Added øvinger
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f094dd3310
commit
948e51aec1
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#!/usr/bin/env python3
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# -*- coding: utf-8 -*-
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"""
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Created on Fri Jan 18 22:30:00 2019
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@author: oyvind
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"""
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import math
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import matplotlib.pyplot as plt
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# Name for the saved graph
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filename = "4b"
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# Tau in microseconds
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tau = 10
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# Square wave freq in kHz
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sqWFreq = 5
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# How many taus you want
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periods = 40
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# The resolution of each tau
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resolution = 1000
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# Calculates the length of each squarewave in taus
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sqTime = (1/sqWFreq) * 10**3
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sqTau = sqTime / tau
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# Creats the square wave
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# Default values are in the argument list
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def CreateSquareWave(amp=0.5, offset=0.5, symetry=0.5):
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squareWave = []
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# Only generates as many datapoints we need
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while len(squareWave) < periods * resolution:
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# First generate the first half
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for r in range(int(resolution*sqTau * symetry)):
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squareWave.append(amp + offset)
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# Then the second
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for r in range(int(resolution*sqTau * (1 - symetry))):
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squareWave.append(offset - amp)
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squareWave = squareWave[:(resolution*periods)]
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return squareWave
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# Generate all the time-ticks
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def GenerateTime():
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times = []
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for t in range(periods * resolution):
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times.append(t/resolution)
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print(len(times))
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return(times)
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def CapVoltage(wave, times):
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cP = wave[0] # Start voltage-supply
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v0 = 0 # Start voltage
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cT = 0 #Start time
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volt = []
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for p in range(len(wave)):
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# If the voltage-supply changes, recalculate startvalues
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if wave[p] != cP:
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v0 = volt[-1] # New start voltage, uses the last voltage calculated
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cP = wave[p] # Variable so the array is not accessed.
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cT = times[p] # Offset time for each period
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# Calculate the voltage over the CAPACITOR with the start values
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volt.append(cP + (v0 - cP) * math.exp(-(times[p] - cT)))
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return(volt)
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SquareWave = CreateSquareWave()
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time = GenerateTime()
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CapWave = CapVoltage(SquareWave, time)
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plt.figure(figsize=(15,5))
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plt.plot(time, SquareWave, time, CapWave)
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plt.xlabel("Time [τ]")
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plt.ylabel("Voltage [V]")
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plt.legend(["Supply voltage", "Capacitor voltage"], loc="lower right")
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plt.savefig(filename + ".png", dpi = 300)
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plt.show()
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\documentclass[10pt]{article}
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\usepackage{pgf,tikz,pgfplots}
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\pgfplotsset{compat=1.15}
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\usepackage{mathrsfs}
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\usetikzlibrary{arrows}
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\pagestyle{empty}
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\begin{document}
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\definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}
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\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=2.0cm,y=3.0cm]
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\begin{axis}[
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x=2.0cm,y=3.0cm,
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axis lines=middle,
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ymajorgrids=true,
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xmajorgrids=true,
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xmin=-0.5,
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xmax=5.5,
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ymin=-0.5,
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ymax=2.5,
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ylabel={Spenning [V]},
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xlabel={Tid [$\tau$]},
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yticklabel={\SI[round-mode=places, round-precision=1]{\tick}{V}},
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xticklabel={\SI[round-mode=places, round-precision=0]{\tick}{\tau}},
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xtick={-0.5,0.0,...,5.5},
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ytick={-0.5,0.0,...,2.5},]
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||||
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||||
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||||
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|
||||
\draw (0.9826971286425398,1.0091692065822461) node[anchor=north west] {$(\tau, 0.79\text{V})$};
|
||||
\draw (1.7132719478893085,0.5951090467957184) node[anchor=north west] {$(2\tau, 0.29\text{V})$};
|
||||
\draw (2.588565094705576,0.46352472684146245) node[anchor=north west] {$(3\tau, 0.11\text{V})$};
|
||||
\draw (3.592848810526346,0.35387112687958255) node[anchor=north west] {$(4\tau, 0.04\text{V})$};
|
||||
\draw (4.597132526347116,0.35387112687958255) node[anchor=north west] {$(5\tau, 0.01\text{V})$};
|
||||
\begin{scriptsize}
|
||||
\draw[color=qqwuqq] (0.14886523614914818,2.1083287262696615) node {$v_c$};
|
||||
\draw [fill=uuuuuu] (1.,0.7872620041068866) circle (2.0pt);
|
||||
\draw [fill=uuuuuu] (2.,0.2896175061263512) circle (2.0pt);
|
||||
\draw [fill=uuuuuu] (3.,0.10654432630722885) circle (2.0pt);
|
||||
\draw [fill=uuuuuu] (4.,0.03919546722189114) circle (2.0pt);
|
||||
\draw [fill=uuuuuu] (5.,0.0144192065780429) circle (2.0pt);
|
||||
\end{scriptsize}
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
Binary file not shown.
Binary file not shown.
|
|
@ -0,0 +1,337 @@
|
|||
\documentclass[11pt,largemargins, norsk]{homework}
|
||||
|
||||
\newcommand{\hwname}{Øyvind Skaaden}
|
||||
\newcommand{\hwemail}{oyvindps@ntnu.no}
|
||||
\newcommand{\hwtype}{Øving}
|
||||
\newcommand{\hwnum}{1}
|
||||
\newcommand{\hwclass}{TTT4260}
|
||||
\newcommand{\hwlecture}{}
|
||||
\newcommand{\hwsection}{}
|
||||
|
||||
\newcommand*{\eq}{=}
|
||||
|
||||
\renewcommand{\questiontype}{Oppgave}
|
||||
\newcommand{\figref}[1]{Figur \ref{#1}}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\question
|
||||
\begin{alphaparts}
|
||||
\item Vi har krets \ref{circ:1a} som vist under med verdiene $R_1 = 1\text{k}\Omega $, $ C_1 = 100\mu\text{F} $ og $V = 5\text{V} $.
