Напряжение и ток в сопротивлении
|
1.2. Закон Ома для участка цепи с ЭДС
На
практике часто встречается задача, когда требуется определить ток в некотор=
ой
ветви при известных ее параметрах и потенциалах ее зажимов.
Пу=
сть в
схеме на рис. 1.8, а заданы R=
span>, =
E, j <=
b>a<=
span
style=3D'font-size:13.5pt;font-family:Arial'>, j <=
b>b<=
sub>, и требуется определить ток. <=
o:p>
Рис. 1.8. Варианты ветви с ЭДС
Ме=
жду R=
span> и =
E<=
span
style=3D'font-size:18.0pt;font-family:Arial'> отметим промежуточную точку =
span>с и выразим ее потенциал через потенциалы точек а и b.
Та=
к как
в резисторе ток протекает слева направо, то потенциал т=
очки
а выше потенциала точки =
с на величину падения напряжения в активном сопротивлении: <=
i>
j a<=
span
style=3D'font-size:13.5pt;font-family:Arial'> =3D j с + IR.
&nb=
sp; =
&nb=
sp; =
(1.4)=
span>
То=
чка b=
span> находится на положительном по=
люсе
источника, а с – на отрицательном. Поэтому=
span>
j b<=
span
style=3D'font-size:13.5pt;font-family:Arial'> =3D j с + E.
&nb=
sp; =
&nb=
sp; =
(1.5)=
span>
Бе=
ря
разность левых и правых частей выражений (1.4) и (1.5), получим <=
i>
j<=
/b> a<=
span
style=3D'font-size:13.5pt;font-family:Arial'> – j <=
b>b<=
b> =3D IR – Е,
откуда
.=
o:p>
Дл=
я цепи
на рис. 1.8, б после аналогичных рассуждений будем иметь
I<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D (j<=
/span> a<=
span
style=3D'font-size:13.5pt;font-family:Arial'> – j <=
b>b<=
b> – E) G.
В =
двух
последних формулах ЭДС записывается с плюсом, е=
сли ее
направление на схеме совпадает с направлением тока, и с минусом – в
противоположном случае.
1.3. Расчет сложных электрических цепей постоянного тока=
=
1.3.1. Метод уравнений Кирхгофа
Эт=
от
метод сводится к решению системы уравнений, количество которых равно числу
неизвестных токов (числу ветвей). Покажем его применение на примере схемы,
изображенной на рис. 1.9.
Рис. 1.9. Сложная электрическая цепь
Пе=
рвый
закон Кирхгофа: в узле электрической цепи алгебраическая сумма токов рав=
на
нулю.
Произвольно задавшись направлениями токов в ветвях и принимая=
токи,
подтекающие к узлу, положительными, а оттекающие от узла – отрицательными,
записываем:
у=
зел а:
узел в:
узел с:
|
<=
v:shape
id=3D"_x0000_i1044" type=3D"#_x0000_t75" alt=3D"" style=3D'width:154.5pt=
;height:67.5pt'>
|
(=
1.6)
|
Чи=
сло
независимых уравнений в первом законе Кирхгофа – на единицу меньше числа уз=
лов,
поэтому для последнего узла d уравнение не пишем.
В
заданной схеме семь ветвей, семь неизвестных токов. Система (1.6) содержит
только три уравнения. Недостающие четыре записываем по второму закону Кирхг=
офа.
Вт=
орой
закон Кирхгофа: в замкнутом контуре электрической цепи алгебраическая су=
мма ЭДС равна алгебраической сумме падений напряжений на =
всех
сопротивлениях контура.
Чи=
сло
уравнений, составляемых по этому закону, равно числу взаимно независимых
контуров. При рассмотрении схемы каждый последующий кон=
тур
является независимым относительно предыдущих, если он отличается от них хот=
я бы
одной новой ветвью. В заданной схеме таких контуров четыре. Они отме=
чены
пронумерованными дугообразными стрелками. Любой другой контур новых ветвей =
не
содержит, поэтому не является независимым. Дугообразные стрелки показывают
произвольно выбранные направления обхода контуров. Если направления ЭДС и токов совпадают с направлением обхода контура, =
то они
записываются с плюсом, если не совпадают – то с минусом.
к=
онтур 1:
контур 2:
контур 3:
контур 4:
|
|
&=
nbsp; &nbs=
p;
(1.7)
|
Си=
стемы
(1.6) и (1.7) дают достаточное количество уравнений для отыскания всех
неизвестных токов.
1.3.2. Метод узловых потенциалов
Уравнения, составляемые по этому методу, называются узловыми уравнениями. В
качестве неизвестных они содержат потенциалы узлов, причем один из них зада=
ется
заранее – обычно принимается равным нулю. Пусть =
таким
узлом будет узел d: j d =3D 0. Равенство нулю какой-то точ=
ки
схемы обычно показывается как ее заземление.
За=
пишем
для каждой ветви выражение закона Ома:
&nb=
sp; =
&nb=
sp;
(1.8=
)
Подставляя формулы (1.8) в систему (1.6) после несложных преобразований
получаем следующие уравнения, количество которых на единицу меньше числа уз=
лов:
&nb=
sp; =
 =
;
(1.9)=
span>
При
решении практических задач указанный вывод не делают,
а узловые уравнения записывают сразу, пользуясь следующим правилом.
Потенциал узла, для которого составляется уравнение (например, в первом
уравнении последней системы – это узел а=
),
умножается на сумму проводимостей ветвей, присоединенных к этому узлу: j<=
/span> а (G1+G2+G3).Это произведение записывается=
в
левой части уравнения со знаком плюс. Потенциал каждого соседнего узла (b=
span> и с) умножается на проводимости ветвей, лежащих между этим
(соседним) узлом и узлом, для которого составляется уравнение.=
span>
Эти
произведения j b (G1 + G<=
span
style=3D'font-size:13.5pt;font-weight:normal;font-style:normal'>2) и j <=
b>сG3 записываются со знаком минус. В правой части уравнения
стоит алгебраическая сумма произведений ЭДС на
проводимости тех ветвей, которые присоединены к рассматриваемому узлу: E=
span>1G1, =
E2G2 и E3G3. Эти произведения записываютс=
я с
плюсом, если ЭДС направлены к узлу, и с минусом=
, если
от узла.
&nb=
sp;
Найдя из (1.9) потенциалы узлов и подставляя их в (1.8), определяем токи
ветвей.
&nb=
sp;
1.3.3. Метод контурных токов
Для
каждого из взаимно независимых контуров назначается так называемый контурный
ток, замыкающийся по всем ветвям контура. Направления этих токов произвольн=
ы.
На=
рис.
1.9 они обозначены дугообразными стрелками, рядом с которыми стоят буквы IK<=
/sub>1=
, =
IK2, IK<=
/sub>3=
и IK<=
/sub>4=
. Для выбранных контурных токов
записываются уравнения по второму закону Кирхгофа. Контур при этом обходитс=
я по
направлению контурного тока. Рассмотрим порядок составления уравнения на
примере третьего контура. Контурный ток <=
span
style=3D'font-size:18.0pt;font-family:Arial'>IK3, протекая по сопротивлениям с=
воего
контура, создает на них падение напряжения
 =
; &n=
bsp;
(1.10)<=
/span>
По
сопротивлению R4, являющемуся элементом третьего контура, протекает
контурный ток IK2. Создаваемое им падение напряжения IK2R4 вычитается из предыдущего, та=
к как
направление тока IK2 в сопротивлении R=
span>4 противоположно току IK3. Сопротивление R=
span>6 также входит в третий контур.
Падение напряжения на нем, создаваемое контурным током IK4, складывается с суммой (1.10)=
, так
как направления IK4 и IK3 в R6 одинаковы. В правой части ура=
внения
записывается алгебраическая сумма всех ЭДС конт=
ура, в
данном случае – единственная ЭДС E=
span>4.
Ит=
ак,
для третьего контура имеем:
Аналогично составляются и остальные контурные уравнения:
По=
сле
решения последней системы действительные токи ветвей определяются по найденным контурным:
&nb=
sp;
 =
;
(1.11)<=
/span>
&nb=
sp;
Контурные уравнения получаются подстановкой формул (1.11) в уравнения второ=
го
закона Кирхгофа (1.7).
&nb=
sp;
1.3.4. Метод наложения
В =
основе
метода лежит принцип суперпозиции (наложения): ток в любой ветви сложной
электрической цепи, содержащей несколько ЭДС, м=
ожет
быть найден как алгебраическая сумма токов в этой ветви от действия каждой =
ЭДС в отдельности.
Это
весьма важное положение, справедливое только для линейных цепей, вытекает из
уравнений Кирхгофа и утверждает независимость действия источников энергии.
Основанный на нем метод сводит расчет цепи, содержащей несколько ЭДС, к последовательному расчету схем, каждая из кото=
рых
содержит только один источник.
Например, токи в схеме на рис. 1.10, а находятся как алгебраические
суммы частичных токов, определяемых из схем 1.10, б и в.
Аналогично:
И,
наконец,
=
a)
&nb=
sp; =
&nb=
sp;
б) =
&nb=
sp;
в)=
Рис. 1.10. Заданная (а=
) и
расчетные (б и в) схемы
При
расчете подобных схем очень удобным оказывается следующий прием. Пусть
требуется определить токи в параллельных ветвях при известном суммарном токе
(рис. 1.11).
Им=
еем:
<=
/span>
=
Рис.
1.11. Токи в параллельных ветвях
|
Из полученной формулы вытекает правило: ток в одной из двух
параллельных ветвей равен произведению общего тока на сопротивление сосед=
ней
ветви, деленному на сумму сопротивлений параллельных ветвей.
Пользуясь этим правилом для тока I2 можно написать:
=
|
&nb=
sp;
Применение этого правила избавляет от необходимости определять напряжения <=
v:shape
id=3D"_x0000_i1070" type=3D"#_x0000_t75" alt=3D"" style=3D'width:31.5pt;he=
ight:27.75pt'>
и
в схемах на рис. 1.10, б и 1.10, в. Так, после
определения тока
, токи
и
можно найти по формулам:
1.3.5. Эквивалентное преобразование треугольника и
звезды сопротивлений
Пу=
сть
требуется рассчитать цепь, показанную на рис. 1.12, а.
Рис. 1.12. Преобразования электрической цепи<=
/span>
Ра=
счет
можно осуществить одним из описанных выше методов. Но так как в цепи имеется
только один источник питания, наиболее простым было бы использование закона
Ома. Однако попытка определения общего сопротивления цепи оказывается
безрезультатной, так как здесь мы не находим ни последовательно, ни паралле=
льно
соединенных сопротивлений. Решить задачу помогает преобразование треугольни=
ка
сопротивлений в эквивалентную звезду.
Треугольник и звезда сопротивлений имеют вид, показанный на рис. 1.13.
Рис. 1.13. Треугольник и звезда сопротивлений
Ес=
ли при
замене одной из этих схем другой не изменяются потенциалы одноименных точек=
и
подтекающие к ним токи, то во внешней цепи также не произойдет никаких
изменений. В этом случае говорят, что схемы эквивалентны.=
Мо=
жно
показать, что условием эквивалентности являются следующие уравнения:=
o:p>
а) при преобразовании
треугольника в звезду:
;
&nb=
sp;
;
;=
o:p>
б) при преобразовании
звезды в треугольник:=
span>
;
;
.=
o:p>
Структура приведенных формул проста и легко запоминается.=
Например, сопротивление звезды R1, присоединенное к узлу 1, получается перемножением
сопротивлений R12 и R31 треугольника, присоединенных к
этому же узлу, и делением полученного произведения на сумму всех сопротивле=
ний
треугольника.
При
обратном преобразовании сопротивление треугольника R12, лежащее между узлами 1 и 2, равно сумме сопротивлений
звезды R=
1 и R2,
присоединенных к этим узлам, плюс их произведение, деленное на сопротивлени=
е третьего
луча звезды R3.
Пример 1.3. Р=
ассчитать
токи в цепи, изображенной на рис. 1.12, а, при следующих числовых
значениях ее параметров: Е =
span>=3D 660 В<=
/span>,
R=
1 =3D 20 Ом, R2 =3D 30 Ом, R3 =3D 5 Ом, R4 =3D 20 Ом, R5 =3D 50 Ом. <=
/p>
а)=
Р е ш=
е н и е п р
е о б р а з о в а <=
span
class=3DSpellE>н и е м т р е у г о=
л ь н и к а в <=
span
class=3DSpellE>з
в е з д
у.
По=
сле
преобразования треугольника, образованного сопротивлениями R=
1, R2 и R5, в звезду, получаем схему, показанную на рис. 1.12, б.
Обращаем внимание на то, что токи в непреобразованной<=
/span>
части схемы (I, =
I3 и I4) остались теми же.=
Сопротивления звезды определяем по сформулированному выше правилу:
6 Ом;
=3D 10 Ом;
=3D 15 Ом.
Те=
перь
общее сопротивление цепи легко находится:
=3D16,5 Ом.
То=
к,
протекающий по источнику (одинаковый в заданной и преобразованной схемах),
равен
40 А.
То=
ки в
параллельных ветвях:
28 A;
12 A.
Возвращаемся к исходн=
ой
схеме (рис. 1.12, а):
26 A;
14 A.
Ток в пятой ветви нах=
одим
из первого закона Кирхгофа: I5 =3D=
I1–I3=
sub> =3D 26–28 =3D –2 A. Знак минус говорит о том, что действител=
ьное
направление тока I5 противоположно указанному
на схеме.
б)=
Р е ш=
е н и е п р
е о б р а з о в а <=
span
class=3DSpellE>н и е м з в е з д =
ы
в т р е у г о л ь н и к.
Преобразуем звезду, образуемую в схеме на рис. 1.12, а сопротивления=
ми R=
1, =
R5 и =
span>R=
3, в эквивалентный треугольник (р=
ис.
1.12, в).
Определяем сопротивления треугольника:
=
o:p>
Те=
перь
рассчитываем преобразованную цепь.
Сн=
ачала
находим эквивалентные сопротивления участков ac =
и cd:
;
За=
тем
определяем общее сопротивление и токи:
Возвращаемся к исходной схеме:
 =
;
Рекомендуем подставить в приведенные формулы числовые значения параметров ц=
епи
и сравнить результаты вычислений с полученными в
примере 1.3а.
1.4. Пассивный и активный двухполюсники. Теорема об акти=
вном
двухполюснике
Двухполюсником называется часть электрической цепи любой сложности и
произвольной конфигурации, выделенная относительно двух зажимов (двух полюс=
ов).
Двухполюсник, не содержащий источников энергии или содержащий скомпенсирова=
нные
источники (суммарное действие которых равно нулю), называется пассивным. Ес=
ли в
схеме двухполюсника имеются нескомпенсированные
источники, он называется активным. На схеме двухполюсник обозначают
прямоугольником с двумя выводами (рис. 1.14). Это обозначение можно условно
рассматривать как коробку, внутри которой находится электрическая цепь.
Рис. 1.14. Пассивный (а) и активный (б)
двухполюсники
Пассивный двухполюсник является потребителем энергии и может быть заменен
эквивалентным сопротивлением, величина которого равна входному сопротивлению
двухполюсника (см., например, рис. 1.15).
.
Рис. 1.15. Замена пассивного двухполюсника сопротивление=
м
Ак=
тивный
двухполюсник ведет себя как генератор. Находящиеся внутри него нескомпенсированные источники отдают энергию во внешн=
юю
цепь (рис. 1.16, а). Можно попытаться подобрать источник энергии с <=
span
class=3DSpellE>ЭДС ЕЭ и внутренним сопротивлением RЭ, который будет эквивалентен двухполюснику, то ес=
ть
будет создавать во внешней цепи тот же самый ток (рис. 1.16, б).
Полученный генератор должен быть эквивалентен двухполюснику в любом режиме,=
в
том числе и в режимах холостого хода и короткого замыкания. Источники энерг=
ии,
входящие в состав активного двухполюсника, в режиме холостого хода создают =
на
его зажимах напряжение <=
i>UХ (рис. 1.17, а), а при коротком замыкании вызывают=
ток
IK<=
/sub> (рис. 1.17, б).
