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используемой литературы}} % Математика @@ -29,11 +30,28 @@ \usepackage{wasysym} -\author{Анна Назарчук Б02-109} -\title{3.6.1 Спектральный анализ электрических сигналов} -\date{} -\begin{document} -\maketitle +\begin{document}\begin{titlepage} + +\thispagestyle{empty} + +\centerline{МОСКОВСКИЙ ФИЗИКО-ТЕХНИЧЕСКИЙ ИНСТИТУТ} +\centerline{(НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ УНИВЕРСИТЕТ)} + +\vfill + +\centerline{\huge{Лабораторная работа 3.6.1}} +\centerline{\large{по теме:}} +\centerline{\LARGE{<<Спектральный анализ электрических сигналов>>}} + +\vfill + +Студент группы Б02-109 \hfill Назарчук Анна + +\vfill + +\centerline{Долгопрудный, 2022} +\clearpage +\end{titlepage} \section{Аннотация} В работе исследованы спектры периодических сигналов: модулированный по амплитуде, прямоугольные импульсы и цуги. Проверены теоретические зависимости параметров спектра на практике. @@ -43,17 +61,17 @@ системы (обозначим его как $f(t)$), и её реакцией на "выходе" $g(t)$). Суть спектрального метода состоит в представлении произвольного воздействия в виде суперпозиции откликов на некоторые элементарные слагаемые. Данный метод используются для анализа многих сигналов, поэтому необходимо экспериментально ознакомиться с ним, сгенерировать и получить на осциллографе спектры различных периодических сигналов, проверить экспериментально соотношение неопределенности и отношения амплитуд гармоник при модулированных по амплитуде сигналах. \section{Методика измерений} -В работе используются генератор сигналов произвольной формы, цифровой осциллограф с функцией быстрого преобразования Фурье. Для произвольного сигнала справедливо соотношение неопределенностей: +Для экспериментальной проверки соотношения неопределенностей необходимо его сформулировать \cite{labnik}: \begin{equation} \Delta \omega \cdot \Delta t \sim 2\pi \end{equation} -Работа разделена на три равноценные части, в каждой из которых генерируется сигнал определенной формы, обрабатывается с помощью цифрового осциллографа, проверяются соотношения неопределенности с помощью курсорных измерений. +Для проверки соотношения неопределенностей работа разделена на три равноценные части, в каждой из которых сгенерирован сигнал определенной формы, обработан с помощью цифрового осциллографа, проверены соотношения неопределенности с помощью курсорных измерений. 1. Первая часть работы заключалась в исследовании спектра периодической последовательности прямоугольных импульсов (пример показан на рисунке \ref{прямоуг}). \begin{figure}[h!] \includegraphics[width=0.4\textwidth]{rect} -\caption{Пример периодической последовательности прямоугольных импульсов} \label{rect} +\caption{Пример периодической последовательности прямоугольных импульсов из \cite{labnik}} \label{rect} \end{figure} -Теоретический расчет дает значение коэффициентов $c_n$: +Теоретически рассчитано значение коэффициентов $c_n$ \cite{labnik}: \begin{equation} c_n = \dfrac{sin(\pi n \tau / T))}{\pi n} \end{equation} @@ -62,10 +80,10 @@ c_n = \dfrac{sin(\pi n \tau / T))}{\pi n} \begin{figure}[h!] \begin{center} \includegraphics[width=0.4\textwidth]{цуги} -\caption{Пример периодической последовательности цуг} \label{цуги} +\caption{Пример периодической последовательности цуг из \cite{labnik}} \label{цуги} \end{center} \end{figure} -Теоретический расчет дает спектр сигнала: +Теоретически известен спектр сигнала \cite{labnik}: \begin{equation} F(\omega) = \dfrac{\tau}{2T}\left[\dfrac{\sin(\omega-\omega_0)\tau /2} {(\omega-\omega_0)\tau /2} @@ -76,10 +94,10 @@ F(\omega) = \dfrac{\tau}{2T}\left[\dfrac{\sin(\omega-\omega_0)\tau /2} \begin{figure}[h!] \begin{center} \includegraphics[width=0.4\textwidth]{Модулированный} -\caption{Пример модулированного сигнала} \label{модулированный} +\caption{Пример модулированного по амплитуде сигнала из \cite{labnik}} \label{модулированный} \end{center} \end{figure} -Теоретический вид сигнала: +Теоретический вид сигнала \cite{labnik}: \begin{equation} f(t) = a_0 \cos (\omega_0 t) +\dfrac{ma_0}{2}\cos (\omega_0 +\Omega)t++\dfrac{ma_0}{2}\cos (\omega_0 -\Omega)t \end{equation} @@ -88,7 +106,7 @@ f(t) = a_0 \cos (\omega_0 t) +\dfrac{ma_0}{2}\cos (\omega_0 +\Omega)t++\dfrac{ma \section{Измерения и обработка данных} \subsection*{Исследования спектра периодической последовательности прямоугольных импульсов} -На генераторе создается сигнал с разными параметрами, по которому на экране осциллографа получается спектр (рис. \ref{прямоуг}) +Для исследования периодической последовательности прямоугольных импульсов на генераторе создан сигнал с разными параметрами, по которому на экране осциллографа получается спектр (рис. \ref{прямоуг}) \begin{figure}[h!] \begin{minipage}[h!]{0.47\linewidth} @@ -106,13 +124,13 @@ f(t) = a_0 \cos (\omega_0 t) +\dfrac{ma_0}{2}\cos (\omega_0 +\Omega)t++\dfrac{ma \begin{minipage}[h!]{0.47\linewidth} \center{\includegraphics[width=1\linewidth]{1_4}} d) $\nu_{повт} = 1000 Гц, \tau = 70 мкс$ \\ \end{minipage} -\caption{Спектры прямоугольных импульсов} +\caption{Спектры последовательностей прямоугольных импульсов при разных частотах повторения и длительности импульса} \label{прямоуг} \end{figure} При $\nu_{повт} = 700 Гц$ проведены измерения ширины спектра. Результаты представлены в таблице \ref{dnu(tau)_tbl} и на рисунке \ref{dnu(tau)_img}. \begin{table}[h!] -\caption{Зависимость ширины спектра от длительности спектра} \label{dnu(tau)_tbl} +\caption{Зависимость ширины спектра от длительности спектра для последовательности прямоугольных импульсов при $\nu_{повт} = 700 Гц$} \label{dnu(tau)_tbl} \begin{tabular}{|l|l|} \hline $\Delta\nu$, Hz & $\tau$, мкс \\ \hline @@ -132,27 +150,27 @@ $\Delta\nu$, Hz & $\tau$, мкс \\ \hline \begin{figure}[h!] \begin{center} \includegraphics[width=\textwidth]{dnu(tau)} -\caption{Зависимость ширины спектра от длительности спектра} \label{dnu(tau)_img} +\caption{Зависимость ширины спектра от длительности спектра для последовательности прямоугольных импульсов} \label{dnu(tau)_img} \end{center} \end{figure} -Рассчитаем коэффициент наклона прямой: +Рассчитан коэффициент наклона прямой: \begin{equation} k = 0.9997 \pm 0.0039 \end{equation} Полученное значение близко к $1$, что подтверждает соотношение неопределенностей. -Для одного из сигналов (a) рассчитаем теоретическую зависимость и изобразим на графике \ref{теор}. Теоретический и экспериментальный спектр похожи. +Для сравнения экспериментальных и теоретических значений спектра для одного из сигналов (a) рассчитана теоретическую зависимость и изображена на графике \ref{теор}. Теоретический и экспериментальный спектр похожи. \begin{figure}[h!] \begin{center} \includegraphics[width=\textwidth]{a(n)} -\caption{Теоретический спектр прямоугольных импульсов} \label{теор} +\caption{Теоретический спектр прямоугольных импульсов из \cite{labnik}} \label{теор} \end{center} \end{figure} \subsection{Исследование спектра периодической последовательности цугов гармонических колебаний} -На генераторе создается сигнал последовательности синусоидальных цугов с разными параметрами, по которому на экране осциллографа получается спектр. (рис. \ref{спектр_цуги}) +Для исследования спектра периодической последовательности цугов гармонических колебаний на генераторе создан сигнал последовательности синусоидальных цугов с разными параметрами, по которому на экране осциллографа получен спектр. (рис. \ref{спектр_цуги}) \begin{figure}[h!] \begin{minipage}[h!]{0.47\linewidth} @@ -174,14 +192,15 @@ k = 0.9997 \pm 0.0039 \begin{minipage}[h!]{0.47\linewidth} \center{\includegraphics[width=1\linewidth]{2_5}} \\e) $\nu = 70 кГц, T = 1 мс, N = 5$ \end{minipage} -\caption{Вид спектра при разных параметрах спектра} +\caption{Вид спектра для периодической последовательности цугов при разных частотах несущей $\nu_0$ = 50 кГц, периодах повторения $T$ = 1 мс, числах +периодов в одном импульсе $N$ = 5} \label{спектр_цуги} \end{figure} -При фиксированной длительности импульсов $\tau$ = 50 мкс измерим расстояния между соседними спектральными компонентами от периода повторения импульсов (табл. \ref{dnu(T)_tbl}, рис. \ref{dnu(T)_img}) +Для проверки соотношения неопределенностей