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Переходные процессы в линейных электрических цепях (конспект)

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18

МОСКОВСКИЙ АВИАЦИОННЫЙ ИНСТИТУТ

(государственный технический университет)

ФИЛИАЛ «ВЗЛЕТ»

Кафедра РЭВС

РАЛДЫГИН И.К.

КОНСПЕКТ ЛЕКЦИЙ ПО КУРСУ

«Основы теории цепей». Часть 3.

Переходные процессы в линейных электрических цепях

Учебное пособие для студентов радиотехнической специальности.

Ахтубинск – 2004

Предисловие к 3-ей части

В третьей части конспекта по Основам теории цепей (ОТЦ) кратко изложены
два метода расчета переходных процессов в линейных электрических цепях:
Классический метод и Операторный метод. И классический и операторный
методы расчета теоретически можно применять для решения задач любой
сложности. Каким из них пользоваться определяется автором.

Однако классический метод физически более прозрачен, чем операторный, в
котором решение уравнений во многом формализовано. Операторный метод
имеет перед классическим явное преимущество если для расчета переходных
процессов использовать прикладную программу Mathcad 2000, особенно в тех
случаях, когда воздействующее напряжение является линейно возрастающее
или в виде всплеска одной или несколько экспонент.

Глава 1. Основные сведения о переходных процессах в линейных
электрических цепях

1.1 Возникновение и общая характеристика переходных процессов

Выше рассматривались цепи, в которых выполнялись два условия: 1)
Источники энергии были подключены к цепи теоретически бесконечно давно;
2) Никаких изменений в состоянии цепи не производилось. Такой режим
работы цепи называется установившимся или принужденным.

Переходным называют электромагнитный процесс, возникающий в
электрической цепи, при переходе от одного установившегося режима к
другому. Этот переход может происходить вследствие преднамеренного или
случайного отключения цепи, а также подключения ее под напряжение,
вследствие обрыва или короткого замыкания в цепи.

Любые изменения в цепи можно представить в виде переключений, которые
называются коммутацией. Коммутация на схемах обозначается в виде ключа
со стрелкой, обозначающей замыкание или размыкание:

Теоретически считается, что коммутация производится мгновенно.

Установившийся режим работы цепи при заданных и неизменных ее параметрах
полностью определяется источником энергии: постоянный ток, переменный
ток.

После коммутации, т.е. во время переходного процесса, токи и напряжения
в цепи определяются не только внешними, но и внутренними источниками
энергии, в качестве которых выступают индуктивности и емкости.

Дело в том, что в режиме, который существовал до коммутации, в катушках
и конденсаторах было накоплено определенное количество энергии:

В момент коммутации (t=0) начинается перераспределение энергии между
внутренними накопителями и внешними источниками; при этом часть энергии
необратимо преобразуется в тепло.

По истечении какого-то времени после коммутации в цепи установится новый
режим, который будет обусловлен только внешними источниками энергии. При
отключении цепи от внешних источников переходной процесс будет
существовать только за счет энергии накопленной в индуктивностях и
емкостях, т.е. только за счет внутренней энергии. Новый установившийся
режим, в этом случае, будет характеризоваться отсутствием тока в цепи.

Задача анализа переходного процесса заключается в том, чтобы установить
по какому закону и как долго будет происходить переход от одного режима
к другому.

В соответствии с законом непрерывности энергии напряжение на емкости и
ток через индуктивность не могут изменяться скачком, т.к. в этом случае
мощность, равная скорости изменения энергии обращалась бы в
бесконечность, что физически невозможно.

На основании изложенного сформулированы два закона коммутации:

· Заряд и напряжение на емкости в момент коммутации остаются такими же,
какими они были непосредственно перед коммутацией, а затем плавно
изменяются;

· Ток через индуктивность в момент коммутации остается таким же, каким
он был непосредственно перед коммуникацией, а затем плавно изменяется.

Математически законы коммутации записываются в следующем виде:

Ток через емкость и напряжение на индуктивности могут изменяться
скачком.

1.2 Начальные условия

Значения токов, напряжений и их производных в момент коммутации называют
начальными условиями.

Начальные условия подразделяются на зависимые и независимые.

Напряжение на емкости и ток через индуктивность, непосредственно перед
коммутацией, называются независимыми начальными условиями, т.к. их
значения не зависят от вида и места коммутации и определяются только
энергетическим состоянием цепи непосредственно перед коммутацией.

Зависимыми начальными условиями являются токи через емкость и напряжение
на индуктивности в момент коммутации. Они зависит от вида и места
коммутации и в общем случае, в момент коммутации, могут изменяться
скачком.

Независимые начальные условия определяются в цепи до коммутации, а
зависимые начальные условия определяются в цепи образовавшейся в момент
коммутации.

1.3 Математические основы анализа переходных процессов

Электромагнитные процессы в линейных электрических цепях в
установившемся режиме описываются законами Кирхгофа для мгновенных или
комплексных значений токов и напряжений.

Для определения законов изменения токов и напряжений в переходном режиме
необходимо линейные уравнения, составленные по законам Кирхгофа для
мгновенных значений, записать в виде дифференциальных уравнений, а затем
решить эти уравнения относительно искомых токов и напряжений.

Таким образом, анализ переходных процессов сводится к решению
обыкновенных дифференциальных уравнений с правой частью.

На практике для решения дифференциальных уравнений применяют
классический и операторный методы расчета.

Суть классического метода расчета рассмотрим на конкретном примере.

Пусть задана электрическая цепь из последовательно соединенных RC
элементов Рис.1.1.

Рис. 1.1. Цепь RC в момент t=0 при нулевых начальных условиях, Uc(0)=0,
подключается к источнику постоянного напряжения

Определим, в общем виде, законы изменения напряжения на емкости и ток в
цепи после коммутации.

Решение. При анализе переходных процессов классическим методом
необходимо составить уравнения по законам Кирхгофа для мгновенных
значений токов и напряжений. В данном случае можно составить только одно
уравнение по второму закону Кирхгофа:

(1.1)

Поскольку исходное алгебраическое уравнение (1.1) можно записать в
дифференциальной форме:

(1.2)

При анализе переходных процессов в качестве искомых функций могут
выступать токи и напряжения на пассивных элементах цепи, образовавшейся
после коммутации. Для общности обозначений, принятых в математике,
условимся в дальнейшем искомую функцию обозначать Y(t).

Тогда дифференциальное уравнение (1.2) можно записать в виде:

(1.3)

где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 – постоянная времени.

Из курса математики известно, что полное решение дифференциального
уравнения (1.3) представляется в виде суммы двух составляющих:

(1.4)

где Yпр(t)- принужденная составляющая;

Yсв(t) – свободная составляющая.