Рис. 1.16. Замена активного двухполюсника эквивалентным
генератором<=
/o:p>
Из=
схем,
приведенных на рис. 1.17, следует:
откуда
Рис. 1.17. Холостой ход (а) и короткое замыкание =
(б)
активного двухполюсника
Итак, любой активный двухполюсник может быть заменен
эквивалентным генератором, ЭДС которого ЕЭ
равна напряжению холостого хода двухполюсника, а внутреннее сопротивление <=
/i>RЭ
напряжению холостого хода, деленному на ток короткого замыкания.=
Это
утверждение и есть теорема об активном двухполюснике (эквивалентном
генераторе).
Пример 1.4.
Заменить активный двухполюсник, выделенный пунктиром на рис. 1.18, а,
эквивалентным генератором (рис. 1.18, б). Численные значения парамет=
ров
цепи составляют: Е1 =3D 200 В<=
/span>, Е=
2 =3D 100 В, R1 =3D 50 Ом, R2 =3D 20 Ом, R3 =3D 20 Ом.
Рис. 1.18. Замена активного двухполюсника эквивалентным
генератором<=
/o:p>
Р е ш=
е н и е. Напряжение холостого хода, определяющее величи=
ну ЭДС эквивалентного генератора, можно найти по схеме н=
а рис.
1.19, а любым известным способом.
Рис. 1.19. Режимы холостого хода (а) и короткого
замыкания (б)
Воспользуемся, например, методом контурных токов. Прини=
мая в
качестве контурных токи I1Х
для левого контура и I3Х
для правого, записываем контурные уравнения, из которых определяем контурные
токи:<=
/o:p>
Напряжение холостого хода – это напряжение между точками m и n.
Оно равно падению напряжения на сопротивлении <=
b>R3:
75 В.
Та=
ким
образом, ЭДС эквивалентного генератора <=
span
class=3DSpellE>ЕЭ<=
/i> =3D 75 В.
Пр=
именим
теперь метод узловых потенциалов.
Пр=
инимая
потенциал узла n равным
нулю (j <=
span
class=3DSpellE>n=
i> =3D 0), для узла m запишем узловое уравнение:
 =
; &n=
bsp;
(1.12)<=
/span>
где
0,02 См;
0,05 См;
0,05 См.
Из
уравнения (1.12) имеем:
75 В.
По=
лучили
тот же самый результат.
Приступаем к расчету режима короткого замыкания. Ток IK в схеме на рис. 1.19, б
найдем методом наложения. При действии только первой Э=
ДС
ее ток проходит по первой ветви и, минуя вторую и третью ветви, замыкается =
по
проводнику, закорачивающему зажимы двухполюсник=
а:
4 А.
Аналогично находим ток, вызываемый второй ЭДС:<=
v:shape
id=3D"_x0000_i1127" type=3D"#_x0000_t75" alt=3D"" style=3D'width:81.75pt;h=
eight:43.5pt'>
5 А.
То=
к в
третьей ветви равен нулю, так как она закорочена. Поэтому
IK =3D I<=
sub>1K =
i>+ I=
2K <=
span
lang=3DPL style=3D'font-size:13.5pt;font-family:Arial;mso-ansi-language:PL'=
>=3D 9 A.=
В
соответствии с теоремой об эквивалентном генераторе
8,33 Ом.
<=
/span>
1.5. Метод эквивалентного генератора
Эт=
от
метод основан на сформулированной выше теореме (см. по=
дразд.
1.4) и применяется в тех случаях, когда требуется рассчитать ток в какой-ли=
бо
одной ветви при нескольких значениях ее параметров (сопротивления и ЭДС) и неизменных параметрах всей остальной цепи.
Су=
щность
метода заключается в следующем. Вся цепь относительно зажимов интересующ=
ей
нас ветви представляется как активный двухполюсник, который заменяется
эквивалентным генератором, к зажимам которого подключается интересующая нас
ветвь. В итоге получается простая неразветвленная цепь, ток в которой
определяется по закону Ома.
=
ЭДС <=
b>ЕЭ эквивалентного генератора и его
внутреннее сопротивление RЭ находятся из режимов холостого хода и короткого
замыкания двухполюсника.
По=
рядок
решения задачи этим методом рассмотрим на конкретном числовом примере.=
Пример 1.5. В=
цепи,
показанной на рис. 1.20, а, требуется рассчитать ток I=
3 при шести различных значениях
сопротивления R3 и по результатам расчета пост=
роить
график зависимости I3(R3<=
/i>).
Чи=
словые
значения параметров цепи: Е1 =3D 225 В; Е3 =3D 30 В; R1 =3D 3 Ом; R2 =3D 6 Ом.
=
&nb=
sp;
а)
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
б)
Рис. 1.20. Схема решения задачи
Р е ш=
е н и е. а) Расчет режима холостого хода.
Уб=
ираем
третью ветвь, оставляя зажимы m и n
разомкнутыми (рис. 1.21, а). Напряжение между ними, равное UX<=
/i>, находится как падение напряж=
ения
на сопротивлении R2:
150 В; =
150 В.
б)
Расчет режима короткого замыкания. Замыкаем накоротко зажимы m и n (рис. 1.21, б). Ток короткого замыкания
75 А.
Внутреннее сопротивление эквивалентного генератора
2 Ом.
Рис. 1.21. Режимы холостого хода (а) и короткого
замыкания (б)
Ве=
личину
RЭ можно найти и другим способом. Оно равно входному
сопротивлению двухполюсника при равенстве нулю всех его ЭДС.
Если на рис. 1.21, а мысленно закоротить зажимы ЭДС
Е=
1, то сопротивления R=
1 и R2 окажутся соединенными параллельно, и входное сопротивле=
ние
цепи относительно зажимов m и n
будет равно
2 Ом.
То=
к в полученной
неразветвленной цепи (рис. 1.20, б) определяется по закону Ома:
 =
; &n=
bsp;
(1.13)<=
/span>
Подставляя в последнюю формулу требуемые значения сопротивления R=
3, вычисляем ток и строим график
(рис. 1.22).<=
/o:p>
Рис. 1.22. Зависимость тока от сопротивления
&nb=
sp;
Данную задачу целесообразно решать именно методом эквивалентного генератора.
Применение другого метода, например метода контурных токов, потребует решать
систему уравнений столько раз, сколько значений тока необходимо найти. Здес=
ь же
всю цепь мы рассчитываем только два раза, определяя ЕЭ и RЭ, а многократно используем лишь одну простую форм=
улу (1.13).
=
1.6. Линия электропередачи постоянного тока=
a>=
Ес=
ли
линия электропередачи имеет небольшую длину, при которой можно пренебречь
утечкой тока через изоляцию, то ее электрическую схему можно представить в =
виде
последовательного соединения сопротивления линии RЛ, равного суммарному сопротивлению прямого и обратного
проводов, и сопротивления нагрузки RН (рис. 1.23).
Рис. 1.23. Линия электропередачи постоянного тока=
При
анализе работы линии нас интересуют, главным образом, три вопроса: напряжен=
ие
на нагрузке, величина передаваемой мощности и коэффициент полезного действия
передачи. Режимы работы линии удобно рассматривать в виде зависимостей
различных величин от тока в линии, равного
Па=
дение
напряжения в линии D =
U и напряжение на нагрузке =
U2 определяются следующими выражениями:
;
Ес=
ли U=
1 и RЛ постоянны, то оба выражения представляют собой
линейные функции тока (рис. 1.24). В режиме холостого хода (при I =3D 0) D =
U =3D 0, а U2 =3D U1<=
/i>. С ростом тока падение напряж=
ения в
линии возрастает, а напряжение на нагрузке уменьшается, и в режиме короткого
замыкания (при RН =3D 0)
,
,
0; все входное напряжение гасится на сопротивлении линии.
Рис. 1.24. Режимы работы линии
Мо=
щность
на входе линии линейно зависит от тока: <=
span
style=3D'font-size:18.0pt'>P1 =3D U=
1I. При холостом ходе она равна н=
улю, а
при коротком замыкании вычисляется по формуле
.<=
/o:p>
По=
тери
мощности в линии
представляют собой квадратичную функцию тока. Ее график – парабол=
а,
проходящая через начало координат. При I =
=3D 0: D<=
/span> <=
b>P =
=3D 0;
при I =3D IK<=
/i>:
, т.е. в режиме короткого замыкания мощность, поступающая в цепь,
полностью теряется в линии.
Мощность, поступающая в нагрузку, равна разности мощности в начале линии и
мощности, теряемой в проводах:
.
&nb=
sp; =
&nb=
sp;
(1.14)=
Последнее выражение представляет собой уравнение параболы со смещенной верш=
иной
и с обращенными вниз ветвями, проходящими через точки I =3D 0 и I =3D IK.
Мо=
щность
нагрузки представляет собой довольно сложную зависимость от сопротивления <=
/span>R=
Н:
.
&nb=
sp; =
(1.15)=
Пр=
и R=
Н =3D 0: Р2 =3D 0; при возрастании RН мощность Р2 сначала возрастает, достигает максимального значения и начинает убыв=
ать,
стремясь к нулю при
RН
®
(рис. 1.25).
Вы=
ясним,
при каком сопротивлении нагрузки передаваемая ей мощность максимальна. Для
этого продифференцируем функцию (1.15) по RН и приравняем ее к нулю:
0.<=
/o:p>
Приравняв к нулю числитель производной, получим:
RH +
RЛ – 2RH =3D 0,
или RH<=
/i> =3D
RЛ. =
o:p>
То=
есть
мощность, получаемая нагрузкой, максимальна, когда сопротивление нагрузки р=
авно
сопротивлению линии.
То=
к,
протекающий при этом по линии, равен
, т.е. составляет половину тока короткого замыкания, а мощность в
конце линии равна
.=
o:p>
Эт=
и же
результаты можно получить, исследуя на экстремум зависимость мощности P2 от тока I =
(1.14).
Коэффициент полезного действия равен отношению мощностей в начале и конце
линии:<=
/span>
.=
o:p>
Он=
представляет
собой линейную функцию тока. При холостом ходе, когда I =3D 0, он равен единице (нет передачи энергии – нет поте=
рь).
При коротком замыкании вся передаваемая мощность теряется в линии, и КПД ра=
вен
нулю.=
span>
Рис.
1.25. Зависимость мощности в конце линии от сопротивления нагрузки
Во=
зможны
и другие формулы для определения КПД:
, (1=
.16)
Из
данной формулы следует, что коэффициент полезного действия передачи
определяется отношением сопротивлений линии и нагрузки.=
p>
Пр=
и их
равенстве, когда нагрузке передается максимальная мощность, h<=
/span> =3D 0,5 =3D 50 %. Этот режим,=
при
котором теряется половина передаваемой энергии, на практике, естественно, не
пригоден. В реальных линиях при передаче больших мощностей КПД составляет
примерно 0,94–0,97. При этом сопротивление нагрузки значительно больше
сопротивления линии.
Для
анализа режимов электропередачи полезной оказывается еще одна формула. Так =
как
, а
, то
.
&nb=
sp; =
&nb=
sp;
(1.17)<=
/span>
То=
есть
при одной и той же мощности нагрузки Р2, потери D <=
span
style=3D'font-size:18.0pt'>Р пропорциональны сопротивлению линии и обратно
пропорциональны квадрату напряжения. Для увеличения коэффициента полезного
действия передачи необходимо повышение напряжения и снижение электрического
сопротивления проводов линии путем увеличения их сечения и применения
материалов с меньшим удельным сопротивлением.
Пример 1.6. Л=
иния
электропередачи с проводами марки А-120 длиной =
l =
=3D 1000
км питает нагрузку мощностью Р2 =3D 50 МВт. Каким должно быть
напряжение в начале линии, чтобы КПД передачи был не ниже 90 %?<=
/span>
Р е ш=
е н и е. Сопротивление одного километра провода марки А=
-120 R0<=
/i> =3D 0,27 Ом/км. Суммарное сопротивление прямого и обратн=
ого
проводов линии составляет <=
b>RЛ =3D 2lR0<=
b> =3D 540 Ом.
Пр=
инимая
h =3D 0,9,=
из
формулы (1.17) получаем:
=3D 4,93 x 105 В
=3D 493 кВ.=
o:p>
Та=
к как
, то
кВ.
Для
выполнения условий задачи напряжение в начале линии должно быть не ниже 548=
кВ.
2. Электрические цепи однофазного синусоидального тока=
span>
2.1. Закон электромагнитной индукции
Пу=
сть
имеется катушка, содержащая w витков (рис. 2.1) и помещенная в переменное магнитное п=
оле.
=
Рис. 2.1. Катушка в магнитном поле
|
В=
каждом
витке такой катушки индуцируется ЭДС электром=
агнитной
индукции. Если магнитный поток, пронизывающий k-тый виток, обозначить <=
span
style=3D'font-size:18.0pt'>Фk, то ЭД=
С
этого витка будет равна
.
|
Та=
к как
все витки катушки соединены последовательно, то их суммарная ЭДС равна
,
&nb=
sp;
(2.1)
где
– потокосцепле=
ние
катушки, определяемое как сумма магнитных потоков всех витков:
.
Ед=
иница
измерения потокосцепления такая же, как и у магнитного потока – вольт-секун=
да (Вс); Ее название – вебер (сокращенно Вб). =
Ес=
ли все
витки катушки, пронизываются одним и тем же магнитным потоком, т.е. если =
span>Ф=
1 =3D Ф2 =3D
... =3D =
Фk =3D Фw =3D
Ф<=
/i>, то
и
.
&nb=
sp; =
&nb=
sp;
(2.2)
<=
/span>
2.2. Получение синусоидальной ЭДС.
Характеристики синусоидальных величин. Обозначения в цепях переменного тока=
Пу=
сть в
однородном магнитном поле, например, между полюсами плоского магнита, под у=
глом
к горизонтальной плоскости расположена плоская катушка, выполненн=
ая в
виде прямоугольной рамки, по периметру которой намотано w витков (рис. 2.2). Площадь се=
чения
рамки – =
S, магнитная индукция – =
В.
Рис. 2.2. Получение синусоидальной=
span> ЭДС
За=
ставим
эту катушку вращаться против часовой стрелки с угловой скоростью w<=
/span> .<=
/span> Если обозначить время полного
оборота катушки через Т, то
, рад/с. За некоторый промежуток времени t =
рамка
повернется на угол w =
t. Площадь проекции рамки в этом
положении
. Рамка и ее проекция на горизонтальную плоск=
ую
поверхность пронизываются одним и тем же числом силовых линий магнитной
индукции, поэтому обусловленный ими магнитный поток равен=
.
При
вращении катушки число силовых линий, охватываемых ее витками, все время
меняется.
Например, при горизонтальном положении рамки это число максим=
ально,
при вертикальном – равно нулю. Другими словами, меняется магнитный
поток, пронизывающий катушку, в результате чего в ней в соответствии с
уравнением (2.2) наводится ЭДС:
.
По=
ясним
величины, входящие в последнее выражение. Еm<=
/i> – максимальное значение или
амплитуда ЭДС. Аргумент синусоидальной функции =
w<=
/span> <=
b>t +y называется фазой. Угол y , определяющий начальное положе=
ние
рамки и равный фазе в начальный момент времени (при t =3D 0), – начальная фаза. Фаза с течением времени (при
вращении катушки) постоянно меняется. Скорость изменения фазы w<=
/span> называется угловой или циклич=
еской
частотой. Время одного цикла изменения фазы (время одного оборота рамки)
называется периодом и обозначается T. Количество полных изменений
синусоидальной ЭДС в секунду определяет частоту=
¦<=
/span> ,<=
/span> измеряемую в герцах (Гц). Оди=
н герц
соответствует одному полному колебанию в секунду. Связь между частотой и
периодом выражается формулой ¦ =3D 1/T . При частоте 50 Гц
314 c-1.=
Графическое изображение синусоидальной функции времени в электротехнике
называют волновой диаграммой. При ее построении на горизонтальной оси
откладывается время t
или пропорциональный ему угол w t.
При нулевой начальной фазе кривая выходит из начала координат и через каждые
четверть периода принимает максимальные значения и переходит через ноль. Гр=
афик
такой функции построен по уравнению е
=3D Еm sinw t на рис. 2.3, а.=
o:p>
Рис. 2.3. Волновые диаграммы
При
ненулевых начальных фазах диаграммы имеют несколько иной вид. Пусть напряже=
ние
и ток на некотором участке цепи определяются выражениями:
&=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
(2.3)
Для
определенности положим y u > 0, а
< 0. Сначала
построим волновую диаграмму напряжения. При =
t =3D 0 u =3D Umsin=
y u=
. При поло=
жительном
y <=
span
class=3DSpellE>u=
i> эта величина положительна, и синусоида отсекает на
вертикальной оси отрезок выше начала координат (рис. 2.3, б). Начало
синусоиды и все ее точки оказываются сдвинутыми влево на величину y<=
/span> <=
b>u. Кривая тока, имея отрицательную начальную фазу,
смещается вправо. Если начальные фазы двух синусоидальных функций, изменяющ=
ихся
с одинаковой частотой, различны, то говорят, что они не совпадают по фазе.
