half-working version 1
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/* Common CSS */
|
||||
html {
|
||||
background-color: var(--bg-primary);
|
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color: var(--grey);
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||||
}
|
||||
|
||||
h1, h2 {
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||||
font-size: 1em;
|
||||
}
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||||
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||||
body {
|
||||
margin: 0;
|
||||
font-family: "Liberation Mono", monospace;
|
||||
}
|
||||
#titlebar {
|
||||
font-family: unifont;
|
||||
}
|
||||
#titlebar h1 {
|
||||
font-size: 2rem;
|
||||
}
|
||||
#subhead {
|
||||
color: var(--cyan);
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||||
font-weight: bold;
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||||
}
|
||||
.hnav a {
|
||||
text-decoration: none;
|
||||
padding: 0 0.5em 0 0.5em;
|
||||
margin: 0 0.5em 0 0.5em;
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||||
color: var(--cyan);
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||||
}
|
||||
.hnav a:hover {
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||||
/*text-decoration: underline;*/
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background-color: var(--bg-secondary);
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color: var(--orange);
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||||
}
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||||
.hnav {
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||||
border-bottom: 2px dashed var(--red);
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||||
}
|
||||
.hnav .current {
|
||||
color: var(--orange);
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||||
}
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||||
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.content p {
|
||||
text-indent: 2em;
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||||
}
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|
||||
#foot {
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||||
border-top: 4px double var(--yellow);
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padding-top: 2em;
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||||
padding-bottom: 1em;
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||||
display: flex;
|
||||
align-items: center;
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||||
justify-content: center;
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||||
}
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a {
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||||
color: var(--cyan);
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}
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a:hover {
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color: var(--orange);
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text-decoration: none;
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||||
cursor: pointer;
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||||
}
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||||
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/* Desktop CSS */
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@media only screen and (min-aspect-ratio: 0.72) {
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body {
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display: flex;
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align-items: stretch;
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flex-direction: column;
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width: 100vw;
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}
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#head {
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display: flex;
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flex-direction: column;
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align-items: center;
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}
|
||||
#titlebar {
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align-self: stretch;
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||||
border-bottom: 4px double var(--yellow);
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display: flex;
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align-items: center;
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justify-content: space-between;
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height: 2em;
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}
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#titlebar h1{
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margin: 0 0 0 0.3em;