Принужденная составляющая искомого тока или напряжения определяется из
анализа установившегося режима в цепи, образовавшейся после коммутации,
для чего применяются любые известные методы расчета: по законам
Кирхгофа, методом контурных токов и др.

Таким образом, принужденная составляющая зависит от вида источника
напряжения и параметров цепи, образовавшейся после коммутации.

Свободная составляющая искомого тока или напряжения представляет собой
решение уравнения (1.3) без правой части:

(1.5)

а именно:

(1.6)

где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 –
постоянная интегрирования, которая зависит от начальных условий;
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 –
корень характеристического уравнения

(1.7)

Свободная составляющая является результатом действия внутренних
источников энергии, когда они не уравновешены внешними источниками.
Свободная составляющая с течением времени затухает и в пределе стремится
к нулю.

Свободная составляющая не зависит от вида воздействующих внешних
источников энергии, и ее характер определяется только свойствами цепи,
образовавшейся после коммутации.

Таким образом, закон изменения искомого тока или напряжения в переходном
режиме определяется двумя факторами: свойствами цепи образовавшейся
после коммутации и приложенным напряжением.

Глава 2. Переходные процессы в цепях первого порядка

2.1 Общий алгоритм расчета переходных процессов в цепях первого порядка
классическим методом

Электрической цепью первого порядка называется цепь, которая включает в
себя только один накопитель энергии (индуктивности или емкость) или
сколько угодно накопителей одного характера, но которые могут быть
заменены одним эквивалентным.

На основании вышеизложенного составлен алгоритм расчета переходных
процессов в цепях первого порядка, который сводится к выполнению
следующих операций.

1. Расчет независимых начальных условий производится в цепи
непосредственно перед коммутацией, в результате чего определяются
значения напряжений на емкости и ток через индуктивность в момент
коммутации:

Независимые начальные условия бывают нулевые, когда UC(0) = 0, iL(0) = 0
и ненулевые, когда UC(0) 0, iL(0) 0.

2. Расчет зависимых начальных условий производится в цепи, которая
образовалась после коммутации. Для этого необходимо составить уравнения
по законам Кирхгофа и рассмотреть их на момент коммутации t=0. После
этого определяются зависимые начальные условия, например, для цепи
Рис.1.1:

3. Расчет принужденных составляющих производится в цепи, которая
образовалась после коммутации, используя известные методы расчета
установившихся процессов. В результате определяются принужденные
составляющие искомых токов и напряжений Yпр(t).

Например, для цепи Рис.1.1:

4. Составление характеристического уравнения и определение его корня.
Для решения дифференциального уравнения первого порядка, например (1.5),
необходимо составить характеристическое уравнение первого порядка и
найти его корень.

Характеристическое уравнение можно составить двумя способами, например,
для цепи Рис.1.1:

· либо путем формальной замены оператора дифференцирования оператором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· либо для цепи, образовавшейся после коммутации, составить комплексное
входное сопротивление, а затем путем формальной замены j*=P получить
операторное сопротивление, которое приравнять к нулю и найти корень
этого уравнения.

Например, для цепи Рис.1.1:

5. Определение свободных составляющих искомых токов и напряжений.

Свободные составляющие всех токов и напряжение в цепях первого порядка
представляют собой решение дифференциального уравнения без правой части
(1.5) и записываются в виде:
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(2.1)

Например, для цепи Рис.1.1 свободные составляющие для тока в цепи и
напряжения на емкости имеют вид:

Они отличаются друг от друга только постоянными интегрирования.

6. Расчет постоянных интегрирования. Для расчета постоянных
интегрирования, входящих в (2.1), необходимо составить полное решение
дифференциального уравнения и рассмотреть его на момент коммутации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:

(2.2)

При t=0 имеем:

(2.3)

Например, для цепи Рис.1.1:

7. Запись полного решения дифференциального уравнения (2.3):

Для цепи Рис.1.1 получим:

(2.4)

2.2 Переходные процессы в цепи RC при подключении ее к источнику
постоянного напряжения и коротком замыкании

Рассмотрим электрическую цепи, изображенную на Рис.2.1, которая в момент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, при нулевых
начальных условиях UC(0)=0, подключается к источнику постоянного
напряжения E, а затем в момент t10 в цепи происходит короткое замыкание,
(ключ K2 замыкается, а ключ K1 размыкается).
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Рис. 2.1. Цепь RC при подключении к источнику постоянного напряжения
(ключ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 замыкается, ключ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 – разомкнут) и коротком замыкании (ключ
picscalex100010009000003c900000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02200200021200000026060f001a00ffffffff000010000000c0ffffff
a6ffffffc0010000c60100000b00000026060f000c004d61746854797065000050001c00
0000fb0220ff0000000000009001000000000402001054696d6573204e657720526f6d61
6e004c53f5775553f5770100000000003000040000002d01000008000000320ae0015901
0100000031791c000000fb0280fe0000000000009001010000000402001054696d657320
4e657720526f6d616e004c53f5775553f5770100000000003000040000002d0101000400
0000f001000008000000320a80014600010000004b790a00000026060f000a00ffffffff
0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d
000000000a000000040000000000000000000100000000003000040000002d0100000400
0000f0010100030000000000 – размыкается, а ключ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 – замыкается).

Определим законы изменения напряжений на емкости и на резисторе после
первой и второй коммутаций.

Законы изменения напряжений на емкости и резисторе после первой
коммутации получены при изложении алгоритма расчета переходных процессов
в цепях первого порядка (2.4).

Определим теперь эти законы после второй коммутации, полагая, что вторая
коммутация произошла в момент t0.

Фактически требуется рассчитать переходные процессы в цепи RC при
ненулевых начальных условиях.

Следуя принятому алгоритму, получим следующее:

1. Независимые начальные условия. Напряжение на емкости в момент t1
будет:

2. Зависимые начальные условия определим из уравнения, составленного по
второму закону Кирхгофа, для цепи образовавшейся после второй
коммутации:

3. Принужденные составляющие после второй коммутации будут равны нулю,
т.к. в цепи нет источника напряжения:

4. Характеристическое уравнение остается таким же как после первой
коммутации:

5. Свободные составляющие не зависят от входного напряжения, поэтому
определяются по формуле (2.1).

6. Постоянные интегрирования определяются по формуле (2.3).

7. Законы изменения напряжений на емкости и на резисторе после второй
коммутации принимают вид:

Из формулы (2.4) видно, что нарастание напряжения на емкости происходит
тем быстрее, чем меньше постоянная времени. Для различных моментов
времени напряжение на емкости, отнесенное к входному напряжению,
характеризуется следующими данными (Таблица 2.1).

