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Теория с=
игналов и
цепей
1.
ВВЕДЕНИЕ
1.1=
. Информация
и сигнал. Общая схема передачи информации
Им=
еется
множество определений понятия информации от наиболее общего философского
(информация есть отражение реального мира) до практического (информа=
ция
есть все сведения (данные), являющиеся объектом хранения, передачи,
преобразования).
Передается информацияв виде сигналов. Сигнал есть
физический процесс, несущий в себе информацию. Сигнал может быть звуковым,
световым, в виде почтового отправления и др. Наиболее распространен сигнал в
электрической форме в виде зависимости напряжения от времени U(t).
Рассмотрим общую схему передачи информации:
ИИ =
-
источник информации (сообщение)
ПС =
-
преобразование в электрический сигнал
Кд =
-
кодирование
М <=
span
style=3D'color:black'>-
модулятор
ГН =
-
генератор несущей
РУ =
-
регистрирующее устройство
ДО =
- декодирование, обработка, выделение=
из
помех
УНЧ -
усилитель низкой частоты
Д <=
span
style=3D'color:black'>-
детектор (демодулятор)=
УВЧ -
усилитель высокой частоты
ИВЦ -
избирательная входная цепь
Канал
передачи информации -
комплекс устройств, используемых для передачи информации от источника до
получателя, а также разделяющая их среда.
Рассмотрим каждый этап канала.
1. Сооб=
щение
может быть в виде знаков (печать), звуковых сигналов (речь, музыка),
светового изображения или сигнала и др.<=
o:p>
2. Прео=
бразование
речи и музыки в электрический сигнал осуществляется с помощью микрофона,
преобразование изображений - с помощью телевизионных передающих
трубок. Письменное сообщение сначала кодируется, когда каждая буква текста
заменяется комбинацией стандартных символов (точки -=
тире,
ноль - единица), которые затем преобразуются в стандартные электриче=
ские
сигналы (например, импульсы разной длительности, полярности и т. д.).
Кодированию могут подвергаться все сообщения, причем одновременно может
производиться их шифровка.
=
span>В радиолокации информация вносится п=
ри
распространении сигнала в свободном пространстве.
3. Генерация высокочасто=
тных
колебаний. Основные требования: диапазон частот, стабильность частоты,
мощность (до миллионов ватт). Общепринято следующее разделение электромагни=
тных
волн по частотам.
Длинные: <=
/span>l=
=3D 10 км
¸ 1 км, f =3D
30 кгц ¸ 300 кгц.=
Средние: <=
/span>l=
=3D 1000
м ¸ 100 м, f =3D 300 кгц ¸=
3 Мгц.
Короткие: =
l=
=3D 100 м
¸ 10 м, f =3D
3 Мгц ¸ 30 Мгц.<=
/span>
Ультракороткие (СВЧ): =
метровые: =
l=
=3D 10 м ¸ 1
м, f =3D 30 Мгц ¸
300 Мгц;
дециметров=
ые: l=
=3D 1,0 м
¸ 0,1 м, f =3D 300 Мгц ¸=
3 Ггц;
сантиметро=
вые: l=
=3D 10 см
¸ 1 см, f =3D
3 Ггц ¸ 30 Ггц;<=
/span>
миллиметро=
вые: l=
=3D 10 мм
¸ 1 мм, f =3D
30 Ггц ¸ 300 Ггц.=
На=
выбор
того или иного диапазона волн для каждой конкретной системы связи оказывают
влияние следующие факторы:
а)
Особенности распространения электромагнитных волн данного диапазона, состоя=
ние
пространства, в котором распространяется волна. Длинные волны сильно
поглощаются землей, короткие и ультракороткие не огибают препятствия. Длинн=
ые,
средние и короткие могут отражаться от верхних слоев атмосферы.<=
/span>
б)
Технические условия: направленность излучения, применение антенной системы
соответствующих размеров, генерирование мощных колебаний и управление ими,
схема приемного устройства.
Направленность излучения можно обеспечить, если антенное устройство по разм=
ерам
существенно превышает длину волны. Направленность имеет большое значение в
радиолокации, радионавигации. Большая мощность колебаний требуется на длинн=
ых
волнах вследствие поглощения землей, а на других диапазонах - при сверхдаль=
ней
космической связи. Освоение новых диапазонов требует новых технических сред=
ств,
вследствие чего переход в коротковолновую область происходил постепенно по =
мере
освоения генерирующих устройств.
в)
Характер шумов и помех в данном диапазоне. Регулярно проводятся исследования
прохождения радиоволн различных диапазонов.
г)
Характер сообщения (количество информации и связанная с этим ширина спектра
(диапазон частот)).
Та=
к,
телевидение ввиду большой передаваемой информации должно иметь широкий спек=
тр
частот, поэтому оно возможно только на УКВ.
4. Моду=
ляция
- изменение одного или нескольких параметров высокочастотного
колебания по закону передаваемого сообщения. Частоты модулирующего сигнала
должны быть малы по сравнению с частотой несущей.
5. Выде=
ление
нужного сигнала в приемнике из всех колебаний в эфире осуществляется
входной избирательной цепью с помощью резонансных колебательных систем Df/f до
10-5¸10-6.
6. Усил=
ение
слабых сигналов в приемнике. Антенна принимает сигнал мощностью 10=
-10¸10-14
Вт ( ~=
10-6 В). На выходе прием=
ника
для надежной регистрации сигнала требуется мощность порядка единиц ватт, т.=
о.
необходимо усиление по мощности до 1010¸1014,
по напряжению - до 107. Это достигается с помощью
многокаскадных усилителей высокой, промежуточной и низкой частот.
7. Дете=
ктирование
(демодуляция) - выделение низкочастотного сообщения
(информационного электрического сигнала) из модулированного высокочастотного
сигнала. Осуществляется с помощью различного рода детекторов (синхронных,
амплитудных, квадратичных).=
span>
8. Деко=
дирование
- восстановление исходной формы информационного сообщения из
электрических сигналов стандартной формы после детектирования. Для
зашифрованных сигналов производится расшифровка. В простейшей системе связи
кодирующее и декодирующее устройства могут отсутствовать. При передаче
сообщения по проводам (телеграф) могут отсутствовать радиопередающее и
радиоприемное устройства.
При
многоканальной связи одно передающее и приемное устройства используются для
параллельной (одновременной) передачи сигналов от нескольких источников
информации с устройствами их смешения до передачи, выделения и разделения п=
осле
приема.
Основными
разделами в теории сигналов и передачи информации вначале были разделы =
по
структуре сигналов, их анализ и синтез, включая вопросы оценки переносимого=
ими
количества информации и помехоустойчивости. Неотъемлемой частью является те=
ория
цепей и прохождение в них сигналов, синтез цепей и оптимальная обработка
сигналов на фоне помех. В дальнейшем развивались такие направления, как
математическое описание каналов передачи информации, создание методов
помехоустойчивого и эффективного кодирования, разделение линий связи,
распределение, накопление и передача информации в сетях, системный оптималь=
ный
синтез объектов информационной техники.
Определения
Инфо=
рмация =
- совокупность сведений об объектах,
рассматриваемая с позиций передачи этих сведений в пространстве и во времен=
и.
Сооб=
щение -=
это
информация, выраженная в определенной форме и предназначенная для передачи =
от
источника к пользователю (тексты, фото, речь, музыка, телевизионное изображ=
ение
и др.).
Сигн=
ал -=
это
физический процесс, распространяющийся в пространстве и времени, параметры
которого способны отображать (содержать) сообщение.
Те=
ория
сигналов и передачи информации изучает процессы формирования, накопления,
сбора, измерения, переработки и преобразования (прохождения через цепи),
хранения, передачи и приема информации, т. е. все процессы, которые имеют м=
есто
при передаче информации на расстояние по определенным физическим средам (ли=
ниям
связи) с помощью электрических сигналов.
1.2. Основные этапы развития радиоэлектро=
ники
и
передачи сигналов<=
/p>
Мы=
можем
гордиться, что именно наш соотечественник Александр Степанович Попов 7 мая =
1895
года впервые в мире продемонстрировал беспроволочную связь. Изобретение рад=
ио
явилось логическим следствием развития науки и техники. В 1831 г. Майкл Фар=
адей
открыл явление электромагнитной индукции. В 1864 г. Джеймс Максвелл
теоретически предсказал электромагнитные волны, распространяющиеся со скоро=
стью
света. В 1887 г. Генрих Герц экспериментально доказал, что колебательный ра=
зряд
вызывает в пространстве волны электромагнитной природы, описываемые теорией
Максвелла. Открытия этих ученых подготовлены множеством других. Независимо =
от
Попова, но позже него Маркони (англичанин итальянского происхождения) в кон=
це
1895 г. повторил опыты Попова по радиотелеграфии. Работы Попова опубликован=
ы в
январе 1896 г., а Маркони - в
1897 г.
Одновременно шло развитие электронной техники. В 1884 г. Эдисоном открыта
термоэлектронная эмиссия, и пока в 1901 г. Ричардсон изучал это явление, уже
были созданы электронно-лучевые трубки. В 1904 г. была изобретена первая
электронная лампа-диод (Флеминг, Англия) и использована для выпрямления
высокочастотных колебаний в радиоприемнике. В 1905 г. Хелл (США) изобрел га=
зотрон,
в 1907 г. Форест (США) ввел в лампу управляющую сетку-триод. Первые
отечественные триоды изготовили в 1914 -1916 г. г. независимо Папалекси Н. =
Д. и
Бонч-Бруевич М. А. Электроника и радиотехника объединились в радиоэлектрони=
ку.
В =
1918
г. декретом В. И. Ленина создана Нижегородская радиолаборатория,
разрабатывающая ламповые приемники и передатчики (до этого использовали
искровые и машинные).
В =
1921
г. А. А. Чернышевым в Ленинграде изобретен косвенный по=
догревной
катод.
В =
1922
г. в Москве вступила в строй 12-киловатная радиотелефонная станция, мощность
которой больше суммарной мощности всех радиостанций мира.
В =
1933
г. введена в строй самая мощная радиостанция им. Коминтерна (500 кВт).=
Возникновение теории передачи информации связывают обычно с появлением
фундаментальной работы американца К. Шеннона «Математическая теория связи»
(1948 г.). Однако элементы теории передачи информации рассматривались в раб=
отах
Р. Хартли по измерению количества информации (1928 г.) В. А. Котельникова о
пропускной способности «эфира» и проволоки в электросвязи (1933 г.). Д. В.
Агеева по основам теории линейной селекции сигналов (1935 г.), а также В. А.
Котельникова по оптимальным методам приема сигналов на фоне помех (1946 г.)=
.
Дальнейшему развитию теории передачи информации способствовало появление те=
ории
случайных функций и статистических решений. Они позволили установить строгие
количественные соотношения в теории передачи информации и сделать ее точной
наукой. К 50-м годам уже стали классическими работы А. Я. Хинчина по теории
корреляции стационарных случайных функций (1938 =
г.),
Н.Н. Колмогорова и Н. Винера по интерполированию и экстраполированию
стационарных случайных последовательностей (1941 г., 1949 г.). Многие теоре=
мы в
теории передачи информации имеют имена указанных ученых. Значительный вклад=
в
развитие отдельных разделов теории внесли: А. А. Харкевич, Д. Миддитон, Р. =
М.
Фано, У. Питерсон, О. Л. Добрушин, Л. Ф. Бородин, Л. М. Финк, Л. С. Гуткин,=
Б.
Р. Левин и др.
ВОПРОСЫ ДЛЯ
САМОПРОВЕРКИ К ГЛАВЕ I.
1. Назовите диапазон частот (длин волн) электромагнитного
излучения, используемый в радиосвязи.
2. Изобразите блок-схему канала передачи информации.
3. Перечислите основные процессы при передаче информации.=
4. Дайте определение информации, сообщению, сигналу.
2. СИГНАЛЫ<=
span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt'>
&nbs=
p; Под
сигналом s(t) будем понимать изменение во времени одного из параметров
физического процесса.
2.1. Классификация сигнало=
в
Детерминированн=
ым
называется сигнал, который точно определен в любой момент времени (например,
задан в аналитическом виде). Детерминированные сигналы могут быть
периодическими и непериодическими.
Периодическим=
i>
называется сигнал, для которого выполняется условие s(t) =3D s(t + кT), где к - любое целое число, Т - период, являющийся
конечным отрезком времени. Пример периодического сигнала - гармоническое
колебание
Любой сложный периодический сигнал может быть
представлен в виде суммы гармонических колебаний с частотами, кратными осно=
вной
частоте
Непериодический=
сигнал, как правило, ограничен во времени.
Случайным с=
игналом
называют функцию времени, значения которой заранее неизвестны и могут быть
предсказаны лишь с некоторой вероятностью. В качестве основных характеристик
случайных сигналов принимают:
а) закон распредел=
ения
вероятности (относительное время пребывания величины сигнала в определенном
интервале);
б) спектральное
распределение мощности сигнала.
Формы представления детерминированных
сигналов
|
|
|
|
ЦИФРОВОЙ
|
<=
/span>Цифровой сигнал представляет
из себя комбинацию узких импульсов одинаковой амплитуды, выражающи=
х в
двоичном виде дискретные отсчеты сигнала.
|
2.2. =
Спектры
периодических сигналов
Известно, что любую
функцию s(t),
кусочно-непрерывную на интервале а£ t &pou=
nd;b и
ограниченную по норме
, можно разложить в ряд, называемый обобщенным рядом Фурье =
по
полному набору (базису) ортогональных функций
Функции
называются ортогональными, если выполняется соотношен=
ие:
при m ¹ к;
- норма.
Для комплексных функций Yк(t),
Yк*(t) есть комплексносопряженная ей функция.
Коэффициенты ряда
определяются умножением поочередно (2.1) на=
и интегрированием по периоду, пр=
и этом
из-за ортогональности функций
справа остается только один член:
Обобщенный ряд Фур=
ье при
заданной системе функций
и при фиксированном числе слагаемых ряда обеспечивает
минимум среднеквадратичной ошибки разложения:
При этом говорят о среднеквадратичной,
энергетической сходимости ряда
к функции s(t).
Для представления
периодических сигналов s(t+T) =3D s(t) с периодом T =3D b-a в=
не
интервала a £ t £ b базисные функции Yк(t)
также должны быть периодическими с периодом
к - целое число.
В радиотехнике в
качестве базисных функций разложения Фурье используют преимущественно
тригонометрические функции. Это объясняется следующими причинами:
а) функции cos wt, sin wt являются простыми, определены=
при
всех значениях t, являются ортогональными и составляют полный набор при кра=
тном
уменьшении периода;
б) гармоническое
колебание является единственной функцией времени, сохраняющей свою форму при
прохождении колебания через линейную систему с постоянными параметрами, мож=
ут
изменяться лишь амплитуда и фаза;
в) для гармоническ=
их
функций и их комплексного анализа имеется мощный математический аппарат,
найдены спектры множества форм сигналов;
г) гармоническое
колебание легко осуществить на практике.
Кроме гармоническо=
го
ряда Фурье применяются и другие типы разложения: по функциям Уолша, Бесселя,
Хаара, полиномам Чебышева, Ляггера, Лежандра и др.
Гармонический ряд =
Фурье
может быть представлен в следующих видах:
где
An - амплитуда гарм=
оник, nw1
- частота гармоник, jn - фаза гармоник,
- комплексная амплитуда гармоник. Все виды разложения
(2.3) тождественны и переходят один в другой.
При выбранном знаке
перед jn фаза гармоник является аргументом комплекс=
ной
амплитуды.
Спектр периодическ=
ого
сигнала является дискретным и представляет набор гармонических колебаний, в
сумме составляющий исходный сигнал. Одним из преимуществ разложения сигнала=
в
спектр является следующее. Сигнал, проходя по цепи,
претерпевает изменения (усиление, задержка, детектирование, изменение фазы,
ограничение и т. д.). Токи и напряжения в цепи под действием сигнала
описываются дифференциальными уравнениями, соответствующими элементам цепи и
способу их соединения. Линейные цепи описываются линейными дифференциальными
уравнениями, причем для линейных цепей верен принцип суперпозиции, согласно
которому действие на систему сложного сигнала, состоящего из суммы простых
сигналов, равно сумме действий от каждого составляющего сигнала в отдельнос=
ти.
Это позволяет при известной реакции системы на какой-либо простой сигнал,
например, на синусоидальное колебание с определенной частотой, определить
реакцию системы на любой сложный сигнал, разложив его в ряд по синусоидальн=
ым
колебаниям.
Распределение мощности в спектре
периодического сигнала
Под функцией s(=
t) будем
понимать электрическое напряжение на резисторе 1 Ом. Энергия сигнала,
выделяемая на этом резисторе за период равна
, средняя мощность -
Подставляя сюда разложение s(t) (2.3=
) и
учитывая, что вследствие ортогональности интегралы от смешанных произведений
равны нулю и остаются только интегралы от квадратов членов ряда, получим
Здесь использовано
Средняя мощность сигнала равна сумме мощнос=
тей
его спектральных составляющих, выделяемых в отдельности. Она не зависит от
сдвига фаз отдельных гармоник.
Спектры простейших периодических сигна=
лов
а) Прямоугольное колебание (м=
еандр)
Для нечетной функц=
ии s(t)=3D
-s(-t) обращаются в ноль все коэффициенты an.
bn=
=3D
0 для четных n =3D 0,2,4...
для нечетных n =3D 1,3,5...
б) Последовательность униполя=
рных
прямоугольных импульсов
bn =3D 0 - четная функция.
;
Чем меньше длительность импульса, тем шире
спектр.
2.3. Спектры непериодическ=
их
сигналов
Пусть задан сигнал=
в
виде ограниченной во времени функции s(t), отличной от нуля в промежутке t<=
sub>1t2.
Выделим произвольный отрезок времени T, включающий промежуток t1=
t2,
далее продолжим аналитически s(t) на всю бесконечную ось с периодом T. Тогд=
а мы
сможем разложить такую периодическую функцию s(t) в гармонический ряд Фурье=
. В
комплексной форме будем иметь:
Полученный ряд на
участке t1t2 будет точно соответствовать нашей функции
s(t). Однако, если нас интересуют моменты времен=
и за
участком t1t2, то необходимо увеличить период Т, т. е.
отодвинуть повторные значения функции s(t). Производя замену переменных и
переходя от суммирования к интегрированию, получим
где
- спектральная плотность сигнала s(t).
Спектр непериодического сигнала сплошной
(непрерывный) и распространяется на отрицательные частоты.
Если
, то
- модуль спектральной плотности – амплитудно-частотная
характеристика.
- фазово-частотная характеристика.
Необходимое условие существования спектраль=
ной
плотности
Пример. Спектр
прямоугольного сигнала
Согласно формуле Эйлера
- площадь под импульсом.
Свойства преобразования Фурье=
h2>
а) Сдвиг сигнала во времени s2(t)=3Ds1(t-t0).
Сдвиг во времени функции s(t) на ±t0
приводит к сдвигу фазы спектра на ±wt0. Это позволяет=
для
удобства разложения в спектр сдвигать сигнал
относительно начала координат.
б) Сжатие и расширение сигнала s<=
sub>2(t)=3Ds1(nt).
При сжатии сигнала в n раз на временной оси=
во
столько же раз расширяется его спектр на оси частот при уменьшении модуля в=
n
раз. Наоборот, при растяжении сигнала во времени имеет место сужение спектр=
а и
увеличение модуля спектральной плотности. Т. о. сжатие спектра импульса с ц=
елью
повышения точности измерения частоты требует удлинения времени измерения. В=
то
же время сжатие импульса по времени с целью, например, повышения точности
измерения времени его появления заставляет расширять полосу пропускания
измерительного устройства. В теории преобразования Фурье доказывается, что =
где
.
