From 9edebab51975c12143dcee7cab9b3fd25b25a7a8 Mon Sep 17 00:00:00 2001 From: anna Date: Sat, 15 Oct 2022 22:09:10 +0300 Subject: [PATCH] 3.2.5 (not edited) --- 3.2.5Вынужденные_колебания/Отчет.aux | 52 ++ 3.2.5Вынужденные_колебания/Отчет.log | 884 ++++++++++++++++++++ 3.2.5Вынужденные_колебания/Отчет.out | 5 + 3.2.5Вынужденные_колебания/Отчет.pdf | Bin 0 -> 371193 bytes 3.2.5Вынужденные_колебания/Отчет.synctex.gz | Bin 0 -> 40057 bytes 3.2.5Вынужденные_колебания/Отчет.tex | 218 +++++ 3.2.5Вынужденные_колебания/затухание.png | Bin 0 -> 40579 bytes 3.2.5Вынужденные_колебания/установка.png | Bin 0 -> 37178 bytes 8 files changed, 1159 insertions(+) create mode 100644 3.2.5Вынужденные_колебания/Отчет.aux create mode 100644 3.2.5Вынужденные_колебания/Отчет.log create mode 100644 3.2.5Вынужденные_колебания/Отчет.out create mode 100644 3.2.5Вынужденные_колебания/Отчет.pdf create mode 100644 3.2.5Вынужденные_колебания/Отчет.synctex.gz create mode 100644 3.2.5Вынужденные_колебания/Отчет.tex create mode 100644 3.2.5Вынужденные_колебания/затухание.png create mode 100644 3.2.5Вынужденные_колебания/установка.png diff --git a/3.2.5Вынужденные_колебания/Отчет.aux b/3.2.5Вынужденные_колебания/Отчет.aux new file mode 100644 index 0000000..bae30b3 --- /dev/null +++ b/3.2.5Вынужденные_колебания/Отчет.aux @@ -0,0 +1,52 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\providecommand\babel@aux[2]{} +\@nameuse{bbl@beforestart} +\catcode `"\active +\babel@aux{russian}{} +\citation{labnik} +\citation{labnik} +\citation{labnik} +\@writefile{toc}{\contentsline {section}{\numberline {1}Аннотация}{1}{section.1}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2}Введение}{1}{section.2}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {3}Методика измерений}{1}{section.3}\protected@file@percent } +\newlabel{кривые}{{1}{1}{}{equation.3.1}{}} +\newlabel{затух}{{2}{1}{}{equation.3.2}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Нарастание и затухание вынужденных колебаний при подаче цугов (синусоидальные сигналы, разделенные временем без сигнала)}}{2}{figure.1}\protected@file@percent } +\newlabel{затухание}{{1}{2}{Нарастание и затухание вынужденных колебаний при подаче цугов (синусоидальные сигналы, разделенные временем без сигнала)}{figure.1}{}} +\newlabel{теор}{{3}{2}{}{equation.3.3}{}} +\citation{labnik} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Схема экспериментальная установка для исследования вынужденных колебаний: ЗГ - 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b/3.2.5Вынужденные_колебания/Отчет.tex new file mode 100644 index 0000000..f36d5ff --- /dev/null +++ b/3.2.5Вынужденные_колебания/Отчет.tex @@ -0,0 +1,218 @@ +\documentclass[a4paper,12pt]{article} % тип документа + +% report, book + +% Рисунки +\usepackage{graphicx} +\usepackage{wrapfig} +\usepackage{mathtext} +\usepackage[left=2cm,right=2cm, + top=2cm,bottom=2cm,bindingoffset=0cm]{geometry} + +\usepackage{hyperref} +\usepackage[rgb]{xcolor} +\hypersetup{ % Гиперссылки + colorlinks=true, % false: ссылки в рамках + urlcolor=blue % на URL +} + +% Русский язык +\usepackage[T2A]{fontenc} % кодировка +\usepackage[utf8]{inputenc} % кодировка исходного текста +\usepackage[english,russian]{babel} % локализация и переносы +\addto\captionsrussian{\def\refname{Список используемой литературы}} + + +% Математика +\usepackage{amsmath,amsfonts,amssymb,amsthm,mathtools} +\usepackage{titlesec} +\titlelabel{\thetitle.