|
||||
|
||||
\begin{figure} [h]
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
|
||||
\draw
|
||||
(0,3) to [V, l=$V$] (0,0)
|
||||
(0,3) to [closing switch, l = $t\eq0$ ] (3,3)
|
||||
to [R, l=$R_1$] (6,3)
|
||||
to [C, l=$C_1$] (6,0) -- (0,0);
|
||||
\end{circuitikz}
|
||||
\caption{Krets til oppgave 1}
|
||||
\label{circ:1a}
|
||||
\end{figure}
|
||||
|
||||
$\tau$ er gitt ved
|
||||
|
||||
$$ \tau = R \cdot C $$
|
||||
|
||||
Da er $\tau$ i denne kretsen er da
|
||||
|
||||
$$ \tau = R_1 \cdot C_1 = 1\text{k}\Omega \cdot 100\mu\text{F} = 100\text{ms}$$
|
||||
|
||||
En funksjon for spenningen over kondensatoren er da
|
||||
|
||||
$$ v_c(t) = 5 \text{V} \cdot ( 1 - e ^ {\frac{-t}{100\text{ms}}}) $$
|
||||
\pagebreak
|
||||
\begin{figure}[!ht]
|
||||
\centering
|
||||
\input{grafer/condisO1a}
|
||||
\caption{Utvikling av spenning over kondensator $v_c$}
|
||||
\label{graph:kondensator1}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\item
|
||||
|
||||
Etter å ha koblet opp kretsen ser vi at spenningen (se \figref{graph:1b}) over kondensatoren når $63\%$ eller $3.16$V etter $\Delta x = 94.83$ms.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{TauMaaling.png}
|
||||
\caption{Spennigsutvikling av krets i oppgave 1, $\tau$ er lik $\Delta x$}
|
||||
\label{graph:1b}
|
||||
\end{figure}
|
||||
\pagebreak
|
||||
|
||||
|
||||
|
||||
\item
|
||||
|
||||
Når det skjer utladning av kondensatoren har ikke strømmen noe sted å gå, eneste er å gå gjennom kondensatoren litt og litt.
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
|
||||
\question
|
||||
For å løse kretsen i oppgave 2, vist i kretsen \figref{circ:krets2} under.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw
|
||||
(-6,3) to [V, v=V] ++(0,-3)
|
||||
(-6,3) to [opening switch, l=$S_1$] ++(3,0)
|
||||
to [R, l=$R_1$] +(3,0) to [short,-*] ++(0,0)
|
||||
(-6,0) to [short,-*] (0,0)
|
||||
(0,0) to [R, l=$R_2$] (0,3)
|
||||
(0,0) -- (2,0) to [short,-*] (2,0) to [R, l=$R_3$] (2,3) to [short,-*] ++(0,0) -- (0,3)
|
||||
(2,3) to [R, l=$R_4$] (6,3)
|
||||
to [C, l=$C_1$, v_=$v_c$] (6,0) -- (2,0);
|
||||
\end{circuitikz}
|
||||
\caption{Krets i oppgave 2}
|
||||
\label{circ:krets2}
|
||||
\end{figure}
|
||||
|
||||
|
||||
Vi må finne spenningen som ligger over $R_2||R_3$ for å finne startspenningen på $C_1$.
|
||||
|
||||
Begynner med å finne $$R_2||R_3 = \frac{470\ohm \cdot 220\ohm}{470\ohm + 220\ohm} = \frac{10340}{69} \ohm $$
|
||||
|
||||
$$ v_{R_2||R_3} = \frac{V}{R_1 + R_2||R_3} \cdot (R_2||R_3) = \frac{5\text{V}}{200\ohm + \frac{10340}{69} \ohm} \cdot \frac{10340}{69} \ohm = \frac{2585}{1207}\text{V} \approx 2.14\text{V} $$
|
||||
|
||||
Dette er da startspenningen på $c_1$.
|
||||
|
||||
Når bryteren brytes, vi vi få en forenklet krets, som vist i \figref{circ:oppgave2}
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw
|
||||
(0,0) to [R, l=$R_2$] (0,3)
|
||||
(0,0) -- (2,0) to [short,-*] (2,0) to [R, l=$R_3$] (2,3) to [short,-*] ++(0,0) -- (0,3)
|
||||
(2,3) to [R, l=$R_4$] (6,3)
|
||||
to [C, l=$C_1$, v_=$v_c$] (6,0) -- (2,0);
|
||||
\end{circuitikz}
|
||||
\caption{Foreklet krets i oppgave 2}
|
||||
\label{circ:oppgave2}
|
||||
\end{figure}
|
||||
|
||||
Vi kan da regne ut $R$ i kretsen
|
||||
|
||||
$$ R = R_4 + R_2||R_3 = 300\ohm + \frac{10340}{69} \ohm = \frac{31040}{69}\ohm \approx 449.9\ohm $$
|
||||
|
||||
$\tau$ er da gitt ved $\tau = R \cdot C_1 = = 4.5\mu\text{s}$.