Таблица 2.1.
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,630,860,9509850,993

Переходной процесс теоретически продолжается бесконечно долго. Однако,
как видно из приведенной таблицы, в цепях первого порядка он
заканчивается через время, равное (4…5)* после коммутации. В связи с
этим принято считать, что длительность переходного процесса в цепях
первого порядка составляет

В заключение отметим, что описанные выше переходные процессы возникают
при подключении цепи RC к одиночному прямоугольному импульсу напряжением
E и длительностью t1. В связи с этим характер изменения напряжений на
емкости и резисторе будет определяться соотношением между постоянной
времени цепи и длительностью импульса.

Ниже приведен пример расчета переходных процессов в цепи RC по
программе Mathcad (Рис.2.2), а также пример электронного моделирования
работы этой цепи по программе Electronics Workbench (Рис.2.3 и Рис.2.4).
Из этих рисунков видно, что результаты расчетов и результаты
моделирования практически совпадают.

2.3 Переходные процессы в цепи RC при подключении ее к источнику
синусоидального напряжения

Рассмотрим цепи RC Рис.1.1, которая при нулевых начальных условиях
UC(0)=0 подключается к источнику синусоидального напряжения

Определим для этой цепи закон изменения напряжения на емкости UC(t)
после коммутации, применив вышеприведенный алгоритм.

1. Независимые начальные условия UC(0)=0.

2. Зависимые начальные условия

На момент коммутации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, получим

3. Амплитуда принужденной составляющей напряжения на емкости
определяется по общему правилу расчета одноконтурных цепей.

Определим модуль входного сопротивления

и его аргумент

Определяем комплексную амплитуду тока в цепи в установившемся режиме

Определим комплексную амплитуду напряжения на емкости

Теперь можно записать принужденную составляющую напряжения на емкости

4.5. Характеристическое уравнение и его корень, а также свободная
составляющая не зависят от вида входного напряжения и определяются по
ранее приведенным формулам

5. Постоянная интегрирования:

6. Закон изменения напряжения на емкости принимает следующий вид:

Ниже приведен пример 2.2 расчета переходных процессов в цепи RC при
подключении ее к источнику синусоидального напряжения при нулевых
начальных условиях Рис.2.5. На Рис.2.6 приведены результаты электронного
моделирования этой цепи при синусоидальном воздействии.

Из этих рисунков видно, что результаты расчетов по программе Mathcad
(Рис.2.5) и результаты электронного моделирования по программе
Elecrronics Workbench (Рис.2.6) практически совпадают.

В первый полупериод после коммутации напряжение на емкости в 1,7 раза
больше принужденной составляющей, что необходимо учитывать при выборе
пробивного напряжения на конденсаторе.

2.4 Переходные процессы в цепи RL при подключении ее к источнику
постоянного напряжения и коротком замыкании

Рассмотрим электрическую цепь, состоящую из последовательно соединенных
RL элементов Рис.2.7.
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Рис. 2.7. Цепь RL в момент t=0 при нулевых начальных условиях i(0)=0
подключается к источнику постоянного напряжения; в момент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 ключ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 размыкается, а ключ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 замыкается.

Определим законы изменения напряжений на резисторе и индуктивности после
первой коммутации и после второй.

Вначале определим U2(t) , UL(t) после первой коммутации: K1 – замкнут,
K2 – разомкнут.

1. Независимые начальные условия

2. Зависимые начальные условия

На момент первой коммутации имеем

Отсюда следует, что индуктивность в момент коммутации представляет собой
разрыв цепи. До коммутации напряжение на индуктивности было равно нулю,
а в момент коммутации оно скачком принимает значение, равное входному
напряжению.

3. Принужденные составляющие.

В установившемся режиме в цепи будет протекать постоянный ток, при
котором индуктивное сопротивление равно нулю и поэтому

4. Характеристическое уравнение и его корень

где – постоянная времени цепи RL.

5. Свободные составляющие

6. Постоянные интегрирования

7. Законы изменения напряжений на индуктивности и резисторе после первой
коммутации:

Переходные процессы в цепи RL при коротком замыкании (после второй
коммутации).

Переходные процессы в цепи RL Рис.2.7 происходят при ненулевых начальных
условиях.

1. Независимые начальные условия на момент t1>0:

2. Зависимые начальные условия

3. Принужденные составляющие.

В цепи после второй коммутации нет источников напряжения, поэтому

4. Характеристическое уравнение и его корень после второй коммутации
такие же, как после первой:

5. Свободные составляющие

6. Постоянные интегрирования

7. Законы изменения напряжений на индуктивности и резисторе после второй
коммутации:

Ниже приведен пример расчета переходных процессов в цепи RL, выполненный
по программе Mathcad (Рис.2.8), а также результат электронного
моделирования переходных процессов (Рис.2.9), который получен по
программе Electronics Workbench (EWB). Из анализа Рис.2.8 и Рис.2.9
видно, что кривые U2(t) и UL(t) первого рисунка практически совпадают с
одноименными кривыми второго рисунка.

2.5 Подключение цепи RL к источнику синусоидального напряжения

Пусть цепь RL Рис. 2.7. при нулевых начальных условиях подключается к
источнику синусоидального напряжения с начальной фазой не равной нулю:

Определим закон изменения тока в цепи после коммутации.

1. Независимые начальные условия i(0)=0.

2. Зависимые начальные условия

На момент коммутации t=0 имеем

Отсюда получаем:

3. Принужденная составляющая тока в цепи:

где

Отсюда амплитуда принужденной составляющей тока

Мгновенное значение принужденной составляющей тока

4.5. Характеристическое уравнение и его корень, а также свободная
составляющая тока не зависят от вида входного напряжения и определяются
по ранее приведенным формулам:

6. Постоянная интегрирования

7. Закон изменения тока в цепи RL при подключении ее к источнику
синусоидального напряжения

Ниже приведен пример 2.4 расчета переходных процессов в цепи RL при
подключении ее к источнику синусоидального напряжения.

Из приведенных формул видно, что при подключении цепи RL к источнику
синусоидального напряжения ток в переходном режиме содержит две
составляющие: синусоиду и экспоненту и его значение, в первый момент
после коммутации, зависит от фазы включения.

Если включение произошло в момент, когда =, то свободная составляющая
будет отсутствовать и в цепи сразу будет установившийся режим (удачное
включение). Наоборот, неудачное включение имеет место, когда начальная
фаза входного напряжения будет = 90.

Если при этом постоянная времени велика, то в начальный момент после
коммутации ток переходного режима может достигнуть почти удвоенной
амплитуды принужденной составляющей, что наглядно показано на Рис.2.10,
где ток переходного режима в 1,66 раза больше амплитуды принужденной
составляющей.