В реальности это проявление принципа
неопределенности:
При
при несреднеквадратичном определении
и
.
в) Дифференцирование и интегрирование си=
гнала
Аналогично спектральная плотность интегра ла
равна
г) Сложение сигналов (линейность
преобразования)
- из-за линейности операции интегрирования.
д) Спектр про=
изведения
двух функций
Изменяем порядок интегрирования:
Спектр произведе=
ния двух
функций равен свертке их спектров (с множителем
).
Аналогично можно
показать, что свертке двух функций
соответствует спектр
являющийся произведением
исходных спектров.
е) Взаимная обратимость s(t) и
.
;
Для четного сигнала s(t)=3Ds(-t), и =
в связи
с симметричностью пределов интегрирования в выражении для
можно поменять знак в экспоненте =
Тогда, если по функциональной зависимости
то
Распределение энергии в спектре
непериодического сигнала
Найдем спектр квадрата функции s(t).
- используем свойства преобразования Фурье для
произведения двух функций.
В частном случае (=
) будем иметь:
. Переходя от
к
и т. к.
, комплексное сопряженние
.
- равенство Парсеваля.
- спектральная плотность энергии (энергия, приходящая=
ся
на единицу полосы частот). Е - полная энергия сигнала.
Для энергии, приходящейся на конечную полосу
частот, получим:
- при симметричной
Примеры. Спектр Гауссова (колокол=
ьного)
импульса
, -¥ < t < ¥, =
а
- условная половина длительности на уровне 0,606.
.
Произведем преобразование в показателях сте=
пени:
где d - определяется из условия:
откуда
.
При d - конечном
т. к.
.
Тогда
т. е. спектр Гауссова импульса имеет Гауссову форму: =
.
Можно показать, что Гауссов импульс обладае=
т наименьшим
при среднеквадратичном их определении.
Спектр d-функции
.
В качестве d-функции может выступать
сигнал любой формы с бесконечно малой длительностью и единичной площадью.=
p>
Свойства d-функции
1) <=
/span>
- фильтрующее свойство.
2) <=
/span>Четность
3) <=
/span>Нормировка
Спектральная плотность=
.
При t0 =3D 0, =
,
при t0 ¹ 0,
.
- это спектральное определение d-функции.
Аналогично
- определение d-функции в частотной области.=
p>
Спектральная плотность
гармонического колебания
|
Одним из условий
применения интегрального преобразования Фурье функции s(t) являетс=
я ее
абсолютная интегрируемость
Применениеd-функции
позволяет получить спектральную плотность и для неинтегрируемых функций.<=
/p>
|
Пусть
Найдем спектральную плотность, формально не
обращая внимания, что сигнал абсолютно не интегрируем.
Произведем замену
.
Но
тогда
.
Гармоническому кол=
ебанию
с конечной амплитудой соответствует бесконечно большая спектральная плотнос=
ть
на дискретных частотах ±w0.
В частности, для
постоянного напряжения w0 =3D 0,
2.4. Корреляционный =
анализ
детерминированных сигналов
Корреляционный анализ - это анализ временной зависимости сигналов.
Взаимно-корреляционная функция
Для
оценки степени связи во времени между двумя различными сигналами
и
используется взаимно-корреляционная функция ВКФ.
|
Найдем спектральную плотн=
ость
ВКФ. Для этого рассмотрим сначала спектральную плотность произведения двух
функций
Каждой из функций сомножителей
соответствуют спектральные плотности
и
Согласно теореме о спектральной плотности произведе=
ния
двух функций,
|
С другой стороны, согласно =
общей
формуле определения спектральной плотности
Положим в обеих формулах w =3D 0,
получим:
Итак, согласно общему
интегральному представлению спектра сигнала спектральная плотность для ВКФ
равна произведению спектральной плотности одной функции на сопряженную
спектральную плотность другой.
Автокорреляционная
функция сигнала (АКФ)
АК=
Ф - это степень связи сигнала =
span>s(t)<=
/span> с его копией, сдвинутой на величину t.=
o:p>
, при t =3D 0=
i>
|
Максимальное значение
автокорреляционной функции (при t =3D 0) равно энергии сигнала, т.=
к.
сигнал полностью коррелирован сам с собой. Полагая в преобразовании Фурье
взаимнокорреляционной функции
получим:
,
|
При t =3D 0 получим
равенство Парсеваля
.
Ит=
ак, в
частотной области имеется две характеристики сигнала: спектральная плотн=
ость
и спектральная плотность энергии.
Спектральной плотности
, содержащей полную информацию о сигнале, соответствует в
преобразовании Фурье сам сигнал s(t). Спектральной
плотности энергии сигнала
, лишенной фазовой информации, в преобразовании Фурье
соответствует автокорреляционная функция сигнала
.
Для периодических функций эн=
ергия
бесконечна, поэтому автокорреляционная функция определяется путем усреднени=
я по
периоду:
- средняя мощность
сигнала.
В более общем случае не
обязательно периодического, но с бесконечной полной
энергией сигнала принимают:
.
Фурье образом такой
автокорреляционной функции будет спектральная плотность мощности =
W(w), что видно=
из
соотношения
. Формально
Од=
нако
практически спектральная плотность мощности определяется через функцию
автокорреляции
Пример=
. Автокорре=
ляционная
функция прямоугольного импульса
при
2.5. Радиосигналы - модули=
рованные
колебания
Амплитудная
модуляция:
s(t) – модулирующий (управляющий) сигнал,
- несущая, k=
–
коэффициент пропорциональности.
Частотная модул=
яция:
Фазовая модуляц=
ия:
.
Между частотой и фазовой модуляцией существ=
ует
связь:
, поэтому их часто объединяют в угловую модуляцию
.
Параметры:
Глубина амплиту=
дной
модуляции - максимальное относительное отклонение амплитуды от
среднего:
Девиация частот=
ы -
максимальное отклонение частоты как при частотно=
й, так
и при фазовой модуляции.
при частотной модуляции;
при фазовой модуляции.
Индекс угловой
модуляции - максимальное изменение фазы:
- фазовая модуляция;
- частотная модуляция.
Для неискаженной
передачи сообщения необходимо, чтобы ширина спектра сообщения была мала по
сравнению с несущей
, а для этого необходимо, чтобы относительное изменение
модулирующей функции A(t), w(t), q(t) были малы за один период несущ=
его
колебания.
Спектр амплитудно-модулированного сигн=
ала
|
- произведение двух функций.
|
- свертка.
- спектральная плотность функции
А(t).
- спектр несущей.=
p>
.
Спектральная плотн=
ость
модулированного сигнала представляет два спектра модулирующей функции,
построенных относительно частот w =3D w0 и w
=3D -w0 (сдвинутых на частоты несущей).
Пример. Спектр
однотональной модуляции
.
Результат можно получить непосредственно:=
p>
.
Второе слагаемое может быть сведено к виду<=
/p>
- это соответствует новым
гармоническим колебаниям с частотами w0 + W и w<=
sub>0
- W, соответствующими верхней и нижней боковым частотам модуляции=
.
Спектр радиосигнала при угловой модуля=
ции
. При фазовой модуляции
, при
частотной
. Преобразуем косинус суммы:, =
p>
.
и
- медленно изменяющиеся амплитуды.
Итак, модулированн=
ое по
углу колебание можно рассматривать как сумму двух амплитудно-модулированных
колебаний (квадрупольных). Для определения спектра каждого из них достаточно
сдвинуть на w0 спектр огибающих амплитуд, т. к. cosq(t=
)
и =
sinq(t)
являются нелинейными функциями своего аргумента q(t), то спектры этих
функций могут существенно отличаться от спектра модулирующей функции s(t=
)
и при однотональной модуляции возможно возникновение кратных и комбинирован=
ных
частот. Однако при малых индексах модуляции
можно положить
, тогда
т. е. Спектр, как и в с=
лучае
амплитудной модуляции, состоит из несущей и двух боковых частей (но сдвинут=
ых
по фазе).
При больших индекс=
ах
модуляции за счет появления новых гармоник спектр расширяется и ширина спек=
тра
равна 2mW вместо 2W при m << 1.
При m >1=
cos(mcosWt)
и sin(mcosWt) разлагают в ряд по Бесселевым функциям.
Автокорреляционная функция
амплитудно-модулированного сигнала
(постоянная фаза не влияет на функцию корреляции).
,
Во втором интеграле
стоит произведение медленно меняющейся функции A(t)A(t-t) и
сильноосцилирующего члена с частотой 2w0.. Вследствие это=
го
второй интеграл существенно меньше первого и им можно пренебречь. Итак, АКФ
амплитудно-модулированного сигнала
равна произведению
и АКФ моделирующей функции. Полная энергия сигнала
за счет усреднения по высокочастотному колебанию. При
бесконечной энергии сигнала выполняется аналогичное соотношение между
корреляционными функциями, соответствующими мощности модулирующего и
амплитудно-модулированного сигнала.
где
Балансная и однополосная амплитудные
модуляции
В
амплитудно-модулированном (АМ) сигнале:
значительная доля мощности сосредоточена в
несущем колебании
Для более эффективного использования мощности
передатчика можно формировать АМ-сигналы с подавленным несущим колебанием, =
реализуя
так называемую балансную амплитудную модуляцию.
Однотональный АМ-сигнал с балансной модуляц=
ией
имеет вид:
Такой сигнал с физической точки зрения явля=
ется
биением двух гармонических сигналов с одинаковыми амплитудами
и частотами
и
. При переходе огибающей биений через нуль фаза
высокочастотного заполнения скачком изменяется на 180о, поскольку
функция
имеет разные знаки справа и слева от нуля. Осуществле=
ние
балансной модуляции, как и обратного процесса демодуляции (детектирования),
технически более сложно, чем при обычной амплитудной модуляции.
Другим
усовершенствованием обычной амплитудной модуляции является удаление всех
гармоник справа или слева от несущей частоты. При этом информация не теряет=
ся,
так как содержится в оставшихся гармониках с другой стороны от
несущей. Такая модуляция называется однополосной и позволяет в два р=
аза
сократить полосу занимаемых частот радиоканала, при этом существенно
усложняется процесс демодуляции с полным восстановлением модулирующего сигн=
ала.
Возможно устранение в однополосной модуляции и несущего колебания с частото=
й w0.
Обобщенное представление радиосигнала в виде
высокочастотного узкополосного колебания. Аналитический сигнал
Используются
радиосигналы, получаемые в результате одновременной модуляции амплитуды и у=
гла
. При этом может возникнуть неоднозначность в выборе функци=
й A(t),
y(t) (можно A(t) представить в виде cos, а y(t) =
-
в виде arccos и поменять местами). Однако если сигнал является узкополосным
(ширина спектра модулирующих функций много меньше центральной частоты w<=
sub>0),
то неопределенностей A(t) и y(t) можно избежать с помощью
следующих соотношений:
; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
(1)
 =
; &n=
bsp;  =
; &n=
bsp;  =
; &n=
bsp;
(2)
где a1(t) - новая функция,
связанная с исходной преобразованием Гильберта;<=
/p>
; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
(3)
 =
; &n=
bsp;  =
; &n=
bsp;  =
; &n=
bsp;
(4)
Мгновенная частота узкополосного сигнала ра=
вна:
.
Выделив в
постоянную часть
, можно написать
, где
не содержит слагаемого, линейно зависящего от времени=
.
Из (1) следует, чт=
о в
точках, где a1(t) =3D 0 имеем
т. к.
.
Следовательно, в э=
тих
точках кривые a(t) и A(t) имеют общие касательные. Кроме тог=
о,
преобразование Гильберта обеспечивает в точках, где a1(t)=3D0=
,
значения a(t), близкие к максимальным
(амплитудным). Таким образом, функция A(t) касается функции a(t)<=
/i>
в ее амплитудных значениях и ее можно считать простейшей огибающей, если она
изменяется медленно по сравнению с быстро осциллирующей функцией a(t),
т. е. выполнено условие узкополосного сигнала.
Аналитическим <=
/i> представлением
сигнала a(t) является комплексный сигнал
где
- комплексная огибающая
узкополосного сигнала.
Для спектральных
плотностей функций a(t) и a1(t) имеют место соотношени=
я
w > 0,
w < 0,
Спектры аналитического сигнала имеют только
положительные частоты.
Корреляционная
функция аналитического сигнала
связана с корреляционной
функцией узкополосного сигнала
соотношениями
Пример. a(t) =3D cosw0=
t.
Здесь использована замена t - t =3D x;=
i>
Известно, что
и
тогда
при
Импульсно-модулированные
сигналы
(манипулированные сигналы)
В импульсной модул=
яции в
качестве носителя управляющих сигналов используется последовательность
прямоугольных импульсов. При радиосигналах эта последовательность наносится=
на
высокочастотное колебание (двойная модуляция).
1) Амплитудно-и=
мпульсная
модуляция (АИМ), когда по закону управляющего сигнала изменяется приращ=
ение
амплитуды импульсов.
2) Модуляция по
длительности импульсов (ДИМ), когда по закону управляющего сигнала меня=
ется
длительность импульса. Иногда этот вид модуляции называется широтно-импу=
льсной
модуляцией( ШИМ).
|
A=3D const, T=3D const, t(t) =3D t0<=
/sub>+ks(t).
|
3) Временная импульсная модуляция (ВИМ), когда по закону
управляющего сигнала происходит смещение импульсов по временной оси (может =
быть
фазовой (ФИМ) или частотной (ЧИМ)).
|
A=3D const, tи=3D const, T =3D const,
Dt =3D ks(t) – ФИМ.<=
o:p>
A=3D const, tи =3D const,
Ti(t) =3D ks(t)+T0 - ЧИМ
|
4)
Кодово-импульсная модуляция (цифровая
|
Каждому уровню сигнала (квантованному)
приписывается определенный номер (код) обычно в двоичной системе. Вместо
передачи величины сигнала s(t) в моменты отсчета функции s(t)=
i>
передается число (в виде комбинации узких импульсов), соответствующее ном=
еру
уровня сигнала в данный момент.
|
Спектр амплитудно-импульсного
модулированного сигнала
где f(t) - периодическая последовательность
прямоугольных импульсов с длительностью t=
и
и частотой F=3D1/T. Спектр f(t) может быть представлен рядом Фурье=
:
;
Т. о., при АИМ около каждой гармонической
составляющей спектра, соответствующей спектру периодической последовательно=
сти
импульсов, появляются боковые составляющие, соответствующие спектру
модулирующей функции (избыточность информации). При дополнительной высокоча=
стотной
модуляции спектр дополнительно сдвигается на частоту несущей w0.
Внутриимпульсная частотная модуляция=
span>
В последнее время в
радиолокации широкое распространение получили сигналы с внутриимпульсной
частотной модуляцией. Чаще всего используется линейное нарастание частоты от
начала к его концу:
при
|
Девиация част=
оты
Практическая
значимость линейно-частотно-модулированных (ЛЧМ) сигналов заключается в
возможности существенного сжатия сигнала при приеме с увеличением его
амплитуды над уровнем помех.
Это достигается
устройством задержки с временем задержки,
уменьшающимся с ростом частоты сигнала. При этом на выходе будет происход=
ить
сложение низкочастотных колебаний, относящихся к началу импульса, и более
высокочастотных, наблюдаемых в его конце
|
Модуль спектральной плотности ЛЧМ-сигнала с
большой «базой»
практически постоянен и сосредоточен в полосе частот.=
со значением в центре
.
2.6. Случайные сигналы=
Случайные величины
Случайная величина
описывается вероятностными законами. Вероятность того, что непрерывная вели=
чина
х при измерении попадет в какой-либо интервал х1
<х <х2, определяется выражением:
, где p(x) - плотность вероятности, прич=
ем
. Для дискретной случайной велич=
ины хi P(x =
=3D xi)=3DPi,
где Pi - вероятность, соответс=
твующая i-у =
уровню
величины х.
Моменты случайн=
ой
величины:
а) среднее значение (математическое ожидани=
е)
б) средний квадрат
в) средний квадрат флуктуаций (дисперсия)=
p>
Вид функции р(х=
) плотности
вероятности для различных случайных величин може=
т быть
различен. Часто выполняется нормальный закон распределения вероятности:
,
где
- среднее значение,
- дисперсия.<=
/p>
Имеет место «це=
нтральная
предельная теорема»: распределение вероятности для суммы независимых
случайных величин с ростом числа слагаемых, при которых нет доминирующих,
стремится к нормальному закону независимо от законов распределения слагаемы=
х.
Плотность
вероятности функции от случайной величины
Пусть y
- случайная величина, связанная с x однозначной
функциональной зависимостью вида у =3D f(x).<=
/i>
Попадание случайной точки х в интервал шириной dx и попадание случайной т=
очки у в отвечающий ему и=
нтервал
шириной
являются эквивалентными событиями, поэтому вероятност=
и их
совпадают:
. Отсюда
.
Если функциональная
связь между х и у неоднозначна, так что имеется несколько
значений
для одного значения у (х=3Dg(у)=
-
функция, обратная по отношению к у=3Df(х)), то выражение для плотности
вероятности ру(у) обобщается:<=
/p>
.
Многомерная
плотность вероятности
Пусть имеем n случайных в=
еличин
х1,х2, …, х=
n. Можно
ввести n-мерную
плотность вероятности p(х1,х2, …, хn),
определяющую вероятность одновременного осуществления событий
,
, ...,
,
причем
. Зная n-мерную плотность вероятности, всегда можно найти m-мерную
(m=
< n) плотность вероятности меньшего порядка, интегрируя по
лишним координатам:
.
Располагая многомерной плотностью вероятнос=
ти,
можно находить среднее значение любых комбинаций этих случайных величин и
определять их моменты. В частности, для двумерной случа=
йной
величины будем иметь:
Новым по сравнению с одномерным случаем явл=
яется
смешанный момент второго порядка - ковариационный момент
или центрированный корреляционный момент=
Вводят также безразмерный коэффициент корре=
ляции
Для статистически независимых случайных вел=
ичин
Статистически независимые случайные величины
некоррелированы между собой:
при
. Обратное утверждение в общем с=
лучае
неверно: из некоррелированности не вытекает автоматически статистическая
независимость случайных величин.
Случайный процесс
Под случайным
процессом понимают множество (ансамбль) случайных функций хк(t), =
называемых
возможными реализациями этого случайного процесса. В каждый выбранный момент
времени t1
конкретная реализация есть случайная величина с плотностью вероятности =
и ее среднее значение определяется усреднением=
по
всем возможным реализациям:
Различают стациона=
рные и
нестационарные случайные процессы. Для стационар=
ных
процессов плотность вероятности от времени не зависит:
. Стационарный процесс называется эргодическим, если
усреднение по множеству реализаций эквивалентно усреднению по времени в
пределах одной реализации.
Кроме одномерной
плотности вероятности
вводят двумерную плотность вероятности
совместной реализации двух значений: х1=
span> в
момент времени t1 и х2 в момент време=
ни t2.
.
Для стационарных процессов двумерная плотно=
сть
вероятности зависит только от разности моментов времени
Двумерная плотность
вероятности определяет дополнительный момент - автокорреляционную функцию
случайного процесса.
Иногда используют нормированные
автокорреляционные функции:
Для стационарного процесса:
.
Если процесс не только стационарный, но и
эргодический, усреднение по множеству может быть заменено усреднением по
времени в пределах одной реализации:
Условие эргодичности для стационарного проц=
есса с
нулевым средним значением:
Это определяет стремление функции корреляци=
и к
нулю с увеличением временного сдвига t. Можно ввести интервал
корреляции
, который определяет время статистической зависимости м=
ежду
мгновенными значениями случайного сигнала.