Отрезок на горизонтальной оси, разделяющий начала синусоидальных кривых (уг=
ол j<=
/span> на рис. 2.3, б), опред=
еляет
угол сдвига фаз. Он равен разности их начальных фаз:
j =3D y u=
– y i=
.
&nb=
sp; =
&nb=
sp; =
(2.4)
В =
случае
напряжения и тока вычисление производится именно в таком порядке: начальная
фаза напряжения минус начальная фаза тока.
Ес=
ли y<=
/span> <=
b>u >
y i<=
/b> и угол j<=
/span> положителен, то говорят, что
напряжение опережает по фазе ток, или ток отстает по фазе от напряжения. На
волновой диаграмме в этом случае кривая напряжения проходит через ноль и
максимальные значения раньше тока; изменения тока отстают от соответствующих
изменений напряжения. Мера отставания – угол j .<=
/span>
Остановимся еще на двух моментах. В цепях синусоидального тока мы будем
встречаться как с переменными, так и с постоянными величинами. Для тех и др=
угих
применяются различные обозначения. Переменные величины –
функции времени – будем обозначать маленькими (строчными) буквами u,
i,
e,
а постоянные – большими (прописными) U=
span>,
I,
Е.
&nb=
sp;
Второй момент касается указания направления тока или напряжения. При постоя=
нном
токе его направление связано с движением положительно заряженных частиц. В
случае переменного тока его стрелка на схеме показывает у с л о в н о в ы б р
а н н<=
/span>
о е положительное направление. Если в какой-то момент времени ток направлен=
по
стрелке, он считается положительным, в противном случае он отрицателен. =
=
2.3. Действующее значение переменного тока=
По=
нятие
действующего значения тока вводится в связи с необходимостью производства
измерений. Что измерять у переменного тока? Если бы мы имели дело только с
синусоидами – кривыми одной формы, то можно было бы измерять амплитуды. Но =
на
практике встречаются самые разные кривые, и может оказаться так, что два
различных по форме тока имеют одинаковые амплитуды, хотя очевидно, что на
электрическую цепь они будут оказывать разное воздействие.
По=
этому
наиболее целесообразно оценивать величину тока по той работе, которую он
совершает. При такой оценке действие переменного тока сравнивается с
аналогичным действием постоянного тока. Например, если некоторый переменный=
ток
выделяет на участке цепи такое же количество тепла, что и постоянный ток си=
лой
10 ампер, то говорят, что величина этого переменного тока составляет 10 амп=
ер.
Это значение тока и называют действующим.
Ит=
ак, действующим
значением переменного тока называется численное значение такого постоянного
тока, который за время, равное одному периоду, выделяет в сопротивлении так=
ое
же количество тепла, что и ток переменный.
Та=
ким
образом, для оценки величины переменного тока мы должны сделать следующее.<=
o:p>
1.
Определить количество теплоты, выделяющейся в сопротивлении R за время Т при протекании переменного тока i. Это количество теплоты равно
.
` 2.
Подобрать такой постоянный ток I, который за то же время <=
i>Т в том же сопротивлении R выделяет такое же количество =
тепла.
При постоянном токе оно равно
.
3.
Приравнять W» и W=3D:
,<=
/o:p>
откуда
.
&nb=
sp; =
&nb=
sp;
(2.5)
Последняя формула и определяет действующее значение переменного тока.<=
/o:p>
Пример 2.1. Н=
а вход
некоторой цепи подается импульсное напряжение треугольной формы (рис. 2.4, =
а).
Чему равно его действующее значение?
Р е ш=
е н и е.<=
/o:p>
.
Рис. 2.4. Переменные напряжения
различной формы
&nb=
sp;
=
=
Пример 2.2. Н=
а рис.
2.4, б показана кривая напряжения на выходе схемы однофазного
однополупериодного выпрямления. Чему равно действующее значение напряжения,
если его амплитудное значение Um=
span> составляет 311 В?
Р е ш=
е н и е.<=
/o:p>
155,5 В=
.
Пример 2.3.
Определить действующее значение синусоидального тока
.
Р е ш=
е н и е.
.<=
/o:p>
Рассмотренные примеры показывают, что действующее значение переменного тока
зависит от его формы.
=
У
синусоидального тока оно равно амплитуде, деленной на=
span>
.
<=
/span>
2.4. Представление синусоидальной функции времени
вращающимся вектором. Векторные диаграммы
Пу=
сть в
прямоугольной системе координат имеется вектор длиной Im<=
/i>, расположенный под углом y<=
/span> к горизонтальной оси (рис. 2.=
5).
Заставим этот вектор вращаться против часовой стрелки =
c
угловой скоростью w . Тогда за время t он повернется на угол w t.
=
Рис.
2.5. Вращающийся вектор
|
Проекцию вектора=
на
вертикальную ось обозначим i. Из треугольника oab она равна
, т.е. представляет собой функцию, определяющую мгновенное знач=
ение
тока. Таким образом, последняя может быть представлена как проекция на
вертикальную ось вращающегося вектора. Изображение тока с помощью вектора
называется его векторной диаграммой. Длина вектора может быть равна
амплитудному Im, либо действующему значению=
I.
|
Обычно вектор при этом
показывается не в произвольный момент времени <=
b>t, а в начальный (t=
span> =3D 0), когда его угол наклон=
а к
горизонтальной оси равен начальной фазе.
Те=
перь
по уравнениям (2.3) построим векторную диаграмму двух векторов – тока и
напряжения (рис. 2.6).
<=
/span>
=
Рис.
2.6. Векторная диаграмма тока и напряжения
|
Длины векторов равны действующим
значениям, углы их наклона к горизонтальной оси – начальным фазам, а угол
между векторами, равный разности начальных фаз y
u=
и y i, в соответствии с уравнением (2.4) определяет
сдвиг фаз напряжения и тока.
Подчеркиваем, что на диаграмме стрелка,
отмечающая угол j , всегда направляется от вект=
ора
тока к вектору напряжения. Сейчас она направлена в положительном направлении – против часов=
ой
стрелки.
|
Векторная диаграмма дает наглядное представление об отставании одних величи=
н и
опережении других. Если вращать картинку, показанную на рис. 2.6, против
часовой стрелки, то вектор тока будет отставать от напряжения на угол j<=
/span> .<=
/span> Так как при вращении длины ве=
кторов
и угол между ними не меняются, то в том случае, когда начальные фазы напряж=
ения
и тока нас не интересуют, мы можем изображать диаграмму без осей и располаг=
ать
ее так, как нам удобно (рис. 2.7).
Рис. 2.7. Варианты построения векторной диаграммы=
o:p>
<=
/span>
2.5. Основные сведения о комплексных числах=
a>=
Комплексным числом называется выражение вида
,
&nb=
sp; =
&nb=
sp; =
(2.6)
где
– обозначение комплексного числа; =
a и b – соответственно действительн=
ая и
мнимая части комплексного числа;
– мнимая единица.
Ве=
личины
a и b часто обозначают следующим об=
разом:
a =3D Re=
, b =3D Im
. Символы Re и Im – есть начальные буквы английских слов Real
– действительный и Imaginary – мнимый.
Геометрически
комплексное число изображается вектором на комплексной плоскости (рис. 2.=
8).
Горизонтальная и вертикальная оси, отмеченные соответственно знаками + и =
+j, называются действительной =
(или
вещественной) и мнимой. Действительная и мнимая составляющие комплексного
числа представляют собой проекции вектора на эти оси.
|
Р=
ис. 2.8.
Вектор на комплексной плоскости
|
На рис. 2.8 с =3D c
c – модуль
комплексного числа, равный длине вектора, а a =3D arg
– аргумент комплексного числа. Так как а =3D c cosa<=
/span> , а
b =3D c<=
span
style=3D'font-size:7.5pt;font-family:Arial'> sina<=
/span> , то
=3D c (cosa<=
/span> + j sina )=
span> – тригонометрическая форма
комплексного числа. С помощью формулы Эйлера
последняя преобразуется в показательную форму
. Применяется еще и полярная форма
, в самой простой форме задающая модуль и аргумент комплексного
числа.
Свойства мнимой единицы (рис. 2.9):
,
,
,
,
и т.д.,
.
|
Рис.
2.9. Единичный вектор
в комплексной плоскости
|
Два
комплексных числа
и
называются сопряженными, если они имеют одинаковые модули и равны=
е по
величине, но разные по знаку аргументы (рис. 2.10):
,
=3D
. =
Изображающие их векторы симметричны относительно вещественной оси.
Действия над комплексными числами.
Сложение и вычитание производится =
над
числами, записанными в алгебраической форме:
|
<=
span
style=3D'font-family:Arial'>Рис. 2.10. Сопряженные комплексные числа
|
=3D
, т.е. складываются по отдельности вещественные и мнимые части
слагаемых: а <=
span
style=3D'font-size:13.5pt;font-family:Arial'>=3D а<=
span
style=3D'font-size:18.0pt;font-family:Arial'>1<=
span
style=3D'font-size:13.5pt;font-family:Arial'>+а2, b<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D b1+b2=
span>. Операции сложения комплек=
сных
чисел соответствует сложение изображающих их векторов.
Су=
мма
сопряженных комплексных чисел равна удвоенному значению вещественной части:=
+
=3D
.
Умножение и деление комплексных чисел удобнее всего производить в показател=
ьной
форме. Модули при этом перемножаются или делятся, а аргументы складываются =
или
вычитаются:
 =
;
(2.7)=
span>
где с =3D с1 с2<=
span
style=3D'font-size:13.5pt;font-family:Arial'>, &nb=
sp;
a =3Da<=
/span> 1+a 2;
,=
o:p>
где
, a =3Da<=
/span> 1 <=
span
style=3D'font-size:18.0pt;font-family:Arial'>– a<=
/span> 2 .
Что
происходит с векторами при перемножении комплексных чисел?
Изобразим на комплексной плоскости два вектора:
1 – первый сомножитель и
– результирующий (рис. 2.11). Последний
получается умножением
1 на комплексное число =
с2е ja<=
/span> 2.
На рис. 2.11 мы видим, что при умножении длина векто=
ра
возросла в с<=
span
class=3DGramE>2 раз, а аргумент
увеличился на a 2.
Рассматривая комплексное число как вектор, мы приходим к следующему
выводу.
При умножении вектора на компл=
ексное
число ае<=
/span> ja , вектор растягиваетс=
я в а раз и поворачивается на уг=
ол a .
|
=
Рис. 2.11. Перемножение комплексных чисел
|
Та=
к как
, то при умножении вектора на ± j он поворачивается на угол ±<=
/span> 90° (рис. 2.12).
Произведение сопряженных комплексных чисел равно квадрату модуля
комплексного числа:
x
,
или
Иногда приходится производить умножение и деление
комплексных чисел в алгебраической форме. Перемножение выполняется по
правилам умножения многочленов с учетом того, что j2=
=3D -1
|
<=
span
style=3D'font-family:Arial'>Рис. 2.12. Умножение вектора на ± j
|
=3D
.
При
делении, чтобы получить результат, необходимо избавиться от мнимой единицы в
знаменателе. Этого можно достичь умножением числителя и знаменателя на
сопряженный знаменатель:
где
;
.
<=
/span>
2.6. Представление синусоидальных функций времени
комплексными числами
Пусть задано вы=
ражение
синусоидального тока i =3D Imsin(w t+=
y ). Как мы видели раньше, этому
выражению соответствует вектор, длина которого равна Im, а угол наклона к горизонта=
льной
оси y . Если
этот вектор изобразить в комплексной плоскости (рис. 2.13), то его можно
обозначить комплексным числом
, которое называется комплексной амплитудой тока.
|
<=
/span>
=
Рис.
2.13. Вектор тока на
комплексной плоскости
|
Комплексное действующее значение тока получается делением последнего выраже=
ния на
:=
o:p>
.=
o:p>
&nb=
sp;
Здесь и дальше буквами с точкой над ними (
) обозначаются комплексные числа, представляющие
синусоидальные функции времени. Это ток, напряжение и =
ЭДС.
Комплексные сопротивление и проводимость обозначаются
прописными буквами Z и Y , а их модули строчными =
z
и y<=
/span>. Комплексная
мощность обозначается буквой S с волнистым значком ~ (тильда) над ней:
.
&nb=
sp;
2.7. Способы задания синусоидального тока=
Как
следует из вышесказанного, синусоидальный ток можно задать четырьмя различн=
ыми
формами: уравнением i =3D Imsin=
(w<=
/span> t=
+ y<=
/span> )=
span>, определяющим мгновенное знач=
ение
тока (значение тока в любой момент времени), волновой диаграммой, вектором и
комплексным числом. При этом мы легко можем перейти от одной формы задания к
другой.
Например:
1) i =3D 20sin(=
w t+110<=
/b>° ),
,
;
|
|
&=
nbsp;
2)
,
,
i=
i> =3D 8,49sin(w t-60=
b>° );
|
|
&=
nbsp;
3)
,
i=
i> =3D 5sin(<=
/i>w t-143,1° ),
,
|
|
u =3D 100 sin (w t + 60° <=
span
style=3D'font-size:18.0pt'>).
|
|
В качестве начальной фазы мы бе=
рем
не 120° , которые
указаны на волновой диаграмме, а тот угол, на который сдвинуто начало
синусоиды. Начальная фаза на волновой диаграмме определяется ближайшей к на=
чалу
координат точкой перехода синусоиды через ноль от минуса к плюсу – это 60=
span>°<=
/span> . Так как начало синусоиды см=
ещено
от точки 0 влево, то начальная фаза поло=
жительна.
2.8.
Законы Кирхгофа в цепях синусоидального тока.
Методы расчета цепей синусоидального тока
Для мгновенных значений ЭДС, токов и напряжений остаются справедливыми
сформулированные ранее законы Кирхгофа.
П
е р в ы й:
в любой момент времени алгебраическая сумма токов в узле электрической ц=
епи
равна нулю:
,
&nb=
sp; =
&nb=
sp; =
(2.8)
где n
– число ветвей, сходящихся в узле.
В т о р о й: в любой м=
омент
времени в замкнутом контуре электрической цепи алгебраическая сумма ЭДС равна алгебраической сумме напряжений на всех ост=
альных
элементах контура:
,
&nb=
sp; =
&nb=
sp; =
(2.9)
где m
– число ветвей, образующих контур.
Токи, напряжения и ЭДС,
входящие в уравнения (2.8) и (2.9), есть синусоидальные функции времени,
которые мы рассматриваем как проекции некоторых векторов на оси координат. =
Так
как сложению проекций соответствует сложение векторов и соответствующих им
комплексных чисел, то справедливыми будут следующие уравнения, которые можно
записывать как для действующих, так и для амплитудных значений.
Законы Кирхгофа
в векторной форме:
|
Законы Кирхгофа
в символической форме:
|
(2.10)
|
(2.11)
|
Из сказанного вытекают три возможных
подхода к расчету цепей синусоидального тока: выполнение операций
непосредственно над синусоидальными функциями времени по уравнениям (2.8) и
(2.9); применение метода векторных диаграмм, основанного на уравнениях (2.1=
0),
использование в расчетах комплексных чисел и уравнений (2.11), являющихся
основой символического метода.
Пример 2.4. В узле электрической цепи сходятся три ветви (рис. 2.14=
).
Токи первых двух ветвей изв=
естны:
i1 =
=3D 8sin(w t+30° ) <=
/b>А,
i2 =
=3D 6sin(w t+120=
i>° ) <=
/b>А.
Требуется записать выражени=
е тока
i3=
и определить показания амперметров электромагнитной системы.
|
<=
/span>
Рис.
2.14. Узел электрической цепи
|
Р е ш е н
и е. 1. Непосредственное сложение синусоид:
i3 =3D=
i1+i2
=3D 8sin(w t+30° )+6sin(=
i>w t+120=
° ) =3D I3m=
sub>sin(w t+y 3).