|
||||
|
||||
Funksjonen for spenningen over $v_c$:
|
||||
|
||||
$$ v_c(t) = 2.14e^{\frac{-t}{4.5\mu\text{s}}} $$
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\input{grafer/condisO2}
|
||||
\caption{Graf for oppgave 2}
|
||||
\label{graph:oppg2}
|
||||
\end{figure}
|
||||
|
||||
\clearpage
|
||||
\question
|
||||
\begin{alphaparts}
|
||||
\item
|
||||
Vi har kretsen som gitt i oppgave 3, men tegnet på en forenklet måte i \figref{circ:3a1}.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw
|
||||
(0,3) to [V, v_=V] (0,0)
|
||||
(0,3) to [R, l=$R_1\eq1\text{k}\ohm$] (3,3)
|
||||
to [closing switch, l=$S_1$] (5,3)
|
||||
to [R, l_=$R_2\eq1\text{k}\ohm$] (5,0) -- (0,0)
|
||||
(5,3) to [short,*-] ++(2,0)
|
||||
to [C, l=$C_1\eq100\mu\text{F}$] ++(0,-3)
|
||||
to [short,-*] (5,0);
|
||||
\end{circuitikz}
|
||||
\caption{Forenklet krets til oppgave 3a}
|
||||
\label{circ:3a1}
|
||||
\end{figure}
|
||||
|
||||
|
||||
Vi skriver om til en Norton ekvivalent ved å regne ut $I_n = \tfrac{V}{R_1}$
|
||||
|
||||
$$ I_n = \frac{1\text{V}}{1\text{k}\ohm} = 1\text{mA} $$
|
||||
|
||||
Vi har da to like motstander i parallell. Siden de er like er den totale motstanden lik halvparten av den ene. Så
|
||||
$$ R_{eq} = 0.5\text{k}\ohm $$
|
||||
Vi regner deretter den nye kretsen tilbake til en thevenin-ekvivalent krets.
|
||||
|
||||
$$ V_{th} = I_n \cdot R_{eq} = 1\text{mA}\cdot 0.5\text{k}\ohm = 0.5\text{V}$$
|
||||
|
||||
Vi har da den nye kretsen under i \figref{circ:3a2}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw
|
||||
(0,3) to [V, v_=V$_{th}$] (0,0)
|
||||
(0,3) to [R, l=$R_{eq}\eq0.5\text{k}\ohm$] ++(3,0)
|
||||
to [C, l=$C_1\eq100\mu\text{F}$, v=$v_{C_1}$] ++(0,-3)
|
||||
-- (0,0);
|
||||
\end{circuitikz}
|
||||
\caption{Thevenin-ekvivalent krets til oppgave 3a}
|
||||
\label{circ:3a2}
|
||||
\end{figure}
|
||||
|
||||
Det er da veldig lett å lage en funksjon som besrkiver spenningen, $v_{C_1}$, over $C_1$.
|
||||
|
||||
\begin{align*}
|
||||
v_{C_1}(t)&= V_{th}\left(1-e^\frac{-t}{R_{eq}C_1}\right) \\
|
||||
v_{C_1}(t)&= 0.5\text{V}\left(1-e^\frac{-t}{0.5\text{ms}}\right)\\
|
||||
v_{C_1}(t)&= 0.5\text{V}\left(1-e^\frac{-t}{\tau}\right)
|
||||
\end{align*}
|
||||
|
||||
\clearpage
|
||||
\item
|
||||
Etter $6\tau$ har kondensatoren nådd ``steady-state'', da er spenningen $v_{C_1} = V_{th} = 0.5\text{V}$. Når bryteren $S_2$ lukkes får vi en veldig lik krets som i opgpave 3a. Se \figref{circ:3b1}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw
|
||||
(0,3) to [V, v_=V$\eq1\text{V}$] (0,0)
|
||||
(0,3) to [R, l=$R_1\eq1\text{k}\ohm$] (4,3)
|
||||
to [R, l_=$R_2\eq1\text{k}\ohm$] ++(0,-3) -- (0,0)
|
||||
(4,3) to [short,*-] ++(3,0)
|
||||
to [R, l_=$R_3\eq1\text{k}\ohm$] ++(0,-3)
|
||||
to [short,-*] (4,0)
|
||||
(7,3) to [short,*-] ++(3,0)
|
||||
to [C, l=$C_1\eq100\mu\text{F}$] ++(0,-3)
|
||||
to [short,-*] ++(-3,0);
|
||||
\end{circuitikz}
|
||||
\caption{Forenklet krets til oppgave 3a}
|
||||
\label{circ:3b1}
|
||||
\end{figure}
|
||||
|
||||
Vi gjør det samme som sist, gjør om til norton-ekvivalent, samler motstandene og går tilbake til en thevenin-ekvivalent.
|
||||
|
||||
Siden det her er tre like motstander i parallell er den totale motstanden lik $1/3$ av en av motstandene. Vi får da $V_{th} = \frac{1}{3}\text{V}=\approx 333.3\text{mV}$ og $R_{eq} \approx 333.3\ohm$.
|
||||
|
||||
Kretsen ser da ut som \figref{circ:3b2}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{circuitikz}
|
||||
\draw
|
||||
(0,3) to [V, v_=V$_{th}\approx 333.3\text{mV}$] (0,0)
|
||||
(0,3) to [R, l=$R_{eq}\approx 333.3\ohm$] ++(3,0)
|
||||
to [C, l=$C_1\eq100\mu\text{F}$, v=$v_{C_1}$] ++(0,-3)
|
||||
-- (0,0);
|
||||
\end{circuitikz}
|
||||
\caption{Thevenin-ekvivalent krets til oppgave 3a}
|
||||
\label{circ:3b2}
|
||||
\end{figure}
|
||||
|
||||
Da er det enkelt å sette opp likningen for spenningen $v_{C_1}$
|
||||
Vi setter $\tau = 1$ for at det skal være lettere å lese grafene. Grafene ser helt like ut men tidsenheten blir da $\tau$ i steden for ms.
|
||||
|
||||
\begin{align*}
|
||||
v_{C_1}(t) &= V_{th}+\left[v_{C_1}(t_0) - V_{th} \right]e^{-\frac{t-t_0}{R_{eq}C_1}}\\
|
||||
&\downarrow \\
|
||||
v_{C_1}(t) &= \frac{1}{3}\text{V} + \frac{1}{6}\text{V} \cdot e^{-\frac{t-6\tau}{\tau}}
|
||||
\end{align*}
|
||||
|
||||
En skisse av spenningsutviklingen kan sees i \figref{fig:3b}.