Спектральный анализ случайных сигна=
лов
Случайную реализац=
ию х(t)=
i> можно
разложить по детерминированным ортогональным функциям
Коэффициенты такого разложения сn
будут случайными величинами. Для гармонического разложения
Ввиду случайности спектральной плотности =
и равенства нулю ее среднего значения при усреднении =
по
всем реализациям при
(ввиду случайности и независимости фаз спектральных
составляющих в различных реализациях) она не используется для характеристики
случайного процесса. Поэтому для случайного процесса x(t) <=
/i>вводится
понятие спектральной плотности мощности, связанной с автокорреляционной
функцией преобразованием Фурье (соотношение Винера - Хинчина).
.
Спектральная плотность мощности определяетс=
я из
последнего соотношения по функции корреляции определяемой для эргодического=
процесса
в пределах одной реализации:
При нулевом среднем значении
имеем:
Чем шире энергетический спектр
случайного процесса, тем быстрее меняется x(t)=
i> и
меньше время корреляции, и наоборот.
Примеры случайных процессов
1. Постоянное напряжение
случайного уровня Ак
|
При равновероятности уровня
что соответствует условию =
Итак, процесс стационарен.
|
;
- процесс не эргодичен.
2. Гармоническое колебание со случайной фазой
, т. к. каждому хк соответствует два знач=
ения
±qк.
- процесс стационарный и эргодический.
3. Белый шум
Это стационарный п=
роцесс
с равномерным на всех частотах спектром мощности
.
Функция корреляции=
, т. е. равна нулю всюду, кроме t =3D 0, где она
бесконечна. Средняя мощность (дисперсия) белого шума неограниченно велика.
Временная реализация х(t) белого шума имеет игольчатую структу=
ру с
бесконечно тонкими выбросами обоих знаков. Многие помехи в технике связи,
вычтехнике и др. обычно рассматривают как белый шум, если ширина частот пом=
ехи
превышает полосу частот пропускания аппаратуры, а амплитуды частот примерно
постоянны. К таким помехам относят флуктуационные шумы, помехи в многоканал=
ьных
системах связи и др.
2.7. Дискретизация
непрерывных сигналов
При
передаче непрерывных сообщений по системам связи c использованием импульсной модуляции или кодирования
возникает необходимость дискретизации сообщений по времени. В последнее вре=
мя
необходимость дискретизации непрерывных сигналов объясняется развитием мето=
дов
квантования, дискретного анализа формы сигналов, развитием цифровой и
вычислительной техники.
|
Сущность дискретизации
заключается в том, что непрерывность во времени функции s(t=
) заменяется последовательностью коротких импульсов,
амплитуды которых (координаты) ск в общем случае
определяются с помощью
|
дискретных весовых функций =
xк(t)
.
Воспроизведение непрерывной функции по ее дискретным координатам производит=
ся с
помощью системы базисных функций
Иногда весовые и базисные функции принимают
одинаковыми
Ввиду сложности определения координатных функций более
широкое распространение получили методы дискретизации, при которых сигнал <=
/span>s(t)<=
/span> заменяется совокупностью его мгновенных значений
, называемых выборками, или отсчетами. Роль весовых функций=
в
этом случае играют d-функции
, Dt =
- шаг
дискретизации (может быть неравномерным).
. Шаг дискретизации должен быть таким, чтобы было возможно
восстановление непрерывной функции по ее отсчетам с допустимой точностью.
Равномерная
дискретизация. Теорема Котельникова
Пр=
авило
выбора предельного шага при равномерной дискретизации с использованием моде=
ли
сигнала с ограниченным спектром сформулировано академиком В. А. Котельников=
ым:
«Любая непрерывная функция s(<=
/span>t), спектр которой ограничен часто=
той Fmax=
sub>
полностью определяется последовательностью своих значений в моменты
времени, отстоящие друг от друга на интервал
» Кроме того, теорема Котельникова дает и способ точного
восстановления сигнала
по его отсчетам.
Доказательство=
span>
причем
при
(1)
Разложим функцию
в частотной области на конечном интервале
(с периодом
) в комплексный ряд Фурье :=
где &nb=
sp;
&nb=
sp; =
&nb=
sp; =
(2)=
=
&nb=
sp; =
&nb=
sp;
(3)
Сравнивая интегралы в (3) и=
(1),
видно, что они равны при
, т. е.
тогда
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
(4)
Подставляем (4) в (2), а за=
тем в
(1)
=
т. к. суммирование по от -¥
до +¥, то можно заменить знак у .
=
&nb=
sp; (5)
Максимальные значения члено=
в ряда
будут при
и равны
, при этом все остальные члены ряда равны нулю, т. е. при
функция s(<=
/span>t) точно переда=
ется
рядом. Во все другие моменты времени необходимо суммировать бесконечное чис=
ло
отсчетов, чтобы передать s(<=
/span>t) точно. =
Представление сигнала в виде ряда Котельникова (5) является частным случаем
разложения
. Роль коэффициента
выполняют отсчеты
Базисными являются функции вида=
span>
Они называются функциями отсчетов. Функции
отсчетов ортогональны, т. к.
|
Спектральная плотность фу=
нкции
отсчета на частотной шкале есть прямоугольник шириной
(идеальный фильтр н.ч.).=
Теорема Котельникова распространяется на непрерывный =
стационарный
случайный процесс с ограниченным спектром
|
Такой процесс представляется
(заменяется) суммой квазидетерминированных процессов, где роль ортогональных
детерминированных функций выполняют функции отсчета, а случайных коэффициен=
тов
- величины выборок:
, где
Т. о.,
при указанных ограничениях случайный процесс полностью определяется счетным
множеством случайных величин - координат процесса.
Практическое
осуществление дискретизации сигнала
рядом Котельникова и дальнейшее его восстановление
сводится к следующему. На передающей стороне через интервалы
определяются мгновенные значения
сигнала и передаются в канал связи в виде d-им=
пульсов
с площадью, равной отсчету
На приемной стороне такая последовательность
импульсов пропускается через идеальный фильтр нижних частот
При длительной передаче сигнал на выходе фильтра будет
точно воспроизводить переданный непрерывный сигнал
Искажения восстановленного (по Котельникову) сигнала могут происходить по следующим причинам. Реальный си=
гнал
имеет конечную длительность и, следовательно, обладает неограниченным спект=
ром.
Дискретизация его с интервалом
ограничивает спектр
и, следовательно, искажает воспроизведение сигнала. С
другой стороны, и при передаче непрерывного сигнала вследствие ограничения
полосы пропускания аппаратуры сигнал искажается. Однако при дискретизации
появляется дополнительное искажение за счет конечности числа отсчетов за
ограниченное время длительности сигнала, в то время как их должно быть
бесконечно много, т. к. ограничению спектра сигнала соответствует увеличение
его длительности до бесконечности. Такое двойное искажение хотя и может
частично компенсироваться, но создает трудности для теоретического анализа
погрешности передачи.
Не=
смотря
на невозможность точного воспроизведения сигнала ограниченной длительности =
(чем
более короткий сигнал, тем больше ошибка воспроизведения), дискретизация и
восстановление по Котельникову используется весьма широко при преобразовании
сигнала в цифровую форму.
Пример 1. Аппроксимировать прямоугольный импульс длительностью=
, амплитудой U р=
ядом
Котельникова, ограничивая его спектр частотой
Затем восстановить сигнал по отсчетам
Котельникова.
|
В нашем случае
и импульс
аппроксимируется тремя отсчетами в моменты
0,
.
|
В моменты отсчетов значения
аппроксимаций совпадают с отсчетами.
Найдем значения сигналов в
промежутках.
а)
;
б)
в)
;
;
г)
;
д)
Пример 2. Сигнал звукового сопровождения в телевизионном канале
ограничен верхней частотой
кГц. Определить интервал Dt между отсчетами этого сигнала, необходимый для
неискаженного воспроизведения сигнала при передаче его дискретным способом.=
Дискретизация по
критерию наибольшего отклонения.=
Адаптивная
дискретизация<=
/p>
За=
дача
обеспечения минимальной погрешности при восстановлении сигнала на практике =
часто
не ставится. Обычно указывается допустимое значение погрешности
В качестве базисных функций выбирают полиномы
(степенные функции нулевого (ступенчатая), первого (линейная) и, реже, втор=
ого
(параболическая) порядка). Аппроксимацию проводят на каждом шаге дискретиза=
ции,
которые выбирают из условия допустимой погрешности. Вследствие того, что
изменение функции
различно в различные моменты времени, шаг дискретизац=
ии
может быть различным, обеспечивая равномерную погрешность на каждом шаге. Т=
акой
тип дискретизации называется адаптивный. Рассмотрим адаптивную дискр=
етизацию
при степенной аппроксимации Тейлора:
Пример 1. Аппроксимация полиномом нулевой степени (ступенчатая=
)
при
,
,
|
На практике производную не
определяют, а интервал дискретизации заканчивают при достижении максималь=
ного
отклонения
затем сигнал скачком увеличивают до значения функци=
и
в момент окончания шага дискретизации. При
восстановлении между отсчетами сигнал равен предыдущему отсчету.
|
Пример 2. Адаптивная аппроксимация степенным полиномом первой
степени (линейная).
|
На момент
каждого интервала аппроксимации принимают
. Момент очередного отсчета определяется равенством =
.
|
Восстанавливают сигнал
прямолинейным соединением отсчета.
Дискретное
преобразование Фурье
Дискретизированный сигнал можно рассматривать как результат умножения
первоначального непрерывного сигнала на ряд единичных импульсов (d-ф=
ункций).
,  =
; &n=
bsp;
(1)
- общее число отсчетов;
- отсчет, т. е. зн=
ачение
при
.
При таком рассмотрении площ=
адь
отсчета равна значению
При этом существенно упрощается нахождение
спектральной плотности дискретного сигнала
непосредственно по совокупности отсчетов
. Действительно, применив преобразование Фурье к (1), получ=
им:
или
=
&nb=
sp; =
&nb=
sp; =
(2)
Сп=
ектр
дискретизированного сигнала есть периодическая послед=
овательность
спектров
исходного непрерывного сигнала
, сдвинутых относительно друг друга на =
При выборе интервала дискретизации, согласно
теореме Котельникова
, наложения спектров этой периодической последовательности =
не
происходит.
Для
цифровой обработки сигналов требуется дискретизация не только во временной,=
но
и в частотной области в основном интервале
(либо
).
Полагая в (2)
, где
, получим:
=
&nb=
sp; =
&nb=
sp; =
(3)
Здесь
(либо
) (=3D0 может не быть, если - четное).
Выражение (3) называется дискретным преобразованием Фурье (ДПФ). Дискретиза=
ция
спектра с интервалом
приводит к периодическому повторению сигнала с период=
ом
. Для обратного дискретного преобразования Фурье (ОДПФ) мож=
но
записать:
=
&nb=
sp; =
&nb=
sp;
(4) либо
.
Ве=
рность
выражения (4) доказывается непосредственной подстановкой в него
из (3) с заменой индекса суммирования k на m,
сменой порядка суммирования и учетом того, что сумма
равна при k
=3D m и равна н=
улю
при k ¹ m как
сумма векторов, которые делят окружность единичного вектора на равные дуги<=
o:p>
Обычно DT выбирают меньше, чем
, увеличивая число отсчетов . Пр=
и этом
крайние значения
вблизи
близки к нулю, что облегчает выделение фильтрами
основного интервала
. Длительность импульсов
должна быть много меньше, чем
, чтобы получилась полная периодичность спектра
с периодом
Площадь импульса должна быть пропорциональна отсчету.=
От=
метим,
что выражение (3) ДПФ при значениях вне основного интервала дает периодичес=
кое
повторение дискретного спектра, а выражение (4) ОДПФ за пределами временного
интервала исходного импульса дает его периодическое продолжение. Аппарат
дискретного преобразования Фурье широко используется в математическом
моделировании и численном математическом преобразовании.<=
/p>
Отметим, что смена знака w в
(2) или номера в (3) не изменяет модуль
, а изменяет только знак фазы, поэтому в (3) достаточно
ограничиться номерами
. Равенство (4) тоже можно упростить:
, т. к. действительные части у членов с положительным
и отрицательным n совпадают=
, а
мнимые имеют противоположный знак.
Быстрое
преобразование Фурье
Рассмотрим дискретное прямое преобразование Фурь=
е.
,  =
; &n=
bsp;  =
; &n=
bsp;  =
; &n=
bsp;
(1)
=3D 0, 1, ..., -1; =
k =3D 0, 1, ..., -1; j - мнимая единица.
- отсчеты сигнала =
через
интервал времени
- отсчеты
спектральной плотности через интервал частоты
- число отсчетов.
Из=
(1)
следует, что для определения одного спектрального коэффициента
требуется умножений, вычислений экспонент и сложений,=
а
на все коэффициентов требуется 2 умножений, сложений и
вычислений экспонент. Алгоритм быстрого преобразования Фурье позволяет
уменьшить число таких операций до
При =3D 210 =3D =
1024
машинное время уменьшается в 100 раз.
Пу=
сть =3D
2p (p - целое число). Разобьем последовательность (1) на сумму двух
последовательностей, составленных из четных и нечетных номеров.
=
&nb=
sp;
(2)
Однако вычисления по (2) мож=
но
ограничить номером
, т. к. функции
и
являются периодическими с периодом
из-за большего (в два раза) интервала отсчетов между
ними. Итак, при
будем иметь:
т.к. =
span>
Аналогично
.  =
;
&nb=
sp; =
&nb=
sp; (3)
Теперь снова разобьем суммы=
и
с четными и нечетными новыми
номерами слагаемых:
;
=3D 0, 1=
, ..., 2P-2-1<=
/i>.
Далее суммы
снова можно разбить. Получим:
=3D 0, 1, ..., 2<=
/i>P-3-1;=
для номеров +2P-3 =
i>знаки перед экспонентами заменяют=
на
минус.
остальным членам
соответствуют следующие отсчеты sk:
k=3D0,1,...2P-3-1.=
Та=
ким
образом, мы имеем все время удваивающееся число сумм X,Y для
одного , однако число членов в этих сумма=
х все
время в два раза уменьшается и, кроме того, в два раза уменьшается число
номеров , так что общее число сумм
остается тем же, но уменьшается число членов в каждой
сумме.
Продолжая такое разбиение l
раз, будем иметь:
;
=
&nb=
sp; =
&nb=
sp;
(4)
.
где
(5)
l =3D =
i>1, 2, ..., p; =
i =3D 1, 2, ...=
, 2l-2 в (4) и <=
/span>i =3D 1, 2, ...=
, 2l-1 в (5).
n =3D 0=
, 1,
..., 2P-l-1. При l =3D p получим
=3D 0, k =3D 0.
i =3D 1, 2, ..., 2P-1.
Ит=
ак,
можно предложить следующий алгоритм быстрого преобразования Фурье.
За=
дано =3D
2P отс=
четов
сигнала
, i =
=3D 0, 1,
2, ..., -1, l =3D =
i>p, =3D 0 <=
/o:p>
for
i =3D 1 to N/2
, end.
* for n =3D 0 to n =3D 2P-l-1=
.
for i =3D 1 to i =3D 2l-2.
end, end.
l=3Dl-1, if l>1 go to *, els=
e for n=3D0
to
.
, end.=
2.8. Информация и сигнал.
Информационная емкость сигналов
Количественная оценка информации
Получение информац=
ии
всегда приводит к уменьшению неопределенности. Рассмотрим дискретный
источник информации, характери-
зуемый дискретным набором (ансамблем) состо=
яний с
заданной вероятностью их появления:
Меру неопределенно=
сти
выбора состояния источника можно рассматривать как меру количества информац=
ии,
получаемой при полном устранении неопределенности (т. е. содержащейся в
источнике). Эта мера должна удовлетворять ряду естественных условий.
1. Монотонное
возрастание с увеличением возможности выбора (оценка по пятибалльной
системе более полная, чем по двухбалльной).
2. Аддитивность
количества информации - неопределенность объединенного источника информ=
ации
должна равняться сумме неопределенностей (а
следовательно, и количества информации) исходных источников. Так, два
независимых источника с числом равновероятных состояний n и m можно
рассматривать как один источник, реализуемый пары состояний nim<=
sub>j
с общим числом состояний N=3Dmn, тогда
Это соотношение выполняется, если в качестве м=
еры
неопределенности источника с равновероятными состояниями принять логарифм ч=
исла
состояний
Для другого примера аналогично: если каждое сообщение
состоит из n символов, выбираемых из объема m, то число всех возможных
сообщений
и, следовательно,
, т. е. информация, полученная при выборке каждого из n
символов, складывается.
Основание логарифма
определяет только масштаб или единицу количества информации, т. к. при N =
=3D 1
(одно состояние не имеет неопределенности), то
минимальное количество информации будет при N =3D 2. Чтобы получить при этом
единицу информации, основание логарифма приравнивают к 2 и такую единицу ин=
формации
называют «бит» - информацию получаем при осуществлении одного из двух возмо=
жных
состояний H =3D 1 бит при
N =3D 2 (Хартли, 1928). Предложенная мера коли=
чества
информации применима только для равновероятных состояний:
тогда
информация, полученная при осуществлении i-го состоян=
ия.
При неравновероятн=
ых
состояниях свобода выбора источника ограничена, что должно привести к
уменьшению неопределенности (и количества информации). Так, если источник и=
нформации
имеет два состояния с вероятностями 0,99 и 0,01, то неопределенность здесь
значительно меньше, чем при равновероятных состояниях с вероятностями 0,5 и
0,5. Значит мера неопределенности должна зависет=
ь от
функции распределения вероятности случайной величины. Еще одно условие сост=
оит
в том, что мера неопределенности не должна зависеть от пути выбора состояни=
я в
ансамбле, в том числе и многоступенчатом по группам состояний. К. Шенноном
высказано утверждение, а Л. Я. Хинчиным математически доказано, что единств=
енный
функционал, удовлетворяющий сформулированным условиям, есть
(Шеннон, 1946 г). Сравнивая это выражение с выражением
Хартли
для равновероятных состояний, видим, что при разных
вероятностях происходит усреднение количества информации по всем возможным
состояниям.
Выражение Шеннона
совпадает с выражением Больцмана для энтропии физической системы. В частнос=
ти,
по второму закону термодинамики энтропия замкнутой системы молекул
, где mN - общее число молекул, m=
i
- количество молекул со скоростями в интервале
, т. к.
- вероятность такого состояния. В обоих случаях величина H
характеризует степень разнообразия состояний системы. Максимальное значение
энтропии при равновероятных состояниях растет с ростом числа этих состояний.
Энтропия объединения нескольких статистически независимых источников равна
сумме энтропии исходных источников:
т. к.
Энтропия характеризует среднюю неопределенн=
ость
выбора одного состояния из ансамбля.
Для статистически
зависимых ансамблей UV, описываемых совместной вероятностью,
где
- условная вероятность.
где
,
- условная энтропия.
Энтропия объединен=
ия
двух статистически связанных ансамблей U и V равна безу=
словной
энтропии одного ансамбля плюс условная энтропия другого относительно первог=
о.
Энтропия непрер=
ывного
источника информации бесконечна, т. к. неопределенность выбора из
бесконечно большого числа возможных состояний бесконечно велика.