для данного сигнала при фиксированной длительности импульсов $\tau$ = 50 мкс измерены расстояния между соседними спектральными компонентами от периода повторения импульсов (табл. \ref{dnu(T)_tbl}, рис. \ref{dnu(T)_img}) \begin{table}[h!] -\caption{Зависимость расстояния между соседними спектральными компонентами от периода повторения импульсов} \label{dnu(T)_tbl} +\caption{Зависимость расстояния между соседними спектральными компонентами от периода повторения импульсов при $\nu_0$ = 50 кГц и $N$ = 5} \label{dnu(T)_tbl} \begin{tabular}{|l|l|} \hline T, ms & $\delta \nu$, Hz \\ \hline @@ -205,24 +224,24 @@ T, ms & $\delta \nu$, Hz \\ \hline \end{center} \end{figure} -Точки должны хорошо ложиться на прямую, однако из графика видно, что это не так. Проблема заключается в снятии данных (был выбран неверный канал при курсорных измерениях). Поэтому подтвердить справедливость соотношения неопределенности невозможно. +Теоретически известно (\cite{labnik}) точки должны хорошо ложиться на прямую, однако из графика видно, что это не так. Проблема заключается в снятии данных (был выбран неверный канал при курсорных измерениях). Поэтому подтвердить справедливость соотношения неопределенности невозможно. \subsection{Исследование спектра гармонических сигналов, модулированных по амплитуде} -На генераторе создается сигнал, модулированных по амплитуде, по которому на экране осциллографа получается спектр (\ref{мод}). +Для исследования спектра гармонических сигналов, модулированных по амплитуде на генераторе создан сигнал, модулированный по амплитуде, по которому на экране осциллографа получается спектр (\ref{мод}). \begin{figure}[h!] \begin{center} \includegraphics[width=\textwidth]{3} -\caption{Спектр сигнала, модулированного по амплитуде} \label{мод} +\caption{Спектр сигнала, модулированного по амплитуде, при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} \label{мод} \end{center} \end{figure} -Измерим с помощью осциллографа глубину модуляции: +Измерена с помощью осциллографа глубину модуляции: \begin{equation} m = \dfrac{A_{max}-A_{min}}{A_{max}+A_{min}} = \dfrac{1.54 - 0.04}{1.54 + 0.04} = 0.5, что сходится с установленным на генераторе \end{equation} -Изменяя глубину модуляции, измерим $\dfrac{a_{бок}}{а_{осн}}$ (табл. \ref{mod_tbl} и рис. \ref{mod_img}). +Для проверки теоретической зависимости, изменяя глубину модуляции, измерена $\dfrac{a_{бок}}{а_{осн}}$ (табл. \ref{mod_tbl} и рис. \ref{mod_img}). \begin{table}[h!] -\caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$} +\caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$ при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} \label{mod_tbl} \begin{tabular}{|l|l|l|} \hline @@ -242,10 +261,10 @@ m & a\_бок & a\_центр \\ \hline \begin{figure}[h!] \begin{center} \includegraphics[width=0.9\textwidth]{a(m)} -\caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$} \label{mod_img} +\caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$ при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} \label{mod_img} \end{center} \end{figure} -Определим коэффициент наклона прямой: +Определен коэффициент наклона прямой: \begin{equation} k = 0.502 \pm 0.002 \end{equation} @@ -254,7 +273,7 @@ k = 0.502 \pm 0.002 \section{Выводы} -1. При исследовании последовательности прямоугольных импульсов получена зависимость ширины спектра от длительности импульса, что подтверждает соотношение неопределенностей: $\tau \cdot \Delta\nu \sim 1$. +1. При исследовании последовательности прямоугольных импульсов получена зависимость ширины спектра от длительности импульса, что подтверждает соотношение неопределенностей для данного вида сигнала: $\tau \cdot \Delta\nu \sim 1$. 2. Проверены теоретические расчеты спектра при прямоугольных импульсах (теоретическая и экспериментальная картины схожи). @@ -262,4 +281,10 @@ k = 0.502 \pm 0.002 4. Получен угол наклона графика зависимости $\dfrac{a_{бок}}{а_{осн}}$ от $m$ (0.5), подтверждено теоретическое значение этого угла (0.5). + +\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} + + \end{document} \ No newline at end of file