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}
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#titlebar .item1 {
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}
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#titlebar .item2 {
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}
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#subhead {
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width: 100%;
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display: flex;
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justify-content: space-between;
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align-items: center;
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}
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#subhead ul {
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list-style-type: none;
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display: flex;
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margin: 0 0 0 0;
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}
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#subhead ul li {
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}
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#motd {
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margin-left: 10px;
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tab-size: 4;
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white-space: pre-wrap;
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}
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.content {
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width: 90vw;
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max-width: 65em;
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margin-top: 1em;
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padding-left: 1em;
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/*align-self: center;*/
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margin-left: 2em;
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min-height: 85vh;
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border-left: 2px solid var(--bg-secondary);
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||||
}
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}
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/* Mobile CSS */
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@media screen and (max-aspect-ratio: 0.71) {
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html {
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font-size: 2vh;
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||||
}
|
||||
#titlebar {
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||||
border-bottom: 4px double var(--yellow);
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display: flex;
|
||||
flex-direction: column;
|
||||
align-items: center;
|
||||
justify-content: space-between;
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}
|
||||
#titlebar h1{
|
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margin: 0;
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padding-bottom: 0.3em;
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}
|
||||
#titlebar .item1 {
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}
|
||||
#titlebar .item2 {
|
||||
}
|
||||
#subhead {
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||||
color: var(--cyan);
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display: flex;
|
||||
flex-direction: column-reverse;
|
||||
justify-content: space-between;
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||||
align-items: stretch;
|
||||
width: 100%;
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||||
}
|
||||
#subhead ul {
|
||||
list-style-type: none;
|
||||
display: flex;
|
||||
justify-content: center;
|
||||
margin: 0 0 0 0;
|
||||
}
|
||||
#motd {
|
||||
margin-left: 0.5em;
|
||||
display: flex;
|
||||
justify-content: left;
|
||||
align-items: center;
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||||
tab-size: 4;
|
||||
white-space: pre-wrap;
|
||||
}
|
||||
.content {
|
||||
margin-left: 15px;
|
||||
font-size: 0.7em;
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||||
}
|
||||
}
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@@ -0,0 +1,38 @@
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/* Color variables */
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:root {
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/* Ayu Dark */
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/* --bg-primary: #0A0E14;
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--red: #FF3333;
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--red2: #FF3333;
