LR-цепь

LR-цепь — электрическая цепь, состоящая из резистора и катушки индуктивности. Её можно рассматривать как делитель напряжения, в котором одно из плеч представляет собой индуктивное сопротивление переменному току.

Цепь дифференцирующего типа

Если входной сигнал подаётся к $V_{in}$ , а выходной снимается с $V_{L}$ , то такая цепь называется цепью дифференцирующего типа (см. рисунок).

LR-цепь дифференцирующего типа является фильтром верхних частот.

Реакция цепи дифференцирующего типа на «ступеньку» определяется следующей формулой:^[1]

V_{R}(t)=V\left(1-e^{-tR/L}\right)

V_{L}(t)=Ve^{-tR/L}

Таким образом, постоянная времени $\tau$ этого апериодического процесса будет равна

\tau ={\frac {L}{R}}

Переходные процессы в LR-цепи. Вывод формул

LR-цепь на источнике постоянного напряжения

Рассмотрим LR-цепь (см. рисунок). Если в начальный момент времени $t=0$ последовательную LR-цепь подключить к источнику постоянного напряжения $V_{s}$ , перекинув переключатель от вывода 1 к выводу 2, в цепи потечёт ток $i(t)$ . Для времени $t\geq 0$ можно записать уравнение цепи:^[2]

$V_{s}=V_{R}(t)+V_{L}(t)$

$V_{s}=R\cdot i(t)+L{\frac {di(t)}{dt}}$

Решением дифференциального уравнения цепи с начальным условием $i(0)=0$ будет функция, описывающая значение тока в момент времени $t$ :

$i(t)={\frac {V_{s}}{R}}\left(1-e^{-tR/L}\right)$

Напряжение на сопротивлении $R$ будет функцией времени:

$V_{R}(t)=R\cdot i(t)=V_{s}\left(1-e^{-tR/L}\right)$

Напряжение на индуктивности $L$ будет функцией времени:

$V_{L}=L{\frac {di(t)}{dt}}=V_{s}e^{-tR/L}$

См. также

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Примечания

↑ Пример расчёта переходного процесса изложен в статье Операционное исчисление.
↑ В. А. Матвиенко. Основы теории цепей : учебное пособие для студентов высших учебных заведений, обучающихся по направлению подготовки 27.03.04 – Управление в технических системах в УрФО. — Издательство УМЦ УПИ, 2016. — ISBN 978-5-8295-0425-0. Архивировано 20 февраля 2020 года.

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