\quad} + +\usepackage{wasysym} + +\begin{document}\begin{titlepage} + +\thispagestyle{empty} + +\centerline{МОСКОВСКИЙ ФИЗИКО-ТЕХНИЧЕСКИЙ ИНСТИТУТ} +\centerline{(НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ УНИВЕРСИТЕТ)} + +\vfill + +\centerline{\huge{Лабораторная работа 3.2.5}} +\centerline{\LARGE{<<Вынужденные колебания в электрическом контуре>>}} + +\vfill + +Студент группы Б02-109 \hfill Назарчук Анна + +\vfill + +\centerline{Долгопрудный, 2022} +\clearpage +\end{titlepage} + +\section{Аннотация} +В работе исследованы вынужденные колебания в электрическом контуре, рассчитана добротность контура различными способами. Измерения проводились на параллельном колебательном контуре с вынуждающей силой, меняющейся гармонически непрерывно или в форме цугов. Получены теоретическое значение добротности, добротность при измерении ширины резонансной кривой, значение при исследовании нарастания колебаний. Рассчитанные значения не совсем идентичны, предложены способы улучшения результатов. + +\section{Введение} +Совсем несложно представить обычные детские качели. В зависимости от их строения, какие-то было легко раскачать, для каких-то требовались усилия. Но если начать двигаться с определенной частотой, то качели быстро начинают колебаться с большой амплитудой. Но, к сожалению, если перестать двигаться, то через некоторое время колебания станут совсем небольшими. Похожие явления наблюдаются и в электрических цепях, состоящих из катушек, резисторов, конденсаторов, то есть основных элементов. Интересно, можно ли численно описать эти эффекты с помощью одной какой-то величины? Ответу на этот вопрос и наблюдению вышеописанных явлений в электрической цепи посвящена работа. + +\section{Методика измерений} +\subsection*{Вводная часть} +<<Раскачкой>> электрической цепи называют вынужденные колебания, в работе рассмотрена вынуждающая сила, гармонически меняющаяся во времени. При подключении к контуру внешнего синусоидального источника в +нём возникают колебания, которые можно представить как суперпозицию +двух синусоид \cite{labnik} первая — с частотой собственных колебаний +контура и амплитудой, экспоненциально убывающей со временем; вторая +— с частотой внешнего источника и постоянной амплитудой. Со временем +собственные колебания затухают, и в контуре устанавливаются +вынужденные колебания. Зависимость амплитуды установившихся +колебаний от частоты внешнего сигнала называется резонансной +кривой. Ширина некоторой высоте (0.707 от максимальной) $\delta \omega$ определяет насколько близко необходимо подобраться к резонансной частоте $\omega_0$, чтобы достаточно сильно раскачать колебания в контуре, и называется добротностью: +\begin{equation} +\label{кривые} +Q=\frac{\omega_0}{\delta \omega} +\end{equation} +После этого необходимо установление колебаний, чтобы понять, как связана добротность контура с этим процессом: +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.55\textwidth]{затухание} +\caption{Нарастание и затухание вынужденных колебаний при подаче цугов (синусоидальные сигналы, разделенные временем без сигнала)} \label{затухание}. +\end{center} +\end{figure} +Нарастание и затухание колебаний (рис. \ref{затухание}) можно наблюдать на +экране осциллографа, если на контур подаются цуги — отрезки синусоиды, +разделённые интервалами, в течение которых сигнал отсутствует. Теоретически известно, что \cite{labnik}, чем выше добротность, тем медленнее нарастают и медленнее затухают колебания в контуре. Получить значение Q можно по формуле \cite{labnik} (обозначения с рисунка \ref{затухание}): +\begin{equation} +\label{затух} +Q = \frac{\pi}{\frac{1}{n}ln(\frac{U_\infty-U_k}{U_\infty-U_{k+n}})} +\end{equation} +Более того, существует теоретическая формула для добротности (рассмотрен параллельный колебательный контур), которая следует из энергетического смысла добротности: +В энергетическом смысле добротность $Q$ колебательной системы (механической, электрической, оптической и т. д.) определяется как отношение запасённой в ней энергии $W_0$ к потере $\Delta W$ энергии за время изменения фазы колебания +на 1 радиан. +\begin{equation} +Q = \frac{1}{R}\sqrt{\frac{L}{C}} +\label{теор} +\end{equation} +\subsection*{Основная часть} +Получено три способа определения добротности (той самой характеристики, которая численно описывает явления затухания, резонанса колебаний), необходимо практически установить, приводят ли эти способы к одинаковому результату. Соответственно, работа состояла из нескольких частей: теоретический расчет добротности по формуле \ref{теор}, исследование резонансных кривых и процесса нарастания и затухания колебаний. + +Исследование резонансных кривых проведено при генерации непрерывной синусоиды, при изменении частоты вынуждающих колебаний в разные стороны от резонансного значения. Построена зависимость амплитуды установившихся колебаний от частоты, из нее по формуле \ref{кривые} получена добротность. + +Затухание и нарастание колебаний наблюдалось при генерации цугов - последовательности синусоидальных сигналов. В условиях резонанса, в которых и исследовалась система, затухание колебаний симметрично их нарастанию. + + +\subsection*{Экспериментальная установка} + +Экспериментальная установка приведена на рис. \ref{установка}. +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.95\textwidth]{установка} +\caption{Схема экспериментальная установка для исследования вынужденных +колебаний: ЗГ - звуковой генератор Г3-112, ген. цугов - генератор цугов ЧЗ-57, V - вольтметр В3-38, $R$ - магазин сопротивлений МСР-60, $L$ - магазин индуктивностей P567, ЭО - осциллограф GOS-620, П - панель} \label{установка}. +\end{center} +\end{figure} + +Колебательный контур состоит из конденсатора $С $ = 0.1 мкФ, катушки с индуктивностью $L$ = 100 мГн и магазина сопротивлений. Для измерения добротности необходимо формировать как непрерывные синусоидальные сигналы (с помощью звукового генератора), так и цуги (с помощью генератора цугов). + +Для визуального наблюдения за процессом колебаний напряжение с конденсатора подаётся на вход электронного осциллографа. Чтобы картина на экране была устойчивой, частота развёртки осциллографа принудительно синхронизуется с частотой повторения цугов. Для этого на генератор развёртки осциллографа подаются следующие с частотой повторения цугов управляющие импульсы, формируемые в блоке электронного реле, клемма «синхр.» которого смонтирована на панели. Эффективное значение напряжение на емкости C измеряется с помощью вольтметра. + +\section{Результаты и их обсуждение} +При непрерывной генерации синусоид получена резонансная кривая в относительных единицах (рис. \ref{график}). + +\begin{figure}[h!] +\begin{center} +\includegraphics[width=0.95\textwidth]{graf} +\caption{Резонансная кривая в относительных единицах: $U/U_m$ - отношение амплитуды к резонансному значению, $\nu/\nu_m$ - отношение частоты к резонансному значению, для параллельного контура $С $ = 0.1 мкФ, $L$ = 100 мГн} \label{график}. +\end{center} +\end{figure} + +Вид зависимости схож с табличными представлениями резонансной кривой для параллельного контура \cite{labnik}. На высоте $U/U_m = 0.707$ вычислена ширина графика, рассчитана добротность системы (по формуле \ref{кривые}). (табл. \ref{кривые_табл}). +\begin{table}[h!] +\caption{Значение добротности системы при различных сопротивлениях, измерения методом резонансных кривых} +\label{кривые_табл} +\begin{tabular}{|l|l|l|} +\hline +R, Ом & Q & $\sigma_Q$ \\ \hline +0 & 20.29 & 0.88 \\ \hline +100 & 6.25 & 0.43 \\ \hline +\end{tabular} +\end{table} + +При генерации цуг на экране осциллографа получены картинки, похожие на рис. \ref{затухание}, однако они не являются абсолютно неподвижными, что не позволило запечатлеть их. Но из измерений некотрых амплитуд колебаний при нарастании колебаний (результаты в приложении в табл. \ref{нарост}). Так как