2.6 Синтез цепи RC с заданными параметрами переходного процесса

Выше рассматривались переходные процессы при заданных параметрах
RC-элементов.

На практике возникает необходимость в решении обратной задачи:
рассчитать потребные значения RC-элементов, при которых обеспечивалась
бы заданная длительность переходного процесса и заданное значение
выходного напряжения, снимаемого с резистора.

Рассмотрим электрическую цепь, изображенную на Рис.2.11.

Рис. 2.11. Исходная схема для расчета неизвестных R1 и C

На вход этой цепи подается последовательность однополярных прямоугольных
импульсов напряжением E B, длительностью ti, с частотой f и скважностью
2.

Для синхронного управления ключевыми схемами с помощью цепи Рис.2.11
необходимо сформировать остроконечные импульсы, длительность которых
была бы во много раз меньше длительности прямоугольных импульсов на
входе цепи. Другими словами, заряд и разряд конденсатора должен
происходить за время много меньше, чем длительность прямоугольного
импульса

где K< Рис.2.11.

При этом заданными величинами должны быть: сопротивление нагрузки R2, с
которого снимается напряжение и максимальное значение U2 этого
напряжение (U2<E). Решение. Из условия задачи следует, что необходимо определить два параметра: R1- балластное (регулировочное) сопротивление; C- емкость конденсатора цепи Рис.2.11. В соответствии с принятым алгоритмом расчета переходных процессов в цепях первого порядка получаем следующее. 1. Независимые начальные условия: где U2 – потребное значение напряжения на нагрузке в момент коммутации. 2. Зависимые начальные условия. К зависимым начальным условиям, в данном случае, относится напряжение на балластном сопротивление picscalex1000100090000034d01000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c022002c0031200000026060f001a00ffffffff000010000000c0ffffff a6ffffff80030000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0205fee300000000009001000000020002001053796d626f6c0077400000008106 0a464c53f5775553f5770100000000003000040000002d01000008000000320a7e01e101 0100000028791c000000fb0205fee300000000009001000000020002001053796d626f6c 007740000000ba060ab94c53f5775553f5770100000000003000040000002d0101000400 0000f001000008000000320a7e011f030100000029791c000000fb0280fe000000000000 9001000000000402001054696d6573204e657720526f6d616e004c53f5775553f5770100 000000003000040000002d01000004000000f001010008000000320a8001530201000000 30791c000000fb0220ff0000000000009001000000000402001054696d6573204e657720 526f6d616e004c53f5775553f5770100000000003000040000002d01010004000000f001 000008000000320ae00147010100000031791c000000fb0280fe00000000000090010100 00000402001054696d6573204e657720526f6d616e004c53f5775553f577010000000000 3000040000002d01000004000000f001010008000000320a800116000100000055790a00 000026060f000a00ffffffff0100000000001c000000fb021000070000000000bc020000 00cc0102022253797374656d000000000a00000004000000000001000000010000000000 3000040000002d01010004000000f0010000030000000000. Для определения этого напряжения составим уравнение по второму закону Кирхгофа и рассмотрим его на момент коммутации (на момент воздействия первого импульса): На момент воздействия первого импульса t=0 имеем Поскольку UC(0)=0, U2(0)=U2, следовательно, U1(0) должно быть равным: С другой стороны на момент t=0 можно записать Разделив левые и правые части этих соотношений, получим: Для определения потребного значения емкости примем во внимание, что длительность переходного процесса в спроектированной цепи должно составлять не более Это означает, что должно выполняться равенство: 5*=K*ti или 5*(R1+R2)*C=K*ti. Отсюда получаем потребное значение емкости Расчет законов изменения напряжений на емкости и резисторах R1, R2 в цепи Рис.2.11 можно производить по формулам, приведенным в примере 2.1. Однако, ограничимся расчетом потребных значений R1, C, но выполним электронное моделирование спроектированной цепи. Результаты расчетов и электронного моделирования приведены в примере 2.5. Из Рис.2.12, где приведены результаты электронного моделирования, видно, что для спроектированной цепи RC переходной режим является «штатным». Данную цепь можно рассматривать как генератор остроконечных импульсов или преобразователь напряжения: прямоугольные импульсы преобразуются в остроконечные. Глава 3. Переходные процессы в цепях второго порядка 3.1 Общая характеристика переходных процессов в цепях второго порядка Цепями второго порядка называются цепи, в которых содержится два накопителя энергии: индуктивность и емкость. Электрические цепи второго порядка бывают разветвленными и неразветвленными. К неразветвленным цепям второго порядка относится последовательный колебательный контур. К разветвленным цепям второго порядка относятся Г-образные фильтры нижних и верхних частот. Электромагнитные процессы в цепях второго порядка описываются дифференциальными уравнениями второго порядка. Например, дифференциальное уравнение относительно тока в последовательном колебательном контуре можно получить из уравнения, составленного по второму закону Кирхгофа, для мгновенных значений тока и напряжений: После дифференцирования (3.2) получим дифференциальное уравнение второго порядка относительно тока Обозначим, как и ранее, искомый ток i(t) через Y и разделим левую и правую части (3.3) на L, получим дифференциальное уравнение второго порядка: где – коэффициент затухания; – резонансная частота контура; – правая часть дифференциального уравнения (3.3). Из курса математики известно, что решение дифференциального уравнения второго порядка, также как и первого, представляется в виде двух слагаемых: где Yпр(t) – принужденная составляющая искомого тока или напряжения, которая зависит от вида источника напряжения, оставшегося в цепи после коммутации; Yсв(t) – свободная составляющая, характер которой определяется только структурой цепи образовавшейся после коммутации. Для решения дифференциального уравнения второго порядка, также как и первого, и по тем же правилам, составляется характеристическое уравнение, которое, в общем, имеет вид: Корни этого уравнения: где . Свободная составляющая искомого тока или напряжения записывается в виде: (3.8) где A1, A2 – неизвестные постоянные интегрирования, которые зависят от начальных условий. Допустим, что для цепи второго порядка составлено характеристическое уравнение (3.6) и определены его корни (3.7), а также найдены принужденные составляющие искомых токов и напряжений. Тогда общее решение дифференциального уравнения можно записать в виде: . (3.9) Для определения двух неизвестных picscalex100010009000003c900000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c022002c0011200000026060f001a00ffffffff000010000000c0ffffff a6ffffff80010000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0220ff0000000000009001000000000402001054696d6573204e657720526f6d61 6e004c53f5775553f5770100000000003000040000002d01000008000000320ae0012501 0100000031791c000000fb0280fe0000000000009001010000000402001054696d657320 4e657720526f6d616e004c53f5775553f5770100000000003000040000002d0101000400 0000f001000008000000320a800158000100000041790a00000026060f000a00ffffffff 0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d 000000000a000000040000000000000000000100000000003000040000002d0100000400 0000f0010100030000000000 и picscalex100010009000003c900000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02200200021200000026060f001a00ffffffff000010000000c0ffffff a6ffffffc0010000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0220ff0000000000009001000000000402001054696d6573204e657720526f6d61 6e004c53f5775553f5770100000000003000040000002d01000008000000320ae0013c01 0100000032791c000000fb0280fe0000000000009001010000000402001054696d657320 4e657720526f6d616e004c53f5775553f5770100000000003000040000002d0101000400 0000f001000008000000320a800158000100000041790a00000026060f000a00ffffffff 0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d 000000000a000000040000000000000000000100000000003000040000002d0100000400 0000f0010100030000000000 необходимо составить два уравнения. В качестве первого уравнения используется уравнение (3.9), а в качестве второго – используется первая производная от (3.9). Рассматривая (3.9) и (3.10) на момент t=0, получим два уравнения с двумя неизвестными A1 и A2: Совместное решение системы (3.11) дает: После подстановки корней характеристического уравнения (3.6) и найденных постоянных интегрирования (3.12) в формулу общего решения (3.9) и преобразования, получим закон изменения искомого тока или напряжения: (3.13) где a=Y(0)-Yпр(0); b=Y’(0)-Y’пр(0) – постоянные коэффициенты, зависящие от начальных условий. Отметим, что формула (3.13) является общей для всех токов и напряжений в данной цепи второго порядка. При этом меняются только коэффициенты a и b и их размерность. При решении конкретных задач могут представиться три случая. Случай 1. Корни характеристического уравнения (3.7) действительные и разные, что возможно при >k.