Производя предельн=
ый
переход в формуле Шеннона
, получим
Вторая часть бесконечна, но она не зависит =
от
плотности вероятности. Первая часть называется дифференциальной энтропией <=
sub>
Ее можно трактовать как среднюю неопределенность выбо=
ра
случайной величины U по сравнению со средней неопределенностью выбора случа=
йной
величины, изменяющейся в диапазоне, равном единице, и имеющей равномерное
распределение.
При статистически
связанных непрерывных источниках, дифференциальная усло=
вная
энтропия
Дифференциальная энтропия зависит от масшта=
ба
(единицы измерения). Для непрерывного источника информации, ввиду бесконечн=
ости
его меры неопределенности, понятие количества информации имеет несколько
подходов. Наиболее простой из них - замена непрерывного сообщения дискретны=
м с интервалом
дискретизации по Котельникову
что позволяет полностью восстановить непрерывное
сообщение, а следовательно, не потерять информац=
ию.
Прямой же подход
определяет количество информации как разность априорной и апостериорной
дифференциальной энтропии, последняя не равна нулю из-за некоторой
неопределенности, оставшейся после регистрации события вследствие помех и
конечной величины точности регистрации. При таком определении количества
информации исчезает бесконечность после вычитания постоянного бесконечного
члена.
Информационная емкость =
дискретного и непрерывного
сигналов
Для двоичного сигн=
ала
имеем два равновероятных состояния
бит-информации при получении одного импульса. Количес=
тво
полной информации в сообщении при длительности сигнала tc
(длительности передачи сообщения):
бит. Предельная скорость передачи информации, считая,=
что
уровень сигнала существенно превышает уровень помех,
бит/сек. При числе возможных равновероятных уровней L:
бит, а предельная скорость передачи информации <=
v:shape
id=3D"_x0000_i1625" type=3D"#_x0000_t75" alt=3D"" style=3D'width:65.25pt;h=
eight:30.75pt'>
, если разница между уровнями больше помехи.
Непрерывный сигнал
Заменяем непрерывн=
ый
сигнал дискретным с интервалом между выборками по Котельникову:
. При полной длительности сигнала
число выборок
Найдем максимально возможное число выборок L п=
ри
наличии шума. Пусть
- средняя мощность сигнала,
- средняя мощность помехи (шума). Тогда
В этом случае информационная емкость сигнала <=
/p>
Разделив последнее выражение на
, получим предельную скорость передачи =
информации
по Шеннону
Информационные возможности сигнала возрастают с
расширением его спектра и превышением его уровня над уровнем помех.
ВОПРОСЫ ДЛЯ САМОПРОВЕРКИ К ГЛАВЕ II.
1. Покажите формы представления детерминированных
сигналов.
2. Что представляет собой спектр периодического сигн=
ала?
3. Чему равны комплексные амплитуды гармони=
ческих
составляющих ряда Фурье?
4. Укажите характерные особенности АЧХ спек=
тра
периодической последовательности импульсов.
5. Напишите спектральную плотность прямоугольного
импульса.
6. Как изобразить спектр видеоимпульса при амплитудн=
ой
модуляции?
7. Какова связь между длительностью импульса и ширин=
ой его
спектра?
8. Дайте определение девиации частоты и индекса угло=
вой
модуляции.
9. Характеризуйте спектр ЧМ сигнала при мал=
ых и
больших индексах одноканальной модуляции.
10. Как определяется автокорреляционная фун=
кция
сигнала и как она связана с его энергетическим спектром?
11. Как определяется амплитуда, угол и част=
ота
радиосигнала при одновременной, амплитудной и частотной модуляции?
12. Укажите типы импульсно-модулированных сигналов.<=
/p>
13. Покажите структуру спектра
амплитудно-импульсного модулированного сигнала при однотональной модуляции.=
14. Чему равен интервал дискретизации сигналов по
Котельникову и как восстанавливается непрерывный сигнал по его отсчетам?
15. Напишите формулу Шеннона для предельной
скорости передачи информации сигнала с шириной спектра fmax<=
/sub> .
Задачи
1. Покажите, что ряд Фурье пилообразного
колебания имеет вид:
, где
.
2. Найти спектральную плотность сигнала: =
p>
при
Ответ:
.
3. Найти спектральную плотность сигнала:
Ответ:
.
4. Найти спектральную плотность сигнала
при
.
Использовать формулу:
.
Ответ:
.
5. Найти спектральную плотность сигнала и е=
го
форму на выходе интегрирующей цепи:
при
.
Ответ:
.
6. Найти спектральную плотность сигнала:
при
.
Ответ:
.
7. Найти спектральную плотность сигнала
при
.
Указание: Использовать формулу:
.
Ответ:
.
8. Использовать ответы задач 1 и 7 подтверд=
ите,
что если периодическая последовательность образована повторением импульса <=
i>s(t) с
известной спектральной плотностью
, то комплексная амплитуда n-го члена ря=
да
Фурье
, где Т – период последовательности,
- основная частота.
9. Найти спектральную плотность сигнала
при t>0.
Указание: Использовать формулу:=
.
Ответ:
.
10. Имеется группа, состоящая из n
одинаковых импульсов. Покажите, что спектральная плотность этой группы
,
- спектр одиночного импульса с длительностью <=
sub>
.
Указание: Воспользуйтесь свойством
преобразования Фурье при сдвиге сигнала во времени и формулой суммирования
геометрической прогрессии
.
3. РАДИОТЕХНИЧЕСКИЕ ЦЕПИ=
b>
3.1.
Квазистационарность тока. Законы Кирхгофа.
Элементы=
цепей
Радиотехнические ц=
епи –
это цепи, по которым распространяются и передаются сигналы, т. е. это
электрические цепи с высокочастотными токами большого диапазона частот.
Электрическая цепь
возникает, если в пространстве создать достаточно узкие пути для электричес=
кого
тока, располагая вдоль этих путей проводники из материалов с высокой
электропроводностью, окруженных хорошо изолирующей средой. Вдоль цепи возмо=
жно
также помещение элементов цепей, т. е. ограниченных по объему проводящих
устройств (сопротивления, электронные лампы, полупроводники), либо так же
ограниченных по объему устройств с локальными концентраторами электрических=
и
магнитных полей (конденсаторы, индуктивности).
Основными пассивны=
ми (т.
е. не содержащими внутри источников энергии) элементами являются:
а) Активное
сопротивление R – элемент, в котором происходит необратимая потеря
электрической энергии:
, т. е. закон Ома выполняется и для переменных токов;
б) Ёмкост=
ь
– элемент, в котором протекание тока сопровождается накапливанием зарядов на
обкладках, а энергия от источников ЭДС переходит в энергию электрического п=
оля
между обкладками.
По определению,
Если емкость постоянна
,
- последнее верно и при
изменяющейся емкости;
в) Индуктивность – элем=
ент, в
котором протекание тока сопровождается переходом электрической энергии в
энергию магнитного поля.
По определению,
, где Ф – создаваемый
током I магнитный поток. Потеря нап=
ряжения
на индуктивности:
;
.
При постоянной индуктивности:
.
Реальные элементы
являются комбинацией рассмотренных элементов R, L, C.
Цепи по характеру
преобразования в них сигналов делятся на линейные с постоянными параметрами,
линейно-параметрические и нелинейные цепи.
Линейные цепи=
i> –
цепи, в которых все элементы линейные, т. е. параметры не зависят от
значений напряжения и тока. Если эти параметры не изменяются во времени, то
цепи называются линейными с постоянными параметрами.
Линейно-парамет=
рические
– цепи, в которых содержатся элементы, зависящие от времени за счет управле=
ния
внешним воздействием, но не зависящие от тока и напряжения.
Нелинейные –
цепи, содержащие хотя бы один нелинейный элемент, параметры которого завися=
т от
процессов, протекающих в них (уровни тока и напряжения). Нелинейные цепи
описываются нелинейными дифференциальными уравнениями.
Принцип суперпозиции для линейной цепи=
Вследствие линейно=
сти
дифференциальных уравнений, описывающих процессы, происходящие одновременно=
в
одной линейной цепи, решения для каждого из этих процессов будут независимы=
от
других. Итак, результаты нескольких одновременных воздействий (сигналов) на
линейную цепь можно представить в виде суммы результатов отдельных воздейст=
вий,
рассматриваемых независимо один от другого. Для нелинейных цепей принцип
суперпозиции (наложения) не выполняется.
Важным условием при
исследовании цепей является условие квазистационарности токов: переменные т=
оки
называются квазистационарными, если они являются замкнутыми и обладают
одинаковой силой во всех сечениях неразветвленных участков цепи в любой мом=
ент
времени. Основным условием выполнения квазистационарности является медленно=
сть
изменения тока за время распространения электромагнитного возмущения по цеп=
и,
т. е. длина волны распространяемого электромагнитного колебания должна быть
больше общей длины цепи. При выполнении этого условия обычно можно пренебре=
чь
током смещения не только внутри проводников, но и вокруг них при условии
замкнутости цепи (пример незамкнутой цепи с током – антенна).
Пример. При=
f =
=3D 30МГц
(в вакууме) – реальные радиотехнические цепи ;
при f =3D 50Гц ,
- сверхдальние линии электропередач.
Законы Кирхгофа
Будем исходить из уравнений
Максвелла в интегральной форме.
- закон полного тока,
- закон электромагнитной индукции.
- линии электромагнитного поля прерываются только на
зарядах.
- непрерывность линий магнитной индукции (отсутствие
магнитных зарядов).
I закон Кирхгофа<=
/i>
Используем второе
уравнение Максвелла в интегральной форме, продифференцируем по времени,
получим:
,
но
,
.
Здесь предполагалась неизменность цепи во в=
ремени
(dS=
span>/dt =
=3D 0).
Итак, получаем зак=
он
непрерывности тока:
, т. е. полный ток через любую замкнутую поверхность равен =
нулю.
|
Охватывая такой повер=
хностью
узел цепи и пренебрегая током смещения по сравнению с током проводимости в
проводах (что верно вплоть до инфракрасной области частот), получим при
выполнении квазистационарности для замкнутой цепи:
.
Сумма токов
проводимости в узлах равна нулю.
|
II Закон Кирхгофа=
Используем второе уравнение
Максвелла для замкнутого контура:
.
Здесь в пра=
вой
части стоит сумма индукционных ЭДС.
Если из полного
интеграла вычесть участки внутри сторонних ЭДС, =
то
;
.
Сумма падений напряжения во всех ветвях люб=
ого
замкнутого контура равна сумме ЭДС (сторонних и индукционных) источников
энергий, действующих в этом контуре. Для практического использования II з=
акона
Кирхгофа при подсчете падения напряжения на участках и
элементах цепи в=
ажно
выполнение условия квазистационарности токов, когда мгновенное значение ток=
ов
на всех участках неразветвленной цепи одинаково.
3.2. Симв=
олическая
форма для синусоидальных токов, действующие значения, мгновенная, полная, а=
ктивная
и реактивная мощности
Для синусоидальных=
токов
и напряжений широко применяется символическая форма записи:
где
- комплексные амплитуды тока и напряжения.
Такая форма записи
совпадает с видом гармоники комплексного ряда Фурье при n > 0. Реальный ток и напряжени=
е есть
действительная часть их комплексных выражений:
.
В электротехнике
синусоидальных токов используется термин действующее значение токов и
напряжений, определяемое как среднеквадратичное значение за период:
;
.
Действующий ток (и
напряжение) численно равен такому постоянному току (и напряжению), при кото=
ром
за один период в активном сопротивлении выделяется такое же количество тепл=
а,
как и при переменном токе:
; =
.
В общем случае при
изменяющемся во времени напряжении U(t) и
токе I(t) мгновенная =
мощность
есть их произведение: =
P(t) =3D I(t)U(t). При положительном
значении P(t) мощность передаетс=
я от
источника к выделенной цепи, при отрицательном значении P(t) мощность возвращается из цепи к источнику.
При периодических =
токах
и напряжениях важно среднее значение мгновенной мощности за период, называе=
мое активной
мощностью:
.
При синусоидальных периодичес=
ких
токах и напряжениях будем иметь:
где
- разность фаз между током и напряжением.
.
Кроме активной мощ=
ности
при периодических токах и напряжениях вводят также полную мощность,
определяемую как произведение действующих значений токов и напряжений:
. Размерность полной мощности – вольт-ампер. Отношение акти=
вной
мощности к полной называется коэффициентом мощности:
. При синусоидальной форме
. Иногда вводят еще реактивную мощность
, измеряемую в ВАР – вольт-ампер реакти=
вные.
Активная, реактивная и полная мощности связаны соотношением:
.
Для повышения акти=
вной
мощности в энергетических целях при тех же токах и напряжениях необходимо
увеличивать коэффициент мощности
.
3.3. Двухполюсники. Комплексное сопротивление двухполюсника<=
/b>
Двухполюсник –
это электрическая цепь, имеющая два выхода. Двухполюсники, содержащие источ=
ники
электрической энергии, называют активными, а без таких источн=
иков
– пассивными. Каждый пассивный двухполюсник характеризуется одним
параметром, устанавливающим связь между потребляемым от источника током и
падением напряжения на нем. В общем случае такая связь может иметь сложную
интегрально-дифференциальную форму. Однако при синусоидальной форме тока (и=
ли
напряжения) линейный двухполюсник обеспечивает такую же синусоидальную форм=
у с
той же частотой и для напряжения на нем (соответственно, тока). Таким образ=
ом,
любой линейный пассивный двухполюсник при
синусоидальных токах может быть заменен комплексным сопротивлением,
устанавливающим связь между комплексными амплитудами напряжения и тока:
, где
;
.
Активный двухполюс=
ник
может быть заменен эквивалентным генератором
с внутренним сопротивлением
.
Эквивалентная
схема двухполюсника:
а
– пассивного, б –
активного
Итак, при синусоид=
альных
токах для комплексных амплитуд тока, напряжения и сопротивления выполняется
закон Ома для участка цепи. Соответственно, для цепи, состоящей из отдельных
двухполюсников, для комплексных амплитуд тока и напряжения выполняются I и II законы
Кирхгофа.
Последовательное сое=
динение
двухполюсников
=
– общий ток по первому закону Кирхгофа.
– по II закону Кирхгофа.=
p>
.
Если =
, то
.
Параллельное соединение двухполюсни=
ков
=
;
;
, где
– комплексная проводимость.
Рассмотрим отдельные типы
двухполюсников:
а)
б)<=
/p>
в) =
г)
|
– последовательный колебательный контур в виде
двухполюсника.
|
;
;
.
При резона=
нсе,
когда
;
, т. е. сильно уменьшается, становится чисто активным, равн=
ым
сопротивлению R.
д) =
|
– паралл=
ельный
колебательный контур в виде двухполюсника.
|
.
При <=
v:shape
id=3D"_x0000_i1757" type=3D"#_x0000_t75" alt=3D"" style=3D'width:78pt;heig=
ht:33pt'>
;
,
и при
,
.
Сопротивле=
ние
параллельного контура при резонансе сильно возрастает и также становится
близким к активному, но существенно большим
сопротивления R.
3.4.
Методы расчета сложных цепей
=
а)
Общий метод
|
Цель:
определение всех токов в ветвях и всех напряжений в узлах относительно
одного. Имеем три независимых узла (выберем 1, 2, 3). Количество неизвест=
ных
токов равно количеству ветвей – 6. Указываем токи с предполагаемыми
направлениями. Если в результате решения получаются отрицательные значени=
я, значит эти токи имеют противоположное направление.=
span>
|
Т. к. урав=
нения
Кирхгофа I
рода более просты, используют все независимые узлы. В качестве недостающих =
трех
уравнений используем уравнения Кирхгофа II рода для трех контуров:
;  =
; &n=
bsp;
.
Получаем систему из шести уравнений для шести токов. =
По
найденным токам легко определить потенциал узлов относительно нулевого.
.
=
б)
Метод контурных токов=
|
Определя=
ем
число независимых контуров n =3D p
– q + 1, где р – число ветвей, q – число узлов. Выбираем контуры т=
ак,
чтобы они охватывали все ветви. В каждом контуре вводятся свои токи. Реал=
ьные
токи в общих (смежных) ветвях будут равны сумме контурных токов соседних
контуров.
|
Для выбран=
ных
контуров записывается уравнение Кирхгофа II рода:
;
- суммы ЭДС в каждом контуре. =
- суммы сопротивлений в каждом контур=
е.
- сопротивление см=
ежных
участков контуров. Знак
положительный, если соседние контурные токи совпадают=
по
направлению.
Таким образом, вместо 6 уравнений решаются 3.
=
в)
Метод узловых напряжений.=
o:p>
|
Данный м=
етод
целесообразно использовать, когда q - 1 меньше n (p > 2(q - 1)).
Узловыми
напряжениями называют напряжения между каждым из q-1
узлов и одним произвольно выбранным опорным узлом.
|
По =
I закону Кирхгофа записывают <=
span
lang=3DEN-US style=3D'font-size:11.0pt;color:black;mso-ansi-language:EN-US'=
>q-1
независимых уравнений, в которых токи заменяют через узловые напряжения и п=
роводимости
ветвей между узлами.=
p>
Обозначим:
-=
задающий
ток 2-го узла (
токов соседних узлов).
. Заменим также
;
.
Введем <=
sub>
;
.
Отсюда нах=
одятся
все узловые напряжения
затем определяются токи в ветвях.
=
г)
Принцип наложения
Пр=
инцип
наложения – следствие принципа суперпозиции для линейных цепей. Контурный т=
ок
равен сумме токов от каждой из ЭДС в отдельности. Остальные источники
замыкаются внутренними сопротивлениями. Контуры без ЭДС заменяют эквивалент=
ными
сопротивлениями.
=
д)
Принцип взаимности
Пр=
инцип
взаимности установлен Максвеллом и следует из метода контурных токов и усло=
вия
Если ЭДС
, действуя в ветви ав, при отсутствии прочих ЭДС
вызывает в другой ветви cd ток
, то такая же ЭДС
, действуя в ветви cd, при отсутствии прочих ЭДС вызывает в ветви ав =
такой
же ток
.
=
е) Метод
эквивалентного генератора=
o:p>
По
отношению к выделенной ветви ав с сопротивлением
вся остальная цепь, содержащая источники ЭДС, может б=
ыть
заменена одним эквивалентным генератором с ЭДС
и внутренним сопротивлением
. ЭДС
есть напряжение
при разрыве цепи, ав – напряжение холостого хо=
да.
определяется по току корот=
кого
замыкания при
Тогда
.
=
ж)
Цепи с взаимной индуктивностью
Ес=
ли
изменение тока в одном из элементов цепи
приводит к появлению ЭДС в другом элементе
, то эти два элемента называются индуктивно связанными.
– коэффициент взаимной инд=
укции,
- число витков в элементе 2, где возникает ЭДС, =
- часть магнитного потока, создаваемого в элементе 1 и
пронизывающего витки второго элемента.
При наличии индуктивной связи
изменение тока во втором элементе, в свою очередь, приводит к такому же
появлению ЭДС самоиндукции в первом элементе:
, причем
.
Ст=
епень
индуктивной связи двух элементов цепи характеризуется коэффициентом связи <=
sub>
, где
-
индуктивности элементов цепи.=
При
синусоидальных токах для комплексных амплитуд будем иметь:
,
а для поте=
рь
напряжения соответственно:
.
При
расчете сложных цепей взаимную индуктивность наиболее просто учесть в методе
контурных токов, добавляя в сопротивления смежных участков цепи
значение
. Плюс ставится, если направление индукционных ЭДС против т=
ока
в данном контуре, а падение напряжения будет положительным. Заметим, что при
последовательном соединении индуктивно-связанных элементов в одной ветви с
одним и тем же током можно ввести общую индуктивность элементов
. Знак минус ставится при встречном включении индуктивносте=
й,
когда их индукционные ЭДС направлены друг против друга.=
p>
3.5.