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--green: #C2D94C;
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--green2: #C2D94C;
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--orange: #FF8F40;
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--orange2: #FF8F40;
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--blue: #59C2FF;
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--blue2: #59C2FF;
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--yellow: #FFEE99;
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--yellow2: #FFEE99;
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--cyan: #95E6CB;
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--cyan2: #95E6CB;
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--grey: #B3B1AD;
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--grey2: #B3B1AD;
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--bg-secondary: #4D5566;
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*/
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/* Elementary */
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||||
--bg-primary: #303030;
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--red: #E1321A;
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--red2: #FF361E;
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--green: #6AB017;
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--green2: #7BC91F;
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--orange: #FFC005;
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--orange2: #FFD00A;
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--blue: #004F9E;
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--blue2: #0071FF;
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--yellow: #EC0048;
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--yellow2: #FF1D62;
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--cyan: #2AA7E7;
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||||
--cyan2: #4BB8FD;
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||||
--grey: #F2F2F2;
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--grey2: #A020f0;
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||||
--bg-secondary: #4D5566;
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||||
}
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||||
@@ -0,0 +1,17 @@
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||||
/* Fonts */
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||||
@font-face {
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||||
font-family: hackmono;
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||||
src: url(fonts/Hack-Regular.ttf);
|
||||
}
|
||||
@font-face {
|
||||
font-family: "Code New Roman";
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||||
src: url("fonts/Code New Roman.woff");
|
||||
}
|
||||
@font-face {
|
||||
font-family: unifont;
|
||||
src: url(fonts/unifont-17.0.04.otf);
|
||||
}
|
||||
@font-face {
|
||||
font-family: "Liberation Mono";
|
||||
src: url(/fonts/liberation-mono.regular.ttf);
|
||||
}
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||||
|
After Width: | Height: | Size: 541 KiB |
|
After Width: | Height: | Size: 9.0 KiB |
|
After Width: | Height: | Size: 86 KiB |
|
After Width: | Height: | Size: 3.2 KiB |
|
After Width: | Height: | Size: 147 KiB |
@@ -0,0 +1,5 @@
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\expandafter\ifx\csname doTocEntry\endcsname\relax \expandafter\endinput\fi
|
||||
\doTocEntry\tocsection{1}{\csname a:TocLink\endcsname{1}{x1-10001}{QQ2-1-1}{Oscillating Circuits}}{1}\relax
|
||||
\doTocEntry\tocsubsection{1.1}{\csname a:TocLink\endcsname{1}{x1-20001.1}{QQ2-1-2}{Passive Oscillators}}{1}\relax
|
||||
\doTocEntry\tocsubsection{1.2}{\csname a:TocLink\endcsname{1}{x1-30001.2}{QQ2-1-3}{Active Oscillators}}{7}\relax
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\par
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@@ -0,0 +1,4 @@
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\expandafter\ifx\csname doTocEntry\endcsname\relax \expandafter\endinput\fi
|
||||
\doTocEntry\tocsection{1}{\csname a:TocLink\endcsname{1}{x1-10001}{QQ2-1-1}{Oscillating Circuits}}{1}\relax
|
||||
\doTocEntry\tocsubsection{1.1}{\csname a:TocLink\endcsname{1}{x1-20001.1}{QQ2-1-2}{Passive Oscillators}}{1}\relax
|
||||
\doTocEntry\tocsubsection{1.2}{\csname a:TocLink\endcsname{1}{x1-30001.2}{QQ2-1-3}{Active Oscillators}}{7}\relax
|
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@@ -0,0 +1,3 @@
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\relax
|
||||
\ifx\rEfLiNK\UnDef\gdef \rEfLiNK#1#2{#2}\fi
|
||||
\gdef \@abspage@last{7}
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@@ -0,0 +1,170 @@
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htfcss: ecbx font-weight: bold;