В этом случае после коммутации в цепи возникает апериодический режим,
при котором все токи и напряжения изменяются плавно, без колебаний, а их
законы описываются уравнением (3.13).

Случай 2. Корни характеристического уравнения (3.7) действительные и
одинаковые, что возможно при =k.

В этом случае в цепи после коммутации возникает критический режим.

При =k , =2-k2 =0, P1=P2=-, формула (3.13) приобретает другой вид,
поскольку

Подстановка полученных результатов в (3.13) дает формулу для расчета
законов изменения токов и напряжений в критическом режиме:

В критическом режиме токи и напряжения, как видно из (3.14), изменяются
также плавно, как в апериодическом режиме.

Критический режим лежит на границе между апериодическим и колебательным,
к рассмотрению которого приступаем.

Случай 3. Корни характеристического уравнения (3.7) комплексные
сопряженные, что возможно при < k.

В этом случае в цепи после коммутации возникает колебательный режим.

При < k имеем:

где – частота свободных колебаний.

Корни характеристического уравнения в этом случае принимают вид:

P1 = – + j* св; P2 = – – j* св.

Если в формуле (3.13) заменить = j* св, то получим:

.

Поскольку, законы изменения токов и напряжений в колебательном режиме
принимают вид:

(3.15)

Подчеркнем еще раз, что формулы (3.13), (3.14) и (3.15) являются общими
для всех токов и напряжений в цепях второго порядка.

3.2 Алгоритм расчета переходных процессов в цепях второго порядка

На основании вышеизложенного можно предложить следующий алгоритм расчета
переходных процессов в цепях второго порядка.

1. Расчет независимых начальных условий производится в цепи до
коммутации, в результате чего определяются ток через индуктивность и
напряжение на емкости:

2. Расчет зависимых начальных условий производиться для цепи,
образовавшейся после коммутации. Для этого составляются уравнения по
первому и второму законам Кирхгофа для мгновенных значений токов и
напряжений. Затем эти уравнения необходимо рассмотреть на момент t=0.

В уравнениях, составленных на момент t=0, известными будут: e(0), iL(0)
и UC(0), а неизвестными UL(0) и iC(0), которые и определяются.

3. Расчет первых производится на момент t=0.

Для определения постоянных коэффициентов a и b, входящих в формулы
(3.13), (3.14) и (3.15), необходимо знать значения не только искомых
функций, но и их первых производных на момент t=0.

Первые производные искомых функций определяются путем дифференцирования
уравнений, составленных по законам Кирхгофа, и рассмотрения их на момент
t=0.

При этом в первую очередь определяются первые производные тока через
индуктивность и напряжение на емкости, для чего используются уравнения
связи между током и напряжением этих элементов.

. (3.16)

После этого определяются первые производные Y’(0).

4. Расчет принужденных составляющих и их первых производных на момент
t=0.

Принужденные составляющие искомых токов и напряжений определяются любым
известным методом расчета установившихся режимов в цепи образовавшейся
после коммутации (по законам Кирхгофа, методом контурных токов и др).

Затем полученные выражения для мгновенных значений принужденных токов и
напряжений необходимо продифференцировать по времени, после чего
определить: Yпр(0) и Yпр’(0).

5. Расчет постоянных коэффициентов a и b.

Постоянные коэффициенты a и b определяются по формулам:

6. Составление характеристического уравнения, определение и анализ его
корней, выбор расчетной формулы для свободной составляющей.

Одним из описанных выше методов составляется характеристическое
уравнение, например, Z(P)=0 и определяются его корни P1 и P2.

На основании анализа корней характеристического уравнения выбирается
одна из трех формул (3.13), (3.14), (3.15) для расчета законов изменения
тока или напряжения после коммутации.

3.3 Переходные процессы в последовательном колебательном контуре при
подключении его к источнику постоянного напряжения

Рассмотрим последовательный колебательный контур (Рис.3.1), который
относится к неразветвленным цепям второго порядка.

Пусть данный контур при нулевых начальных условиях подключается к
источнику постоянного напряжения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.

Рис. 3.1. Последовательный колебательный контур

Определим для этих условий законы изменения тока в цепи и напряжений на
пассивных элементах после коммутации.

Расчет законов будем вести по принятому выше алгоритму.

1. Независимые начальные условия.

Цепь до коммутации была обесточена, поэтому

2. Зависимые начальные условия.

Зависимым начальным условием, в данном случае, является напряжение на
индуктивности. Для его определения составим единственно возможное
уравнение по второму закону Кирхгофа для мгновенных значений напряжений
и рассмотрим его на момент t=0.

.

На момент t=0 получим:

Поскольку i(0)=0 и UC(0)=0, постольку

Напряжение на индуктивности в момент коммутации скачком принимает
значение, равное приложенному напряжению.

3. Первые производные и их значения на момент t=0.

В соответствии с (3.16), имеем:

(3.18)

.

