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zXwGo*y8dX7kcO*!1f_T{OlVcux$Y!n_(kymHQH<3d4K-fR1us$1UvQ!Hxp0T!~56&s`UerT;sC|XZVYtvd`!XX=4?P=WEztDJYL?urXNKq2yCzK>}XxPv{@^_mvv0H}c_5hl%dLJ@iu8s&~U#QhorGr)9BmsO7;g6k3d z5=-J|huZ!2cL?=La>?`kVsET^>-}WoO>?Wu6Mw7I>+N*oEn8pYJzszSsIyf3@94iT{|DoCzf1rC diff --git a/3.6.1Спектр/Спектр.tex b/3.6.1Спектр/Спектр.tex index ba3616d..43a3fde 100644 --- a/3.6.1Спектр/Спектр.tex +++ b/3.6.1Спектр/Спектр.tex @@ -40,7 +40,6 @@ \vfill \centerline{\huge{Лабораторная работа 3.6.1}} -\centerline{\large{по теме:}} \centerline{\LARGE{<<Спектральный анализ электрических сигналов>>}} \vfill @@ -53,15 +52,15 @@ \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. +В работе исследованы спектры периодических сигналов: модулированный по амплитуде, прямоугольные импульсы и цуги. Проверены теоретические зависимости параметров спектра и соотношения неопределенности на практике . \section{Введение} Многие практические задачи описания поведения некоторой системы во времени зачастую сводятся к выяснению связи между сигналом, подаваемым на <<вход>> -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 используется 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. +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 используется для анализа многих сигналов, поэтому необходимо экспериментально ознакомиться с ним, сгенерировать и получить на осциллографе спектры различных периодических сигналов, проверить экспериментально параметры спектра и некоторые теоретические соотношения между ними. \section{Методика измерений} -Для экспериментальной проверки соотношения неопределенностей необходимо его сформулировать \cite{labnik}: +Основным соотношением для спектра является соотношение неопределенностей, верное для любого сигнала \cite{labnik}: \begin{equation} \Delta \omega \cdot \Delta t \sim 2\pi \end{equation} @@ -72,10 +71,10 @@ \caption{Примеры сигналов а) периодической последовательности прямоугольных импульсов, б) периодической последовательности цуг, в) модулированного по амплитуде сигнала из \cite{labnik}} \label{пример} \end{center} \end{figure} -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. +Рассмотреть все сигналы невозможно, поэтому работа проводилась на трех относительно простых сигналах. Для каждого из них 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{пример}). -Теоретически рассчитано значение коэффициентов $c_n$ \cite{labnik}: +Теоретически рассчитано значение коэффициентов $c_n$ \cite{labnik}, которое проверено экспериментально: \begin{equation} c_n = \dfrac{sin(\pi n \tau / T))}{\pi n} \end{equation} @@ -88,17 +87,26 @@ F(\omega) = \dfrac{\tau}{2T}\left[\dfrac{\sin(\omega-\omega_0)\tau /2} + \dfrac{\sin(\omega+\omega_0)\tau /2}{(\omega+\omega_0)\tau /2}\right] \end{equation} -3. Последняя часть заключалась в исследовании спектра гармонических сигналов, модулированных по амплитуде (пример показан на рисунке \ref{пример}). +3. Последняя часть заключалась в исследовании спектра гармонических сигналов, модулированных по амплитуде (пример показан на рисунке \ref{пример}). 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 +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} +где $m$ - глубина модуляции. +Модулированное колебание представляется в виде: +\begin{equation} +f(t)=f_0(t)+f_1(t)+f_2(t),\hspace{3mm} f_0(t)=a_0 \cos (\omega_0 t),\hspace{3mm} f_1(t)=\dfrac{ma_0}{2}\cos (\omega_0 +\Omega)t,\hspace{3mm} f_2(t)=\dfrac{ma_0}{2}\cos (\omega_0 -\Omega)t +\end{equation} +$f_0$ - основная гармоника, $f_1, f_2$ - боковые гармоники +Для модулированного по амплитуде сигнала существует теоретическое соотношение между амплитудами гармоник, которое можно проверить экспериментально: +\begin{equation} +\dfrac{a_{бок}}{а_{осн}}=\frac{1}{2} +\end{equation} - -\section{Измерения и обработка данных} +\section{Результаты и их обсуждение} \subsection*{Исследования спектра периодической последовательности прямоугольных импульсов} -Для исследования периодической последовательности прямоугольных импульсов на генераторе создан 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 получен спектр (рис. \ref{прямоуг}) \begin{figure}[h!] \begin{minipage}[h!]