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htfcss: ecsx font-weight: bold;
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htfcss: ecbi font-weight: bold; font-style: italic;
|
||||
htfcss: ecit font-style: italic; font-family: monospace,monospace;
|
||||
htfcss: ecss font-family: sans-serif;
|
||||
htfcss: ecff font-family: fantasy;
|
||||
htfcss: ecti font-style: italic;
|
||||
htfcss: ectt font-family: monospace,monospace;
|
||||
htfcss: ecbx font-weight: bold;
|
||||
htfcss: ecsx font-weight: bold;
|
||||
htfcss: ecbi font-weight: bold; font-style: italic;
|
||||
htfcss: ecit font-style: italic; font-family: monospace,monospace;
|
||||
htfcss: ecss font-family: sans-serif;
|
||||
htfcss: ecff font-family: fantasy;
|
||||
htfcss: ecti font-style: italic;
|
||||
htfcss: ectt font-family: monospace,monospace;
|
||||
htfcss: ecbx font-weight: bold;
|
||||
htfcss: ecsx font-weight: bold;
|
||||
htfcss: ecbi font-weight: bold; font-style: italic;
|
||||
htfcss: ecit font-style: italic; font-family: monospace,monospace;
|
||||
htfcss: ecss font-family: sans-serif;
|
||||
htfcss: ecff font-family: fantasy;
|
||||
htfcss: ecti font-style: italic;
|
||||
htfcss: ectt font-family: monospace,monospace;
|
||||
htfcss: cmmi font-style: italic;
|
||||
htfcss: cmmib font-style: italic; font-weight: bold;
|
||||
File: oscillators.html
|
||||
File: oscillators.css
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||||
File: oscillators.tmp
|
||||
Css: p{margin-top:0;margin-bottom:0}
|
||||
Css: p.indent{text-indent:0;}
|
||||
Css: p + p{margin-top:1em;}
|
||||
Css: p + div, p + pre {margin-top:1em;}
|
||||
Css: div + p, pre + p {margin-top:1em;}
|
||||
Css: a { overflow-wrap: break-word; word-wrap: break-word; word-break: break-word; hyphens: auto; }
|
||||
Css: @media print {div.crosslinks {visibility:hidden;}}
|
||||
Css: a img { border-top: 0; border-left: 0; border-right: 0; }
|
||||
Font_Css("4"): .small-caps{font-variant: small-caps; }
|
||||
Font_Css("10"): .htf-cmbx {font-weight: bold; font-style:normal;}
|
||||
Font_Css("12"): .htf-calligraphy {font-family:cursive}
|
||||
Font_Css("14"): .htf-italic {font-style: italic;}
|
||||
Font_Css("16"): .htf-bold {font-weight: bold;}
|
||||
Font_Css("12"): .htf-calligraphy-bold {font-family:cursive ; font-weight: bold; }
|
||||
Css: center { margin-top:1em; margin-bottom:1em; }
|
||||
Css: td center { margin-top:0em; margin-bottom:0em; }
|
||||
Css: .Canvas { position:relative; }
|
||||
Css: img.math{vertical-align:middle;}
|
||||
Css: div.par-math-display, div.math-display{text-align:center;}
|
||||
Css: li p.indent { text-indent: 0em }
|
||||
Css: li p:first-child{ margin-top:0em; }
|
||||
Css: li p:last-child, li div:last-child { margin-bottom:0.5em; }
|
||||
Css: li p~ul:last-child, li p~ol:last-child{ margin-bottom:0.5em; }
|
||||
Css: .enumerate1 {list-style-type:decimal;}
|
||||
Css: .enumerate2 {list-style-type:lower-alpha;}
|
||||
Css: .enumerate3 {list-style-type:lower-roman;}
|
||||
Css: .enumerate4 {list-style-type:upper-alpha;}
|
||||
Css: div.newtheorem { margin-bottom: 2em; margin-top: 2em;}
|
||||
Css: .obeylines-h,.obeylines-v {white-space: nowrap; }
|
||||
Css: div.obeylines-v p { margin-top:0; margin-bottom:0; }
|
||||
Css: .overline{ text-decoration:overline; }
|
||||
Css: .overline img{ border-top: 1px solid black; }
|
||||
Css: td.displaylines {text-align:center; white-space:nowrap;}
|
||||
Css: .centerline {text-align:center;}
|
||||
Css: .rightline {text-align:right;}
|
||||
Css: pre.verbatim {font-family: monospace,monospace; text-align:left; clear:both; }
|
||||
Css: .fbox {padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }
|
||||
Css: div.fbox {display:table}
|
||||
Css: div.center div.fbox {text-align:center; clear:both; padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }
|
||||
Css: div.minipage{width:100%;}
|
||||
Css: div.center, div.center div.center {text-align: center; margin-left:1em; margin-right:1em;}
|
||||
Css: div.center div {text-align: left;}
|
||||
Css: div.flushright, div.flushright div.flushright {text-align: right;}
|
||||
Css: div.flushright div {text-align: left;}
|
||||
Css: div.flushleft {text-align: left;}
|
||||
Css: .underline{ text-decoration:underline; }
|
||||
Css: .underline img{ border-bottom: 1px solid black; margin-bottom:1pt; }
|
||||
Css: .framebox-c, .framebox-l, .framebox-r { padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }
|
||||
Css: .framebox-c {text-align:center;}
|
||||
Css: .framebox-l {text-align:left;}
|
||||
Css: .framebox-r {text-align:right;}
|
||||
Css: span.thank-mark{ vertical-align: super }
|
||||
Css: span.footnote-mark sup.textsuperscript, span.footnote-mark a sup.textsuperscript{ font-size:80%; }