Производную напряжения на индуктивности найдем после дифференцирования
уравнения (3.17) и рассмотрения его на момент t=0.

На момент t=0, имеем

Таким образом, первые производные для рассматриваемого случая:

4. Принужденные составляющие и их первые производные на момент t=0.

В цепи образовавшейся после коммутации (Рис.3.1), через какое-то время
конденсатор зарядится до уровня приложенного напряжения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, ток в
цепи прекратится, т.к. постоянный ток через емкость не проходит.
Принужденные составляющие и их первые производные соответственно будут
равны:

5. Постоянные коэффициенты a и b определяются по общей формуле:

.

6. Характеристическое уравнение и его корни.

Характеристическое уравнение, соответствующее дифференциальному
уравнению второго порядка (3.4), получим из комплексного входного
сопротивления цепи Рис.3.1 путем формальной замены j* P.

Приравняв к нулю Z(P) и выполнив очевидные преобразования, получим
характеристическое уравнение (3.6), корни которого зависят от конкретных
значений RLC-элементов рассматриваемого контура Рис.3.1.

(3.20)

В данном случае корни (3.20) определяются по формулам:

где – коэффициент затухания свободной составляющей;

– резонансная частота последовательного контура Рис.3.1;

– постоянная величина.

Как было сказано выше, в цепях второго порядка, в зависимости от вида
корней характеристического уравнения (3.19), после коммутации может
возникнуть один из трех возможных режимов: апериодический, критический
или колебательный.

Найдем закон изменения напряжений на пассивных элементах цепи Рис.3.1
при подключении ее к источнику постоянного напряжения при нулевых
начальных условиях для упомянутых случаев.

3.3.1 Апериодический режим

Апериодический режим наступает, если корни характеристического уравнения
(3.20) действительные и разные, а это возможно если

где – волновое сопротивление контура;

– добротность контура.

Таким образом, в последовательном колебательном контуре (Рис.3.1)
апериодический режим наступает при Q<0.5.

В связи с этим при анализе переходных процессов в последовательном
колебательном контуре отпадает надобность в составлении
характеристического уравнения и определении его корней.

В апериодическом режиме законы изменения тока и напряжений на пассивных
элементах описываются формулой (3.13).

Если подставить в (3.13) найденные значения коэффициентов a и b (см.
п.5) и выполнить простейшие преобразования, то получим законы изменения
напряжений на пассивных элементах последовательного колебательного
контура (3.1) в апериодическом режиме:

Для иллюстрации законов изменения напряжений на пассивных элементах
последовательного колебательного контура в апериодическом режиме
зададимся произвольными значениями E, L, C, , а сопротивление нагрузки R
выберем таким, чтобы Q соответствующие графики.

Пример таких расчетов приведен на Рис.3.2.

3.3.2 Критический режим

Критический режим в последовательном колебательном контуре наступает,
если корни характеристического уравнения действительные и одинаковые, а
это возможно, если:

Таким образом, критический режим в последовательном колебательном
контуре наступает при Q0.5 в последовательном контуре
Рис.3.1 возникают затухающие колебания, при которых происходит
непрерывный обмен энергией между индуктивностью и емкостью.

Затухание свободных колебаний происходит вследствие необратимых потерь
энергии в активном сопротивлении R.

Длительность переходного процесса в колебательном режиме определятся
коэффициентом затухания

Чем больше Q, т.е. чем меньше R, тем дольше продолжается переходной
процесс.

Частота свободных колебаний всегда меньше резонансной частоты контура

и при

Из Рис.3.4 видно, что напряжение на емкости в начале переходного
процесса почти в два раза превышает приложенное напряжение, что
необходимо учитывать при выборе пробивного напряжения конденсатора.

Таким образом, режим переходного процесса в колебательном контуре, при
подключении его к источнику постоянного напряжения, целиком определяется
комбинацией значений RLC-элементов:

При Q0.5 – колебательный режим.

Глава 4. Операторный метод расчета переходных процессов в линейных
электрических цепях

4.1 Общие сведения

В предыдущих главах был изложен классический метод расчета переходных
процессов в линейных электрических цепях.

Такие процессы описываются линейными дифференциальными уравнениями с
постоянными коэффициентами. Для их решения классическим методом
необходимо определить постоянные интегрирования, зависящие от начальных
условий. По мере усложнения электрических схем и возрастания порядка
дифференциальных уравнений трудности, связанные с нахождением постоянных
интегрирования, увеличиваются.

Решение упомянутых уравнений может быть выполнено операторным методом,
где не требуется дополнительно определить постоянные интегрирования.

При использовании операторного метода действительные функции времени,
называемые оригиналами, заменяются их операторными изображениями. В
результате чего исходные дифференциальные уравнения заменяются
алгебраическими; затем после решения алгебраических уравнений
производится обратный переход в область функций действительного
переменного.

Связь между оригиналом f(t) и его изображением устанавливается прямым
преобразованием Лапласа:

где P = + j* – комплексное число.

Из определения изображения (4.1) следует, что каждый оригинал имеет
единственное изображение. В свою очередь, оригинал вполне определяется
своим изображением.

Фразу «оригинал f(t) имеет своим изображением F(P)» принято записывать в
виде знака соответствия :

или

Существует обратное функциональное преобразование, дающее возможность
определить оригинал по его изображению (4.1):

Формула (4.2) называется обратным преобразование Лапласа.

4.2 Изображения простейших функций

При исследовании переходных процессов в электрических цепях чаще всего
возникает необходимость в определении изображений единичной функции
l(t), линейной функции a*t, экспоненциальной функции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, синусоидальной и
косинусоидальной функции, их производных и интегралов.

Единичная функция задается условием:

Изображение единичной функции:

Изображение постоянной величины E:

Изображение линейной функции:

Изображение экспоненты:

Изображения тригонометрических функций:

Изображение производной от функции f(t):

Изображение интеграла от функции f(t):

Операция дифференцирования оригинала заменяется операцией умножения на P
изображения, а операция интегрирования оригинала заменяется операцией
деления изображения на P.

4.3 Операторное сопротивление. Закон Ома в операторной форме

Рассмотрим вначале пассивные RLC-элементы и определим их операторные
сопротивления.

Пусть через индуктивность при нулевых начальных условиях i(0)=0
протекает ток i(t), изображение которого I(P).

По закону электромагнитной индукции напряжение на индуктивности:

Умножим обе части этого равенства на множитель
picscalex1000100090000030501000002001c0000000000050000000902000000000500
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b7ffffffc0020000b70100000b00000026060f000c004d61746854797065000030001c00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 и выполним прямое
преобразование Лапласа:

По теореме дифференцирования оригинала, при i(0)=0, получим:

Отсюда получаем выражение для операторного сопротивления индуктивности:

(4.3)

Рассмотрим теперь емкость C, которая при нулевых начальных условиях
подключается к источнику напряжения UC(t), изображение которого UC(P).