|
||||
|
||||
\pagebreak
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{grafer/3b.png}
|
||||
\caption{Spenningen $V_{C_1}$ som graf, der $S_2$ lukkes etter $6\tau$}
|
||||
\label{fig:3b}
|
||||
\end{figure}
|
||||
|
||||
\item
|
||||
|
||||
For å lage en funksjon for kretsen når bryter $S_2$ lukkes når $t=0.5\tau$, tar vi utgangspunkt i funksjonen fra oppgave 3b og spenningen $v_{C_1}(0.5\tau)\approx \frac{1}{5}\text{V}$.
|
||||
|
||||
Funksjonen for spenningen over $C_1$ fra $t=0.5\tau$ blir da
|
||||
|
||||
\begin{align*}
|
||||
v_{C_1}(t) &= V_{th}+\left[v_{C_1}(t_0) - V_{th} \right]e^{-\frac{t-t_0}{R_{eq}C_1}}\\
|
||||
&\downarrow \\
|
||||
v_{C_1}(t) &=\frac{1}{3}\text{V}+\left[\frac{1}{5}\text{V} - \frac{1}{3}\text{V} \right]e^{-\frac{t-0.5\tau}{\tau}}\\
|
||||
v_{C_1}(t) &= \frac{1}{3}\text{V} - \frac{2}{15}\text{V} \cdot e^{-\frac{t-0.5\tau}{\tau}}
|
||||
\end{align*}
|
||||
|
||||
En skisse av spenningsutviklingen kan sees i \figref{fig:3c}.
|
||||
|
||||
\pagebreak
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{grafer/3c.png}
|
||||
\caption{Spenningen $V_{C_1}$ som graf, der $S_2$ lukkes etter $0.5\tau$}
|
||||
\label{fig:3c}
|
||||
\end{figure}
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
\question
|
||||
\begin{alphaparts}
|
||||
\item
|
||||
Tidskonstanten $\tau$ er gitt ved
|
||||
|
||||
$$ \tau = R \cdot C $$
|
||||
|
||||
I denne kretsen vil $\tau$ bli følgende.
|
||||
|
||||
$$ \tau = 1\text{k}\ohm \cdot 1\text{nF} = 1\mu\text{s} $$
|
||||
|
||||
\pagebreak
|
||||
\item
|
||||
Graf ved $f=5$kHz
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Oppgave4Python/4b.png}
|
||||
\caption{Graf for kretsen i oppgave 4, ved $f=5\text{kHz}$}
|
||||
\label{graph:4b}
|
||||
\end{figure}
|
||||
\item
|
||||
Graf ved $f=30$kHz
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Oppgave4Python/4c2.png}
|
||||
\caption{Graf for kretsen i oppgave 4, ved $f=30\text{kHz}$}
|
||||
\label{graph:4c}
|
||||
\end{figure}
|
||||
|
||||
\item
|
||||
|
||||
Etter oppkobling av kretsen ser vi at kondensatoren oppfører seg veldig likt som regnet ut i oppgave 4b. 4c ($30$kHz) er litt mer ulik da kondensatoren lades og utlades litt raskere enn beregnet. Den når litt høyere og litt lavere spenninger enn beregnet.
|
||||
|
||||
\pagebreak
|
||||
\item
|
||||
Vi ser fra \figref{graph:4b} at firkantpulsen er $1$V i $10\tau = 10 \cdot 10\mu\text{s} = 100\mu\text{s}$.
|
||||
|
||||
Vi ønsker da at likningen $v_{C_1}(100\mu\text{s}) = 0.8$V. Vi setter kondensatorverdien konstant og regner ut motstanden $R$ i kretsen.
|
||||
|
||||
\begin{align*}
|
||||
v_{C_1}(t) &= V_1+\left[v_{C_1}(t_0) - V_1 \right]e^{-\frac{t-t_0}{R\cdot C_1}}\\
|
||||
0.8\text{V} &= 1\text{V}\cdot\left(1-e^{-\frac{100\mu\text{s}}{R\cdot10\text{nF}}}\right) \\
|
||||
R &= \frac{10000}{\ln 5}\\
|
||||
R&\approx 6213 \ohm
|
||||
\end{align*}
|
||||
|
||||
Tester dette og ser at den lader seg litt for mye opp.
|
||||
|
||||
Etter å har justert til $6300\ohm$ ser det ut som at spenningen når ca $0.8$V på firkantpulsen.
|
||||
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
|
||||
\end{document}
|
||||
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|
|
@ -0,0 +1,128 @@
|
|||
\documentclass[11pt,largemargins, norsk]{homework}
|
||||
|
||||
\newcommand{\hwname}{Øyvind Skaaden}
|
||||
\newcommand{\hwemail}{oyvindps@ntnu.no}
|
||||
\newcommand{\hwtype}{Øving}
|
||||
\newcommand{\hwnum}{2}
|
||||
\newcommand{\hwclass}{TTT4260}
|
||||
\newcommand{\hwlecture}{}
|
||||
\newcommand{\hwsection}{}
|
||||
|
||||
\renewcommand{\questiontype}{Oppgave}
|
||||
\newcommand{\figref}[1]{Figur \ref{#1}}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\question
|
||||
\begin{alphaparts}
|
||||
\item
|
||||
Når $A$ er logisk høy, er $C$ logisk lav. Når $A$ er logisk lav, er $C$ logisk høy.
|
||||
|
||||
\item
|
||||
|
||||
Kretsen i oppgave 1 er en inverter fordi den tar inn et logisk signal, og sender ut det motsatte ut etter kretsen. Dersom inngangen er 1 er utgangen 0, og når inngangen er 0 er utgangen 1.
|
||||
|
||||
\item
|
||||
Når det er $0$V på inngangen $A$ er det $5$V på utgangen $C$.