Сумма двух синусоид одинаковой
частоты есть тоже синусоида той же частоты. Ее амплитуд=
а и
начальная фаза могут быть найдены по известным из математики формулам:
A,
,<=
/o:p>
откуда y 3 =
span>=3D 66,87°<=
/span> . Итак, i3 =3D 10sin=
span> (w t+66,87° ).
2. Применение метода векторных диагра=
мм.
В соответствии с первым законом Кирхгофа в векторной фор=
ме
для цепи на рис. 2.14 имеем
. В прямоугольной системе координат строим векторы
и
и находим вектор
, равный их сумме (рис. 2.15).
Так как треугольник oab прямоугольный, а сторона ab=
span>
равна длине вектора I2m,
то
=3D
А.
Если треугольник получается не прямоугольным, то применяется теорема
косинусов.
Начальная фаза третьего тока равна углу наклона: вектора I3m
к горизонтальной оси:
|
<=
/span>
Рис. 2.15. Векторная диагра=
мма
токов
|
3.
Решение символическим методом.
Записываем комплексные
амплитуды первого и второго токов:
A,
A.<=
/o:p>
По
первому закону Кирхгофа в символической форме
А.<=
/o:p>
Мо=
дуль
последнего комплексного числа равен амплитуде третьего тока, а аргумент –
начальной фазе.
Определяем показания амперметров. Приборы электромагнитной системы показыва=
ют
действующие значения токов и напряжений, поэтому
A,
A,
A.=
Об=
ращаем
внимание на то, что
. Это не ошибка. В цепях синусоидального тока для показаний
приборов законы Кирхгофа не справедливы. Можно складывать мгновенные
значения токов (синусоидальные функции времени), векторы и комплексные числ=
а,
но не численные значения токов и напряжений, не показания приборов.
Сл=
едует
заметить, что первый из рассмотренных в примере методов из-за громоздкости
вычислительных операций с синусоидами практически не применяется.
Ме=
тод
векторных диаграмм удобен при решении относительно несложных задач.
В
символической форме, как будет показано ниже, можно рассчитать сколь угодно
сложную линейную цепь.
=
2.9. Понятие об активном
сопротивлении. Синусоидальный
ток в активном сопротивлении
При
протекании электрического тока выделяется энергия в виде тепла или механиче=
ской
работы. Параметр электрической цепи, характеризующий этот процесс, называет=
ся
активным сопротивлением. Количественно он определяется следующим образом. П=
усть
на некотором участке цепи за время Т, равное периоду переменного т=
ока,
действующее значение которого I, необратимо преобразуется в тепло или механическую рабо=
ту
электрическая энергия WТ. Тогда активное сопротивление рассматриваемого
участка цепи по определению равно
. &nb=
sp; =
&nb=
sp; =
(2.12)<=
/span>
Н=
а схеме
активное сопротивление обозначается точно так же, как и сопротивление
постоянному току (рис. 2.16). Последнее, называемое еще омическим,
определяется структурой кристаллической решетки проводника и состоянием
свободных электронов. Наличие вблизи каких-либо проводящих тел и
ферромагнитных сердечников на омическое сопротивление не влияет.=
p>
|
Рис. 2.16. Активное сопротивление
|
Ин=
аче
обстоит дело при переменном токе.
При
невысоких частотах сопротивление проводника мало отличается от сопротивления
постоянному току. Но с повышением частоты все сильнее и сильнее сказывается
поверхностный эффект, заключающийся в вытеснении переменного тока из середи=
нных
областей проводника к его поверхности. Это приводит к уменьшению сечения,
занимаемого током, к увеличению сопротивления и возрастанию тепловых потерь=
. К
аналогичным последствиям приводит и эффект близости, выражающийся в
возникновении неравномерности распределения электрического тока по сечению
проводника из-за действия магнитного поля соседних проводов.
Ес=
ли
вблизи катушки имеются ферромагнитные сердечники и какие-либо другие провод=
ящие
тела, то магнитное поле переменного тока индуцирует в них вихревые токи, что
вызывает дополнительные потери энергии на нагрев. Кроме того, в переменном
магнитном поле происходит непрерывное периодическое перемагничивание
ферромагнитного сердечника, требующее энергетических затрат на изменение
направления магнитных моментов доменов.
Та=
ким
образом, понятие активного сопротивления является более широким, по сравнен=
ию с омическим. Числитель в формуле (=
2.12)
при переменном токе всегда больше, чем при постоянном, так как он включает в
себя все перечисленные потери электромагнитной энергии на тепло. Поэ=
тому
для одной и той же электрической установки активное сопротивление переменно=
му
току всегда оказывается больше чем сопротивление постоянному току.
Мгновенные значения напряжения и тока в активном сопротивлении связаны зако=
ном
Ома:
или
,
где
– активная проводимость:
.
При
изменении тока по синусоидальному закону
,
&nb=
sp; =
&nb=
sp;
(2.13)
напряжение тоже
синусоидально и имеет с током одинаковые начальные фазы:<=
/p>
.
&nb=
sp;
(2.14)
От=
сюда
или
.
&nb=
sp;
Разделив два последних уравнения на
, получим
и
.
Че=
тыре
последних уравнения представляют собой различные формы записи закона Ома для
активного сопротивления.
По
уравнениям (2.13) и (2.14) можно записать комплексные амплитуды тока и
напряжения:=
o:p>
,
, откуда=
или
.=
o:p>
По=
сле
деления последних двух уравнений на
будем иметь:
и
.
Получили те же самые
выражения закона Ома, но в символической форме.
На=
рис.
2.17 показаны волновая и векторная диаграммы, построенные по формулам (2.13=
) и (2.14).
Рис.
2.17. Волновая и векторные диаграммы для активно=
го
сопротивления
=
В активном сопротивлении напряжение и ток совпадают по фазе; их начальные фазы одинаковы,=
угол
сдвига фаз равен нулю, векторы на векторной диаграмме направлены в одну сто=
рону
(параллельны).
&nb=
sp;
2.10. Самоиндукция. Индуктивность. Синусоидальный ток в
индуктивности
Ес=
ли в
катушке, изображенной на рис. 2.1, магнитное поле создается собственным ток=
ом i=
span>, то магнитный поток называется
потоком самоиндукции и обозначается ФL, а индуцируемая в кат=
ушке ЭДС еL – ЭДС самоиндукции. В соответствии с фо=
рмулой
(2.1) она равна
,
&nb=
sp; =
&nb=
sp; =
(2.15)<=
/span>
где
–
потокосцепление самоиндукции, величина, пропорциональная протекающему по
катушке току:
=
=3D Li.
Коэффициент пропорциональности L между потокосцеплением и током
называется собственной индуктивностью или просто индуктивностью катушки
(контура). Она зависит от формы и размеров катушки, а также от магнитной
проницаемости сердечника. Ее размерность В x с/А=3DОм x
с. Эта единица измерения называется генри (Гн).
Подставляя последнее выражение в (2.15) и полагая L<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D cons=
t, получаем следующую формулу,
определяющую ЭДС самоиндукции:
.
&nb=
sp; =
&nb=
sp; =
(2.16)<=
/span>
Н=
а рис.
2.18 показано изображение индуктивности на электрической схеме; uL – напряжение на зажимах кат=
ушки,
обусловленное электродвижущей силой самоиндукции, или другими словами,
напряжение, наведенное в катушке собственным переменным магнитным полем.<=
/span>
|
Рис. 2.18. Обозначение
индуктивности
|
Вс=
е три
стрелки на схеме (i, eL, uL<=
/sub>) принято направлять в одну ст=
орону.
Раньше мы видели, что при одинаковых направлениях стрелок напряжения и ЭДС они имеют разные знаки. Поэтому=
.
&nb=
sp; =
&nb=
sp; =
(2.17)
Знак минус в правой части формулы (2.16) обусловлен принципом
Ленца, определяющим направление индуцированной ЭДС.
В рассматриваемом случае он может быть сформулирован следующим образом:
=
ЭДС самоиндукции направлена так,
что своим действием препятствует причине, вызвавшей ее появление.
Пр=
ичина
появления ЭДС самоиндукции – изменение тока. По=
этому
при возрастании тока она направлена ему навстречу, при уменьшении тока – в =
одну
с ним сторону.
Препятствуя изменению тока, ЭДС самоиндукции оказывает ему сопротивление, которое
называется индуктивным и обозначается х<=
span
style=3D'font-size:18.0pt;font-family:Arial'>L<=
/span>. В соответствии с формулой (2=
.16)
его величина определяется индуктивностью и скоростью изменения тока, т.е.
частотой. Формула, определяющая индуктивное сопротивление, имеет вид:
.=
o:p>
В цепях постоянного тока такого понят=
ия
мы не встречали, так как при постоянных магнитных полях ЭДС
самоиндукции не возникает.
Пусть ток, протекающий по индуктивнос=
ти,
определяется выражением (2.13). Тогда напряжение на ее зажимах, в соответст=
вии
с формулой (2.17), равно
.=
o:p>
Это – мгновенное значение
напряжения. Его амплитуда равна
.
Аналогичное выражение получается (пос=
ле
деления на
) и для действующих значений
, откуда
,
где ВL – индуктивная проводимость;
.
Запишем соответствующие формулы в
символической форме:
.
Так как
, то
.
Отсюда
.
Аналогично для действующих значений:<=
o:p>
&n=
bsp;  =
; &n=
bsp;
,
.
&nb=
sp; =
&nb=
sp;
(2.18)
Уравнения, связывающие напряжение и ток в индуктивно=
сти,
как в вещественных, так и в комплексных числах, представляют собой закон =
Ома
для индуктивности.
Начальная фаза напряжения больше начальной фазы тока на 90° . В индуктивности ток отстает от напряжения =
на
четверть периода. Выражение закона Ома, записанно=
е в
символическое форме, указывает на этот сдвиг фаз. Вспомним, что
умножение вектора на j приводит=
к его
повороту на угол 90° против часовой стрелки.<=
/p>
|
<=
/span>
Рис. 2.19. Векторная диагра=
мма
напряжения и тока в индуктивности
|
Согласно уравн=
ениям
(2.18)
получается путем умножения произведения
на j, в результате чего вектор
оказывается повернутым относительно вектора
.
Пример 2.5.
Мгновенное значение напряжения на индуктивности определяется выражением uL =3D 200 sin(w t+60°<=
/span> )<=
/span> В. Записать выражение мгновен=
ного
значения тока, если L =3D 63,67 мГн, а частота питающего напряжения f=
span> =3D 50 Гц. Построить векторные
диаграммы напряжения и тока.
Р е ш е н и е. При частоте =
f =3D 50 Гц циклическая частота w =3D 314 с-1, и
индуктивное сопротивление xL=
=3D w L =3D 20 Ом. Амплитуда тока равна
А.
Так как в индуктивн=
ости
ток отстает от напряжения на четверть периода, его начальная фаза меньше
начальной фазы напряжения на 90° : y i =3D y u – 90° =3D 60–90–30=
° .
Итак, i =3D =
10sin (w t–30° ). Векторная диаграмма показа=
на на
рис. 2.20.
|
Рис. 2.20. Векторная диаграмма напряжения итока в индуктивности
|
2.11. Синусоидальный ток в емкости
Система из двух
проводящих тел, разделенных диэлектриком, образует конденсатор. Эти прово=
дящие
тела называются обкладками. Если к ним подключить источник энергии, то на=
них
будет накапливаться заряд q, пропорциональный напряжению на конденсаторе uC (рис. 2.21):
|
<=
/span>
<=
span
style=3D'font-size:13.5pt;font-family:Arial'>Рис. 2.21. Обозначение
конденсатора
|
q<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D CuC=
..
&nb=
sp; =
&nb=
sp; =
&nb=
sp; =
(2.19)<=
/span>
Коэффициент пропорциональности C между зарядом и напряжением
называется емкостью конденсатора. Единица измерения емкости – фарада (Ф). О=
на
имеет следующую размерность:
. Емкость зависит от формы, размеров конденсатора и от
диэлектрической проницаемости диэлектрика между обкладками.
Пу=
сть
напряжение, подаваемое источником на конденсатор, изменяется по закону
uC=
span> =3D U<=
sub>Cm
sin (w<=
/span> t+y<=
/span> )=
span>.
&nb=
sp; =
&nb=
sp; =
(2.20)=
Пр=
и его
возрастании от нуля до максимального значения конденсатор заряжается, на его
обкладки от источника поступает электрический заряд. При уменьшении напряже=
ния
от максимума до нуля, заряд стекает с конденсатора, он разряжается. Таким
образом, в проводах, соединяющих конденсатор с остальной цепью, постоянно
движется электрический заряд, т.е. протекает электрический ток. Вывод о нал=
ичии
электрического тока мы делаем, совершенно не касаясь вопроса о том, какие
процессы происходят между обкладками конденсатора. Величина тока определяет=
ся
зарядом, прошедшим в единицу времени через поперечное сечение проводника:
.
&nb=
sp; =
&nb=
sp; =
(2.21)
Она
зависит от емкости и скорости изменения питающего напряжения, т.е. от часто=
ты.
От этих же факторов зависит и электрическая проводимость участка цепи с
конденсатором. Ее называют емкостной проводимостью и определяют по формуле<=
/span>
BC=
span> =3D w <=
b>C<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D 2p <=
b>fC.
Величина, обратная емкостной проводимости, называется емкостным сопротивлен=
ием:
.=
o:p>
Подставляя в (2.21) приложенное к конденсатору напряжение из (2.20), получа=
ем
 =
; &n=
bsp;
(2.22)<=
/span>
где Im =3D w<=
/span> CUC=
m
=3D BcUCm<=
span
style=3D'font-size:18.0pt;font-family:Arial'>.
Действующее значение тока
I<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D w<=
/span> CUC=
=3D BCUC ,
От=
сюда
.
Последние три уравнения представляют разные формы записи закона Ома для
конденсатора. Запишем их в символической форме. На основании (2.20) и (2.22=
):
,=
o:p>
, <=
/o:p>
или
.=
o:p>
От=
сюда
.
Векторная диаграмма, построенная по приведенным выше уравнениям, показана на
рис. 2.22.
Уг=
ол
наклона каждого вектора к положительному направлению вещественной оси
определяется начальными фазами в выражениях (2.20) и (2.22). Так как при
определении напряжения
мы умножаем
на –j, то вектор
оказывается повернутым относительно вектора тока на угол 90°<=
/span> в отрицательном направлении, =
по
часовой стрелке. Как отмечалось раньше, направление угла j<=
/span> на диаграмме показывается от
вектора тока к вектору напряжения.
Рис. 2.22. Векторная диаграмма напряжения и тока в емкос=
ти
Пример 2.6.
Напряжение на конденсаторе uC=
span> =3D 100sin (1000t –30° ). Написать выражение мгновенно=
го
значения тока через конденсатор. Каким станет ток, если частота питающего
напряжения увеличится вдвое? Емкость конденсатора С=
span> =3D 50 мкФ.
Р е ш=
е н и е. Определяем емкостное сопротивление:
Ом.
Амплитуда тока
A.
Та=
к как
, а
и
, то начальная фаза тока
.
Та=
ким
образом,
.
При
возрастании частоты вдвое емкостное сопротивление уменьшается также вдвое: =
Ом.
Амплитуда тока при этом увеличивается:
A.
Та=
к как
угол сдвига фаз не меняется, то мгновенное значение тока будет равно=
o:p>
А.
2.12.
Последовательное соединение активного
сопротивления, индуктивности и емкости
В =
схеме,
состоящей из последовательно соединенных активного сопротивления, индуктивн=
ости
и емкости (рис. 2.23), заданы приложенное напряжение U, частота f и числовые значения параметров R, L и С=
. Требуется найти ток и напряж=
ения
на элементах
При анализе электрических цепей синусоидального тока
типична ситуация, когда метод решения незнакомой задачи неизвестен. Во мн=
огих
случаях помогает следующий подход. По установленным ранее правилам строит=
ся
векторная диаграмма, из анализа которой выводятся необходимые расчетные
формулы. Так же поступим сейчас и мы.
В последовательной цепи общим для всех элементов является протекающий =
по
ним ток, поэтому именно с него начинаем построение векторной диаграммы.