Ток и напряжение на емкости связаны уравнением:

Применим к левой и правой частям этого выражения прямое преобразование
Лапласа, в результате получим:

Отсюда находим операторное сопротивление емкости:

.

Рассмотрим теперь последовательный колебательный контур Рис.3.1, который
при нулевых начальных условиях подключается к источнику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, изображение которого E(P).

По второму закону Кирхгофа можем записать (3.17):

(4.5)

Применим к этому уравнению прямое преобразование Лапласа, в результате,
с учетом (4.3) и (4.4), получим:

(4.6)

или

где – операторное сопротивление последовательного
колебательного контура.

Формула (4.6) представляет собой закон Ома в операторной форме. Нетрудно
заметить, что структура операторного и комплексного сопротивлений
подобны по форме:

Для перехода от комплексного сопротивления к операторному достаточно
заменить j* на P.

Сопротивление цепи в операторной форме есть новая более общая форма
сопротивления, которая может применяться для решения задач, относящихся
к любому режиму цепи при любой форме внешнего воздействия. Тогда как
комплексное сопротивление применимо лишь при синусоидальном воздействии
на цепь.

Наряду с операторным сопротивлением применяется операторная
проводимость, которая, по определению, является величиной обратной
сопротивлению.

4.4 Законы Кирхгофа в операторной форме

Первый закон Кирхгофа для мгновенных значений токов в узле электрической
цепи:

Полагая, что каждый ток, входящий в узел или выходящий из него, имеет
свое изображение Ik(P), получим первый закон Кирхгофа в операторной
форме:

который формулируется так: алгебраическая сумма изображений токов в узле
электрической цепи равна нулю.

Соответственно второй закон Кирхгофа для любого замкнутого контура

где ek(t), Uk(t) – мгновенные значения э.д.с. и напряжений на пассивных
элементах данного замкнутого контура, записывается в операторной форме:

Естественно, что при составлении уравнений по законам Кирхгофа в
операторной форме необходимо задаться положительными направлениями всех
токов и э.д.с., а также соблюдать все правила при составлении уравнений
по законам Кирхгофа для действительных функций времени.

4.5 Эквивалентные операторные схемы

Вышеприведенные формулы (4.7) и (4.8), выражающие законы Кирхгофа в
операторной форме справедливы при нулевых начальных условиях:

iL(0)=0 и UC(0)=0.

Если до возникновения переходного процесса цепь обладала запасом энергии
в виде электрического и магнитного полей, то, естественно, этот запас
энергии необходимо учесть при составлении операторных уравнений. Надо
ожидать, что законы Ома и Кирхгофа в этом случае изменяются в своей
записи и примут более общую форму, из которой, как частный случай,
должны вытекать формулы для нулевых начальных условий.

При ненулевых начальных условиях формула (4.3) принимает следующий вид:

где i(0) – ток через индуктивность в момент коммутации (t=0).

Этому операторному уравнению соответствует следующая эквивалентная
операторная схема замещения (Рис.4.1)
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picscalex1000100090000034101000002001c0000000000050000000902000000000500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picscalex1000100090000039101000002001c0000000000050000000902000000000500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а) в)

Рис. 4.1. Исходная а) и в) операторная схема замещения индуктивности: EL
= L*i(0) – внутренний источник напряжения, направление которого
совпадает с направлением тока.

При ненулевых начальных условиях уравнение (4.4) принимает вид:

где UC(0) – напряжение на емкости в момент коммутации.

Операторному уравнению (4.10) соответствует следующая эквивалентная
операторная схема замещения (Рис.4.2):

Рис. 4.2. Исходная а) и в) эквивалентная операторная схема замещения
емкости

– внутренний источник напряжения, направление которого противоположно
направлению тока.

Расчет переходных процессов операторным методом сводится к выполнению
следующих операций:

· вместо источников напряжений, оставшихся в цепи после коммутации,
вводятся их операторные изображения e(t) E(P);

· вместо всех искомых токов и напряжений на пассивных элементах вводятся
пока неизвестные их изображения:

· вместо индуктивности и емкости рисуются их операторные схемы
замещения, как показано на Рис.4.1 и Рис.4.2; при этом комплексные
сопротивления заменяются операторными, а при ненулевых начальных
условиях в операторную схему замещения вводятся внутренние источники
напряжений

Активное сопротивление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остается без изменений.

Искомые изображения токов и напряжений могут быть определены любым
известным методом расчета установившихся режимов (по законам Кирхгофа,
методом контурных токов и др.).

Рассмотрим, для примера, электрическую схему Рис.4.3а, в которой до
коммутации был установившийся режим. В момент коммутации t=0 происходит
короткое замыкание резистора R1 и в цепи возникает переходной процесс.

Определим изображения токов в ветвях с индуктивностью и емкостью.
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а) в)

Рис. 4.3. Исходная а) и ее операторная схема замещения в)

Операторный метод, как и классический, предусматривает, в первую
очередь, определить независимые начальные условия.

В исходной схеме до коммутации был установившийся режим, при котором:

Для определения неизвестных изображений токов через индуктивность и
емкость составим операторную схему замещения (рис.4.3 в), а затем
составим два уравнения по второму закону Кирхгофа:

где

Отсюда определяем неизвестные изображения токов

(4.11)

4.6 Определение оригинала по известному изображению

Из вышеизложенного следует, что по законам Ома и Кирхгофа в операторной
форме всегда можно найти изображения искомых токов и напряжений. После
этого возникает обратная задача: по известному изображению тока или
напряжения, например (4.11), найти соответствующий ему оригинал i(t),
т.е. найти закон изменения тока или напряжения в функции времени.

Для нахождения оригинала пользуются готовыми таблицами, которые
приводятся в учебниках и справочниках, где приводятся изображения и
соответствующие им оригиналы.

Однако, в настоящее время расчет переходных процессов операторным
методом можно выполнять с помощью программы Mathcad, которая позволяет
производить прямое и обратное преобразование не прибегая в таблицам:
laplace, invlaplace.

На нескольких примерах покажем, как производится расчет переходных
процессов операторным методом в среде Mathcad.

4.6.1 Расчет переходных процессов в цепях первого порядка операторным
методом

Пример 4.1. Рассчитать реакцию цепи RC (Рис.4.4) при воздействии на нее
одиночного прямоугольного импульса.

Рис. 4.4. Одиночный прямоугольный импульс с напряжением E и
длительностью T воздействует на цепь RC в момент t=0

Для описания единичной функции в среде Mathcad имеется встроенная
функция Хевисайда, Ф(t) которая представляет собой источник постоянного
напряжения в 1 В.