Преобразование Лапласа
Преобразование Фурье облада=
ет
следующими недостатками:
1. Оно допустимо только для
абсолютно-интегрируемых функций
. Для этого =
S(t) должно
экспоненциально спадать к бесконечности.
2. Обратное преобразование Фурье трудно
интегрируемо в частотной области.
Анализ прохождения
сигналов через линейные цепи значительно облегчается при использовании
преобразования Лапласа, в котором обратное преобразование осуществляется
методом контурного интегрирования на плоскости комплексной переменной =
, где
- некоторое действительное положительное число,
выбираемое так, чтобы функция
была абсолютно интегрируема, т. е.
.
Сравним с преобразованием Фурье:
Как видно, формаль=
но
переход от преобразования Фурье к преобразованию Лапласа осуществляется зам=
еной
, при этом рассматривают только положительные значения врем=
ени
t. За счет множителя
под интегралом в выражении для S(P) преобразование Лапласа
возможно и для неинтегрируемых функций s(t). Обратное преобразование Лапласа
легко осуществляется с помощью вычетов:
.
Если функция
имеет в точке
полюс кратности m, то
.
В частности, если =
и имеет полюс первого порядка
m =3D 1, т=
о
.
Свойства преобразования Лапласа аналогичны
свойствам преобразования Фурье.
В частности, если
, то
;
при
,
при
.
Таким образом, при
дифференцировании функций и нулевых начальных условий образ умножается на <=
span
class=3DGramE>Р, при интегрировании – делится на Р.
Некоторые
примеры (
)
3.6. Расчет линейных цепей п=
ри
несинусоидальных ЭДС
и
переходных процессах
При
несинусоидальных, но периодических ЭДС, действую=
щих в
ветвях, их раскладывают в ряд Фурье:
.
Затем производится расчет комплексных сопро=
тивлений
ветвей для каждой гармоники:
.
Далее производится
расчет уже рассмотренными методами всех узловых напряжений и токов в ветвях=
для
каждой гармоники в отдельности:
Здесь «m» означает амплитудное значение напряжения и тока.<=
/p>
Реальные мгновенные
значения токов и напряжений вследствие верности принципа суперпозиции для
линейных цепей будут равны сумме значений всех гармоник:
При импульсных сигн=
алах
или переходных процессах в цепях (например, включение и выключение ЭДС) спе=
ктр
таких сигналов сплошной, определяемый спектральной плотностью. В этом случае возможно находить спектральную плотность напряж=
ений в
узлах и токов в ветвях при известной зависимости комплексных сопротивлений
ветвей от частоты и затем с помощью обратного преобразования Фурье определя=
ть
временные зависимости токов и напряжений. Здесь замена преобразования
Фурье преобразованием Лапласа (так называемый операторный метод)
позволяет расширить область временной зависимости сигналов (или ЭДС) на
абсолютно не интегрируемые функции с помощью вычетов. Рассмотрим сопротивле=
ние
элементов цепи в операторной форме:
.
а) Активное
сопротивление
б) Индуктивность <=
/p>
, здесь Z(P) =3D PL, но появляется дополнительная ЭДС «LI(0)», определяемая запасенной энергией магнитног=
о поля
в индуктивности.
в) Емкость
,
,
, здесь
, но также появилась дополнительная
ЭДС «
» за счет запасенной
энергии электрического поля в конденсаторе.
Таким образом,
операторное сопротивление элементов совпадает с комплексным сопротивлением =
при
замене
на
, но при ненулевых начальных значениях токов и напряжений к=
роме
внешних операторных ЭДС необходимо учитывать и внутренн=
ие
ЭДС:
.
Для относительно простых цепей возможно непосредственное решение
интегро-дифференциальных уравнений, получаемых при применении I и II
законов Кирхгофа или уравнений для контурных токов и узловых напряжений (так
называемый классический метод).
Рассмотрим примеры
расчетов переходных процессов в цепях.
Включение
цепи R, =
L, C под постоянное напряжение
|
Пусть в момент времени t<=
span
style=3D'font-size:12.0pt'> =3D 0 включается постоянн=
ое
ЭДС
на цепь R, L<=
span
style=3D'font-size:12.0pt'>, C с нуле=
выми
начальными условиями, т. е. I(0) =3D=
0;
.
|
Запишем II уравнение Кирхгофа для мгновенных значений
напряжения:
.
Дифференцируя обе части, получим уравнение второго пор=
ядка
для тока:
.
Обозначив
для соответствующего однородного уравнения, получим <=
sub>
. Характеристическое уравнение
имеет корни
, решение однородного уравнения:
Общее решение неоднородного уравнения есть
, где
- ток установившегося режима (частное решение). В наш=
ем
случае
, т. к. постоянный ток через емкость пройти не может. Постоянные
найдем из начальных условий:
;
,
т. к.
.
Тогда
;
.
При <=
v:shape
id=3D"_x0000_i1872" type=3D"#_x0000_t75" alt=3D"" style=3D'width:47.25pt;h=
eight:20.25pt'>
, т. е.
,
;
,
.
Применим к этой за=
даче
операторный метод.
При
,
,
,
.
Корни знам=
енателя
. Найдем вычеты от функции
, имеющей два полюса первого порядка:
.
Наглядно в=
идное
преимущество операторного метода особенно проявляется при расчете сложных
цепей.
Размыкание ключом части сложной цепи=
span>
|
Пусть
. В момент t<=
span
style=3D'font-size:11.0pt'> =3D 0 размыкается ключ и вместо него подключа=
ется
сопротивление
|
Составим уравнения для двух
контуров по методу контурных токов в операторной форме.
.
Здесь
,
,
,
,
.
Начальные значения тока в индуктивности и напряжения на емкости
,
,
.
Начальные значения=
тока
в индуктивности и напряжения на емкости определяются при замкнутом ключе, т=
ак
как при его размыкании они изменяться не могут. =
определяется непосредствен=
но.
Для комплексной амплитуды напряжения на емкости можно получить:
где
,
или <=
v:shape
id=3D"_x0000_i1904" type=3D"#_x0000_t75" alt=3D"" style=3D'width:113.25pt;=
height:35.25pt'>
, т. к.
Решив
систему уравнений для контурных токов получим:
;
, где
.
Подставляя=
сюда значения
операторных образов, сопротивлений и ЭДС, получим, например, следующие
выражения для тока
:
где <=
v:shape
id=3D"_x0000_i1912" type=3D"#_x0000_t75" alt=3D"" style=3D'width:306.75pt;=
height:33.75pt'>
.
В полученном выраже=
нии
для
первые два члена определяют ток в переходном процессе=
при
включении цепи под действием ЭДС е1 и
е2. Последние два члена определяют ток переходного процесса,
возникающего в цепи за счет ненулевых начальных значений тока в индуктивнос=
ти Lи напряжения на конденсаторе С. Для пер=
ехода
к временной зависимости тока с помощью вычетов необходимо найти нули
знаменателей:
определяются решением уравнения
.
.
При вычислении вычетов от рациональных дроб=
ей
выражения
значения нулей знаменателя
поочередно подставляются в числители и производные
знаменателей:
3.7. Четырехполюсники<=
/p>
|
Четырехполюсником называют электрическую це=
пь,
имеющую два ввода и два вывода.
|
Четырехполюсники и=
цепи,
состоящие из нескольких четырехполюсников, являются основой тракта передачи=
и
преобразования сигналов, несущих информацию. Теория четырехполюсников дает
возможность единым методом анализировать системы, самые различные по структ=
уре
и принципу действия.
Четырехполюсники
называются активными, если внутри них содержатся источники энергии, и пасси=
вными,
если в них нет источников энергии. Активный четырехполюсник может быть заме=
нен
эквивалентным ему пассивным и вынесенными за зажимы пос=
леднего
эквивалентными ЭДС. Рассмотрим параметры четырехполюсников при установивших=
ся
синусоидальных токах. Несинусоидальные токи могут быть разложены с помощью
преобразования Фурье на гармонические составляющие. Установим зависимости,
связывающие между собой входные и выходные напряжения и токи:
В зависимости от того, какая пара из них будет заданной, мо=
жно
записать шесть различных по форме, но эквивалентных по существу пар уравнен=
ий,
связывающих эти четыре величины. Более всего распространены 4 системы уравн=
ений
четырехполюсника, в каждой из которых используются четыре параметра. При
указанных на схеме направлениях токов эти системы будут иметь вид:
Все системы параметров выражаются друг через
друга, например:
;
;
;
.
Иногда направление=
тока
при рассмотрении четырехполюсника не как части тракта
передачи сигнала, а как самостоятельной части сложной цепи меняют на противоположное к четырехполюснику. При этом параме=
тры
четырехполюсника
, стоящие коэффициентами при токе
, меняют знак.
Заметим,
что параметры
имеют размерность сопротивления, параметры
- размерность проводимости, параметры
- различные размерности:
- сопротивления,
- проводимости,
- безразмерные.
Для пассивных
четырехполюсников выполняется принцип взаимности, устанавливающий связь меж=
ду
входными и выходными напряжениями и токами, а
следовательно и между параметрами четырехполюсника:
либо
либо
.
Таким образом, независимыми остаются только=
три
компонента. Докажем записанное соотношение для параметров
. Замкнем накоротко выход, и для тока на выходе получим:
, т. к.
, теперь перенесем источник питания из входной цепи, закоро=
тив
ее, в выходную цепь. Получим из системы уравнений
, т. к.
. Но, согласно принципу взаимности, эти токи должны быть ра=
вны
по величине и направлению, а с учетом указанных на схеме направлений <=
span
lang=3DEN-US style=3D'font-size:11.0pt;mso-ansi-language:EN-US'>
, что и приводит к соотношению
, т. к. источник тот же самый.
Другим упрощением
является симметричный четырехполюсник, одинаковый по отношению к входным и
выходным выводам. Перенося источник при разомкнутых
противоположных выводах и меняя направления токов, из уравнений для =
Z-па=
раметров
можно получить
, при этом
.
Симметричный пасси=
вный
четырехполюсник имеет независимыми всего два
параметра.
Входные сопротивления
четырехполюсника – это отношение
при заданном сопротивлении нагрузки
легко определяемые экспериментально. В качестве таких
параметров применяют:
1) сопротивление
холостого хода
при разомкнутой выходной цепи. Разделив уравнения для
параметров А первое на второе при
, получим
;
2) сопротивление
короткого замыкания
при замкнутой выходной цепи, т. е. при
. Тогда
.
Для симметричного
пассивного четырехполюсника этих двух параметров достаточно, т. к. существу=
ют
соотношения
.
Для несимметричного
пассивного четырехполюсника необходимы дополнительные измерения, например,
отношение напряжений
при разомкнутой выходной цепи
, либо отношение
также при
, либо оба эти измерения, если четырехполюсник активный.
Интересным парамет=
ром
симметричного четырехполюсника является характеристическое сопротивление
, обладающее свойством повторности, т. е. если его включить=
на
выход в качестве нагрузки, то входное сопротивление также окажется равным <=
span
class=3DGramE>характеристическому. Полагая
, можно получить:
и при
.
Совместным с
характеристическим сопротивлением параметром симметричного четырехполюсника
является постоянная передачи, равная логарифму отношений входного и выходно=
го
напряжений при сопротивлении нагрузки, равном характеристическому:
,
, где
(непер) - коэффициент затухания,
(децибел);
- коэффициент фазы.<=
/span>
Можно показать, чт=
о
.
Соединения четырехполюсников. Матричная
запись уравнений
Уравнения четырехполюсника в матричной форме имеют в=
ид:
;
и др.
Матричная запись оказывается целесообразной при анал=
изе различных
соединений четырехполюсников.
Каскадное соединение
|
Здесь выходные значения сигнала для перво=
го
четырехполюсника являются входными для второго. Происходит умножение матр=
иц:
|
.
При каскадном соед=
инении
симметричных четырехполюсников складываются постоянные передачи
. При одинаковых четырехполюсниках
.
Параллельное соединение
|
При параллельном соединении складываются =
токи,
следовательно, суммируются “Y”-матри=
цы:
|
Последовательное
соединение
|
Здесь складываются напряжения, следовател=
ьно,
складываются “Z”-матрицы.
|
3.8. Передаточная функция и
импульсная
характеристика цепи
|
В радиотехнических цепях сопротивления на=
грузки
обычно велики и не влияют на четырехполюсник либо сопротивление нагрузки
стандартно и уже учтено в схеме четырехполюсника.
|
Тогда четырехполюс=
ник
может характеризоваться одним параметром, устанавливающим связь между выход=
ным
и входным напряжениями при пренебрежении током нагрузок. При синусоидальном
сигнале такой характеристикой является передаточная функция цепи (коэффицие=
нт
передачи), равная отношению комплексной амплитуды сигнала на выходе к
комплексной амплитуде сигнала на входе:
, где
– фазово-частотная характеристика,
- амплитудно-частотная характеристика цепи.
Передаточная функц=
ия
линейной цепи вследствие справедливости принципа суперпозиции позволяет ана=
лизировать
прохождение сложного сигнала через цепь, разлагая его на синусоидальные
составляющие. Другой возможностью использования принципа суперпозиции являе=
тся
разложение сигнала на сумму сдвинутых во времени
d-функций d(t).
Реакцией цепи на действие сигнала в виде d-функций явля=
ется импульсная
характеристика g(t), т. е. это сигнал на выходе, если сигнал на входе есть
d-функция.
при
. При этом g(t) =3D 0 при t < 0 – выходной сигнал не может возникнуть ранее момента
появления входного сигнала.
Экспериментально
импульсную характеристику можно определить подав=
ая на
вход короткий импульс площадью единица и уменьшая длительность импульса при
сохранении площади до тех пор, пока сигнал на выходе перестанет изменяться.=
Это
и будет импульсная характеристика цепи.
Так как независимый
параметр, связывающий напряжения на выходе и входе цепи, может быть
только один, то между импульсной характеристикой и передаточной функцией
имеется связь.
Пусть на вход пода=
ется
сигнал в виде d-функции
со спектральной плотностью
. На выходе цепи будет импульсная характеристика
, при этом все спектральные составляющие входного сигнала
умножаются на передаточную функцию соответствующей частоты:
. Таким образом, импульсная характеристика цепи и передаточ=
ная
функция связаны преобразованием Фурье:
. Иногда вводят так называемую переходную характерис=
тику
цепи =
h(t),
являющуюся откликом на сигнал, называемый единичным скачком:
|
I(t) =3D 1 при t ³ 0
I(t) =3D 0 при t < 0
|
при этом
, h(t) =3D 0 &nb=
sp; при t &l=
t; 0.
Ввиду связи между
передаточной функцией и импульсной характеристикой, на передаточную функцию
накладываются ограничения:
· Условие, что g(t) д=
олжна
быть вещественной, приводит к требованию, что
, т. е. модуль передаточной функции (АЧХ) есть четная, а
фазовый угол (ФЧХ) – нечетная функция частоты.
· Условие, что при t &l=
t; 0, g(t) =
=3D 0
приводит к критерию Пэли-Винера:
.
Например, рассмотр=
им
идеальный фильтр низких частот ФНЧ с передаточной функцией
.
Здесь интеграл в критерии Пэли-Винера расхо=
дится,
как и для любой
, обращающейся в нуль на конечном отрезке оси частот.
Импульсная
характеристика такого фильтра есть
,
g(t) не равна нулю при t < 0, тем сил=
ьнее,
чем меньше время задержки
, которое определяет ее угол наклона
. Это указывает на нереализуемость идеального ФНЧ, имеющего
близкое приближение при достаточно больших
.
3.9. Пере=
даточная
функция соединений четырехполюсников.
Обратная связь
При каскадном соединении, к=
огда
сигнал последовательно проходит несколько четырехполюсников, их передаточные
функции перемножаются:
.
Это непосредственно следует при замене
коэффициентов передачи отношением комплексных амплитуд выходных и входных
напряжений каждого четырехполюсника.
Обратная связь в
четырехполюсниках заключается в том, что часть выходного сигнала подается
обратно на вход и суммируется с входным сигналом.
|
Пусть K(iw) – передаточная функция основного
четырехполюсника,
- передаточная функция четырехполюсника обратной св=
язи.
|
;
.
Отсюда
.
Итак, общий коэффициент пер=
едачи
системы с обратной связью есть
.
Если на заданной ч=
астоте
w выполняется неравенство
, то введение обратной связи уменьшает модуль общего
коэффициента передачи системы. Такую связь называют отрицательной обратной
связью. При обратном неравенстве
в системе реализуется положительная обратная связь,
которая увеличивает амплитуду выходного сигнала. Положительная обратная свя=
зь
ограничена условием
, т.к. при переходе этого неравенства в равенство общий
коэффициент передачи становится бесконечным, а это означает, что система
самовозбуждается, т. е. появляется выходной сигнал в отсутствие сигнала на
входе.
3.10. Примеры расчета передато=
чных
функций и импульсных
характеристик четырехполюсников
Скомпенсированный делит=
ель
напряжения
Мы имеем
последовательное соединение двух сложных двухполюсников
, каждый из которых состоит из двух параллельно соединенных
элементов R
и C=
span>.
;
;
;
;
, отсюда
;
при
и не зависит от частоты.
Интегрирующая цепь<=
/b>
.
Здесь t =3D RC – постоянная интегрирующей цепи.
- АЧХ цепи.
- ФЧХ цепи.
При
. Деление в спектре на “iw” означает интегрирование сигнала.
Интегрирующая цепь является простейшим фильтром низких частот с полосой
пропускания
.
Импульсную
характеристику можно найти через передаточную функцию с помощью обратного
Фурье-преобразования, но можно ее определить и непосредственно, подавая на =
вход
очень короткий прямоугольный сигнал площадью единица (
). Во время действия сигнала емкость заряжается током =
, т. к. напряжение на емкости во время зарядки
. За время импульса емкость получит заряд
и зарядится до напряжения
, т. к.
.
После прохождения
входного импульса емкость разряжается через источник сигнала и сопротивлени=
е R по
экспоненциальному закону: . Действите=
льно,
с решением
. Другим видом интегрирующей цепи является L-R цепь
Д=
ифференцирующая
цепь
, где t =3D RC.
;
.
Действительно, при
подаче на вход сигнала в виде дельта-функции d(t) она проходит ч=
ерез
емкость на выход, при этом емкость заряжается до напряжения
за счет поступления положительного заряда
на противоположную пластину емкости. После прохождени=
я входного
сигнала емкость разряжается по экспоненциальному закону с постоянной
RC.
Последовательный колебательный контур=
&nbs=
p;
Внутреннее сопротивление источника входит в R.
.
Введем параметры контура:
- резонансная частота,
- характеристическое сопротивление,
- добротность, обычно Q>>1.
Тогда
при
,
.
, т. к.
.
,
где
- полоса пропускания контура.
Импульсную
характеристику можно найти, взяв обратное преобразование Фурье от передаточ=
ной
функции. Переходя к преобразованию Лапласа, будем иметь:
,
где
- коэффициент затухания.
Нули знаменателя
, где
, т. к.
.
Импульсная характеристика
имеет вид затухающей синусоиды. За время
, т. е. амплитуда колебаний
уменьшается в двадцать раз, что близко к разрешающей способности осциллогра=
фа.
Таким образом, видимое число колебаний равно добротности контура.
|
За число периодов n =3D Q/p амплитуда
колебаний уменьшается в е=3D2,72 раза.
|
Параллельный
колебательный контур как четырехполюсник
Сопротивление нагрузки
и сопротивление источника сигнала включим в схему
контура.