{0.47\linewidth} @@ -108,7 +116,8 @@ 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_2}} \\b) $\nu_{повт} = 1400 Гц, \tau = 50 мкс$ \end{minipage} -\vfill +\end{figure} +\begin{figure}[h!] \begin{minipage}[h!]{0.47\linewidth} \center{\includegraphics[width=1\linewidth]{1_3}} c) $\nu_{повт} = 700 Гц, \tau = 50 мкс$ \\ \end{minipage} @@ -119,18 +128,19 @@ f(t) = a_0 \cos (\omega_0 t) +\dfrac{ma_0}{2}\cos (\omega_0 +\Omega)t++\dfrac{ma \caption{Спектры последовательностей прямоугольных импульсов при разных частотах повторения и длительности импульса} \label{прямоуг} \end{figure} +\newpage При $\nu_{повт} = 700 Гц$ проведены измерения ширины спектра. Результаты представлены на рисунке \ref{dnu(tau)_img}. \begin{figure}[h!] \begin{center} -\includegraphics[width=0.85\textwidth]{dnu(tau)} +\includegraphics[width=\textwidth]{dnu(tau)} \caption{Зависимость ширины спектра от длительности спектра для последовательности прямоугольных импульсов при частоте повторения $\nu_{повт} = 700 Гц$} \label{dnu(tau)_img} \end{center} \end{figure} \begin{figure}[h!] \begin{center} -\includegraphics[width=0.85\textwidth]{a(n)} +\includegraphics[width=\textwidth]{a(n)} \caption{Теоретический спектр прямоугольных импульсов при частоте повторения $\nu_{повт} = 1000 Гц$ и длительности импульса $\tau = 50 мкс$ из \cite{labnik}} \label{теор} \end{center} \end{figure} @@ -141,10 +151,12 @@ k = 0.9997 \pm 0.0039 Полученное значение близко к $1$, что подтверждает соотношение неопределенностей. -Для сравнения экспериментальных и теоретических значений спектра для одного из сигналов (a) с рис. \ref{прямоуг}) рассчитана теоретическую зависимость и изображена на графике \ref{теор}. Теоретический и экспериментальный спектр похожи. - +Для сравнения экспериментальных и теоретических значений спектра для одного из сигналов (a) с рис. \ref{прямоуг}) рассчитана теоретическую зависимость и изображена на графике \ref{теор}. Теоретический и экспериментальный спектр похожи, что показывает справедливость теоретического расчета в данном случае. +\newpage +\newpage +\newpage \subsection*{Исследование спектра периодической последовательности цугов гармонических колебаний} \begin{figure}[h!] @@ -183,25 +195,24 @@ k = 0.9997 \pm 0.0039 \end{center} \end{figure} -Теоретически известно (\cite{labnik}), что 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}), что 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 по данным экспериментальным значениям невозможно. \subsection*{Исследование спектра гармонических сигналов, модулированных по амплитуде} 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=0.85\textwidth]{3} +\includegraphics[width=0.75\textwidth]{3} \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} -Измерена с помощью осциллографа глубина модуляции: +Измерена с помощью осциллографа глубина модуляции $m$: \begin{equation} m = \dfrac{A_{max}-A_{min}}{A_{max}+A_{min}} = \dfrac{1.54 - 0.54}{1.54 + 0.54} = 0.5, что \hspace*{1mm} сходится \hspace*{1mm}с\hspace*{1mm} установленным \hspace*{1mm}на\hspace*{1mm} генераторе \end{equation} -Для проверки теоретической зависимости, изменяя глубину модуляции, измерена $\dfrac{a_{бок}}{а_{осн}}$ (рис. \ref{mod_img}). +Для проверки теоретической зависимости, изменяя глубину модуляции, измерена $\dfrac{a_{бок}}{а_{осн}}$ - отношение амплитуд боковой и центральной полос спектра (рис. \ref{mod_img}). \begin{figure}[h!] \begin{center} -\includegraphics[width=0.9\textwidth]{a(m)} +\includegraphics[width=0.75\textwidth]{a(m)} \caption{Зависимость $\dfrac{a_{бок}}{а_{осн}}$ от $m$ для сигнала, модулированного по амплитуде, при частоте несущей $\nu_0$ = 50 кГц, частоте модуляции $\nu_{мод}$ = 2 кГц} \label{mod_img} \end{center} \end{figure} @@ -214,7 +225,7 @@ 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 для данного вида сигнала: $\tau \cdot \Delta\nu \sim 1$. +\hspace{4mm} 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).