Одиночный прямоугольный импульс, изображенный на Рис.4.4, описывается
так:

U1(t)=E*(Ф(t)-Ф(t-T)).

где Ф(t-T) – функция Хевисайда, смещенная по оси времени вправо на T.

Результаты расчетов переходных процессов в цепи RC, выполненные
операторным методом в среде Mathcad, представлены на Рис.4.5.

Сравнение законов изменения напряжений на резисторе и емкости,
полученные классическим методом Рис.2.2 и операторным методом Рис.4.5,
показывает, что они совпадают.

Пример 4.2. Рассчитать реакцию цепи RC (Рис.4.4) при подключении ее к
источнику синусоидального напряжения.

Результаты расчетов представлены на Рис.4.6.

Сравнение Рис.4.6 с рис.2.6 показывает, что они совпадают.

Пример 4.3. Рассчитать реакцию цепи RC (Рис.4.4), если входное
напряжение представляет затухающую экспоненту.

Результаты расчетов представлены на Рис.4.7.

4.6.2 Расчет переходных процессов в цепях второго порядка операторным
методом

Пример 4.4. Рассчитать переходные процессы в последовательном
колебательном контуре (Рис.4.8) операторным методом при подключении его
к источнику постоянного напряжения U1(t).

Рис. 4.8. Последовательный колебательный контур при нулевых начальных
условиях подключается к источнику напряжения U1(t).

Реакция цепи (Рис.4.8) при подключении ее к источнику постоянного
напряжения U1(t)=E представлена на Рис.4.9 и Рис.4.10 для критического и
колебательного режимов соответственно.

Сравнение этих рисунков с результатами расчетов, выполненными
классическим методом (Рис.3.3 и рис.3.4), показывает, что они совпадают.
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0002d010f000500000014022b0023020500000013022a002402050000001302330027020
5000000130208002c0205000000130208004502040000002d010c0004000000f0010f000
5000000090200000000040000002d01000005000000090200000000040000002d0106000
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0000000040000002d010000040000002e01000005000000090200000000040000002d010
700040000002d010d0004000000f0010c00040000002d010b0004000000f0010e0005000
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000000000022200040000002d010c0007000000fc020000ffffff020000040000002d010
e0005000000090200000000050000000102ffffff0204000000020101000500000009020
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70000002105010031001700b70105000000090200000000040000002d010000050000000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3001900700105000000090200000000040000002d0103000500000009020000000004000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Пример 4.5. Реакция цепи RLC (Рис.4.8) при воздействии одиночного
прямоугольного импульса (опер ХЕВИСАЙД LRC) показана на Рис.4.11.
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50000000902000000001c000000fb02f6ff0000000000009001000000000400000054696
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500050000000902000000001c000000fb02f6ff000000000000900100000000040000005
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d01060005000000090200000000050000000902000000000500000009020000000004000
0002d010500050000000902000000001c000000fb02f6ff0000000000009001000000020
700000053796d626f6c007740000000f7030a624c53f5775553f57701000000000030000
40000002d01070005000000090200000000040000002d010500050000000902000000000
40000002d010300050000000902000000001c000000fb02f3ff000000000000900100000
0020700000053796d626f6c00774000000011050a374c53f5775553f5770100000000003
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0001c000000fb020c00090000000000900100000000000202024d532053616e732053657
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000500000028020e00100004000000f0010e00050000000c026a02bb0204000000f00110
00030000000000

ЛИТЕРАТУРА

1. Атабеков Г.И. Основы теории цепей. Учебник для вузов. М., «Энергия»,
1969 г. 424с. с ил.

2. Г.В. Зевеке Г.В., П.А. Ионкин, А.В. Нетушил, С.В. Страхов. Основы
теории цепей. Учебник для вузов. Изд. 4-е, переработанное. М., «Энергия»
1975 г. 752с. с ил.

3. Бессонов Л.А. Теоретические основы электротехники: Электрические
цепи: Учебник для электротехн., энерг., приборостроит. спец. вузов.-8-е
изд., перераб. и доп.-М.: Высш. шк., 1984.-559с., ил.

4. Нейман Л.Р., Демирчян К.С. Теоретические основы электротехники: В 2-х
т. Учебник для вузов. Том 1.-3-е изд., перераб. и доп.-Л.: Энергоиздат.
Ленингр. отд-ние, 1981.-536с., ил.

5. Дьяконов В. Д. Mathcad 2000: Учебный курс-СПб: Питер, 2000.-592с.:
ил.

6. М. Херхагер, Х. Партоль Mathcad 2000 полное руководство: перевод с
нем.-К.: Издательская группа BHV. 2000.-416с.

7.Карлащук В.И. Электронная лаборатория на IBM PC. Программа Electronics
Workbench и ее применение. «Салон-Р», 2000.-506с.

СОДЕРЖАНИЕ

Предисловие к 3-ей части

Глава 1. Основные сведения о переходных процессах в линейных
электрических цепях

1.1 Возникновение и общая характеристика переходных процессов

1.2 Начальные условия

1.3 Математические основы анализа переходных процессов

Глава 2. Переходные процессы в цепях первого порядка

2.1 Общий алгоритм расчета переходных процессов в цепях первого порядка
классическим методом

2.2 Переходные процессы в цепях RC при подключении ее к источнику
постоянного напряжения и коротком замыкании

2.3 Переходные процессы в цепях RC при подключении ее к источнику
синусоидального напряжения

2.4 Переходные процессы в цепях RL при подключении ее к источнику
постоянного напряжения и коротком замыкании

2.5 Подключение цепи RL к источнику синусоидального напряжения

2.6 Синтез цепи RC с заданными параметрами переходного процесса

Глава 3 Переходные процессы в цепях второго порядка

3.1 Общая характеристика переходных процессов в цепях второго порядка

3.2 Алгоритм расчета переходных процессов в цепях второго порядка

3.3 Переходные процессы в последовательном колебательном контуре при
подключении его к источнику постоянного напряжения

3.3.1 Апериодический режим

3.3.2 Критический режим

3.3.3 Колебательный режим

Глава 4. Операторный метод расчета переходных процессов в линейных
электрических цепях

4.1 Общие сведения

4.2 Изображения простейших функций

4.3 Операторное сопротивление. Закон Ома в операторной форме

4.4 Законы Кирхгофа в операторной форме

4.5 Эквивалентные операторные схемы

4.6 Определение оригинала по известному изображению

4.6.1 Расчет переходных процессов в цепях первого порядка операторным
методом

4.6.2 Расчет переходных процессов в цепях второго порядка операторным
методом

Литература

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