Проводим его изображение горизонтально (рис. 2.24).
|
<=
/span>
Рис. 2.23. Последовательная=
цепь
переменного тока
|
Во=
обще,
направление первого вектора при построении диаграмм произвольно. Оно диктуе=
тся
соображениями удобства. Дальше мы должны показать векторы напряжений на всех
элементах и в соответствии со вторым законом Кирхгофа в векторной форме
получить вектор входного напряжения. Сложение векторов можно
выполнять по правилу параллелограмма, однако удобнее применять правило мног=
оугольника,
когда каждый последующий вектор пристраивается к концу предыдущего.
Рис. 2.24. Векторная диаграмма последовательной цепи
Нам
известно, что напряжение на активном сопротивлении совпадает по фазе с токо=
м,
поэтому вектор UR мы направляем по вектору I. К его концу пристраиваем вектор UL и направляем его вверх, так к=
ак
напряжение на индуктивности опережает ток на 90° . Напряжение U=
С находится в противофазе с UL, т.е. отстает от тока на тот =
же
угол 90° , поэтому
вектор U=
С, пристроенный к концу вектора=
UL<=
/i>, направлен вниз. Векторная су=
мма UR<=
/i>, =
UL и UС дает вектор приложеного
напряжения U.
Ве=
личины
напряжений на отдельных элементах цепи нам известны:
, &nb=
sp;
,
.
&nb=
sp;
(2.23)
Из
треугольника
(рис. 2.24) по теореме Пифагора находим
.
Вынося
из под знака радикала, записываем последнее выражение в виде:
,
&nb=
sp; =
&nb=
sp;
(2.24)
где z – полное сопротивление цепи, равное
.
&nb=
sp;
(2.25)=
В
последней формуле разность индуктивного и емкостного сопротивлений мы
обозначили буквой х.=
i> Это общее реактивное сопротив=
ление
цепи: х<=
/span> =3D хL –
xC. Сами индуктивность и емкость называются реактив=
ными
элементами, и их сопротивления хL<=
/i> и xC тоже носят названия реактивны=
х.
Выражение (2.24) называется законом Ома для всей цепи. Оно может быть запис=
ано
и так:
,
&nb=
sp; =
&nb=
sp; =
(2.26)
где
– полная проводимость цепи, представляющая величину, обратную пол=
ному
сопротивлению:
.
Ес=
ли
необходимо определить угол сдвига фаз между напряжением и током, то это мож=
но
сделать из треугольника напряжений
(рис. 2.24):
.=
o:p>
Векторная диаграмма на рис. 2.24 построена для случая, когда
, что имеет место при
, когда в цепи преобладает индуктивность, и цепь носит
активно-индуктивный характер. Общий ток отстает по фазе от входного напряже=
ния.
Возможны также режимы, когда
и
.
Пример 2.7. Параметры цепи на рис. 2.23, =
имеют
следующие числовые значения: R =3D 60 Ом, xL =3D 40 Ом, xC =3D 120 Ом, U =3D 200 В.<=
/p>
Определить ток, напряжения на элемент=
ах и
угол сдвига фаз между напряжением и током. Построить векторную диаграмму.
Р
е ш е н и е. Опреде=
ляем
полное сопротивление цепи
Ом.=
Действующее значение тока
А.
Напряжения на элементах:=
span>
B,
В,
В.<=
/o:p>
Угол сдвига фаз
Векторная диаграмма показана на рис. 2.25. Так =
как
, цепь носит активно-емкостный характер, ток опережает напряжен=
ие,
угол сдвига фаз j отрицателен и на диаграмме направлен от тока к напряжению в
отрицательном направлении – по часовой стрелке.Типичным для цепей переменного тока является
рассмотренный в задаче режим, когда напряжение на одном из реактивных
элементов (в данном случае на емкости) превышает входное напряжение. При
определенных соотношениях параметров цепи это превышение может быть довол=
ьно
значительным – в десятки и сотни раз.
|
<=
span
style=3D'font-size:13.5pt;font-family:Arial'>Рис. 2.25. Векторная диаграм=
ма
последовательной цепи
|
Об=
ратим
внимание также на то, что сумма падений напряжений на элементах в вольтах (=
120
+ 80 + 240) не равна 200 – напряжению, приложенному к цепи. Еще раз повторя=
ем,
что для численных значений токов и напряжений законы Кирхгофа неприменимы. =
Они
справедливы только для мгновенных значений, векторов и комплексных чисел. <=
i>Показания
приборов в цепях переменного тока складывать нельзя. =
Пример 2.8.
Определить величины сопротивления R и емкости=
С=
span>,
если при частоте f =3D 50 Гц приборы в схеме на рис. 2.26, а имеют следующие показания:=
U=
span> =3D 125 В<=
/span>, UR =3D 100 В, I<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D 2,5 А. <=
/p>
Р е ш=
е н и е. По показаниям вольтметра и амперметра определя=
ем
величину активного сопротивления
Ом.=
Для
определения емкости конденсатора необходимо знать напряжение
на его зажимах. Его можно найти из векторной диаграммы (рис. 2.26=
, б).
При ее построении вектор
направляем по вектору тока – они совпадают по фазе, а вектор
– под углом 90° к току в сторону отставания. Вектор входного напряжения
находим как сумму векторов
и
:
.
Из
векторной диаграммы определяем напряжение на конденсаторе=
В.<=
/o:p>
Те=
перь
находим емкостное сопротивление:
Ом.=
Из
формулы
определяем емкость
мкФ.
Рис.
2.26. Электрическая цепь и ее векторная диаграмма
Пример 2.9. К
зажимам катушки с параметрами R<=
span
style=3D'font-size:18.0pt;font-family:Arial'>
=3D 30 Ом и L =3D 40 мГн приложено
напряжение
В. Записать выражение мгновенного значения тока. Каким оно
станет, если частота питающего напряжения уменьшится вдвое?
Р е ш=
е н и е. Вычисляем индуктивное и полное сопротивления:<=
/span>
Ом,
Ом.=
По
закону Ома находим амплитуду тока
А.
Определяем угол сдвига фаз
.
При
отстающем токе (при активно-индуктивном характере цепи) этот угол положител=
ен.
Из (2.4) находим начальную фазу тока:=
.
Итак
А.
При
уменьшении частоты, т.е. при
с-1:
Ом,
Ом,
А,
,
,
А.
В =
схемах
на рис. 2.27 предлагается определить показание вольтметра V=
span> на входе цепи, если во всех с=
лучаях
показания вольтметров на отдельных элементах составляют V1 =3D 40 В<=
/span>, V2 =3D 30 В. Ответы на поставлен=
ный
вопрос приводятся под каждой схемой.
Рис.
2.27. Измерение напряжения в электрической цепи
Предлагаем учащемуся для каждого случая построить векторную диаграмму и
проверить правильность приведенных ответов.
Ве=
рнемся
к векторной диаграмме, представленной на рис. 2.24. Изобразим отдельно
треугольник oab, являющийся
ее частью (рис. 2.28, а). Этот треугольник называется треугольником
напряжений.=
o:p>
Рис. 2.28. Треугольники напряжений (а) и сопротивлений (=
б)
Проекция вектора напряжения на вектор тока называется активной составляющей
напряжения. О=
на
обозначается U<=
sub>а и равна падению напряжения на активном сопротивл=
ении
. Реактивная составляющая напряжения U=
span>r<=
/span> – это проекция вектора напр=
яжения
на направление, перпендикулярное вектору тока. Она равна падению напряж=
ения
на суммарном реактивном сопротивлении цепи
. Как видно из рис=
.
2.28, если все стороны треугольника напряжений разделить на ток, то получит=
ся
подобный ему треугольник сопротивлений (рис. 2.28, б). Ему соответст=
вуют
следующие формулы:
,
,
,
.
&nb=
sp;
(2.27)
Ре=
шим
задачу, поставленную в начале подразд. 2.12,
символическим методом.
За=
пишем
для цепи, изображенной на рис. 2.23, второй закон Кирхгофа в символической
форме:<=
/span>
.=
o:p>
Подставляя в него напряжения на элементах, записанные ранее в комплексных
числах
,
и
, получим:
или
, &nb=
sp; =
&nb=
sp;
(2.28)=
где
– комплексное сопротивление цепи, равное
.<=
/o:p>
Выражение (2.28) представляет собой закон Ома в символической форме.=
o:p>
=
Пример
2.10. Определ=
ить
показание амперметра в цепи, состоящей из последовательно соединенных
сопротивлений – активного R<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D 60 Ом и индуктивного xL =3D 80 Ом, если напряжение на =
входе
цепи U =
=3D 220
В.
Р е ш=
е н и е. Комплексное сопротивление цепи равно
Ом.
Для
применения формулы (2.28) входное напряжение необходимо записать в символич=
еской
форме, в виде комплексного числа
. Так как нам задано только его численное значение, начальную фазу
(аргумент комплексного числа
) мы выбираем по своему усмотрению.
Пр=
имем
, тогда
В.
Ко=
мплекс
действующего значения тока
A.
Показание амперметра равно модулю комплексного числа
A.
Пример 2.11. В
цепи, состоящей из последовательно соединенных активного сопротивления R=
span> и катушки с параметрами RK =3D 34 Ом и хК =3D w LK =3D 48 Ом (рис. 2.29, а=
),
известны UR =3D 100 B=
и UK =3D 120 В. Требуется найти нап=
ряжение
на входе цепи U.
Рис.
2.29. Электрическая цепь и её векторная диаграмма
Р е ш=
е н и е начинаем с построения векторной диаграммы (рис.=
2.29,
б). Вектор напряжения U равен векторной сумме напряже=
ний
и
:
. Напряжение на катушке
представляем как сумму напряжений на ее активном и реактивном
сопротивлениях
. Векторы
и
направляем по току, а
проводим перпендикулярно вверх.
Величину входного
напряжения, равную длине вектора U, можно найти одним из трех спо=
собов:
а) графически, постро=
ив
векторную диаграмму в масштабе, для чего предварительно нужно рассчитать
следующие величины:
– полное сопротивление
катушки
Ом;
– ток в цепи
А;
– составляющие напряж=
ения
на катушке
.
Из=
мерив
длину вектора U и умножив ее на масштаб, получим нужное нам значение;
б)
решением треугольников, применяя известную из тригонометрии теорему косинус=
ов:
.
Угол a определяем из векторной диаграммы:
, где
,
.
Вычисляем напряжение:
В;=
в)
символическим методом, записав
и
в комплексном виде и использовав ура=
внение
второго закона Кирхгофа в символической форме
. Комплексные выражения
и
мы получим, изобразив векторы этих напряжений в комплексной
плоскости, направив их из одного начала (рис. 2.30):
В;
В.
|
=
Рис. 2.30. Векторы в комплек=
сной
плоскости
|
На=
ходим
В.
Ве=
личина
напряжения
В.
=
2.13. Параллельное соединение
активного сопротивления,
индуктивности и емкости
Рассмотрим схему, состоящую из параллельно соединенных =
активного
и реактивных элементов (рис. 2.31, а).
Требуется по известным G, ВL, ВC, =
U рассчитать токи. Как и прежде,
задачу будем решать двумя методами.
1.=
М е т
о д в е к т о р н ы =
х
д и а г р а =
м м.
То=
ки
ветвей находятся сразу:
,
,
.
Для
определения общего тока
необходимо построить векторную диаграмму (рис. 2.31, б).
Построение начинаем с вектора напряжения, так как оно является общим для вс=
ех
ветвей. Из векторной диаграммы имеем
или
,=
o:p>
где
– полная проводимость цепи, равная
.
Ра=
зность
индуктивной и емкостной проводимостей представляет собой общую реактивную
проводимость цепи
.<=
/o:p>
=
o:p>
Рис. 2.31. Электрическая цепь и ее векторная диаграмма
Ве=
кторы
токов на диаграмме образуют треугольник токов. Его горизонтальный катет,
представляющий проекцию вектора тока на вектор напряжения, называется актив=
ной
составляющей тока и равен току в активном элементе цепи:
(рис. 2.32, а). Проекция вектора тока на направление,
перпендикулярное напряжению, – это реактивная составляющая тока. Она равна
суммарному току реактивных элементов
и определяется как разность длин векторов:
(см. рис. 2.31, б и 2.32, а).
Рис.
2.32. Треугольники токов и проводимостей
Ра=
зделив
все стороны треугольника токов на
, получим треугольник проводимостей (рис. 2.32, б), стороны
которого связаны следующими соотношениями:
&nb=
sp; =
,
,
,
. (2.29)
2.=
С и м в о л и ч е с к и й =
; м е т
о д.
Ра=
ньше
были получены следующие формулы:
,
,
.
Подставляя их в уравнение первого закона Кирхгофа, получаем:
или
,
где
– комплексная проводимость цепи, равная
Пример 2.12. =
Для
цепи, показанной на рис. 2.33, а, рассчитать токи, угол сдвига фаз м=
ежду
током и напряжением на входе цепи, построить векторную диаграмму. Числовые
значения параметров цепи:
В,
Ом,
мкФ,
с-1.
=
o:p>
Рис. 2.33. Электрическая цепь и ее векторная диаграмма
Р е ш=
е н и е.
А,
Ом,=
А,
А.
Векторная диаграмма приведена на рис. 2.33, б.
Уг=
ол
сдвига фаз
.
Ве=
личину
общего тока можно найти иначе:
См,
&nb=
sp;
См,=
См,
&nb=
sp;
А.<=
/o:p>
Пример 2.13.
Начертить цепь, векторная диаграмма которой изображена на рис. 2.34, а=
i>.
Р е ш=
е н и е задачи показано на рис. 2.34, б.<=
span
style=3D'font-size:7.5pt;font-family:Arial'>
Рис.
2.34. Векторная диаграмма и соответствующая ей электрическая цепь=
Пример 2.14. =
Чему
равно показание амперметра А на входе цепи в схем=
ах
рис. 2.35, если амперметры А1 и А2 во всех случаях показывают
соответственно 4 и 3 А?=
Рис.
2.35. Измерение тока в электрической цепи
&nb=
sp;
Предлагаем для каждого случая самостоятельно построить векторную диаграмму и
убедиться в правильности приведенных ответов: а) 5А,
б) 7А, в) 1А.
&nb=
sp;
2.14.
Пассивный двухполюсник в цепи синусоидального тока.
Эквивалентные сопротивления и проводимости
На=
рис.
2.36 показан пассивный двухполюсник, состоящий из активных и реактивных
элементов. Действующие значения напряжения
, тока
и угол сдвига фаз между ними
известны.
Построим по этим значениям векторную диаграмму и, спроектировав вектор
напряжения на вектор тока и перпендикулярное к нему направление, получим
треугольник напряжений, образованный сторонами
,
и
(рис. 2.37, а).
Как и раньше,
и
будем называть активной и реактивной составляющими напряжения.
Изображенная в таком виде диаграмма соответствует схеме, показанной на ри=
с.
2.37, б. Действительно, для нее
,
и
.
|
=
Рис. 2.36. Пассивный двухполюсник
|
Схема называет=
ся
последовательной схемой замещения или последовательной эквивалентной схемой
пассивного двухполюсника, а ее параметры
,
и
– эквивалентными сопротивлениями двухполюсника.=
Рис.
2.37. Векторная диаграмма и соответствующая ей последовательная эквивалентн=
ая
схема<=
/span>
Треугольник, образованный сторонами
,
и
и подобный треугольнику напряжений,
представляет собой треугольник сопротивлений (рис. 2.28, б), для кот=
орого
справедливы формулы (2.27).
Те=
перь
разложим в е к т о р=
т о к а
на две составляющие – активную
, направленную по вектору напряжения, и реактивную
, перпендикулярную к нему (рис. 2.38, а). Такой векторной диаграм=
ме
соответствует параллельная схема замещения двухполюсника (рис. 2.38, б=
i>).
Ее параметры
,
и
называются эквивалентными проводимостями. Токи в элементах
и
мы и представляем как активную и реактивную составляющие общего т=
ока:
,
. Из треугольника токов (рис. 2.38, а) получается треуголь=
ник
проводимостей
(рис. 2.32, б), стороны которого связаны между собой формулами (2.29=
).
а)
&nb=
sp; =
&nb=
sp; =
б)
Рис.
2.38. Параллельная эквивалентная схема и ее векторная диаграмма
По=
лучим
условия эквивалентности приведенных схем.
Для последовательной цепи
, для параллельной
, а так как токи и напряжения в обеих схемах одинаковы, то=
и
,
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
(2.30)=
т.е. в любой
электрической цепи полная проводимость есть величина, обратная полному
сопротивлению.