|
||||
Css: div.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }
|
||||
Css: table.tabular td p{margin-top:0em;}
|
||||
Css: table.tabular {margin-left: auto; margin-right: auto;}
|
||||
Css: td p:first-child{ margin-top:0em; }
|
||||
Css: td p:last-child{ margin-bottom:0em; }
|
||||
Css: div.td00{ margin-left:0pt; margin-right:0pt; }
|
||||
Css: div.td01{ margin-left:0pt; margin-right:5pt; }
|
||||
Css: div.td10{ margin-left:5pt; margin-right:0pt; }
|
||||
Css: div.td11{ margin-left:5pt; margin-right:5pt; }
|
||||
Css: table[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }
|
||||
Css: td.td00{ padding-left:0pt; padding-right:0pt; }
|
||||
Css: td.td01{ padding-left:0pt; padding-right:5pt; }
|
||||
Css: td.td10{ padding-left:5pt; padding-right:0pt; }
|
||||
Css: td.td11{ padding-left:5pt; padding-right:5pt; }
|
||||
Css: table[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }
|
||||
Css: .hline hr, .cline hr{ height : 0px; margin:0px; }
|
||||
Css: .hline td, .cline td{ padding: 0; }
|
||||
Css: .hline hr, .cline hr{border:none;border-top:1px solid black;}
|
||||
Css: .tabbing-right {text-align:right;}
|
||||
Css: div.float, div.figure {margin-left: auto; margin-right: auto;}
|
||||
Css: div.float img {text-align:center;}
|
||||
Css: div.figure img {text-align:center;}
|
||||
Css: .marginpar,.reversemarginpar {width:20%; float:right; text-align:left; margin-left:auto; margin-top:0.5em; font-size:85%; text-decoration:underline;}
|
||||
Css: .marginpar p,.reversemarginpar p{margin-top:0.4em; margin-bottom:0.4em;}
|
||||
Css: .reversemarginpar{float:left;}
|
||||
Css: table.equation {width:100%;}
|
||||
Css: .equation td{text-align:center; }
|
||||
Css: td.equation { margin-top:1em; margin-bottom:1em; }
|
||||
Css: td.equation-label { width:5%; text-align:center; }
|
||||
Css: td.eqnarray4 { width:5%; white-space: normal; }
|
||||
Css: td.eqnarray2 { width:5%; }
|
||||
Css: table.eqnarray-star, table.eqnarray {width:100%;}
|
||||
Css: div.eqnarray{text-align:center;}
|
||||
Css: div.array {text-align:center;}
|
||||
Css: div.pmatrix {text-align:center;}
|
||||
Css: table.pmatrix {width:100%;}
|
||||
Css: span.pmatrix img{vertical-align:middle;}
|
||||
Css: div.pmatrix {text-align:center;}
|
||||
Css: table.pmatrix {width:100%;}
|
||||
Css: span.bar-css {text-decoration:overline;}
|
||||
Css: img.cdots{vertical-align:middle;}
|
||||
Css: .partToc a, .partToc, .likepartToc a, .likepartToc {line-height: 200%; font-weight:bold; font-size:110%;}
|
||||
Css: .index-item, .index-subitem, .index-subsubitem {display:block}
|
||||
Css: div.caption {text-indent:-2em; margin-left:3em; margin-right:1em; text-align:left;}
|
||||
Css: div.caption span.id{font-weight: bold; white-space: nowrap; }
|
||||
Css: h1.partHead{text-align: center}
|
||||
Css: p.bibitem { text-indent: -2em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }
|
||||
Css: p.bibitem-p { text-indent: 0em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }
|
||||
Css: .paragraphHead, .likeparagraphHead { margin-top:2em; font-weight: bold;}
|
||||
Css: .subparagraphHead, .likesubparagraphHead { font-weight: bold;}
|
||||
Css: .verse{white-space:nowrap; margin-left:2em}
|
||||
Css: div.maketitle {text-align:center;}
|
||||
Css: h2.titleHead{text-align:center;}
|
||||
Css: div.maketitle{ margin-bottom: 2em; }
|
||||
Css: div.author, div.date {text-align:center;}
|
||||
Css: div.thanks{text-align:left; margin-left:10%; font-size:85%; font-style:italic; }
|
||||
Css: div.author{white-space: nowrap;}
|
||||
Css: div.abstract p {margin-left:5%; margin-right:5%;}
|
||||
Css: div.abstract {width:100%;}
|
||||
Css: .abstracttitle{text-align:center;margin-bottom:1em;}
|
||||
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\documentclass{article}
|
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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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\graphicspath{ {./img/} }
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\usepackage[a4paper, margin=35mm]{geometry}
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\begin{document}
|
||||
|
||||
\section{Oscillating Circuits}
|
||||
|
||||
A discussion of synthesizer oscillators requires an introduction to simple oscillating circuits.
|
||||
We will discuss the most basic oscillating circuits, then we will move on to oscillators with easily-tunable
|
||||
frequencies. Then we will tackle the most complex issue for the purposes of musical synthesis: \textit{voltage control.}
|
||||
This is the important part! If we can control an oscillator's frequency by voltage, then we can make another circuit
|
||||
change its voltage, like a sequencer for example. Let's check it out!