|
||||
|
||||
\item Vi gjør målinger på kretsen, setter spenning på $A$ lik $v_A $ og måler spenningen $v_C $ på utgangen $C$. 43
|
||||
|
||||
\begin{table}[h]
|
||||
\centering
|
||||
\begin{tabular}{|c|c|}
|
||||
\hline
|
||||
$v_A$ & $v_C $ \\ \hline
|
||||
\hline
|
||||
0 & 4.98 \\
|
||||
0.5 & 4.95 \\
|
||||
1 & 4.95 \\
|
||||
1.5 & 4.95 \\
|
||||
2 & 4.93 \\
|
||||
2.1 & 4.89 \\
|
||||
2.2 & 4.79 \\
|
||||
2.3 & 4.56 \\
|
||||
2.4 & 4.13 \\
|
||||
2.5 & 3.4 \\
|
||||
2.6 & 2.32 \\
|
||||
2.7 & 1.11 \\
|
||||
2.8 & 0.23 \\
|
||||
2.9 & 0.12 \\
|
||||
3 & 0.08 \\
|
||||
3.5 & 0.03 \\
|
||||
4 & 0.02 \\
|
||||
4.5 & 0.018 \\
|
||||
5 & 0.014 \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{Målte spenninger på $C$, alle verdier har enhet V}
|
||||
\label{tab:oppg1}
|
||||
\end{table}
|
||||
|
||||
\begin{figure}[h!]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{grafOppg1.png}
|
||||
\caption{Spenning $v_C$ som funksjon av $v_A$}
|
||||
\label{graph:oppg1}
|
||||
\end{figure}
|
||||
|
||||
\item
|
||||
Vi ser i \figref{graph:oppg1} at transistoren begynner å lede rundt $2.2$V til $2.3$V. Den er som en kortslutning ved ca $3.0$V.
|
||||
|
||||
\item
|
||||
Dioden begynner å lyse når $A$ er ca $2$V. Da er $C$ lik $2.42$V.
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
\question
|
||||
\begin{alphaparts}
|
||||
\item
|
||||
|
||||
Her er begge grafene skisserte.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{bilder/oppg2_a1.png}
|
||||
\caption{Graf ved $T=10\tau$}
|
||||
\label{graph:2a1}
|
||||
\end{figure}
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{bilder/oppg2_a2.png}
|
||||
\caption{Graf ved $T=2\tau$}
|
||||
\label{graph:2a2}
|
||||
\end{figure}
|
||||
|
||||
\clearpage
|
||||
\item
|
||||
|
||||
Vi ønsker at kretsen skal nå $2$V. Vi ønsker å finne tiden det tar.
|
||||
|
||||
$$ 2\text{V} = 5V(1-e^{\frac{-t}{\tau}})$$
|
||||
Som gjør at
|
||||
$$ t = \frac{1}{2}\tau $$
|
||||
|
||||
Vi vet også at perioden er $T=1$ms. Vi vet også at $v_1 $ er $5$V i $1/2$ periode. Som betyr at
|
||||
$$ \frac{1}{2}T = \frac{1}{2}\tau \Leftrightarrow \tau = 1\text{ms}$$
|
||||
|
||||
Vi må lage $\tau$. Velger kondensator lik $100\mu$F.
|
||||
|
||||
$$ \frac{1\text{ms}}{100\mu\text{F}} = 10\ohm$$
|
||||
|
||||
|
||||
\item
|
||||
|
||||
Kobler opp kretsen og oppladning når maksimalt $2$V.
|
||||
|
||||
\item
|
||||
Ved frekvensen $1$kHz vil dioden lyse, og samme for frekvenser over.
|
||||
|
||||
For lave frekvenser, feks $1$Hz vil dioden blinke med frekvensen $1$Hz.
|
||||
|
||||
\item
|
||||
Dioden lyser hele tiden egentlig. Ved veldig lave frekvenser blinker dioden, desto høyere frekvenser jo serkere lys, men veldig lite forskjell.
|
||||
|
||||
|
||||
|
||||
\end{alphaparts}
|
||||
\end{document}
|
||||
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|
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|
|
@ -0,0 +1,37 @@
|
|||
#!/usr/bin/env python3
|
||||
# -*- coding: utf-8 -*-
|
||||
"""
|
||||
Created on Sun Jan 13 16:30:58 2019
|
||||
|
||||
@author: oyvind
|
||||
"""
|
||||
|
||||
import csv
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
header = []
|
||||
data = []
|
||||
|
||||
filename = "0.1V1K"
|
||||
|
||||
|
||||
with open(filename + ".csv") as csvfile:
|
||||
csvreader = csv.reader(csvfile)
|
||||
|
||||
header = next(csvreader)
|
||||
|
||||
for dataplot in csvreader:
|
||||
values = [float(value) for value in dataplot]
|
||||
|
||||
data.append(values)
|
||||
|
||||
time = [p[0] * 1000 for p in data]
|
||||
ch1 = [p[1] for p in data]
|
||||
ch2 = [p[2] for p in data]
|
||||
|
||||
plt.plot(time,ch1, time,ch2)
|
||||
plt.xlabel("Tid (ms)")
|
||||
plt.ylabel("Spenning (V)")
|
||||
plt.legend(["Forsterket signal","Inngangssignal"])
|
||||
plt.savefig(filename + ".png", dpi=200)
|
||||
plt.show()
|
||||
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|
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|
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|
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|
|
@ -0,0 +1,177 @@
|
|||
\documentclass[11pt,largemargins, norsk]{homework}
|
||||
|
||||
\newcommand{\hwname}{Øyvind Skaaden}
|
||||
\newcommand{\hwemail}{oyvindps@ntnu.no}
|
||||
\newcommand{\hwtype}{Øving}
|
||||
\newcommand{\hwnum}{10}
|
||||
\newcommand{\hwclass}{TTT4260}
|
||||
\newcommand{\hwlecture}{}
|
||||
\newcommand{\hwsection}{}
|
||||
|
||||
\renewcommand{\questiontype}{Oppgave}
|
||||
\newcommand{\figref}[1]{Figur \ref{#1}}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\question
|
||||
\begin{alphaparts}
|
||||
\item
|
||||
For å finne utgangsspenningen $v_2 $ må vi først finne spenningen over $R_i $, $v_1 $. Den er
|
||||
|
||||
$$ v_1 = \frac{R_i}{R_i + R_s} v_s \quad\Rightarrow\quad v_1 = \frac{100\text{k}\Omega}{100\text{k}\Omega + 33\Omega} \cdot 0.6\text{mV} \approx 0.6\text{mV}$$
|
||||
|
||||
Spenningen $Av_1 $, der $A = 10^4 $ blir $Av_1 = 6$V.