Из
сопоставления формул (2.27) и (2.29) можно записать:
и
.=
o:p>
Рассматривая последние выражения совместно с (2.30), можно получить две гру=
ппы
формул:=
Формулы перехода от
последовательной эквивалентной схемы к паралле=
льной:
|
Формулы перехода от параллельной эквивалентной схемы <=
span
class=3DGramE>к последовательной:
|
(2.31)
|
(2.32)
|
Об=
ращаем
внимание на то, что каждая из проводимостей G и B зависит от обоих сопротивлени=
й –
активного и реактивного.
В свою очередь, каждое из сопротивлений определяется обеими проводимостями.
Соотношения G =3D 1/R и B =3D 1/x справедливы только в частном случае, первое – при х =3D 0,
второе – при R =3D 0.
Сл=
едует
отметить, что активная и реактивная составляющие напряжения и тока физическ=
и не
существуют, измерить их нельзя. Они относятся только к соответствующим
эквивалентным схемам замещения и находятся расчетом. Более того, проектируя,
например, вектор тока на различные напряжения, мы получим для него разные
составляющие.
Пример 2.15. =
Найти
общее сопротивление цепи, состоящей из параллельно соединенных активного =
span>R=
span> =3D 30 Ом и индуктивного х =3D 40 Ом сопротивлений (рис. 2.39, <=
i>а).
Рис.
2.39. Схемы к примерам 2.15–2.17
Р е ш=
е н и е. Так как в левой ветви реактивного сопротивлени=
я нет,
то ее проводимость в соответствии с (2.31) равна G<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D 1/R. Аналогично, во второй ветви =
B=
span> =3D 1/x=
. Полная проводимость цепи
. В соответствии с (2.30) полное сопротивление цепи
Ом.
Пример 2.16.
Рассчитать общее сопротивление цепи, состоящей из параллельно соединенных
индуктивности L =3D 0,478 Гн и емкости С=
span> =3D 31,85 мкФ (рис. 2.39, б). Частота питающего
напряжения f =3D 50 Гц.
Р е ш=
е н и е. Определяем сопротивления ветвей:
Ом,
Ом.=
Та=
к как
в ветвях отсутствуют активные сопротивления, то их проводимости соответстве=
нно
равны BL =3D 1/xL и BC=
span> =3D 1/xС<=
/span>. Полная эквивалентная проводи=
мость
цепи не содержит активной составляющей и равна
.=
o:p>
По=
лное
эквивалентное сопротивление
Ом.
В
рассматриваемой цепи активных элементов нет, она носит чисто реактивный
характер. Он может быть индуктивным или емкостным. Знак=
минус
в ответе свидетельствует о последнем, т.е. вся цепь может быть заменена
конденсатором емкостью
мкФ.
Пример 2.17.
Амперметр А, вольтметр V=
span> и фазометр j<=
/span> ,<=
/span> включенные в цепь катушки (ри=
с.
2.39, в), дали следующие показания: <=
i>U<=
span
style=3D'font-size:18.0pt;font-family:Arial'> =3D 220 В, I=
=3D 4,4 А, cos j<=
/span> =3D 0,8. Частота питающего нап=
ряжения
50 Гц. Определить параметры последовательной и параллельной схем замещения
катушки.
Р е ш=
е н и е. Находим параметры последовательной эквивалентн=
ой
схемы:
Ом,
Ом,=
Ом.=
Рассчитываем элементы параллельной эквивалентной схемы:
См,
См,
См.
По=
сле
определения эквивалентных сопротивлений эквивалентные проводимости можно бы=
ло
найти иначе, по формулам (2.31):
См,
См,
См.
Пример 2.18. Рассчитать токи в схеме, приве=
денной
на рис. 2.40.
В,
Ом,
Ом,
Ом,
Ом,
Ом,
Ом,
Ом.
|
<=
span
style=3D'font-family:Arial'>Рис. 2.40. Расчетная схема
|
=
Р е ш=
е н и е. Определяем полные сопротивления второй и треть=
ей
ветвей:
Ом,
Ом.
=
Преобразуем эти ветви в эквивалентные параллельные (рис. 2.41=
, а).
Рис.
2.41. Преобразования электрической цепи
=
Их проводимости:
См,
См,
См,
См.
Суммируем активные и реактивные проводимости параллельных ветвей:
См,
См (см. рис. 2.41, б).
Определяем эквивалентные сопротивления участка
(рис. 2.41, в):
Ом,
Ом,
Ом,
и полное сопротивл=
ение
цепи:<=
/o:p>
Ом.
=
Ток
на входе цепи I1 =3D U/z =3D 220/41,53 =3D 5,297 A.
Напряжение на участке
Uab=3D I1zab=3D 119,7 В.
=
Токи
второй и третьей ветвей:
А,
&nb=
sp;
А.
=
Еще
раз напоминаем, что для численных значений токов и напряжений законы Кирхго=
фа
неприменимы:
.
=
2.15. Закон Ома в символической
форме для произвольной цепи
Пу=
сть
мгновенные значения напряжения и тока на зажимах произвольного пассивного
двухполюсника определяются выражениями (2.3). Тогда комплексы их действующих
значений соответственно равны
 =
; &n=
bsp;
,<=
/o:p>
а их отношение опреде=
ляет
комплексное сопротивление двухполюсника
Величина, обратная комплексному сопротивлению, – комплексная проводимость
.
Сопротивления z, R, x и проводимости y, =
G =
и B, входящие в два последних
выражения, есть не что иное, как эквивалентные параметры двухполюсника, о
которых говорилось в подразделе 2.14.
Пример 2.19.
Определить эквивалентные активное и реактивное сопротивления цепи, если
мгновенные значения напряжения и тока на ее входных зажимах соответственно
равны<=
/span>
В,
А.
Р е ш=
е н и е.
Ом,
т.е. R =3D 40 Ом, x =3D 30
Ом.
Зн=
ак
минус перед мнимой частью комплексного сопротивления говорит о том, что
суммарное реактивное сопротивление цепи носит емкостный характер. Это видно=
и
из условия задачи. Ток опережает напряжение, его начальная фаза больше.
Пример 2.20.
Определить комплексную проводимость цепи, состоящей из последовательно
соединенных активного R
и реактивного x сопротивлений.
Ре=
шение.
где
,
.
2.16. О расчете цепей синусоидального тока=
Как
следует из изложенного теоретического материала и приведенных примеров, при
анализе цепей синусоидального тока широко применяются векторные диаграммы и
комплексные числа. Сами по себе векторные диаграммы зачастую служат для
иллюстрации результатов теоретических исследований и решения задач. Они
помогают лучше понять сущность изучаемых процессов и наглядно представить
соотношения и связи напряжений и токов на различных участках с параметрами
цепи.
Во
многих случаях векторные диаграммы, построенные предварительно по изложенным
выше правилам без каких-либо вычислений, являются основой для вывода из них
конкретной методики решения данной задачи. Возможны так=
же
привязка векторной диаграммы к комплексным осям, выражение векторов
комплексными числами и дальнейший расчет в символической форме.
Принципиального отличия между методом векторных диаграмм и символическим не=
т. Как
мы видели раньше, за аналитическими действиями с комплексными числами кроют=
ся
определенные геометрические операции с векторами.
Сл=
едует
также помнить, что никакого физического содержания векторы и комплексные чи=
сла
в себе не несут. Это чисто математические абстракции, необходимые для анали=
за.
Символический метод базируется на законах Ома и Кирхгофа, которые в
символической форме записываются точно так же, как в цепях постоянного тока.
Поэтому все изложенные ранее методы расчета цепей постоянного тока, вытекаю=
щие
из этих законов, применимы и для расчета в символической форме цепей
синусоидального тока.
Пример 2.21.
Рассчитать комплексные сопротивления цепей, изображенных на рис. 2.39, а=
и б.
Ре=
шение.
Сопротивление каждой ветви записываем в символической форме и применяем
формулу, известную из теории цепей постоянного тока.
Для схемы, изображенн=
ой на
рис. 2.39, а:
Ом,
Ом,=
См=
ысл
полученного результата заключается в том, что рассматриваемая параллельная =
цепь
может быть заменена эквивалентной последовательной с активным сопротивлением
19,2 Ом и индуктивным 14,4 Ом.
Для
схемы на рис. 2.39, б:=
Ом.
Пример 2.22. =
Рассчитать
цепь, приведенную на рис. 2.40.
Р е ш=
е н и е. Находим комплексные сопротивления участков:
Ом,
Ом,
Ом
Ом=
Определяем комплексные токи ветвей:
А,
А,
А.
Численные значения токов:
А,
А,
А.<=
/o:p>
Для
проверки правильности расчета используем первый закон Кирхгофа в символичес=
кой
форме
.
См=
отрим:
А.
&nb=
sp;
В пределах точности расчета закон выполняется.
&nb=
sp;
2.17. Резонансы в электрических цепях
=
Резонансом
называют режим, когда в цепи, содержащей индуктивности и емкости, ток совпа=
дает
по фазе с напряжением.
Входные реактивные сопротивление и проводимость =
равны
нулю: x<=
/span> =3D ImZ =3D =
0 и B =3D ImY =3D =
0. Цепь носит чисто активный характер: Z =3D R; сдвиг ф=
аз
отсутствует (j =3D 0).
В =
цепи,
содержащей последовательно соединенные участки с индуктивным и емкостным
характерами сопротивлений, резонанс называется резонансом напряжений.
Рассмотрим простейшую цепь такого вида (рис. 2.23), которую часто называют
последовательным контуром. Для нее резонанс наступает при x =3D xL –=
xC =3D 0 или xL =3D xC=
span>, откуда
&=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
(2.33)<=
/span>
Напряжения на индуктивности и емкости в этом режиме равны по величине и,
находясь в противофазе, компенсируют друг друга. Все приложенное к цепи
напряжение приходится на ее активное сопротивление (рис. 2.42, а).=
span>
Рис. 2.42. Векторные диаграммы при резонансе напряжений =
(а)
и токов (б)<=
/o:p>
Напряжения на индуктивности и емкости могут значительно превышать напряжени=
я на
входе цепи. Их отношение, называемое добротностью контура Q, определяется величинами
индуктивного (или емкостного) и активного сопротивлений
.
`Добротность показыва=
ет, во
сколько раз напряжения на индуктивности и емкости при резонансе превышают
напряжение, приложенное к цепи. В радиотехнических цепях она может достигать
нескольких сотен единиц.
Из
условия (2.33) следует, что резонанса можно достичь, изменяя любой из
параметров – частоту, индуктивность, емкость. При этом меняются реактивное и
полное сопротивления цепи, а вследствие этого – ток, напряжение на элемента=
х и
сдвиг фаз. Не приводя анализа формул, показываем графические зависимости
некоторых из этих величин от емкости (рис. 2.43). Емкость
, при которой наступает резонанс, можно определить из формулы (2.=
33):
.
Ес=
ли,
например, индуктивность контура L =3D
0,2 Гн, то при частоте 50 Гц, резонанс наступит при
емкости=
мкФ.
Рис. 2.43. Зависимости параметров режима от емкости
Аналогичные рассуждения можно провести и для цепи, состоящей из параллельно=
соединенных R, L и C (рис. 2.31, а). Векторная диаграмма ее резонансн=
ого
режима приведена на рис. 2.42, б.
Рассмотрим теперь более сложную цепь с двумя параллельными ветвями, содержа=
щими
активные и реактивные сопротивления (рис. 2.44, а).
Рис. 2.44. Разветвленная цепь (а) и ее эквивалент=
ная
схема (б)<=
o:p>
Дл=
я нее
условием резонанса является равенство нулю ее реактивной проводимости: ImY =3D =
0. Это равенство означает, что мы должны мнимую часть комплексного
выражения Y приравнять к нулю.=
Оп=
ределяем
комплексную проводимость цепи. Она равна сумме комплексных проводимостей
ветвей:=
.
Приравнивая к нулю выражение, стоящее в круглых скобках, получаем:
или
.
&nb=
sp;
(2.34)=
Ле=
вая и
правая части последнего выражения представляют собой не что иное, как
реактивные проводимости первой и второй ветвей =
B1 и B2. Заменяя схему на рис. 2.44, =
а
эквивалентной (рис. 2.44, б), параметры которой вычисляем по формуле
(2.31), и используя условие резонанса<=
i> (B =3D B=
1
– B2 =3D <=
span
style=3D'font-size:13.5pt;font-family:Arial'>0), снова приходим к выражению
(2.34).
Сх=
еме на
рис. 2.44, б соответствует векторная диаграмма, приведенная на рис.
2.45.
Рис. 2.45. Векторная диаграмма резонансного режима разве=
твленной
цепи
Резонанс в разветвленной цепи называется резонансом токов. Реактивные составляющие токов
параллельных ветвей противоположны по фазе, равны по величине и компенсируют
друг друга, а сумма активных составляющих токов ветвей дает общий ток.=
Пример 2.23. =
Считая
R2<=
/sub> и x3 известными, определить величи=
ну x1<=
/i>, при которой в цепи наступит
резонанс напряжений (рис. 2.46, а). Для резонансного режима построить
векторную диаграмму.
Рис. 2.46. Электрическая цепь и ее векторная диаграмма=
span>
Ре=
шение.
При резонансе напряжение U1 на индуктивном сопротивлении =
x1<=
/i> равно реактивной составляющей
напряжения Uab: =
I1x1=
=3D I1xab<=
/i>, откуда x1 =3D xab. Последнее есть реактивное
сопротивление последовательной эквивалентной схемы замещения участка ab:
.
За=
дача
может быть решена и символическим методом. В соответствии с условием резона=
нса
напряжений, мы должны приравнять к нулю мнимую часть комплексного сопротивл=
ения
цепи. Величина последнего равна
.
Су=
мму
всех коэффициентов при мнимой единице приравниваем к нулю:
, откуд=
а
.
&nb=
sp;
Построение векторной диаграммы начинаем с вектора I1<=
/sub> (рис. 2.46, б). В том =
же
направлении проводим вектор приложенного к цепи напряжения U=
span> – при резонансе они совпадают=
по
фазе. Напряжение на индуктивности опережает ток на 90°<=
/span> , его вектор U1 направляем вверх. Вектор Uab проводим так, чтобы он в сумме=
с
вектором U1 давал вектор U. Ток I2=
span> совпадает по фазе с Uab, а I3=
span> опережает последний на 90°<=
/span> . В сумме векторы I2 и =
I3=
span> дают вектор I1.
&nb=
sp;
2.18. Энергия и мощность в цепи синусоидального тока=
Пу=
сть на
некотором участке цепи, напряжение на зажимах ко=
торого
равно u<=
/span>, током i за время dt переносится электрический заряд dq =3D idt. Затрачиваемая источником энергия равна при этом=
dw =3D udq =3D uidt, а развиваемая мощность =
p =3D dw=
span>/dt =3D ui<=
span
style=3D'font-size:13.5pt;font-family:Arial'>. Эта величина называется мгно=
венной
мощностью и определяет скорость и направление движения энергии на
рассматриваемом участке. Если энергия поступает в цепь и накапливается в не=
й,
функция w(t)
возрастает, и мгновенная мощность положительна как производная
возрастающей функции. Напряжение u и ток i в эти моменты времени имеют о=
динаковые
знаки. Процесс накопления энергии в цепи наблюдается, например, при заряде
конденсатора. В те моменты времени, когда u и i имеют разные знаки, мгновенная
мощность отрицательна, функция w<=
span
style=3D'font-size:18.0pt;font-family:Arial'>(t=
), определяющая энергию, поступ=
ающую
в цепь, убывает, так как только убывающая функция имеет отрицательную
производную. Убыль энергии в электрической цепи означает возврат ее источни=
ку.
Такая ситуация возникает при разряде конденсатора.
Эн=
ергия,
поступающая в цепь, может не возвращаться к источнику, а необратимо
преобразовываться в тепло или механическую работу. Количество этой энергии
определяется законом Джоуля–Ленца и за время, ра=
вное
периоду синусоидального тока, равно
.
Эта
величина, отнесенная ко времени Т, определяет среднее значение
мгновенной мощности за период и называется активной мощностью
.
&nb=
sp; =
&nb=
sp;
(2.35)=
Физически активная мощность представляет собой энергию, выделяющуюся в виде
тепла или механической работы в единицу времени.
Пу=
сть
ток и напряжение на входе произвольного пассивного двухполюсника описываются
выражениями<=
/o:p>
,
.