в резонансе (при котором и проведено измерение) затухание и нарастание симметричны, нет необходимости вычислять амплитуды из обоих участков графика. + +Из полученных данных по формуле \ref{затух} рассчитаны значения добротности (табл. \ref{затух_тбл}). +\begin{table}[h!] +\caption{Значение добротности системы при различных сопротивлениях, измерения методом исследования нарастания колебаний} +\label{затух_тбл} +\begin{tabular}{|l|l|l|} +\hline +R, Om & Q & $\sigma_Q$ \\ \hline +0 & 22.49 & 2.11 \\ \hline +100 & 10.09 & 1.46 \\ \hline +\end{tabular} +\end{table} + +По формуле \ref{теор} рассчитано теоретическое значение добротности контура: +\begin{table}[h!] +\caption{Значение добротности системы при различных сопротивлениях, теоретический расчет} +\label{теор_тбл} +\begin{tabular}{|l|l|l|} +\hline +R, Om & Q & $\sigma_Q$ \\ \hline +0 & 33.48 & 1.01 \\ \hline +100 & 7.7 & 0.5 \\ \hline +\end{tabular} +\end{table} + +Сведем рассчитанные значения добротности в единую таблицу \ref{един}. +\begin{table}[h!] +\caption{Результаты измерения добротности различными методами} +\label{един} +\begin{tabular}{|l|l|l|l|} +\hline +R, Om & $Q_{рез.крив.}$ & $Q_{затух.}$ & $Q_{теор.}$ \\ \hline +0 & 20.29 $\pm$ 0.88 & 22.49 $\pm$ 2.11 & 33.48 $\pm$ 1.01 \\ \hline +100 & 6.25 $\pm$ 0.43 & 10.09 $\pm$ 1.46 & 7.7 $\pm$ 0.5 \\ \hline +\end{tabular} +\end{table} + +Из таблицы видно, что данные не совсем совпадают, это можно объяснить методическими проблемами: резонансное значение частоты определяется не совсем точно (ручным способом), поэтому точки на резонансных кривых теряют в точности, еще число точек невелико, поэтому трудно определить ширину резонансной кривой на той или иной высоте. Также при нарастании колебаний амплитуды определяются из изображения на осциллографе, что также добавляет проблемности результату. + +\section{Выводы} + +\hspace{5mm}1. Из эксперементально полученных резонансных кривых, которые видом похожи на теоретически предсказанные, найдены добротность системы, из которых заметна высокодобротность контура ($Q\ll 1$). + +2. Из исследования нарастания колебаний при резонансной частоте рассчитана добротность системы с большей погрешностью, нежели первым методом. + +3. Сравнение экспериментальных значений добротности и теоретического ее расчета показал, что результаты не совсем близки друг к другу. Это объясняется проблемами в методике работы: большая часть измерений проведена вручную, из-за чего их небольшое количество и они имеют, возможно, не до конца верные значения. Данная проблема решается при внедрении автоматизации снятия данных. + + + +\begin{thebibliography}{} + \bibitem{labnik} Никулин М.Г., Попов П.В., Нозик А.А. и др. Лабораторный практикум по общей физике : учеб. пособие. В трех томах. Т. 2. Электричество и магнетизм +\end{thebibliography} + + +\section*{Приложение} +\begin{table}[h!] +\caption{Амплитуды некоторых колебаний при нарастании колебаний при генерации цугов} +\label{нарост} +\begin{tabular}{|l|l|l|l|l|} +\hline +$U_k$, mV & $U_{k+n}$, mV & $U_\infty$, mV & n & R=0 Om \\ \hline +6 & 28 & 56 & 4 & \\ \hline +5 & 21 & 43 & 4 & \\ \hline +4 & 18 & 37 & 4 & \\ \hline +3 & 12 & 26 & 4 & \\ \hline +2 & 10 & 19 & 4 & \\ \hline +$U_k$, mV & $U_{k+n}$, mV & $U_\infty$, mV & n & R=100 Om \\ \hline +2 & 5 & 7 & 3 & \\ \hline +3 & 7 & 10.5 & 3 & \\ \hline +4 & 10 & 13 & 3 & \\ \hline +5 & 13 & 17.5 & 3 & \\ \hline +7 & 18 & 25 & 3 & \\ \hline +\end{tabular} +\end{table} + +\end{document} diff --git a/3.2.5Вынужденные_колебания/затухание.png b/3.2.5Вынужденные_колебания/затухание.png new file mode 100644 index 0000000000000000000000000000000000000000..da0ba62dbb82912f85e00664580aadd2045c91a2 GIT binary patch literal 40579 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