|
||||
|
||||
Spenningen $v_2 $ blir da spenningen over $R_L $. Den er
|
||||
|
||||
$$ v_2 = \frac{R_L}{R_L + R_0} Av_1 \quad\Rightarrow\quad v_2 = \frac{1\text{k}\Omega}{1\text{k}\Omega + 200\Omega} = 5\text{V}$$
|
||||
|
||||
\item Se \figref{graph:oppgave1b}
|
||||
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{pic/SkissetTegning.png}
|
||||
\caption{Skissert spenning $v_2 $ med ikke-ideell op-amp}
|
||||
\label{graph:oppgave1b}
|
||||
\end{figure}
|
||||
|
||||
\item
|
||||
Signalet er klippet fra 5V og oppver. Dette kan forhindres ved å senke amplituden til inngangssignalet ned til $0.5$mV. Dette kan gjøres ved å øke motstanden $R_s $ og senke $R_i $.
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
|
||||
\question
|
||||
|
||||
\begin{alphaparts}
|
||||
\item
|
||||
Kretsen i figur 3 er en buffer. Den vil kunne ta inn et inngangssignal og levere akkurat det samme tilbake til kretsen. Den har en forsterkning på 1, altså det samme signalet inn som ut. Den brukes ofte der kretsen som leverer signalet ikke klarer å levere nok strøm til det den leverer til. Bufferen klarer da å levere nok strøm.
|
||||
|
||||
|
||||
\item
|
||||
|
||||
Kretsen i figur 4 er en inverterende forsterker. Det går ingen strøm gjennom forsterkeren, men det går en strøm fra $v_i $ til $v_o $. Vi kan da sette opp KVL, basert på at det går en strøm fra $v_i $ til $v_o $.
|
||||
|
||||
\begin{align}
|
||||
-v_i + R_1 \cdot i + R_2 \cdot i + v_o = 0
|
||||
\label{eq:2b}
|
||||
\end{align}
|
||||
Vi har også at spenningen til terminalene er like mellom seg, og at den ikke inverterende er koblet til jord.
|
||||
|
||||
$$ -v_i + R_1\cdot i = 0 \qquad\Leftrightarrow\qquad i = \frac{v_i}{R_1} $$
|
||||
|
||||
Setter vi dette inn i (\ref{eq:2b}), får vi
|
||||
|
||||
\begin{align*}
|
||||
-v_i + R_1 \cdot \frac{v_i}{R_1} + R_2 \cdot \frac{v_i}{R_1} + v_o &= 0 \\
|
||||
\frac{v_o}{v_i} &= -\frac{R_2}{R_1}
|
||||
\end{align*}
|
||||
|
||||
\item
|
||||
|
||||
Kretsen i figur 5 er en ikke inverterende forsterker. Spenningen over terminalene er lik. Bruker nodespenning.
|
||||
|
||||
\begin{align*}
|
||||
\frac{v_i}{R_1} + \frac{v_i + v_o}{R_2} &= 0 \\
|
||||
\frac{R_2}{R_1} &= \frac{-v_i + v_o}{v_i} \\
|
||||
\frac{v_o}{v_i} - 1 &= \frac{R_2}{R_1} \\
|
||||
\frac{v_o}{v_i} &= \frac{R_2 + R_1}{R_1}
|
||||
\end{align*}
|
||||
|
||||
\item
|
||||
|
||||
Kretsen i figur 6 er en derivator. Vi vet at strømmen gjennom en kondensator er $ i_c = C\frac{dv_c}{dt} $ Vi vet også at det ikke går noe strøm gjennom forsterkeren, så all strøm må gå gjennom motstanden $R_1 $. Siden det ikke er noen spenning mellom terminalene på forsterkeren, og den ikke inverterende er koblet til jord vil spenningen over motstanden $R_1 $ være $-v_o $
|
||||
|
||||
Setter dette lik hverandre.
|
||||
|
||||
\begin{align*}
|
||||
C\frac{dv_i}{dt} &= \frac{-v_o}{R_1}\\
|
||||
v_o &= -RC\frac{dv_i}{dt}
|
||||
\end{align*}
|
||||
|
||||
\item
|
||||
|
||||
Kretsen i figur 7 er en integrator. Her er det tilsvarende som oppgaven over.
|
||||
|
||||
Finner strømmen gjennom $R $ og $C $.
|
||||
|
||||
\begin{align*}
|
||||
\frac{v_i}{R_1} &= -C\frac{dv_o}{dt} \\
|
||||
\frac{dv_o}{dt} &= -\frac{v_i}{RC} \\
|
||||
v_o &= -\frac{1}{RC}\int v_i\ dt
|
||||
\end{align*}
|
||||
|
||||
\item
|
||||
|
||||
Kretsen i figur 8 er en komparator. Den har en terskelspenning som kan settes på den inverterende inngangen. Dersom inngangssignalet er mindre enn terskelspenningen vil utgangssignalet trekkes ned mot det nedre spenningsforsyningen.
|
||||
Dersom den er større, vil utgangssignalet trekkes til den øvre spenningsforsyning.