&nb=
sp; =
&nb=
sp;
(2.36)=
Подставляя их в (2.35) и интегрируя, получаем
.
Используя соотношения между сторонами в треугольниках напряжений и токов,
сопротивлений и проводимостей, можно написать цепочку формул для вычисления=
активной
мощности:
.
Рассмотрим теперь энергетические процессы, происходящие в отдельно взятых
элементах.
В
активном сопротивлении напряжение и ток совпадают по фазе (j<=
/span> =3D 0); в любой момент времени их знаки одинаковы, мгновенная
мощность положительна, т.е. в него постоянно поступает энергия электрическо=
го
тока, преобразуясь в тепловую или механическую.
Активная мощность равна:
P =3D UI =3D I=
2R
=3D U2G.
В
реактивных элементах угол сдвига фаз по величине равен 90°<=
/span> . В индуктивности, при отстаю=
щем
токе, он положителен, в емкости, при опережающем токе, – отрицателен.
Подставляя j <=
b>=3D ±=
90° в выражение напряжения на вхо=
де
цепи (2.36), получим u =3D =
Um sin (w t ± 90°<=
span
style=3D'font-size:13.5pt'> )=
=3D±=
Um cos =
w t. При таком напряжении мгновен=
ная
мощность колеблется с двойной частотой, изменяясь по синусоидальному закону=
р =3D ±=
U I =
sin
2w t,
т.е. дважды за полпер=
иода
меняет знак. Подстановка этого выражения в (2.35) приводит к результату: =
span>P =3D 0. Равенство нулю активной мощности означает, что в
реактивных элементах не происходит необратимого преобразования электромагни=
тной
энергии в тепловую и механическую. =
span>
Мо=
жно
показать, что в индуктивности в течение первой четверти периода, при
возрастании тока от нуля до Im, в магнитном поле индуктивнос=
ти
накапливается энергия
. В течение следующей четверти периода, когда ток уменьшается до
нуля, эта энергия из магнитного поля возвращается во внешнюю цепь.
В
емкости – аналогично: в течение одной четверти периода, когда напряжение на
обкладках конденсатора возрастает от нуля до Um, конденсатор заряжается, в его
электрическом поле накапливается энергия:
. В следующую четверть периода конденсатор разряжается, его
напряжение уменьшается до нуля, и накопленная в электрическом поле энергия
возвращается в цепь. Энергию, которой электрическое поле конденсатора и
магнитное поле катушки обмениваются с цепью, будем называть энергией обмена=
.
Для
энергии магнитного поля WM
и
электрического поля <=
span
style=3D'font-size:18.0pt'>WЭ можно записать следующие формулы:
,
.
Ве=
личины
и
имеющие размерность мощности, называются соответственно реактивной
мощностью индуктивности и реактивной мощностью емкости. К работе, совершаем=
ой
переменным током, они отношения не имеют, а являются величинами,
пропорциональными энергии магнитного и электрического полей:
,
.
В =
цепи,
содержащей одновременно и индуктивность и емкость, колебания энергии происх=
одят
таким образом, что в те моменты времени, когда магнитное поле индуктивности
накапливает энергию, электрическое поле емкости энергию отдает, и наоборот.
Т.е., когда энергия магнитного поля положительна, энергия электрического по=
ля
отрицательна. Суммарная энергия электрического и магнитного полей за четвер=
ть
периода равна=
,
г=
де
– реактивная мощность цепи, она пропорциональна суммарной энергии
электрического и магнитного полей и может быть определена через реактивные
сопротивления:=
span>
=
o:p>
При
резонансе, когда
, равны реактивные мощности
и
и энергии
и
, накапливаемые в магнитном и электрическом полях. В этом случае
обмен энергией между индуктивностью и емкостью происходит без участия
источника.
Для
вычисления реактивной мощности можно написать цепочку формул, аналогичную
(2.36):
.<=
/o:p>
При
анализе электрических цепей часто используется треугольник мощностей, котор=
ый
можно получить, умножив стороны треугольника сопротивлений на квадрат тока
(рис. 2.47). Для него справедливы следующие соотношения:
,
&nb=
sp;
,
&nb=
sp;
.
Бу=
квой
, стоящей рядом с гипотенузой треугольника, обозначается полная
мощность. Ее можно вычислить по одной из следующих формул:
Рис. 2.47. Треугольник мощностей
По=
лная
мощность определяется той электрической энергией, которая вырабатывается
генератором и отдается в цепь. Она характеризует габариты электрических маш=
ин и
аппаратов. Величина напряжения определяет уровень изоляции – ее толщину и
расстояние между токоведущими частотами, а ток – поперечное сечение проводн=
ика,
условия охлаждения машины.
Пр=
и
=3D 1 полная мощность равна наибольшему значению активной мощност=
и,
которую можно получить при заданных напряжении и=
токе.
Ед=
иницы
измерения мощности, имея одну и ту же размерность, называются по-разному. <=
span
class=3DGramE>Единица активной мощности – ватт (Вт), реактивной – вольт-амп=
ер
реактивный <=
/span>(вар),
полной – вольт-ампер (ВА).
Комплексная мощность определяется произведением комплекса напряжения и
сопряженного комплекса тока:
.
Пример 2.24. Рассчитать активную, реактивн=
ую и
полную мощности цепи, если напряжение и ток на ее зажимах определяются
выражениями:
,
&nb=
sp;
.
Р е ш е н
и е.
=3D 100 В,
=3D 0,5 А,
=3D 36,9° ,
50 ВА,
40 Вт,
30 вар.
Ил=
и
В,
А,
ВА.=
<=
/span>
3. ЦЕПИ С ВЗАИМНОЙ ИНДУКТИВНОСТЬЮ
3.1. Явление взаимной индукции
Рис. 3.1. Индуктивно связанные элементы
Пу=
сть
имеются две катушки, намотанные в виде тонких колец. Их
активные сопротивления равны нулю, числа витков
и
. Катушки находятся достаточно близко друг от друга, так что
магнитное поле каждой из них какой-то своей частью охватывает соседнюю.
Схематическая картина магнитных потоков, создаваемых токами
и , показана на рис. 3.1. Каждый поток изображен в виде одн=
ой
силовой линии, обозначенной буквой
с двумя индексами. Первый – указывает номер катушки, током которо=
й он
создается (происхождение магнитного потока), второй – номер катушки,
охватываемой этим потоком (объект его воздействия). Рассмотрим магнитные по=
токи
первой катушки. Ток
создает поток , называемый потоком самоиндукции. Его=
часть
охватывает только первую катушку, а
захватывает и витки второй. В сумме они равны=
. Кроме того, витки первой катушки охватываются потоком
, называемым потоком взаимной индукции и составляющим часть поток=
а
, создаваемого током второй катушки
. Суммарный магнитный поток
, пронизывающий первую катушку, складывается из потоков самоиндук=
ции
и взаимной индукции
. Сумма берется алгебраическая
, так как эти потоки могут быть направлены одинаково, либо
противоположно друг другу. Изображение на рис. 3.1 соответствует вто=
рому
случаю.
При
принятой форме катушки можно считать, что все ее витки охватываются одним и=
тем
же магнитным потоком. Умножив последний на число
витков
, получим полное потокосцепление первой катушки
,<=
/o:p>
где
– собственное потокосцепление первой катушки (потокосцепление
самоиндукции);
– потокосцепление взаимной индукции.
Ка=
ждое
из этих потокосцеплений пропорционально создающему его току:
и
. Поэтому
. При изменении магнитного потока в катушке индуцируется эдс электромагнитной индукции, и на ее зажимах появля=
ется
напряжение
или в символической форме
.
Аналогичное уравнение можно записать и для второй катушки.
Пе=
рвое
слагаемое в правой части последнего уравнения
представляет собой напряжение, обусловленное током самой катушки
(напряжение самоиндукции), а второе
– напряжение, наведенное на зажимах первой катушки изменяющимся
магнитным полем второй катушки (напряжение взаимной ии=
ндукции).
Эти напряжения имеют одинаковые знаки при согласном направлении магнитных
потоков и разные – при встречном.
Для
решения задачи о характере включения катушек и о направлении их магнитных
потоков вводят понятие одноименных зажимов, отмечая их на схеме одинаковыми
значками. Разметку делают руководствуясь следующ=
им
определением.
Одноименными зажимами двух катушек называются такие зажимы, когда при
одинаковых направлениях токов относительно этих зажимов магнитные потоки
самоиндукции и взаимной индукции в каждой катушке складываются.<=
span
style=3D'font-size:13.5pt;font-family:Arial'>
Др=
угими
словами, если мы имеем две катушки, у которых отмечены начала и концы намот=
ки,
и если токи в них протекают одинаково, например =
от
начала к концу в обеих катушках, то оба магнитные потока в каждой из них бу=
дут
направлены согласно.
На=
личие
магнитной связи между катушками обозначается на схемах двухсторонней
дугообразной стрелой, рядом с которой ставится буква
.
Пример 3.1.
Записать выражения, определяющие напряжения на зажимах каждого из элементов,
изображенных на рис. 3.2.
Рис. 3.2. Индуктивно связанные элементы
Р е ш=
е н и е. Схема а. Включение катушек согласное, т=
ак как
токи в них направлены одинаково – от звездочки в катушку. Поэтому
и
.=
o:p>
Зн=
ак
перед напряжением самоиндукции (перед первым слагаемым) определяется взаимн=
ыми
направлениями этого напряжения и тока. В первой катушке напряжение
направлено по току, поэтому
пишется с плюсом. При определении напряжения
от
к
мы идем против тока, поэтому перед
ставим минус. Так как включение катушек согласное, оба слагаемые в
каждом уравнении имеют одинаковые знаки.
В =
схеме б
катушки включены встречно – в первой ток направлен от точки в катушку, во
второй – из катушки к точке. Напряжения самоиндукции и взаимной индукции им=
еют
противоположные знаки:
и
.<=
/o:p>
Ст=
епень
индуктивной связи двух элементов цепи характеризуют коэффициентом магнитной
связи
, определяемым как среднее геометрическое отношений потока взаимн=
ой
индукции к потоку самоиндукции каждой катушки:
или после умножения
числителя и знаменателя на
:=
o:p>
,
где
– сопротивление взаимной индукции.
Та=
к как
является частью
, а
– частью
, то коэффициент магнитной связи всегда меньше единицы. Он равен
единице в том идеальном случае, когда поток, создаваемый током одной катушк=
и,
полностью сцепляется с витками другой катушки. А это возможно лишь при
пространственном совмещении катушек.
3.2. Последовательное соединение индуктивно связанных
элементов
Пу=
сть
две катушки, обладающие сопротивлениями
и
, индуктивностями
и
и взаимной индуктивностью
, соединены последовательно (рис. 3.3).
Рис. 3.3. Последовательное соединение индуктивно связанн=
ых
элементов
Во=
зможны
два вида их соединения – согласное и встречное. Если считать, что звездочка=
ми
отмечены начала обмоток, то при согласном включении начало второй подключае=
тся
к концу первой (рис. 3.3, а). Токи в обеих катушках направлены одина=
ково
относительно одноименных зажимов: от начала к концу. При встречном включении
катушек конец второй присоединяется к концу первой (рис. 3.3, б).
Напряжение на каждой из катушек содержит три составляющих: падение напряжен=
ия
на активном сопротивлении, напряжение самоиндукции и напряжение взаимной
индукции:
&=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
(3.1)=
span>
Последние имеют одинаковые знаки при согласном включении и разные при
встречном. Напряжение на входе цепи равно сумме этих двух напряжений=
.
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
(3.2)=
span>
Вх=
одное
комплексное сопротивление цепи получим из совместного рассмотрения трех
последних уравнений
,
&nb=
sp; =
(3.3)=
span>
где
и
– комплексные сопротивления катушек, а
– комплексное сопротивление взаимной индукции:
, =
,
&nb=
sp;
.<=
/o:p>
Из=
(3.3)
вытекают формулы, определяющие общую индуктивность цепи и суммарное индукти=
вное
сопротивление:
,
&nb=
sp;
,<=
/o:p>
причем
,
, =
т.е.
. <=
/o:p>
Мо=
жно
определить результирующее индуктивное сопротивление каждой катушки. У первой
оно равно
. И здесь при согласном включении оно больше чем при встречном.
Физически это объясняется тем, что в первом случае магнитный поток,
охватывающий каждую катушку, больше чем во втором; например, для первой кат=
ушки
, а
. Вследствие этого ЭДС электромагнитн=
ой
индукции, оказывающая току индуктивное сопротивл=
ение,
при согласном включении больше, чем при встречном.
На=
рис.
3.4 изображены векторные диаграммы, построенные по уравнениям (3.1) и (3.2)=
.
Рис. 3.4. Векторные диаграммы последовательной
цепи при согласном (а) и встречном (б) включениях<=
/span>
При
встречном включении возможен так называемый "емкостный" эффект, к=
огда
у одной из катушек напряжение на зажимах отстает по фазе от тока (напряжени=
е
на рис. 3.4, б). Это имеет место, когда индуктивность кату=
шки
меньше величины взаимной индуктивности. В этом случае результирующая
индуктивность рассматриваемой катушки (с учетом взаимной индукции) отрицате=
льна:
. Для всей цепи такой эффект невозможен. Ее индуктивность всегда
положительна, и цепь носит активно-индуктивный характер.<=
/p>
Пример 3.2. В
схемах рис. 3.3 производились измерения напряжения, тока и активной мощност=
и на
входе цепи. При согласном (с) и встречном (в) включениях приборы дали следу=
ющие
показания:
220 В,
150 Вт,
234,5 Вт,
1 А,
1,25 А. Чему равна взаимная индуктивность катушек, если частота
питающего напряжения
50 Гц?
Р е ш=
е н и е. Определим полные, активные и реактивные
сопротивления обеих схем
Ом,&nbs=
p;
Ом,=
Ом,&nbs=
p;
Ом.=
При переключении кату=
шек с
согласного соединения на встречное общее активное сопротивление цепи не
меняется. Оно остается равным сумме их активных сопротивлений.=
span>
161 Ом,=
92,1 Ом.
Та=
к как
, а
, то сопротивление взаимной индукции
, а
=3D 0,0548 Гн =
=3D 54,8 мГн.
=
Пример
3.3. Катушки,
включенные по схеме рис. 3.3, б, имеют следующие параметры
18,5 Ом,
40 мГн,
10,3 Ом,
90 мГн,
48 мГн. Частота источника
50 Гц, его напряжение
220 В. Определить показания вольтметра, подключенного к зажимам
второй катушки.
Р е ш=
е н и е. Определим сопротивления элементов и полное
сопротивление цепи
314&nbs=
p; x 0,04 =3D =
span>12,6 Ом,
314 x 0,09 =3D 28,3 =
Ом,
314 x 0,048 =3D <=
/span>15,1 Ом,
28,8 Ом,
10,7 Ом=
,
30,7 Ом=
.
То=
к,
протекающий по цепи:
7,17 А. Сопротивление второй катушки с учетом взаимной индукции <=
v:shape
id=3D"_x0000_i1769" type=3D"#_x0000_t75" alt=3D"" style=3D'width:175.5pt;h=
eight:28.5pt'>
16,7 Ом; напряжение на ее зажимах
119,7 В.
Пример 3.4. П=
оследовательно
с двумя соединенными согласно катушками, индуктивности которых
90 мГн и
40 мГн, включен конденсатор емкостью
65 мкф. Частота источника
50 Гц. Взаимную индуктивность катушек можно менять, меняя их взаи=
мное
расположение (такие устройства, состоящие из двух взаимно перемещающихся
катушек, называются вариометрами). При какой магнитной связи в цепи наступит
резонанс напряжений?
Р е ш=
е н и е. Из условия резонанса напряжений (2.33) находим
индуктивность цепи при резонансе:<=
o:p>
0,156 Г=
н.
При
согласном включении катушек их общая индуктивность равна
, отсюда
(0,156–0,04–0,09)/2 =3D 0,026 Гн.
&nb=
sp;
Коэффициент магнитной связи
=
<=
b>3.3. Параллельное соединение
индуктивно связанных элементов
Пусть две индуктивно связанные катушки с параметрами=
и M=
соединены параллельно (рис. 3.5). =
Оба
вида соединения будем рассматривать одновременно. Сог=
ласное
соединение получается при подключении к одному и тому же узлу одноименных
зажимов, встречное - разноименн=
ых.