|
||||
|
||||
\subsection{Passive Oscillators}
|
||||
|
||||
The simplest oscillators are those which rely on \textit{passive components}, electrical components which do not
|
||||
generate power or 'add amplitude' to a signal. These are components like resistors, inductors, and capacitors which
|
||||
only dissipate, store, or release already-existing power introduced by another component. An
|
||||
example of a non-passive component would be a power supply or a transistor. Passive components tend to be governed by
|
||||
simpler rules that are easier to understand and exploit.
|
||||
|
||||
The simplest oscillating circuit to my knowledge is the Resistor-Inductor-Capacitor or RLC circuit. It's not an
|
||||
'RIC' circuit because the letter I commonly represents current in electrical engineering, so we use L to indicate
|
||||
inductors or inductance.
|
||||
|
||||
The diagram above depicts an RLC-circuit, with each component in series. We can come up with an equation to describe its behavior,
|
||||
but first we need to know how each component responds to voltage and current. \textbf{Resistors} are governed by \textbf{Ohm's Law}:
|
||||
\[
|
||||
V = IR, \hspace{8mm} I = \frac{V}{R},
|
||||
\]
|
||||
where $R$ is resistance, measured in Ohms ($\Omega$). Resistors are called \textbf{linear components} because their voltage-current
|
||||
response is linear i.e. an increase in voltage or current causes a linear increase in the other.
|
||||
\textbf{Inductors} are governed by:
|
||||
\[
|
||||
V = L\frac{di}{dt}, \hspace{8mm} i = \frac{1}{L} \int V \,dt
|
||||
\]
|
||||
where $L$ is inductance, measured in Henries (H). Note that this means that the voltage across an inductor responds to a change
|
||||
in current. If a current is constant, then the voltage vanishes. But if we change the current, a voltage is generated across the
|
||||
inductor. That means if we send an \textit{alternating current} which is always changing through the inductor, then we will get a
|
||||
voltage across the inductor. This is unusual because an inductor is essentially just a short-circuit and yet when a changing
|
||||
current passes through it, it will have a voltage like a resistive element! You could say that inductors \textit{resist a change
|
||||
in current} in this sense of resistance.
|
||||
|
||||
\pagebreak
|
||||
|
||||
Finally, \textbf{Capacitors} are governed by the equation:
|
||||
\[ i = C\frac{dv}{dt}, \]
|
||||
where $C$ is the capacitance of the component, measured in Farads (F). Here, the capacitor's behavior is similar but the relationship is sort of reversed or
|
||||
flipped, if you will. Now, if we have a constant voltage across the capacitor then no current will flow. This makes sense because
|
||||
a capacitor is essentially made from two conductive plates seperated from one another by a non-conductive material. This is
|
||||
effectively a break in the circuit, as indicated by the standard electrical symbol for a capacitor.
|
||||
But if we change the voltage across the capacitor, it starts to conduct! changing the voltage
|
||||
somehow forces current to flow between the plates! It's no mystery, this is due to some complicated rules of physics known generally
|
||||
as \textit{electrodynamics}, but that's for another time.
|
||||
|
||||
Okay. We can connect these three equations mathematically by utilizing \textbf{Kirchoff's Laws of Voltage and Current}. This sounds
|
||||
a little complicated but it relies on some straightforward principles. The core idea is that any current which enters a wire junction
|
||||
(usually called a \textit{node}) must exit the junction in some way. For many reasons, another logical rule that follows from this
|
||||
is that the voltages across each component in a loop must sum to zero. If this were not true, it would result in charge pooling
|
||||
somewhere in the wire, which is almost always impossible. Here's the laws:
|
||||
|
||||
\begin{itemize}
|
||||
\item \textbf{KIRCHOFF'S CURRENT LAW:} All current entering a node must exit i.e. the sum of currents entering/leaving a node
|
||||
must always be \textit{zero.}
|
||||
\item \textbf{KIRCHOFF'S VOLTAGE LAW:} The sum of component voltages over any loop of wire must be zero.
|
||||
\end{itemize}
|
||||
|
||||
If the current entering each node must also be leaving it, in our RLC circuit this means the current through each node is the same.