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
\question
|
||||
|
||||
\begin{alphaparts}
|
||||
\item Kretsen i figur 9 fungerer ikke på samme måte som kretsen i figur 4 (inverterende forsterker). Denne kretsen vil vokse veldig fort oppover når inngangsspenning er positiv og omvendt når inngangen er negativ.
|
||||
|
||||
Etter litt søking på internettet er dette en ``Schmitt-trigger'' \footnote{Wikipedia contributors. (2019, January 20). Operational amplifier. In Wikipedia, The Free Encyclopedia. Retrieved 10:43, February 7, 2019, from \url{https://en.wikipedia.org/w/index.php?title=Operational_amplifier&oldid=879387924}}
|
||||
|
||||
\item
|
||||
Dersom $v_1 $ er et trekantsignal vil signalet ut på $v_2 $ bli et firkantsignal.
|
||||
|
||||
\item
|
||||
|
||||
Dersom vi integrerer et firkantsignal vil vi få en kurve som alternerer mellom et konstant stignigstall som er positivt og et negativt. Den eneste kurven som passer dette, er en trekantbølge.
|
||||
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
|
||||
\question
|
||||
|
||||
\begin{alphaparts}
|
||||
\item Krets koblet opp. Inngangsamplitude er på $0.1$V. Forventet utgangsamplitude er $1$V, forsterkingen er på -10. Valgte motstander $R_1 = 1\text{k}\Omega $ og $R_2 = 10\text{k}\Omega$
|
||||
Vi kan se forsterkningssignalet i \figref{graph:oppgave4a}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{pic/vanligForsterker.JPG}
|
||||
\caption{Oppkoblet krets etter Figur 4 i oppgavetekten, en inverterende forsterker}
|
||||
\label{pic:oppgave4a}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{graf/0.1V1K.png}
|
||||
\caption{OP-Amp med inngangsspenning $0.1$V og forventet utgangsspenning på $1$V}
|
||||
\label{graph:oppgave4a}
|
||||
\end{figure}
|
||||
|
||||
\item
|
||||
Forsterkeren blir mettet når inngangssignalet overstiger 0.5V. Vi kan se dette i grafen i \figref{graph:oppgave4b}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{graf/0.5V1K.png}
|
||||
\caption{OP-Amp med inngangsspenning $0.5$V. Her klipper forsterkeren på ca 4V}
|
||||
\label{graph:oppgave4b}
|
||||
\end{figure}
|
||||
|
||||
\end{alphaparts}
|
||||
|
||||
\clearpage
|
||||
\question
|
||||
|
||||
Kobler opp kretsen i oppgave 5. Bruker inngangsspenning 1V og spenningskilde 5V og -5V. Bruker et 10k potmeter. Kan variere forsterkningen fra 3.45V til 0.18V, eller i dB, ca +10db til -14.9dB
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{pic/varierendeForsterker.JPG}
|
||||
\caption{Fysisk krets for en varierende inverterende forserker}
|
||||
\label{pic:oppgave5}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
Binary file not shown.
|
|
@ -0,0 +1,28 @@
|
|||
#!/usr/bin/env python3
|
||||
# -*- coding: utf-8 -*-
|
||||
"""
|
||||
Created on Thu Feb 28 11:00:18 2019
|
||||
|
||||
@author: oyvind
|
||||
"""
|
||||
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
def H(f):
|
||||
w = 2 * np.pi * f
|
||||
|
||||
return (w * 10**(-4))/np.sqrt(1+(w*10**(-4))**2)
|
||||
|
||||
frq = []
|
||||
values = []
|
||||
for i in range(2*10**5):
|
||||
frq.append(i)
|
||||
value = 20 * np.log10(abs(H(i)))
|
||||
values.append(value)
|
||||
|
||||
plt.plot(frq, values)
|
||||
plt.xscale('log')
|
||||
plt.gca().xaxis.grid(True)
|
||||
|
||||
|
||||
|
|
@ -0,0 +1,44 @@
|
|||
#!/usr/bin/env python3
|
||||
# -*- coding: utf-8 -*-
|
||||
"""
|
||||
Created on Thu Feb 28 11:00:18 2019
|
||||
|
||||
@author: oyvind
|
||||
"""
|
||||
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
def H13(f):
|
||||
w = 2 * np.pi * f
|
||||
|
||||
return (w * 10**(-4))/np.sqrt(1+(w*10**(-4))**2)
|
||||
|
||||
def H2(f):
|
||||
w = 2 * np.pi * f
|
||||
|
||||
return (1/np.sqrt(1+(w*100*10**(-9)*10**4)**2))
|
||||
|
||||
def H4(f):
|
||||
w = 2 * np.pi * f
|
||||
|
||||
R = 99.76
|
||||
L = 100 * 10**(-3)
|
||||
|
||||
t = L / R
|
||||
|
||||
return (1/np.sqrt(1+(w*t)**2))
|
||||
|
||||
frq = []
|
||||
values = []
|
||||
for i in range(2*10**4):
|
||||
frq.append(i)
|
||||
value = 20 * np.log10(abs(H4(i)))
|
||||
values.append(value)
|
||||
|
||||
plt.plot(frq, values)
|
||||
plt.xscale('log')
|
||||
plt.gca().xaxis.grid(True)
|
||||
plt.gca().yaxis.grid(True)
|
||||
|
||||
|
||||
|
|
@ -0,0 +1,44 @@
|
|||
#!/usr/bin/env python3
|
||||
# -*- coding: utf-8 -*-
|