где
- сопротивление параллельного контура как
двухполюсника
.
, т. к. при
,
.
.
Получаем аналогичн=
ую
частотную зависимость, как и для последовательного колебательного контура, =
но с
эквивалентной добротностью:
, где
– добротность самого контура.
Как видно, все
проводимости, подключенные параллельно к контуру - суммируются и уменьшают
эквивалентную добротность. Параллельный контур можно использовать в качестве
избирательной цепи, если сопротивления источника и нагрузки больше величины=
.
Мост Вина
;
, где Q=3D1/3,
.
Таким образом, мост Вина, не
имеющий индуктивностей, эквивалентен колебательному контуру с резонансной
частотой
, но с низкой добротностью
Тро=
йная
фазосдвигающая RC-цепь
Применим метод узл=
овых
напряжений для первого и второго узлов:
.
Исключим
из системы, предварительно умножив ее на R. П=
олучим:
,
отсюда следует:
,
.
При
, т. е. число действительное и отрицательное.
Такая цепь использ=
уется
для смены знака (переворачивания) синусоидального сигнала, т. е. сдвига фаз=
ы на π.
Двойной Т-образный мост
|
.
Для определения
применим метод узловых напряжений для узлов 3 и 4:<=
/p>
,
|
где
;
;
, или
.
Отсюда легко наход=
ятся
и
, а затем
.
Найдем передаточную
функцию при
:
Двойной Т-образный=
мост
имеет частотную зависимость, обратную колебательному контуру, и не пропуска=
ет
синусоидальный сигнал с частотой
(заградительный фильтр).
Система идентичных индуктивно-связанных контуров=
Запишем
уравнения Кирхгофа второго рода для первичного и вторичного контуров:
.
Здесь М – коэффициент взаимоиндукции. Степе=
нь
связи между контурами определяется коэффициентом связи
.
Исключая ток =
из системы, получим:
;
, где
.
Здесь
;
- расстройка,
.
Подставляя значение
в передаточную функцию, получим:
,
где
.
Найдем экстремумы К(w),
приравнивая к нулю производную подкоренного выражения:
.
При
имеем только один максимум при
,
. При К>d при
будет минимум, а при
- боковые максимумы:
. Провал в центре станет равным
при
, при этом полоса пропускания
, т. е. при той же добротности в три раза шире, чем у
одиночного контура, но при этом будут более крутые спады (за счет зависимос=
ти
), что ближе подходит к идеальному полосовому фильтру.
Резистивный (апериодический) каскад ус=
иления
Сопротивлениями R1 и R2
подбирается постоянное напряжение на базе (затворе), соответствующее
прямолинейному участку характеристики транзистора
, где
– входное переменное напряжение малой амплитуд=
ы в
пределах линейного участка, s -
крутизна характеристики. Напряжение на коллекторе (стоке) транзистора
определяется эквивалентным сопротивлением, равным обратной величине сумм вс=
ех
проводимостей, соединяющим коллектор с источником питания Е и землей, т. к.=
для
переменного сигнала источник питания эквивалентен земле. Емкость С ставится=
для
отделения нагрузки от постоянного напряжения на коллекторе.
– паразитная емкость.
Эквивалентная схема усилителя приведена справа, имеем:
&=
nbsp;
.
Знак «минус» стоит, т. к. при увеличении тока напряж=
ение
на коллекторе уменьшается. Заменяя сопротивления проводимостями
;
;
, получим:
Здесь пренебрежено членом
, т. к.
.
;
, где
;
.
Минимальная частота определяется разделител=
ьной
емкостью «С», максимальная частота – паразитной емкостью «С 0».
ВОПРОСЫ ДЛЯ САМОПРОВЕРКИ К ГЛАВ=
Е III.
1. Что
представляет собой комплексная амплитуда гармонического сигнала?
2. Какова =
связь
между током и напряжением на емкости и индуктивности в общем случае зависим=
ости
ЭДС от времени и при гармонической ее форме?
3. Что так=
ое
комплексное сопротивление двухполюсника и чему оно равно для емкости и
индуктивности?
4. Укажите
условия применимости законов Кирхгофа и Ома при расчетах сложных цепей.
5. В чем
заключается идея методов контурных токов и узловых напряжений при расчетах
сложных цепей?
6. Дайте
определение передаточной функции и импульсной характеристики четырехполюсни=
ка.
Какова связь между ними?
7. Как вли=
яет
внутреннее сопротивление источника сигнала на добротность последовательного=
и
параллельного контура?
Задачи
1. Найти
передаточную функцию цепи:
Ответ:
.
2. Найти передаточную функцию цепи:
Ответ:
.
3. Найти передаточную функцию цепи:
Применить метод контурных токов.
Ответ:
.
4. Найти передаточную функцию цепи:
Применить метод узлового напряжения.
Ответ:
.
5. Найти передаточную функцию цепи:
Применить метод контурных токов.
Ответ:
.
6. Найти передаточную функцию цепи:
Применить метод контурных токов.
Ответ:
.
7. Рассчитать параметры колебательного конт=
ура
,
R=3D10 ом; L=3D10-3Гн; C=3D0,4 мкф.
Ответ:
.
8. Найти передаточную функцию цепи.
Ответ:
.
9. Найти передаточную функцию цепи.
Ответ:
.
10. Найти передаточную функцию цепи.
Ответ:
.
4. МЕТОДЫ АНАЛИЗА
ПРОХОЖДЕНИЯ СИГНАЛОВ
ЧЕРЕЗ ЦЕПИ
4.1. Спектральный ме=
тод
На
основании принципа суперпозиции, сигнал на входе линейной цепи можно разлож=
ить
на сумму более простых сигналов заданной формы, проанализировать прохождение
каждого из них через цепь, а на выходе цепи сложить. В спектральном методе
сигнал на входе представляется в виде суммы синусоидальных сигналов, реакци=
ей
цепи на каждый из них является передаточная функция цепи. Тогда анализ
прохождения сигнала через цепь состоит в следующем:
1. Находим спектр входного си=
гнала.
- интеграл Фурье.<=
o:p>
- ряд Фурье.<=
/o:p>
2. Умножаем спектраль=
ную
функцию сигнала на входе цепи на передаточную функцию цепи, получаем спектр
сигнала на выходе:
3. Определяем сигнал на выходе по его спектру.
или
.
Частные случаи:
а) Полоса пропускания цепи
значительно уже спектра сигнала:
, где
- центральная частота полосы пропускания цепи.=
o:p>
б)=
Полоса
пропускания цепи значительно шире спектра сигнала:
, где
- центральная частота спектра сигнала.
Условия отсутствия
искажений
Ес=
ли
считать, что информация, заключенная в сигнале, отображается его формой, то=
для
неискаженной передачи сигнала необходимо, чтобы
, где К – коэффициент
пропорциональности,
- время запаздывания сигнала. Коэффициент передачи це=
пи в
этом случае должен удовлетворять следующим требованиям:=
p>
1)
, т. е. амплитуды всех частотных составляющих сигнала должны
изменяться в одинаковое число раз.
2)
, т .е. фазы всех спектральных
составляющих при запаздывании сигнала должны изменяться пропорционально их
частоте.
Докажем эти условия:=
o:p>
;
;
;
.
4.2=
. Метод
интеграла наложения Дюамеля
|
В
методе интеграла наложения в качестве характеристики цепи используется
импульсная характеристика цепи g=
(t), являющейся реакцией цепи на сигнал в виде
дельта-функции δ(t=
).
Сигнал произвольной формы разбивается на узкие элементарные импульсы
.
|
Площадь такого элементарного
сигнала
.
Ве=
личина
отклика в момент
на такой элементарный импульс будет
. Для определения полного значения выходного сигнала в моме=
нт t нужно просуммировать действия всех импульсов в проме=
жутке от
до
.
- интеграл Дюамеля=
.
К
сожалению, вычисление интеграла Дюамеля не всегда представляет легкую задач=
у,
поэтому спектральный метод используют чаще.
4.3=
. Корреляция
сигнала на входе и выходе четырехполюсника
Пу=
сть на
входе четырехполюсника с передаточной функцией
и импульсной характеристикой g(t)
действует сигнал
, обладающий спектральной плотностью
и автокорреляционой функцией
. Требуется найти автокорреляционную функцию выходного сигн=
ала
и взаимно корреляционную функцию
.
.
Произведению спектральных ф=
ункций
соответствует свертка их оригиналов
, где=
span>
- автокорреляционн=
ая
функция сигнала на входе.
- автокорреляционн=
ая
функция импульсной характеристики цепи.
Найдем взаимно корреляционн=
ую
функцию выходного и входного сигнала:
- согласно выражен=
ию для
спектра взаимно корреляционной функции.
- комплексно-сопря=
женная
функция
. Заменяя
, получим
. Произведению двух спектральных функций соответствует свер=
тка
оригиналов
.
Рассмотрим два кр=
айних
случая
1. Полоса пропускания цепи
значительно уже спектра сигнала:
;
.
2. Полоса пропускания цепи значительно шире спектра сигн=
ала.
.
4.4. =
Анализ
прохождения прямоугольного импульса
через=
интегрирующую
цепь
а. Спектральный=
метод
Коэффициент передачи цепи
;
.
1. Спектр =
сигнала
на входе:
.
2.<=
span
style=3D'font-size:7.0pt'> Спектр сигнала на выходе:
.
3. =
Импульс на выходе:
.
Нули знаме=
нателя:
.
При
- сигнал на в=
ыходе
равен интегралу сигнала на входе.
Здесь
- единичная функция,
действующая при
:
при
=3D0; при
=3D1. Появление множителя
в спектре сигнала согласно свойству преобразования Фу=
рье
и Лапласа указывает на задержку сигнала во времени на=
span>
.
б. Метод интеграла
Дюамеля
Цепь
характеризуется импульсной характеристикой
.
.
а) =
.
б) =
.
4.5.
Прохождение прямоугольного импульса через =
p>
=
дифференцирующую
цепь
Применим спектральный метод:<=
/p>
;
.
.
.
;
.
При
возникают только короткие всплески на
фронтах импульса, т. е. происходит дифференцирование сигнала.
4.6.
Воздействие контура на единичный скачок (передача
пр=
ямоугольного
сигнала через колебательный контур)
;
.
.
Используем табличный интеграл
.
.
при n =3D Q
.
Время установления
(затухания):
. При передаче прямоугольного импульса
затухающие колебания возникают на обоих его фронтах.
|
|
4.7. Включение на ко=
нтур
синусоидального напряжения
с частотой, равной
собственной частоте контура
(передача радиоимпул=
ьса
через контур)
при
.
.
.
Первый интеграл много мен=
ьше
второго, т. к. в нем под интегралом находится быстро меняющаяся
знакопеременная функция, а во втором – положительная.=
p>
|
|
Во втором интеграле при
можно усреднять по=
:
При прямоугольном радиоимпу=
льсе
после окончания импульса возникают затухания колебаний с тем же временем
установления
.
4.8.
Прохождение амплитудно-модулированного сигнала
через колебательный =
контур
|
Рассмотрим однотональную
модуляцию
.
.
|
.
;
.
При
прохождении через настроенный контур амплитудно-модулированного сигнала с
однотональной модуляцией искажения формы огибающей спектра не происходит,
уменьшается глубина модуляции. Если сигнал имеет несколько гармоник, то буд=
ет
происходить неодинаковое ослабление колебаний различных пар боковых частот.=
В
результате возникнут частотные искажения сигнала после детектирования, т. е.
изменяются соотношения колебаний различных частот, входящих в управляющий
сигнал. Однако нелинейных искажений, т. е. новых частот после детектировани=
я не
появится.
|
При расстройке контура
относительно несущей кроме изменения глубины модуляции возникают искажения
формы огибающей, она становится отличной от
синусоидальной. Это приводит к возникновению нелинейного искажения
управляющего сигнала после детектирования (появление новых низких частот)=
.
|
4.9.
Прохождение частотно-модулированного сигнала через колебательный контур
При
однотональной модуляции
, где
- девиация частоты=
.
;
.
При m << 1<=
o:p>
|
При m >> 1<=
o:p>
|
=
1. При m
<< 1 и настроенном контуре уменьшается m
и
. При многих частотах
возникают частотные (линейные) искажения.
|
|
2. При любых m возникает паразитная амплитудная модуляция.
Огибающая
.
3. При m
³ 1 даже при настроенном контуре возникают нелинейные искажения, т. к.=
по разному ослабляются составляющие спектра, соответст=
вующие
одной Ω.
4. При расстроенном контуре=
в
любом случае возникают нелинейные искажения.
4.=
10.
Импульсный трансформатор
|
Импульсный трансф=
орматор
применяется для повышения или понижения напряжения широкополосных сигнало=
в.
Он представляет собой частный случай связанных контуров с коэффициентом с=
вязи
,
|
в которых вместо конденсаторов играют роль паразитные ем=
кости.
Для увеличения коэффициента связи обе индуктивности наматывают на один желе=
зный
или ферритовый сердечник, если же сердечник не используется, то обмотки дол=
жны
быть близки друг к другу. В общем случае схема сложна для анализа, поэтому =
рассмотрим
упрощенные частные случаи.
а. Идеальный трансформатор
&=
nbsp;
Пренебрегаем паразитными емкостями, сопротивлением обмоток и индуктивностью
рассеяния.
,
,
,
- коэффициент трансформации.
Т. к.
.
.
б. Низкие ч=
астоты
(передача трансформатором вершины импульса)=
i>
На низких
частотах будем пренебрегать индуктивностью рассеяния и паразитными емкост=
ями.
|
|
Введением
сопротивления
получим следующую
эквивалентную схему:
;
,
где
; =
; =
.
Найдем сигнал на выходе при п=
одаче
на вход длинного
прямоугольного импульса 1(t).
При
:
;
;
.
В общем случае <=
v:shape
id=3D"_x0000_i2355" type=3D"#_x0000_t75" alt=3D"" style=3D'width:189pt;hei=
ght:35.25pt'>
, где
,
- суммарное сопротивление источника и первичной
обмотки.
в. Высокие частоты (передача фронта импульса)=
.
;
.
Введем коэффициент затухания
.
1) d =3D 1
;
, где
или
,
.
,
.
2) d >> 1 , что будет при
, т. е. при
.
;
; =
,
;
С учетом емкости
при d > 1
.
,
.
3) d << 1. Во=
зникает колебательный
процесс, но длительность фронта остается примерно равной
с учетом затухания колебаний.
В общем случае подачи
прямоугольного импульса:
Индуктивность рассея=
ния:
t1; t2 – толщины первичной и
вторичной намотки, Ñ - зазор между обмотками, n1 – число первичных ви=
тков, lm – длина
намотки, Ls=
0
– включает также индуктивность проводов. Полная индуктивность рассеян=
ия
включает также индуктивность нагрузки:
,
sж и lж
- сечение и длина железного сердечника.
&nbs=
p;
Паразитная емкость:
; =
.
При повышающем
трансформаторе иногда считают
.
&nbs=
p;
Насыщение сердечника:
; =
.
Для железа
, для ферритов
.
4.11.
Передача прямоугольного импульса резистивным
ап=
ериодическим
усилителем
Эквивалентная схема (источник тока)
. =
Тогда:
;
;
- постоянная коллек=
торной
цепи;
- постоянная времени
нагрузки.
;
;
.
.
.
.
ВОПРОСЫ ДЛЯ САМОПРОВЕРКИ К ГЛАВЕ IV.
1. В чем состоит сущно=
сть
спектрального метода анализа прохождения сигналов через линейные цепи?=
2. Запишите интеграл Д=
юамеля
и изложите суть метода.
3. При
каком условии интегрирующая цепь с постоянной времени
не искажает прямоугольный импульс с длительностью
?
4. Какова длительность
переходных процессов (время установления) при прохождении видео- и
радиоимпульсов через колебательный контур?
5. Какие искажения
амплитудно-модулированного сигнала возможны при прохождении прямоугольного
импульса через апериодический усилитель?
Задачи
1. Найти сигнал <=
v:shape
id=3D"_x0000_i2431" type=3D"#_x0000_t75" alt=3D"" style=3D'width:47.25pt;h=
eight:18pt'>
при
на выходе интегрирующей це=
пи
спектральным методом.
2. Решить задачу 1 мет=
одом
интеграла наложений.
3. Найти сигнал задачи=
1 на
выходе дифференцирующей цепи спектральным методом.
4. Решить задачу 3. ме=
тодом
интеграла наложений при импульсной характеристике дифференцирующей цепи
.
Указание:
, так как t’ не достигает <=
i>t.
5. а) Радиостанция раб=
отает
на волне 300 м и ее сигнал модулирован по амплитуде
огибающей
. Какой добротностью должен обладать к=
онтур,
настроенный на эту станцию, чтобы боковые частоты сигнала вмещались в полосу
пропускания контура.
б) К последовательному
колебательному контуру L=3D1 мГн, С=3D1 нф, =
span>Q=
=3D100 в
момент подключается сигнал
, где wo-
резонансная частота контура. Найти резонансную частоту, время установления и
амплитуду сигнала на выходе после установления.
6. а) К колебательному
контуру с резонансной частотой 1 мГц и полосой пропускания 20 кГц подключен
амплитудно-модулированный сигнал
(вольты). Найти глубину модуляции на выходе контура.=
span>
б) Добротность контура=
Q
задана. Определить после скольких периодов свободных колебаний амплитуда
уменьшится в n раз.
7. а) Простой параллел=
ьный
контур подключен к источнику ЭДС
. Параметры контура: L=3D1 мГн, С=3D200 пф, =
r=3D4
ом. Внутреннее сопротивление источника ЭДС 100 ком. Найти резонансную часто=
ту и
глубину модуляции на выходе контура, настроенного на частоту несущей.<=
/o:p>
б) После 25 периодов
свободных колебаний колебательного контура амплитуда уменьшилась с 80 В до 3 В. Определить добротность контура.<=
/span>
8. а) На колебательный
контур воздействует телеграфный сигнал, представляющий собой последовательн=
ость
высокочастотных импульсов длительностью 2 мс. Частота заполнения импульсов =
1000
кГц и совпадает с резонансной частотой контура. Определить максимальную
допустимую добротность контура, если за время передачи импульса ток должен
увеличиться до 95 максимального значения.
б) Коэффициент модуляц=
ии
ЭДС, подключенной к контуру, равен 100 Коэффициент модуляции на выходе конт=
ура
– 71 . Контур построен в резонанс с несущей частотой и имеет параметры: L=
=3D2 мГн,
С=3D500 пф, r=3D20 ом. Определить модулирующую частоту.<=
/o:p>
9. Найти сигнал (
при
, 0 при
) на выходе интегрирующей цепи спектральным методом.=
o:p>
Ответ:
при
, , 0 при
.
10. Решить задачу 9 ме=
тодом
интеграла наложений.
5. НЕЛИНЕЙНЫЕ ЦЕПИ И МЕТОД=
Ы ИХ
АНАЛИЗА
5.1. Нелинейные элементы=
b>
Параметры нелинейн=
ых
элементов зависят от уровня сигнала. Нелинейные элементы разделяются на =
резистивные
(сопротивления) и индуктивные (индуктивности и емкости). Наиболее
распространенными резистивными нелинейными элементами являются
полупроводниковые, ламповые и другие приборы, имеющие нелинейную
вольт-ампернуюхарактеристику
. При этом дифференциальная кру=
тизна
зависит от положения точки характеристики.
Нелинейная емкость=
имеет
нелинейную вольт-кулоновскую характеристику q(U), =
при этом
емкость
зависит от положения точки характеристики. Иногда вво=
дят
дифференциальную емкость
. Если
, то
и
.