Первый случай отмечен на схеме
звездочками, второй - точками.
Запишем уравнения Кирхгофа для рассматриваемой цепи и=
решая их, получим выражения, определяющие токи:
|
<=
span
style=3D'font-family:Arial'>Рис. 3.5. Параллельное соединение индуктивно
связанных элементов
|
.
В =
этих
уравнениях
,
,
.
Вх=
одное
комплексное сопротивление цепи равно отношению напряжения к току на ее зажи=
мах:
.
При
отсутствии магнитной связи между катушками, полагая
, получаем известную формулу для определения общего сопротивления
двух параллельных ветвей:
Во всех приведенных выражениях у слагаемых с двойным знаком верхний знак
отн=
осится к
согласному соединению, нижний к встречному.
На рис. 3.6 представл=
ены
векторные диаграммы рассматриваемой цепи при согласном (а) и встречн=
ом (б)
соединениях катушек. При построении векторы
и
проводятся перпендикулярно току
, а векторы
и
перпендикулярно току
. При согласном соединении напряжения взаимной индукции опережают
соответствующие токи, при встречном отстают от них.
Рис. 3.6. Векторные диаграммы для параллельного соединен=
ия
индуктивно связанных элементов
<=
/span>
3.4. Разметка зажимов индуктивно связанных катушек
Ес=
ли
разметка осуществляется в процессе изготовления катушек, то одноименные заж=
имы
можно указать, проследив направления намотки. Для двух катушек это сделать
очень легко (рис. 3.7, а).
Рис. 3.7. Определение одноименных зажимов по характеру
намотки катушек=
p>
Поступаем следующим образом. Отмечаем один из зажимов первой катушки каким-=
либо
значком, например, звездочкой. Предположим, что это – начало обмотки. Напра=
вим
ток в ней от начала к концу и, пользуясь правилом правой руки, определяем
направление магнитного потока: правой рукой охватываем катушку так, чтобы
четыре пальца показали направление тока в ее витках, тогда отогнутый большой
палец покажет направление магнитного потока. Во второй катушке ток
направляем так, чтобы его магнитный поток имел то же самое направление. Заж=
им,
от которого ток уходит в катушку, также является началом. Его тоже отмечаем
звездочкой.
Бо=
лее
сложный случай показан на рис. 3.7, б. Оказывается, здесь невозможно=
указать
одноименные зажимы сразу для всех трех катушек. Приходится рассматривать их
попарно и действовать так, как только что описано. При этом, рассматривая
отдельно какую-то пару катушек, на стержень магнитопро=
вода
с третьей катушкой не обращаем внимания.
Рекомендуем самостоятельно проверить правильность разметки зажимов, выполне=
нной
на рис. 3.7, б.
В =
том
случае, когда направление намотки катушек неизвестно и установить его без
разрушения катушки невозможно, прибегают к помощи электроизмерительных
приборов.
Од=
ин из
возможных способов заключается в следующем. Обе катушки поочередно собирают=
в
схемы, показанные на рис. 3.8, и подключают к источнику синусоидального
напряжения одной и той же величины.
Рис. 3.8. Разметка зажимов с источником синусоидального
напряжения=
o:p>
Очевидно, что в одном случае получается согласное соединение, в другом –
встречное. Вид соединения определяем по показаниям амперметра. Вспомним, что
при согласном соединении катушек их полное сопротивление больше, а,
следовательно, при одной и той же величине входного напряжения ток меньше, =
чем
при встречном. А определив вид соединения, легко
делаем разметку: при согласном последовательном соединении катушки
подключаются друг к другу разноименными зажимами (начало второй к концу
первой). Если на рис. 3.8 при одинаковых показаниях вольтметров амперме=
тр
показывает 1,5 А в левой схеме и 1,1 А – в право=
й, то
слева имеем встречное соединение, справа согласное, и поэтому одноименными
зажимами являются первый и четвертый, а также второй и третий.=
span>
По=
кажем
еще один способ разметки. Первую катушку присоединяем через ключ к источнику
постоянного напряжения, например, к аккумуляторной батарее; к зажимам второй
катушки подключаем гальванометр (или вольтметр) магнитоэлектрической системы
(рис. 3.9, а).
Рис. 3.9. Разметка зажимов с помощью источника постоянно=
го
напряжения=
o:p>
За=
жим
первой катушки, подключаемый к положительному полюсу источника, помечаем
каким-нибудь способом, например, прикрепляем к нему бирку. Затем замыкаем к=
люч.
Если стрелка прибора при этом отбрасывается на шкалу, вешаем такую же бирку=
на
тот зажим второй катушки, который присоединен к положительной клемме прибора
(зажим 3). Если стрелка отклоняется влево, за пределы шкалы, то одноименным=
с
зажимом 1 является зажим 4.
С =
целью
теоретического обоснования метода проведем этот опыт с катушками, направлен=
ия
намотки и одноименные зажимы которых известны (рис. 3.9, б).
При
замыкании ключа в первой катушке возникает возрастающий по величине ток
, который создает магнитный поток
, также возрастающий по величине. Последний
индуцирует во второй катушке ЭДС электромагнитн=
ой
индукции. Создаваемый ею ток
возбуждает магнитный поток
, направление которого противоположно направлению
, так как п о п р и н
ц и п у Л е н ц а он должен
противодействовать его возрастанию. А магнитный поток такого направления
создается током, направление которого показано на схеме. Напоминаем, что
направления тока в катушке и создаваемого им магнитного потока связаны прав=
илом
правой руки. Ток
в рассматриваемой схеме протекает через гальванометр от его плюсо=
вой
клеммы к минусовой. При таком направлении тока ч=
ерез
прибор его стрелка отбрасывается на шкалу.
Результатом проведенных рассуждений является следующее практическое правило=
: если
в процессе эксперимента при замыкании ключа стрелка прибора
магнитоэлектрической системы отклоняется в сторону шкалы, то одноименными
являются зажимы, присоединенные к плюсу батареи и плюсу прибора.=
o:p>
3.5. Сложная цепь с взаимной индуктивностью=
a>=
Пу=
сть
задана двухконтурная цепь, содержащая индуктивно связанные элементы (рис.
3.10). Для ее расчета необходимо составить три (по числу неизвестных токов)
уравнения по законам Кирхгофа. Первое уравнение, для верхнего узла, затрудн=
ений
не вызывает:
Еще два уравнения напишем по второму закону Кирхгофа=
для
контуров, обозначенных закругленными стрелками
I и II<=
/span>, показывающими направление об=
хода
контура при написании уравнений. Но предварительно необходимо определить =
вид
включения катушек. Для каждой их пары одноименные зажимы отмечены своими
значками. Предположим, что это начала обмоток. Первая и вторая катушки,
одноименные зажимы которых отмечены звездочками, включены встречно, так к=
ак в
первой ток протекает от начала к концу, а во второй от конца к началу.
Поставим для памяти рядом со стрелкой
букву в (встречное
включение). У второй и третьей катушек начала обмоток обозначены точками.=
В
обеих катушках токи протекают одинаково относительно этих зажимов – от на=
чала
к концу, значит катушки включены согласно; ста=
вим
рядом со стрелкой букву с (согласное включение). Аналогично поступ=
аем
и с остальными катушками.
|
Рис.
3.10. Сложная цепь с
взаимной индуктивностью
|
Записываем уравнение =
для
первого контура:
 =
; &n=
bsp;
(3.4)
Дадим некоторые пояснения. Напряжение на зажимах катушки,
индуктивно связанной с другой катушкой, складывается из напряжения самоинду=
кции (
) и напряжения взаимной индукции (
). При согласном включении эти напряжения имеют одинаковые знаки,=
при
встречном – разные. Для лучшего восприятия индексы у буквы M=
span> поставлены так, чтобы они ука=
зывали
катушку, создающую магнитное поле (первый индекс), и катушку, в которой
наводится ЭДС (второй индекс). Например, обозна=
чение
показывает, что мы определяем влияние третьей катушки на вторую.
Рассмотрим составляющие напряжения на элементе
. В уравнении (3.4) они объединены фиг=
урной
скобкой
. Первое слагаемое
– это напряжение самоиндукции. Оно записано с минусом, так как при
обходе контура мы идем по этому элементу против тока. Второе слагаемое
– это напряжение, наведенное на зажимах второй катушки магнитным
потоком, создаваемым током первой катушки. Его знак (плюс) из-за встречного
включения противоположен знаку напряжения самоиндукции. Напряжение, которое
наводится во второй катушке со стороны третьей (
), имеет тот же знак (минус), что и напряжение самоиндукци=
и,
так как вторая и третья катушки соединены согласно.
Пр=
иводим
уравнение, записанное для второго контура:
–
&=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
(3.4)
<=
span
style=3D'font-size:13.5pt;font-family:Arial'>
3.6. Эквивалентная замена
индуктивных связей
Им=
еется
возможность избежать составления таких сложных уравнений, как в предыдущем
подразделе. Для этого нужно произвести так называемую развязку электрической
цепи, заменив схему с индуктивно связанными элементами эквивалентной схемой=
без
индуктивных связей. Делается это по следующему правилу: если два элемент=
а
и
, имеющие взаимную индуктивность
, присоединены к узлу электрической цепи одноименными зажимами, то
при переходе к эквивалентной схеме к этим элементам добавляется –M, а в третью, отходящую от узл=
а,
ветвь включается M
(рис. 3.11, а).
Ес=
ли
характер подключения катушек меняется, т.е. они присоединяются к узлу
разноименными зажимами, то в эквивалентной схеме знак перед M меняется на противоположный (рис. 3.11, б).=
Для
доказательства приведенных утверждений необходимо в каждой паре схем
произвольно указать направления токов (одинаковые для одной и той же ветви)=
и
записать выражения напряжений
и
. Для обеих схем они оказываются одинаковыми, что подтверждает их
эквивалентность.
а)
б)
Рис. 3.11. Эквивалентная замена индуктивных связей
Пример 3.5.
Требуется произвести развязку схемы, изображенной на рис. 3.10.<=
/span>
Р е ш=
е н и е. Ограничимся рассмотрением только верхнего узла.
Пусть катушки, присоединенные к нему, имеют следующие параметры:
После развязки схема приобретает вид, показанный на =
рис.
3.12.
Эквивалентные индуктивности ветвей
определяем с использованием сформулированного правила:
|
=
Рис. 3.12. Узел электрической цепи после разв=
язки
|
=
o:p>
Зн=
ак
минус перед последней индуктивностью говорит о том, что для физической
реализации эквивалентной схемы в третью ветвь необходимо включить не катушк=
у, а
конденсатор, емкостное сопротивление которого равно индуктивному сопротивле=
нию
эквивалентной третьей ветви.
Пример 3.6. П=
ри
какой частоте в цепи, представленной на рис. 3.13, а, наступит резон=
анс
напряжений? Числовые значения параметров цепи:
а)
&nb=
sp; =
&nb=
sp; =
б)
Рис. 3.13. Схема до (а) и после (б) развяз=
ки
Р е ш=
е н и е. Заменяем заданную схему эквивалентной без магн=
итной
связи (рис. 3.13, б) и находим ее индуктивность:
.
Из
условия резонанса напряжений (w L =3D =
) определяем искомую частоту:
Гц.
<=
/span>
3.7. Трансформатор без стально=
го
сердечника
Простейший трансформатор представляет собой совокупность двух обмоток,
размещенных на общем магнитопроводе (рис. 3.14,=
а).
а)
б)
Рис. 3.14. Трансформатор и его схема замещения
К =
его
первичной обмотке подводится напряжение источника питания, а ко
вторичной – подключается нагрузка. Одноименными зажимами обмоток являются их
верхние выводы. Ток первичной обмотки
создает в магнитопроводе магнитный по=
ток
, который в свою очередь во вторичной обмотке вызывает появление =
тока
. Создаваемый им магнитный поток
в соответствии с принципом Ленца препятствует потоку
, т.е. направлен ему навстречу. Направление тока
, соответствующее показанному на схеме потоку
, определяем по правилу правой руки.
Мы=
будем
рассматривать трансформатор, не имеющий ферромагнитного сердечника. Такие
трансформаторы применяются при высоких частотах и в специальных
электроизмерительных устройствах. Катушки с ферромагнитными сердечниками им=
еют
нелинейные характеристики и здесь не рассматриваются.
Электрическая схема замещения трансформатора изображена на рис. 3.14, б<=
/i>.
На схеме указаны:
и
– сопротивления первичной и вторичной обмоток трансформатора,
и
– сопротивления нагрузки. Введем обозначения:
и
– суммарные активное и реактивное сопротивления вторичной цепи
трансформатора,
,
,
,
– компл=
ексные
сопротивления соответствующих участков.
За=
пишем
уравнения второго закона Кирхгофа для первичной и вторичной цепей
трансформатора, учитывая, что его обмотки имеют встречное включение:=
o:p>
 =
; &n=
bsp;
(3.5)
или=
 =
; &n=
bsp;  =
; &n=
bsp;  =
;
(3.6)=
span>
Обозначив
, второе уравнение системы (3.5) можно записать так:=
span>
 =
; &n=
bsp;  =
; &n=
bsp;  =
;
(3.7)=
span>
Физически
– это ЭДС, котора=
я
наводится во вторичной обмотке переменным магнитным полем первичной обмотки=
. С
учетом этого уравнение (3.7) можно прочитать так: ЭДС<=
/span>,
наведенная во вторичной обмотке трансформатора, =
равна
сумме падений напряжений на всех элементах его вторичного контура. Подставл=
яя в
(3.7)
, получим:
. Смысл последнего уравнения заключается в
следующем: напряжение на вторичных зажимах трансформатора меньше эдс, наведенной во вторичной обмотке, на величину пад=
ения
напряжения на ее сопротивлении.
На рис. 3.15 изображе=
на
векторная диаграмма трансформатора. Ее построение начинаем со
вторичного тока
. Ориентируясь на его направление, проводим векторы напряжений на
всех элементах вторичной цепи. Их сумма равна ЭДС
. Так как в формуле, определяющей ее величину, присутствует множи=
тель
, поворачивающий вектор на четверть оборота, то ток
проводим под углом 90° к
в сторону отставания. Определив направление
, строим векторы
и
, которые в сумме с –
дают
.
Рис. 3.15. Векторная диаграмма трансформатора
Для
анализа работы трансформатора применяют различные эквивалентные схемы.
Рассмотрим некоторые из них.
Со=
единив
между собой два нижних зажима трансформатора (режим его работы при этом не
изменится) и произведя развязку индуктивных связ=
ей,
придём к Т-образной эквивалентной схеме (рис. 3.16=
).
Рис. 3.16. Получение двухконтурной (Т-образной)
эквивалентной схемы трансформатора=
Из второ=
го
уравнения системы (3.6) выразим ток
и подставим в первое уравнение той же системы:<=
/p>
&nb=
sp;
. (3.8)
Последнему выражению соответствует схема, изображенная на рис. 3.17. Соедин=
енное
последовательно с
сопротивление
называется вносимым (из вторичной цепи трансформатора в первичную).
Как
следует из формулы (3.8), оно равно
Его
активная и реактивная составляющие соответственно равны:<=
/p>
Те=
перь
полное входное сопротивление цепи будет складываться из сопротивления
первичного контура и вносимого сопротивления:
&nb=
sp;
.
Появление в первичном контуре активного сопротивления, вносимого из первичн=
ого
контура, физически означает следующее. Энергия, подводимая к трансформатору,
потребляется не только сопротивлением
, но и сопротивлениями вторичной цепи
и
, куда она передается через переменное магнитное поле между
обмотками.
Из=
-за
минуса в формуле вносимого реактивного сопротивления общее реактивное
сопротивление всей цепи, равное сумме
и
, оказывается меньше индуктивного сопротивления первичной обмотки=
.
Это
хорошо согласуется со сказанным ранее. При встречном соединении обмоток
трансформатора поток
, направленный противоположно потоку
, уменьшает последний, что приводит к уменьшению общего индуктивн=
ого
сопротивления.
СПИСОК ЛИТЕРАТУРЫ
1.=
Зевеке Г.В., Ионкин П.А.,=
Нетушил А.В., Страхов С.В. Основы теории цепей: Учебн=
ик для
вузов. – М.: Энергия, 1989. – 528 с.
2.=
Попов
В.П. Основы теории цепей: Учебник для вузов. – М.: Выс=
ш.
шк., 1998. – 575 с.
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