||||
"""
|
||||
Created on Thu Feb 28 11:00:18 2019
|
||||
|
||||
@author: oyvind
|
||||
"""
|
||||
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
def H13(f):
|
||||
w = 2 * np.pi * f
|
||||
|
||||
return np.arctan(1/(w * 10**-4)) * 180 / np.pi
|
||||
|
||||
|
||||
def H2(f):
|
||||
w = 2 * np.pi * f
|
||||
|
||||
return np.arctan(w * 100*10**(-9) * 10**4) * 180 / np.pi
|
||||
|
||||
def H4(f):
|
||||
w = 2 * np.pi * f
|
||||
|
||||
R = 99.76
|
||||
L = 100 * 10**(-3)
|
||||
|
||||
t = L / R
|
||||
|
||||
return np.arctan(w * t) * 180 / np.pi
|
||||
|
||||
frq = []
|
||||
values = []
|
||||
for i in range(1, 2*10**4):
|
||||
frq.append(i)
|
||||
value = H4(i)
|
||||
values.append(value)
|
||||
|
||||
plt.plot(frq, values)
|
||||
plt.xscale('log')
|
||||
plt.gca().xaxis.grid(True)
|
||||
|
||||
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 30 KiB |
|
|
@ -0,0 +1,172 @@
|
|||
#Digilent WaveForms Network Analyzer - Bode
|
||||
#Device Name: Discovery2NI
|
||||
#Serial Number: SN:210321A36D3D
|
||||
#Date Time: 2019-03-04 13:47:34.600
|
||||
#Start: 10 Hz
|
||||
#Stop: 20000 Hz
|
||||
#Steps: 151
|
||||
#Wavegen: Wavegen1
|
||||
#Amplification: 1 X
|
||||
#Settle: 10 ms
|
||||
#MinPeriods: 16
|
||||
#Channel: Channel 1
|
||||
#Range: 5.47148 V
|
||||
#Offset: -3.64348e-05 V
|
||||
#Relative: no
|
||||
#Channel: Channel 2
|
||||
#Range: 5.48107 V
|
||||
#Offset: -6.42632e-05 V
|
||||
#Relative: yes
|
||||
|
||||
Frequency (Hz),Channel 1 Magnitude (dB),Channel 2 Magnitude (dB),Channel 2 Phase (°)
|
||||
10,-0.00537326,-0.00481132,-0.3339
|
||||
10.5198,-0.00550897,-0.00468579,-0.350273
|
||||
11.0666,-0.00553953,-0.00497733,-0.366327
|
||||
11.6418,-0.00540155,-0.00537231,-0.38602
|
||||
12.2469,-0.00550687,-0.00508884,-0.406273
|
||||
12.8835,-0.00509335,-0.00539532,-0.427304
|
||||
13.5532,-0.00553483,-0.00544864,-0.448588
|
||||
14.2577,-0.00527835,-0.0057729,-0.469659
|
||||
14.9987,-0.00519837,-0.00589985,-0.495311
|
||||
15.7784,-0.00532765,-0.00593774,-0.520151
|
||||
16.5985,-0.00538368,-0.00622925,-0.547678
|
||||
17.4613,-0.00528162,-0.00670784,-0.57488
|
||||
18.3689,-0.00546862,-0.00668217,-0.604199
|
||||
19.3237,-0.00555204,-0.00687947,-0.634738
|
||||
20.3281,-0.00533271,-0.00723183,-0.666323
|
||||
21.3847,-0.00538782,-0.00742054,-0.701453
|
||||
22.4962,-0.00546445,-0.00789399,-0.735715
|
||||
23.6656,-0.00519679,-0.00848513,-0.774108
|
||||
24.8957,-0.00538932,-0.00843235,-0.812998
|
||||
26.1897,-0.00523701,-0.00917148,-0.853232
|
||||
27.551,-0.00518192,-0.00980661,-0.898256
|
||||
28.9831,-0.00530315,-0.00982177,-0.943569
|
||||
30.4896,-0.00545199,-0.0100736,-0.991705
|
||||
32.0744,-0.00528373,-0.0106913,-1.0408
|
||||
33.7415,-0.00529,-0.0111226,-1.09468
|
||||
35.4954,-0.0052797,-0.0118656,-1.14995
|
||||
37.3404,-0.00554357,-0.0121321,-1.20827
|
||||
39.2813,-0.00531512,-0.0130159,-1.27031
|
||||
41.323,-0.00543937,-0.013854,-1.33445
|
||||
43.471,-0.00540697,-0.0144741,-1.40109
|
||||
45.7305,-0.00532245,-0.0151413,-1.47166
|
||||
48.1075,-0.00547021,-0.0159066,-1.54728
|
||||
50.6081,-0.00517615,-0.0173125,-1.62645
|
||||
53.2386,-0.0050528,-0.0183198,-1.70792
|
||||
56.0059,-0.00538225,-0.0192564,-1.79434
|
||||
58.917,-0.0053449,-0.0202372,-1.88548
|
||||
61.9794,-0.00544656,-0.0216106,-1.98053
|
||||
65.201,-0.00550144,-0.022905,-2.08071
|
||||
68.59,-0.00564915,-0.0243751,-2.1851
|
||||
72.1552,-0.00544825,-0.0261777,-2.29628
|
||||
75.9057,-0.00531167,-0.027924,-2.41336
|
||||
79.8512,-0.0052448,-0.0299534,-2.53401
|
||||
84.0018,-0.00539383,-0.0318696,-2.66404
|
||||
88.368,-0.00566807,-0.0338469,-2.79711
|
||||
92.9613,-0.00558168,-0.0361629,-2.93891
|
||||
97.7933,-0.00577305,-0.0389504,-3.08796
|
||||
102.876,-0.00527313,-0.0417832,-3.2438
|
||||
108.224,-0.00559215,-0.0446737,-3.40746
|
||||
113.849,-0.00547246,-0.048086,-3.58008
|
||||
119.767,-0.00530848,-0.0517561,-3.75967
|
||||
125.992,-0.00557531,-0.0552902,-3.95007
|
||||
132.541,-0.00536839,-0.0594675,-4.15219
|
||||
139.43,-0.00523049,-0.0641689,-4.36139
|
||||
146.678,-0.00526805,-0.068836,-4.58326
|
||||
154.302,-0.00551183,-0.0742347,-4.81298
|
||||
162.322,-0.00520949,-0.0801541,-5.05723
|
||||
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Reference in New Issue