Added øvinger
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Øvinger/Ø1/ESDAI_Ø1.pdf
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Øvinger/Ø1/Fukka.ggb
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Øvinger/Ø1/Fukka.ggb
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Øvinger/Ø1/LF_ø1.pdf
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Øvinger/Ø1/LF_ø1.pdf
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Øvinger/Ø1/Oppgave1.ggb
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Øvinger/Ø1/Oppgave1.ggb
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Øvinger/Ø1/Oppgave1_a.ggb
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Øvinger/Ø1/Oppgave2.ggb
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Øvinger/Ø1/Oppgave2bc.ggb
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Øvinger/Ø1/Oppgave4Python/4b.png
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Øvinger/Ø1/Oppgave4Python/4b2.png
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Øvinger/Ø1/Oppgave4Python/4c.png
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Øvinger/Ø1/Oppgave4Python/Graftegning.py
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Øvinger/Ø1/Oppgave4Python/Graftegning.py
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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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Øvinger/Ø1/Oppgave4Python/cap.png
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Øvinger/Ø1/TauMaaling.png
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Øvinger/Ø1/grafer/3b.png
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Øvinger/Ø1/grafer/condisO1.tex
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Øvinger/Ø1/grafer/condisO1.tex
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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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401
Øvinger/Ø1/grafer/condisO1a.tex
Normal file
401
Øvinger/Ø1/grafer/condisO1a.tex
Normal file
@@ -0,0 +1,401 @@
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|
||||
433
Øvinger/Ø1/grafer/condisO2.tex
Normal file
433
Øvinger/Ø1/grafer/condisO2.tex
Normal file
@@ -0,0 +1,433 @@
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\end{tikzpicture}
|
||||
BIN
Øvinger/Ø1/oppgave4e.ggb
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Øvinger/Ø1/oppgave4e.ggb
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Øvinger/Ø1/Øving1.pdf
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Øvinger/Ø1/Øving1.pdf
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337
Øvinger/Ø1/Øving1.tex
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Øvinger/Ø1/Øving1.tex
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|
||||
\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}
|
||||
Reference in New Issue
Block a user