|
||||
This is because the circuit is a closed loop! If the current were not the same across each node, then it would have to pool somewhere
|
||||
or escape into thin air. We can express this mathematically:
|
||||
\[i_{R} + i_{L} + i_{C} = 0.\]
|
||||
We know the equations for the current through a resistor and capacitor:
|
||||
\[\frac{v}{R} + i_{L} + C\frac{dv}{dt} = 0.\]
|
||||
We know that voltage across an inductor is proportional to the change in current. If we integrate with respect to time,
|
||||
we can get a figure for the current through an inductor:
|
||||
\[i_{L} = \frac{1}{L} \int v \,dt\]
|
||||
\[\frac{v}{R} + \frac{1}{L} \int v \,dt + C\frac{dv}{dt} = 0.\]
|
||||
Now we just differentiate:
|
||||
\[C\frac{d^{2}v}{dt^{2}} + \frac{1}{R}\frac{dv}{dt} + \frac{1}{L} v =0,\] or
|
||||
\[av'' + bv' +cv = 0; \hspace{6mm} a=C,\hspace{4mm}b=\frac{1}{R},\hspace{4mm}c= \frac{1}{L}.\]
|
||||
I would like to state here that the variable $v$ is a function dependent on time, so it should be written as $v(t)$ for clarity. We
|
||||
prefer to be unclear here, because it is less cluttered to write equations that way. Just keep this in mind.
|
||||
|
||||
What we have stumbled upon here is a truth that I find quite exciting but it has been the nightmare of many underclassmen electrical
|
||||
engineers forced to learn this before finishing their math coursework (who could I possibly be talking about here?): This circuit
|
||||
is governed by a \textbf{homogenous second-order ordinary differential equation!}
|
||||
\pagebreak
|
||||
Unfortunately we cannot go over the basics of ODE's here. Like many stressed undergraduate engineers before you, you will have
|
||||
to take my word as gospel. A homogenous second-order ODE is basically an equation where the function ($v$ in this case)
|
||||
is not defined directly in terms of an independent variable like time or $x$ or space etc,
|
||||
but in terms of its own differentials. This equation is \textit{second-order} because the function is defined in terms of its
|
||||
second differential. If you are very clever, you may already be thinking of how one might solve such an equation to find
|
||||
$v(t)$ in terms of $t$ alone and not in terms of its differentials. If you are even cleverer, you may be thinking about
|
||||
the classic function $f(t) = e^{x},$ because its differential/integral is itself. If you are some kind of genius, you may even
|
||||
be considering also the trigonometric functions (sine, cosine, not so much tangent here), because they have a similar property.
|
||||
This will put you on the right track. Going
|
||||
forward, you will find that the equations that govern these circuits involve many natural exponentials and sine waves
|
||||
because of this property.
|
||||
Most ordinary differential equations are solved using complicated techniques that involve relating differentials with
|
||||
natural exponentials and waves. I cannot get into specifics here but if you are interested, I would keep this property in mind.
|
||||
|
||||
OKAY. Back to the equation. Our equation can be represented by something called its \textit{characteristic equation:}
|
||||
\[ar^{2} + br + c = 0.\]
|
||||
This is just a representation of the equation that represents the order of each differential by a power of the variable \textit{r}.
|
||||
If we solve it like a polynomial we get the quadratic formula:
|
||||
\[r_{1,2} = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}.\]
|
||||
You might realize that it is possible for the root of our characteristic equation to be complex ($r=\lambda + \mu i$).
|
||||
When this is the case, our solution is of the form:
|
||||
\[v(t) = C_{1}e^{\lambda t}\cos(\mu t) + C_{2}e^{\lambda t}\sin(\mu t),\]
|
||||
where $C_{1}, C_{2}$ are constants determined by the \textit{initial conditions} of the system. Once all the math is done,
|
||||
we will take a break to build some intuition for all this nonsense. The important part is that the reason this is a solution has
|
||||
to do with the fact that the differentials of exponentials and sine waves are equal to themselves or related to themselves.