Примером нелинейной
индуктивности является катушка с ферромагнитным сердечником с током, доводя=
щим
сердечник до магнитного насыщения.
; здесь также вводят дифференциальную индуктивность =
.
Рассмотрим соотнош=
ения
между мгновенными значениями тока и напряжения при нелинейных емкостях и
индуктивностях:
=3D
.
.
С другой стороны,
,
, т. е. как и при линейных L и =
С.
5.2. Аппр=
оксимация
нелинейных характеристик
В случае сложного =
вида вольт-амперной характеристики ее аппроксимируют суммой=
более
простых характеристик.
Аппроксимация степенным полиномом
Для транзистора: <=
span
lang=3DEN-US style=3D'font-size:11.0pt;mso-ansi-language:EN-US'>I – =
ток
коллектора, U - напряжение на базе; для л=
ампы: I – =
анодный
ток, =
U
– напряжение на сетке.
;
;
.
При заданной форме=
вольт-амперной характеристики коэффициенты
определяются положением рабочей точки на характеристи=
ке
.
|
Примеры:
1. Рабочая точка, как и весь сигнал, нахо=
дится
на квадратичном участке (обычно начало характеристики).
|
,
здесь
- дифференциальная крутизна в рабочей точке. Если
- начало характеристики, то при
, I =3D 0.
Тогда
, откуда
.
Таким образом,
характеристика определяется тремя параметрами:
|
2. Рабочая точка является точкой перегиба
характеристики. В точке перегиба все четные производные равны нулю. Для
упрощения часто ограничиваются членом третьей степени:
|
- пунктир на схеме при
отрицательном.
Пусть
- насыщающий сигнал.
При
,
,
отсюда
.
Характеристика определяется тремя параметрами:
Кусочно-линейная аппроксимация
|
Применяется, ког=
да
рабочая точка находится в начале характеристики или левее ее при достаточ=
но
большой амплитуде сигнала. При этом реальная характеристика заменяется дв=
умя
отрезками прямых и определяется рабочей точкой=
, напряжением пересечения прямых
и наклоном характеристики
.
|
Аппроксимация показательной функцией=
span>
В некоторых случаях
(например, кремниевые диоды) начальный участок вольт-ам=
перной
характеристики аппроксимируют показательной функцией.
<=
v:shape
id=3D"_x0000_i2486" type=3D"#_x0000_t75" alt=3D"" style=3D'width:114.75p=
t;height:87pt'>
|
,
где
- обратный ток насыщения,
- температурный потенциал. Для кремниевых приборов<=
sub>
25мВ при Т =3D 300К.
|
Логарифмируя выраж=
ения
для тока, получим
, полагая
,
, получим линейную аппроксимацию
.
Все описанные прие=
мы
аппроксимации применимы и к соответствующим характеристикам реактивных
нелинейных элементов.
5.3. Воздействие гармониче=
ского
сигнала
на нелинейные элементы.
Степенная аппроксимация.
Подставим гармонич=
еский
сигнал
в аппроксимирующий полином
.
, тогда
'>Таким образом=
, ток
содержит кратные частоте воздействия ω1 гармоники, причем
четные гармоники обусловлены четными степенями полинома, нечетные гармоники=
–
нечетными степенями. Наивысший порядок гармоник совпадает со степенью полин=
ома.
Общее выражение для амплитуды n-й гармоники при высшей степени полинома m
следующее:
.
В верхнем пределе суммирования используется
только его целая часть.
Кусочно-линейная аппроксимация
При гармоническом
возбуждении ток принимает импульсную форму. Угол Θ, соответствующий
изменению тока от максимума до нуля, называется углом отсечки.
;
.
, или
.
Найдем коэффициент=
ы ряда
Фурье для периодической последовательности таких импульсов тока.
При n>1 коэффициенты Берга <=
v:shape
id=3D"_x0000_i2511" type=3D"#_x0000_t75" alt=3D"" style=3D'width:60.75pt;h=
eight:35.25pt'>
равны:
, число гармоник бесконечно.
В частности,
.
С повышением номера гармоники, максимум фун=
кций
перемещается в область малых значений Θ =3D 120<=
sup>0/n.
Показательная аппроксимация<=
/i>
I(U)=
=3DI0[exp<=
/i>(aU)=
-1]
Для вычислений спе=
ктра
используют формулу:
, где Jn(m) – модифицированная функция
Бесселя К-го порядка. Если
, то
.
5.4.
Воздействие суммы двух гармонических колебаний
на не=
линейный
элемент со степенной характеристикой
.
Подставляем U=
i> в
выражение для нелинейной характеристики:
+…
Первые два члена
характеристики являются линейными и передают частоты сигнала и постоянную
составляющую. Рассмотрим квадратичный член:
Частоты, образуемые
квадратичным слагаемым, можно записать в форме
:
m =3D 0, n =3D 0,w =3D 0 – постоянная
составляющая;
m =3D 2, n =3D 0, w =3D 2w1=
sub>; m =3D 0,
n =3D 2, w =3D 2w2 – гармоники второго порядка;
m =3D 1, n =3D 1, w =3D w1 =
sub>± w2
– комбинационные частоты второго порядка.
Проводя аналогичные
преобразования над кубическим членом, получим частоты w =3D mw1 ±
nw2 при следующих значениях m и n:
m =3D 1, n =3D 0, w =3D w1; m=
=3D 0, n =3D 0,
w =3D w2 – гармоники первого порядка;
m =3D 3, n =3D 0, w =3D 3w1; =
m =3D 0, n =3D
3, w =3D 3w2 – гармоники третьего порядка;
m =3D 1, n =3D 2, w =3D w1 ± =
2w2;
m =3D 2, n =3D 1, w =3D 2w1 ± w2
– комбинационные частоты третьего порядка.
Выводы
1. Слагаемые четной
степени приносят в спектр тока гармоники четных порядков (как и в случае од=
ного
гармонического колебания) и, кроме того, комбинационные частоты четных
порядков.
2. Слагаемые нечет=
ной
степени кроме кратных гармоник нечетных порядков приносят в спектр
комбинационные колебания нечетных порядков.
3. Максимально воз=
можный
порядок колебаний p =3D m + n равен степени полинома K, т. е. pmax =3D
K.
5.5. Нелинейное
резонансное усиление
В
качестве примера использования нелинейных цепей рассмотрим нелинейный
резонансный усилитель. Линейные усилители имеют малый КПД из-за небольшой д=
оли
переменной составляющей тока I1 от постоянного тока I0, отбираемо=
го от
источника питания. Применяются для усиления слабых сигналов в приемных
устройствах. В передающих устройствах из-за большой мощности важно поднять =
КПД,
что осуществляется переводом в сильно нелинейный режим с отсечкой тока.
Для сохранения структуры усиливаемого сигнала на выходе ставят фильтр=
ы (резонансный или связанные контуры), которые для основн=
ой
частоты имеют большое сопротивление и, следовательно, дадут на выходе больш=
ую
амплитуду напряжения, а для кратных гармоник – малое сопротивление, сравним=
ое с
коротким замыканием, что определяет малую амплитуду высших гармоник в выход=
ном
напряжении.
Принципиальная схема
замещения для первой гармоники<=
/o:p>
резонансного усилите=
ля
Ес=
ли не
учитывать обратной реакции выходного напряжения на ток I1 (внутр=
еннее
сопротивление транзистора R i>> Z=
экв), то=
;
;
где
- эквивалентное сопротивление контура. Введем среднюю
крутизну характеристики для первой гармоники:
|
, причем
.
=
При
учете выходного напряжения на ток (конечность Ri) часть тока ответвляется и мы =
получим:
|
,
где
есть внутренняя проводимость нелинейного элемента,
приведенная к току первой гармоники. Заменяя здесь
, получим
. Отсюда коэффициент передачи
. Здесь Scp и Ri’ зависят от у=
гла
отсечки Θ и, следовательно, амплитуды входного напряжения Е. КПД
усилителя
.
=
span>
за счет увеличения
сопротивления контура может быть доведено до значения, близкого к
. Отношение
максимально (близко к двум) при м=
алых
Θ, однако при этом снижается амплитуда первой гармоники I1. Практичес=
ки
принимают Θ ≈ π/2, при этом угле α1, а следовательно и
, слабо зависит от Θ, а следовательно, и от ампли=
туды
входного сигнала. В этом случае
,
. При наличии амплитудной модуляции КПД снижается на множит=
ель
1/(1 + M), где М – глубина модуляции. Уг=
ловая
модуляция на КПД не влияет.
5.6.
Умножение частоты<=
/p>
На=
личие
в составе тока усилителя, работающего в нелинейном режиме, гармоник, кратных
основной частоте возбуждения, позволяет использовать его в качестве умножит=
еля
частоты. Для этого необходимо настроить нагрузочный колебательный контур на
частоту выделяемой гармоники. Амплитуды высших гармоник растут при уменьшен=
ии
угла отсечки.
, где αn =3D In/Im – n-й=
коэффициент
Берга, определяющий In п=
ри
заданном максимальном токе через прибор Im. Максимум n-го
коэффициента Берга αn max=
sub> достигается при Θ =3D 2π/3n. Для каждой гармоники вводится своя средняя крутизна=
. Соответственно, и внут=
реннее
сопротивление электронного прибора приводится к используемой гармонике
, коэффициент передачи
.
5.7.
Амплитудная модуляция
Модуляция амплитуды высокочастотного колебания может быть достигнута
воздействием модулирующего напряжения на нелинейный резонансный усилитель.<=
o:p>
На=
вход
нелинейного резонансного усилителя подается сумма сигналов: модулирующее
низкочастотное колебание – UМОД (сообщен=
ие) и
высокочастотное несущее колебание с частотой ω0 от независи=
мого
источника (автогенератора).
Модулирующее колебание изменяет положение рабочей точки на вольт-амперной
характеристике и в результате изменяется амплитуда несущих колебаний на вых=
оде.
.
(при пренебрежении=
внутренним
сопротивлением Ri, =
т. к.
).
|
=
Так
как изменение Uмод(t=
) во
времени сопровождается изменением угла отсечки Θ и, соответственно,
α1(Θ), то форма I1(<=
span
lang=3DEN-US style=3D'font-size:11.0pt;color:black;mso-ansi-language:EN-U=
S'>t) отличается от формы еω(t).
|
Искажения могут быть достат=
очно
малыми при правильном выборе пределов изменения угла отсечки (при
Θ≈90о α1(Θ) изменяется слабо) и
работе с не слишком глубокой амплитудной модуляцией (до 40-50 %).
, где
определяется по углам отсечки Qmax=
sub> и Qmin.
Амплитудная модуляция будет осуществляться и при степенной аппроксимации
нелинейного элемента, необходимо только чтобы среднее положение рабоч=
ей
точки U0 находилось на нелинейном участке характеристики. В
результате подачи на вход суммы колебаний несущего ω0 и
модулирующего Ω, на выходе кроме этих частот, будут комбинационные час=
тоты
ω0+ Ω и ω0-Ω, что и
соответствует спектру амплитудно-модулированного сигнала.=
5.8.
Амплитудное детектирование
Амплитудный детектор состоит из нелинейного элемента и =
низко-частотного
фильтра в качестве нагрузки. На детектор подается моду-=
p>
лированный сигнал
.
=
Квадратичное детектирование
осуществляется при малой амплитуде сигнала, полностью укладывающемся на ниж=
нем
квадратичном участке характеристики.
.
Высокочастотные составляющие ω0 и 2ω0
отфильтровываются. Информация содержится в последнем низкочастотном слагаем=
ом.
при условии <=
v:shape
id=3D"_x0000_i2568" type=3D"#_x0000_t75" alt=3D"" style=3D'width:90pt;heig=
ht:33.75pt'>
. Если же вывести U0 на линейный
участок характеристики, детектирования вообще не будет. Рассмотрим тональную
модуляцию:
,
Как
видно, при квадратичном детектировании появляется паразитная двойная частот=
а.
При модуляции несколькими частотами возникают еще комбинированные частоты с
амплитудами, пропорциональными произведению М1М2. Для
уменьшения высших комбинационных гармоник нужно уменьшать глубину модуляции=
М.
Если требуется неискаженное воспроизведение сигналов (музыка), то квадратич=
ное
детектирование нецелесообразно.
Линейное детектирование осуществляет=
ся при
большой амплитуде сигнала при кусочно-линейной аппроксимации детектора.
Пу=
сть
Θ =3D 90о, UH =3D U0,
,
.
Последовательность импульсов коллекторного тока оказывается промодулированн=
ой
по амплитуде. Нулевая составляющая медленно (с частотой Ω) изменяется =
во
времени, причем
.
Вы=
ходное
напряжение детектора
,
т. е. амплитуды сигналов на входе и выходе связаны пр=
ямой
пропорциональностью.
5.9.
Преобразование частоты сигнала<=
/o:p>
Преобразование частоты – сдвиг спектра сигнала по оси частот при сохранении
структуры сигнала. При воздействии на нелинейный элемент двух напряжений из
гетеродина
и от источника
нелинейную характеристику будем аппроксимировать
полиномом второй степени.
.
Фи=
льтр
выделяет необходимую разностную ω0 - ωг или
суммарную ω0 + ωг частоту, при этом спектр
сигнала смещается на частоту гетеродина. Полоса пропускания фильтра должна =
быть
больше ширины спектра модулированного колебания.
5.=
10.
Синхронное детектирование=
Осуществляется при частоте гетеродина, равной частоте несущей сигнала ω=
;0
=3D ωг , при этом разностная час=
тота
ω0 - ωг =3D 0 и спектр выходного сигнала
перемещается на нулевую частоту, т. е. совпадает со спектром модулирующей
функции. Таким образом совершаются преобразовани=
я,
обратные амплитудной модуляции. Получающаяся здесь двойная частота ω0
+ ωг =3D 2ω0 не пропускается фильтром.=
Основные свойства
синхронного детектора:=
1. При слабых сигналах Es <<
EГ, детекторная характеристика остается линейной.<=
/o:p>
2.
Повышенная частотная избирательность радиоприема слабых сигналов на фоне шу=
ма,
т. к. устраняется взаимодействие сигнала с помехой в нелинейном элементе.
3. Напря=
жение на
выходе детектора зависит от разности фаз колебаний гетеродина и сигнала =
20;
=3D Θг - Θs, что требует синхронизации гетеродина с несущей сигна=
ла.
ВОПРОСЫ ДЛЯ
САМОПРОВЕРКИ К ГЛАВЕ V.
1. Чем характеризуется нелинейная цепь?
2. Каковы соотношения между мгновенн=
ыми
значениями токов и напряжений на нелинейных сопротивлениях, емкостях и
индуктивностях?
3. Назовите основные методы
аппроксимации нелинейных характеристик.
4. Какие составляющие возникают при
воздействии гармонического сигнала и суммы двух гармонических сигналов на
нелинейном элементе, аппроксимируемой квадратичным полиномом?
5. Какие составляющие тока будут при
подаче на нелинейный резистивный элемент с кусочно-линейной аппроксимацией =
ВАХ
гармонического напряжения заданной частоты и как вычисляются их амплитуды?<=
o:p>
6. Принцип работы нелинейного
резонансного усилителя, его преимущества. Каков типичный угол отсечки?=
7. Как получить
частоту, кратную заданной, в схеме нелинейного резонансного усилителя?
Каков оптимальный угол отсечки для n–ой гармон=
ики?
8. Объясните принцип получения
амплитудной модуляции в схеме нелинейного резонансного усилителя.
9. Как осуществляется квадратичное и
линейное детектирование амплитудно-модулированного сигнала на нелинейном
элементе?
10. Что такое преобразование частоты и как оно осуществляется?=
11. Объясните принцип работы синхронного детектора.=
6. ЛИНЕЙНО-ПАРАМЕТРИ=
ЧЕСКИЕ
ЦЕПИ
6.1. Особенности
линейно-параметрических цепей=
o:p>
В
линейно-параметрической цепи параметры элементов не зависят от уровня сигна=
ла,
но могут независимо изменяться во времени. Реально параметрический элемент
получают из нелинейного элемента (резистивного, емкостного или индуктивного=
),
на вход которого подается сумма двух сигналов. Один из них – сигнал, несущий
информацию и имеющий малую амплитуду, так что в области изменения сигнала
параметры цепи практически не изменяются. Вторым независимым сигналом
является управляющий сигнал большой амплитуды, который изменяет положение
рабочей точки нелинейного элемента, а следовател=
ьно,
его параметр. Для резистивного элемента таким управляемым параметром являет=
ся
дифференциальная крутизна.
|
.
Малость амплитуды сигнала=
ec=
, сохраняющей линейность цепи, позволяет
как и для линейной цепи ввести импульсную характеристику и передаточную
функцию, которые дополнительно зависят от момента времени t, при котором фиксируется параметр цепи:=
|
, где x – сдвиг времени относительно момента времени =
t. Так как для линейной цепи принцип суперпозиции вере=
н, то
будут верны спектральный метод и метод наложения для анализа прохождения
сигналов через цепь.
;
, причем
и
связаны преобразованием Фу=
рье:
.
В =
то же
время зависимость передаточной функции от времени приводит к появлению в
выходном сигнале новых частот. Действительно, пусть передаточная функция
изменяется по гармоническому закону с частотой Ω:
.
При входном сигналe
;
Та=
ким
образом, как и на нелинейном элементе, появляются комбинированные частоты&n=
bsp;
(ω0+Ω) и (ω0-Ω). Однако кратные
частоты сигнала не возникают. Не возникает также взаимодействие между
отдельными гармониками сложного входного сигнала. В связи с этим в схемах
преобразования частоты сигнала, амплитудной модуляции и синхронного
детектирования, где необходимы только разностные или суммарные частоты,
используют малую амплитуду сигнала. В этом случае эти цепи можно считать
линейно-параметрическими. Другим побочным явлением параметрических цепей
является появление на выходе цепи преобразованного управляющего сигнала, в =
том
числе и при отсутствии входного сигнала, что требует установки соответствую=
щих
фильтров.
6.2=
. Схема
замещения емкости, изменяющейся
по гармоническому за=
кону
По=
дадим
на нелинейную емкость слабый сигнал e(t) с частотой ω и сигнал управления (накачки) бол=
ьшой
амплитуды.
,
, E
<< Ey.<=
/span>
Фи=
льтр Ф1 преграждает путь току частотой Ω в ц=
епь
источника сигнала, фильтр Ф2 – току частоты сигнала ω в цепь
управления. При малой амплитуде сигнала Е цепь считаем линейно-параметричес=
кой.
При dC/dU =3D const,
закон изменения емкости будет также гармоническим:
, где
.
Ток источника сигнала будет равен:<=
/p>
.
Предположим, что Ω =3D 2ω, тогда частота ω + Ω =3D 3 =
69;
через фильтр Ф1 не пройдет. Таким образом остается два члена с одинаковой частотой ω=
; для
тока. Первый член – ωС0Е sinωt=
span>, сдвинутый по фазе на 900 относительно =
span>e(t)=
=3DE cos ωt создает нулевую среднюю активную=
мощность.
Второй член
создает следующее значение мощности, равное полной
мощности, отдаваемой источником сигнала.
где
– эквивалентная
активная проводимость.
&nb=
sp; =
Схема
замещения
Рассмотрим три частных случая значения сдвига фазы =
966;:
1) φ =3D 0, Gэкв =3D 0 - схема работает как, с постоянной емкостью.
2) φ =3D π/2, Gэкв =3D
ΔСω/2>0=
-
здесь источник сигнала отдает энергию источнику накачки. =
3) φ=3D - π/2, =
span>Gэкв=3D - ΔСω/2<0 - энергия передается от ис=
точника
накачки к источнику сигнала.