|
||||
|
||||
Now we can start getting specific with our constants. If we assume our system begins with the components already charged
|
||||
(meaning we have allowed a current source to keep a constant voltage across the components for enough time for the capacitor
|
||||
to be charged to the positive voltage $V_{+}$), then we can say $v_{0} = V_{+}.$ If it has sat for a long long time, then
|
||||
there will be no change in voltage, so also we can say $v'_{0} = 0.$ So:
|
||||
\[v(0) = V_{+} = C_{1}, \hspace{8mm} v'(0) = 0 = \lambda C_{1} + \mu C_{2},\]
|
||||
\[C_{1} = V_{+}, \hspace{8mm} C_{2} = - \frac{\lambda}{\mu} V_{+}.\]
|
||||
|
||||
We can keep defining constants and it will quickly cause your mind to cloud over. The point here is that if we charge up this
|
||||
circuit and then let it run, the voltage and current will oscillate back and forth.
|
||||
I have included below a graph of a potential oscillation. The frequency of this oscillation is determined
|
||||
by the constant $\mu$, which is equal to the inverse of the root of the
|
||||
product of inductance and capacitance ($\mu=\frac{1}{\sqrt{LC}}$).
|
||||
In electrical engineering, it is usually called the \textbf{angular frequency} and is more commonly
|
||||
denoted with the greek character $\omega$. We found the constants $C_{1},C_{2}$ to show that they are dependent on the
|
||||
initial voltage we charge the circuit to, as well as the properties of the three components. We have shown mathematically
|
||||
that the behavior of the circuit under certain parameters (such that $b^{2} < 4ac$) will be oscillatory in nature. But why does
|
||||
the circuit oscillate sometimes?
|
||||
|
||||
\pagebreak
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.3]{natural-RLC}
|
||||
\end{center}
|
||||
|
||||
Well, recall that an inductor generates a voltage once the current through it changes, and that a capacitor
|
||||
begins conducting current once the voltage across it changes. When we charge the capacitor to some voltage and then close the
|
||||
circuit, the voltage across the capacitor suddenly becomes the voltage across the resistor as well. When there is a voltage
|
||||
across a resistor which is connected in a loop, then a current must flow. Conversely, the moment the circuit is closed it
|
||||
forces the voltage to drop because a current must flow through the resistor. If the capacitor were alone in series with the resistor,
|
||||
then it would simply discharge to a voltage of 0.
|
||||
|
||||
As the capacitor pushes current through the inductor, the inductor begins to respond. Initially it acts as a short, but as the
|
||||
current through it changes, it begins to generate a voltage. That voltage causes the inductor to push current
|
||||
into the capacitor again, charging it. Then once the inductor has discharged its stored energy, the capacitor is recharged
|
||||
and it begins to conduct again. This continues until all of the electrical energy is dissipated through the resistor
|
||||
(and realistically also through the resistances in the capacitor and inductor) as heat until there is none left. Kind of cool!
|
||||
|
||||
If we tuned the capacitance and inductance properly, we could get one of these circuits to oscillate at an audible frequency.
|
||||
This would not be an ideal circuit for music making, and there are two big reasons for this:
|
||||
|
||||
\begin{itemize}
|
||||
\item{The circuit will only oscillate for a short period after it is triggered, preventing us from ever using it
|
||||
to play any sustained note.}
|
||||
\item{YOU CAN'T TUNE THE FREQUENCY! You would need a variable inductor or transformer, and those solutions quickly
|
||||
become impractical. Unless you like plucky drone music, you're out of luck here.}
|
||||
\end{itemize}
|
||||
|
||||
I'm sure there are many other kinds of passive oscillating circuits, but I think we have done enough here. Next we will
|
||||
consider \textit{active oscillators.}
|
||||
|
||||
\pagebreak
|
||||
|
||||
\subsection{Active Oscillators}
|
||||
|
||||
\end{document}
|
||||
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