Аналогичные результаты можно
получить и для периодически изменяющейся индуктивности:=
p>
;
, где
.
6.3.
Одноконтурный параметрический усилитель
Сп=
особность
управляемых реактивных двухполюсников играть роль активных элементов цепи
(источников энергии) послужила основой для создания параметрических усилите=
лей,
используемых чаще всего в СВЧ-диапазоне как входные ступени
высокочувствительных радиоприемных устройств. Основное достоинство
параметрических усилителей – низкий уровень собственных шумов, что связано с
отсутствием в них дробовых флуктуаций тока.
В
простейшей схеме одноконтурного параметрического усилителя нагрузка включае=
тся
параллельно нелинейной реактивной емкости, изменяющейся с частотой W =3D 2w=
при
фазе
Сх=
ема
включает проводимость источника сигнала Gi. Проводимость нагрузки Gн учитывает =
также
проводимость индуктивности L. =
Входной контур L, C0 настроен на частоту сигнала w =3D W/2. =
span>
Пр=
и j =3D
- p/2, Gэкв=3D -DCw/2. Средняя мощность, выделяемая в нагрузке при исто=
чнике
сигнала в виде генератора тока с амплитудой
:
.
Ма=
ксимум
мощности, выделяемой в проводимости нагрузки при отсутствии усиления (Gэкв =3D 0), достигается при GН =3D Gi.
. При включении <=
span
class=3DGramE>Gэкв и GH =3D Gi ,
. Отсюда коэффициент усиления мощности:
.
Условие устойчивости
параметрического усилителя (т. к. Gэкв < 0) =
будет
. Отсюда критическое зна=
чение
коэффициента параметрической модуляции:
.
Зд=
есь Qэкв – добротность контура с учетом Gi и GH =3D Gi. При Gэкв=3D - GH усиление по мощности равно четырем. Недостатком
одноконтурного усилителя является необходимость строгого слежения частоты W=
и
фазы j генератора накачки за частотой и фазой сигнала. Имеется более сложная
схема двухконтурного параметрического усилителя, в которой этот недостаток
устраняется.
Зд=
есь
резонансная частота первого контура равна частоте сигнала, а частота второго
контура должна от нее отличаться
. Частота накачки
Вспомогательный контур вместе с переменной емкостью дает эквивалентную
проводимость, равную
, не зависящую от сдвига фазы между накачкой и сигналом. Кр=
оме
того, при изменении частоты сигнала
можно не изменять частоту накачки, если одновременно в
другую сторону изменять частоту
.Коэффициент усиления по мощности двухконтурного
усилителя
. Здесь
- проводимость первого контура вместе с проводимостью источ=
ника
сигнала,
- проводимость второго контура вместе с проводимостью нагру=
зки.
6.4.
Параметрический генератор-параметрон=
Параметрический генератор можно получить из параметрического усилителя при
модуляции нелинейной емкости выше критического значения.<=
/p>
|
Пусть закон изменения емк=
ости
определяется выражением
, где ω0 –
резонансная частота контура при отсутствии накачки
.
|
Сопротивление R включает в себя сопротивление
индуктивности и сопротивление нагрузки. Запишем дифференциальное уравнение =
для
тока в контуре:
, или переходя к заряду:=
,
, где
.
При
высокой добротности контура Qэкв <=
/i>>> 1 решение будет в виде колебания с частотой =
ω0
и медленно меняющейся амплитудой:
, при этом ξ =3D 0,=
либо
ξ =3D π – два устойчивых решения, отличающихся знаком,
, что легко проверить
подстановкой.
Для
нарастания амплитуды должно выполняться условие µ > 0;
, что совпадает с опреде=
лением
mКР.
Со
временем амплитуда колебаний ограничивается заходом амплитуды на нелинейный
участок характеристики C(t),
расстройкой контура относительно частоты ω0 и ухудшением
условий преобразования энергии накачки.
Параметроны могут применяться в устройствах обработки дискретной информации
из-за двузначности фазы генерируемых колебаний ξ =3D 0 и ξ =3D =
960; при
задании в момент запуска генератора начальной фазы с помощью специального
сигнала (плюс или минус). В емкостном параметроне в качестве переменной емк=
ости
используются два полупроводниковых диода, а индуктивностью контура служит
первичная обмотка трансформатора. Напряжение накачки еу(<=
span
lang=3DEN-US style=3D'color:black;mso-ansi-language:EN-US'>t) подается на диоды синфазно. При этом исключается
прохождение частоты накачки 2ω0 на выход. Благодаря симметр=
ии
устраняется также прохождение колебания ω0 из контура в цепь
накачки. Положение рабочей точки на характеристике p-n переходов задается постоянным напряжением смещения <=
/span>E0.
В индуктивном параметроне к=
онтур
состоит из постоянной емкости и катушек, насаженных на ферритовые сердечник=
и,
магнитная проницаемость которых периодически изменяется током накачки IН. Катушки
включаются встречно.
ВОПРОСЫ ДЛЯ
САМОПРОВЕРКИ К ГЛАВЕ VI.
1. Какие цепи называются линейно-параметрическими?<=
/span>
2. Какие новые частоты возникают при
прохождении гармонического сигнала через линейно-параметрические цепи с
гармоническим управлением?
3. Покажите возможность усиления сла=
бого
гармонического сигнала на параметрической емкости.
4. Каково условие устойчивости
одноконтурного параметрического усилителя?
7. НЕКВАЗИСТАЦИОНАРН=
ЫЕ
ТОКИ.
ЦЕПИ С РАСПРЕДЕЛЕННЫ=
МИ
ПАРАМЕТРАМИ
7.1. Уравнение
телеграфистов. Синусоидальный сигнал=
Гл=
авным
условием выполнения квазистационарности тока, кроме замкнутости цепи являет=
ся
медленность изменения тока по сравнению со скоростью распространения
электромагнитного возмущения по цепи, т. е. длина волны распространяющегося
электромагнитного колебания должна быть много больше общей длины цепи. В эт=
ом
случае токи во всех сечениях неразветвленных участков цепи одинаковы, для
анализа цепи можно использовать законы Ома и Кирхгофа и, кроме того, можно
считать некоторые распределенные параметры в эквивалентной схеме локально
сосредоточенными, например, активные сопротивления, индуктивности даже при =
их
совместном распределении.
Од=
нако
часто встречаются длинные цепи передач сигналов (сотни километров) или лини=
и не
очень длинные, но служащие для передачи сигналов высокой частоты. В этих це=
пях
мгновенные значения тока в различных точках цепи различны, здесь нельзя
применять законы Ома и Кирхгофа, нельзя считать распределенные параметры
сосредоточенными в одном месте, кроме того, здесь становится существенной и
распределенная емкость отдельных элементов цепи друг к другу. Непосредствен=
ное
применение уравнений Максвелла представляет сложную задачу и выполнимо лишь=
в
отдельных частных случаях.
|
В качестве примера рассмо=
трим
коаксиальный кабель, подключенный к источнику синусоидального сигнала
|
Вследствие симметрии задачи токи текут вдоль оси z, при пренебрежении сопротивлением проводников <=
v:shape
id=3D"_x0000_i2648" type=3D"#_x0000_t75" alt=3D"" style=3D'width:36pt;heig=
ht:15.75pt'>
, и остается одна составляющая электрического поля
Магнитное поле имеет одну составляющую
Уравнения Максвелла
 =
; &n=
bsp;  =
; (1=
)
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
(2)
При зад=
анной
симметрии
Тогда =
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp; =
(3)
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
(4)
Беря производные по =
z и заменяя потом первые производные с помощью соседних
уравнений, получим уравнение только для Е и
В.
с решением:=
o:p>
=
&nb=
sp; =
&nb=
sp; =
(5)
=
&nb=
sp;
&nb=
sp; =
&nb=
sp; (6)
где
Таким образом, получаем прямую и обратную волн=
ы,
распространяющующиеся по z со
скоростью
Согласно закону по=
лного
тока
или
Т. о. волнам электромагнитного поля соответствуют вол=
ны
электрического тока в проводнике.
При
учете активного сопротивления проводников анализ на основании уравнений
Максвелла существенно усложняется, в результате амплитуды бегущих волн будут
затухать:
где
- константа, определяемая радиусами и материалом
проводников, а также зависящая от частоты из-за скин-эффекта. Фазовая скоро=
сть
волны с увеличением сопротивления начнет несколько снижаться. При малой тол=
щине
скин-слоя
.
В случае более сложной цепи
(двухпроводная, некоаксиальная линии) задача на основании уравнения Максвел=
ла
точно не решается.
По=
этому
в электротехнике быстрых токов прибегают к упрощению.
|
Распределенную линию разб=
ивают
на участки dz, меньшие д=
лины
волны, и для таких участков применяют теорию квазистационарных токов, т.е.
вводят сосредоточенные
,
|
где
- распределенные параметры на единицу длины, и записы=
вают
для такого элемента законы Ома и Кирхгофа:
.  =
; &n=
bsp;  =
; &n=
bsp;
(7)
С другой стороны
;
,  =
; &n=
bsp;  =
; &n=
bsp;  =
; &n=
bsp;  =
;
(8)
т. к. часть тока ответвляет=
ся на
емкость
.
=
&nb=
sp; =
&nb=
sp; =
(9)<=
/o:p>
.  =
; &n=
bsp;  =
; &n=
bsp;  =
; &n=
bsp;  =
;
(10)
Условие применения принятого
приближения кроме
- малое расстояние между
проводниками
, чтобы не влияла взаимная индуктивность и емкость соседних
элементов.
Дифференцируя (9) по
и подставляя
из (10), получим:
Наоборот, дифференцируя (9)=
по
, а (10) по dz,
получим
Это и есть уравнения телеграфистов.
Рассмотрим синусоидальный с=
игнал
Общее решение линейного ура=
внения
. Имеем прямую и обратную распространяющие волны.
Положим
отсюда
;
Тогда
т. е. a - коэффициент затухания, k - волновой вектор, фазовая скорость волны
При малых потерях на длине волны
,
=
- характеристическое сопротивление линии,
Пример коаксиального каб=
еля:
С - скорость света.
т. е. получаем, как и в точном решении, через уравнен=
ия
Максвелла.
7.2. Частный случай
пренебрежения затуханием =
(короткие линии)
.
Ток найдем из соотношения <=
sub>
,
Пу=
сть
нагрузка находится в точке
Отсюда =
p>
;
, где
- коэффициент согласования.
Зн=
ак у z изменен, т. к. ось направлена в другую сторону.
Коэффициент отражения по напряжению
1)=
.
Согласованная нагрузка
M=3D1;
- отраженная волна
отсутствует.
Входное сопротивление =
.
2)=
. Линия,
замкнутая на конце,
.
Входное сопротивление =
В случае полного отражения =
имеем
точки
где амплитуда
и точки, где
и амплитуда напряжения максимальна - стоячие волны.
|
=
При
заданной длине линии входное сопротивление зависит от ее длины и равно ну=
лю
при
и бесконечности при
|
При
Короткозамкнутая линия малой длины имеет индуктивное
сопротивление.
3)=
. Линия,
разомкнутая на конце
M =3D 0.
Входное сопротивление при д=
лине
линии
.
При
- емкостный характер.
Итак, входное сопротивление=
во
втором и третьем случаях зависит от длины линии. В общем случае =
.
Только при согласованной на=
грузке
Ес=
ли
сопротивление нагрузки сильно отличается от волнового сопротивления линии, =
то
для согласования используют «четвертьволновый трансформатор». Рассмотрим
входное сопротивление линии при длине
т. к.
.
Ес=
ли
выбрать
то данный отрезок согласует нагрузку
с длиной линии, у которой
|
При этом условии на всей
длинной линии вплоть до начала вставки отражений не будет, по линии будет
распространяться бегущая волна.
|
Вс=
тавка
изменяет напряжение на нагрузке, согласуя его с концом линии как согласующий
трансформатор. Отметим, что короткозамкнутая четвертьволновая линия имеет
бесконечно большое входное сопротивление и может использоваться как опора
двухпроводной линии без изоляторов.
|
Другой способ согласования
нагрузки с линией состоит в том, что к некоторому сечению линии (недалеко=
от
конца) подключают короткозамкнутую петлю, представляющую отрезок линии с =
тем
же волновым сопротивлением.
|
Изменяя длину петли =
l и расстояние от места ее подключения до конца линии =
х,
можно полностью согласовать линию с нагрузкой.
|
Проводимость в сечении, г=
де подключена
петля:
|
Отделяя здесь действительны=
е и
мнимые части и приравнивая их поочередно к
и нулю, можно получить:
.
7.3=
. Общий
случай временной зависимости входного сигнала
Для
синусоидального источника постоянной амплитуды и частоты мы получим решение
уравнения телеграфистов в виде двух противоположно направленных волн:<=
/o:p>
где
Амплитуда отраженной волны
зависит от отношения сопротивления нагрузки
к волновому сопротивлению линии
:
Об=
щий
вид уравнения для напряжения показывает, что от источника распространяется
волна со скоростью
с затухающей амплитудой. На конце линии волна отражае=
тся
и распространяется назад, по-прежнему затухая. От переднего конца линии вол=
на
вновь может отразиться и т. д.
Те=
перь
предположим, что на линию, нагруженную на сопротивление Rн, от источника с внутренним сопротивлением Ri, подается сигнал U1(t). Тогда для напряжения в фиксированной точке на лини=
и z можно записать:
или
Напряжение на нагрузке можно
получить, положив z =3D l.
При согласовании линии либо=
на конце
либо вначале =
на нагрузке будет только один сигнал, задержанный на
время
и пропорциональный входному сигналу.
В =
связи
с возможностью появления отраженных сигналов согласование линии с нагрузкой=
при
передаче импульсных сигналов часто является обязательным. Проще
всего это достигается установкой добавочных сопротивлений последовательно (=
при
либо параллельно (при
нагрузке. При несогласованной с нагрузкой линии
отраженные сигналы могут возникать и при переходных процессах, например, при
включении и выключении постоянного напряжения:
|
Rн =3D ¥
|
Rн =3D r
|
Rн =3D 0
|
7.4. Подключение
заряженной линии к активному
сопротивлению=
Формирующие
линии
|
Пусть линия без потерь дл=
иной l, предварительно заряженная до напряжения E, подключается к активному сопротивлению
.
|
Сопротивление нагруз=
ки равно
волновому сопротивлению линии
|
При замыкании ключа в мом=
ент
t =3D 0 в линии возникает и начинает распространяться в ст=
орону
ее начала волна напряжения с амплитудой
и полярностью, противоположной полярности зарядного
напряжения линии. С ней связана волна тока с амплитудой
|
Такой же ток течет через
нагрузку. В момент
волны
и
достигнут разомкнутого начала линии. Волна напряжения
отражается от него без изменения величины и полярности, волна же тока измен=
яет
при отражении свою полярность на противоположную=
. В
момент
отраженные от разомкнутого начала линии волны достига=
ют
сечения, в котором включена нагрузка Rн. Линия=
при
этом полностью разряжается, ток становится равным нулю и переходный процесс
заканчивается.
|
Параметры формируемого на
нагрузке импульса тока следующие:
|
где L и C =
- полная
индуктивность и емкость линии. При сопротивлении Rн=3Dr=
- десятки Ом и E=
- десятки кВ можно получить импульсный ток до 1 кА. Длительность
импульса при L0<=
/sub> =3D 0,5 мкГн/=
м, С0 =3D 50 пФ/м,
мкс/м.
Несогласованная лини=
я
Мо=
щность
первой волны на нагрузочном сопротивлении
|
Даже при сравнительно бол=
ьших
рассогласованиях коэффициент передачи мощности уменьшается незначительно.=
При
изменении
от 0,7 до 1,42
не уменьшается ниже 0,97.
|
С
кабельной линии невозможно получить достаточно мощные импульсы тока
длительностью десятки и более микросекунд и амплитудой выше единиц кА. В эт=
ом
случае используют искусственные формирующие линии, состоящие из сосредоточе=
нных
индуктивностей и емкостей.
, где n - число звеньев.
При этом форма импульса ток=
а на
нагрузке искажается, появляются завал фронтов и колебания на вершине: =
7.5.
Волноводы
С
увеличением частоты потери энергии в длинных линиях возрастают из-за
скин-эффекта в проводниках, увеличения потерь в диэлектрике кабеля и излуче=
ния
двухпроводных линий. Волновод имеет существенно большую=
,
чем коаксиальная линия, поверхность, что уменьшает потери в проводнике при
скин-эффекте, кроме того, отсутствуют диэлектрик и потери в нем. Чаще всего=
применяют
волновод прямоугольного сечения, который можно представить как двухпроводную
линию, закрепленную на четвертьволновых опорах a/2»l/4, расста=
вленных
так часто, что они образуют сплошную трубу.
|
Симметричная форма волнов=
ода,
как и в случае коаксиального кабеля, позволяет получить точное решение
уравнений Максвелла для распространения электромагнитной волны в нем, не
прибегая к уравнениям телеграфистов.
|
Та=
к для
синусоидальной временной зависимости и при пренебрежении потерями комплексн=
ые
амплитуды проекций векторов
и
(обозначим их f=
) удовлетворяют волновому уравнению
где
(при отсутствии диэлектрика). Решение этого уравнения
ищут в виде
и после подстановки получают отдельные уравнения для
функций
Интегрирование их с использованием граничных условий
(равенство нулю перпендикулярных к границе составляющих
и параллельных границе составляющих
) показывает, что в прямоугольном волноводе могут
распространяться два типа волн: поперечно-электрические
-волны
(
и поперечно-магнитные
-волны (
половина длины волны которых должна укладываться целое
число раз в одном из поперечных размеров волновода. Появление конкретной во=
лны
зависит от способа возбуждения волновода. Полностью поперечные волны=
o:p>
(
) в волноводе (в отличие от коаксиальных кабелей)
распространяться не могут. Основной волной является
-волна, являющаяся единственной, если размеры волновода
удовлетворяют условиям
Для этой волны проекции
и
равны:
=
где =
А
- постоянная, определяемая мощностью возбуждения волны;
- волновое сопротивление волновода;
Ом - волновое сопротивление среды (вакуума);
- коэффициент
распространения волны,
Фа=
зовая
скорость распространения волны
,
- скорость волны в среде.
Групповая скорость
т. е.
При
h =
- мнимый
и волна быстро затухает, при
E и H имеют одну и ту же фазу по всей длине волновода и он
работает как поперечный резонатор. Для передачи мощности по волноводу
необходимо, чтобы
.
Конфигурация магнитн=
ого
поля (H --- >) и
электрического тока (I&a=
grave;)
для волны TE10
Возбуждение в волнов=
оде
волны ТЕ10 при помощи штыря (а)
и петли с током (б).=
Для
уменьшения потерь в волноводе его внутренняя поверхность на толщину скин-сл=
оя
покрывается хорошо проводящим серебром. Недостатком волноводов является
фиксированная длина передаваемой электромагнитной волны, т. е. для эффектив=
ной
передачи каждой частоты необходим свой волновод.
ВОПРОСЫ ДЛЯ
САМОПРОВЕРКИ К ГЛАВЕ VII.
1. Какие используют приближения для вывода телеграфных уравнен=
ий?
2. Какие особенности возникают при
передаче гармонических сигналов по длинным линиям в зависимости от нагрузки?
Какая нагрузка называется согласованной?
3. Что происходит при передаче
импульсного сигнала по длинным цепям?
4. Каковы параметры сформированного =
на
длинной однородной линии прямоугольного сигнала =
на согласованной
нагрузке?
5. Как устроен волновод?
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