Lao Tzu
334 lines
17 KiB
HTML
Executable File
334 lines
17 KiB
HTML
Executable File
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<h3 class="sectionHead"><span class="titlemark">1 </span> <a
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id="x1-10001"></a>Oscillating Circuits</h3>
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<!--l. 11--><p class="noindent" >A discussion of synthesizer oscillators requires an introduction to simple oscillating circuits. We
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will discuss the most basic oscillating circuits, then we will move on to oscillators with
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easily-tunable frequencies. Then we will tackle the most complex issue for the purposes of musical
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synthesis: <span
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class="ecti-1000">voltage control. </span>This is the important part! If we can control an oscillator’s frequency by
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voltage, then we can make another circuit change its voltage, like a sequencer for example. Let’s
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check it out!
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<!--l. 17--><p class="noindent" >
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<h4 class="subsectionHead"><span class="titlemark">1.1 </span> <a
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id="x1-20001.1"></a>Passive Oscillators</h4>
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<!--l. 19--><p class="noindent" >The simplest oscillators are those which rely on <span
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class="ecti-1000">passive components</span>, electrical components which
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do not generate power or ’add amplitude’ to a signal. These are components like resistors,
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inductors, and capacitors which only dissipate, store, or release already-existing power introduced
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by another component. An example of a non-passive component would be a power supply or a
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transistor. Passive components tend to be governed by simpler rules that are easier to understand
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and exploit.
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<!--l. 25--><p class="indent" > The simplest oscillating circuit to my knowledge is the Resistor-Inductor-Capacitor or RLC
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circuit. It’s not an ’RIC’ circuit because the letter I commonly represents current in electrical
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engineering, so we use L to indicate inductors or inductance.
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<!--l. 29--><p class="indent" > The diagram above depicts an RLC-circuit, with each component in series. We can come up
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with an equation to describe its behavior, but first we need to know how each component responds
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to voltage and current. <span
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class="ecbx-1000">Resistors </span>are governed by <span
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class="ecbx-1000">Ohm’s Law</span>:
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<div class="math-display" >
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<img
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src="oscillators0x.png" alt="V = IR, I = V-,
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R
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" class="math-display" ></div>
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<!--l. 33--><p class="nopar" > where <span
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class="cmmi-10">R </span>is resistance, measured in Ohms (<span
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class="cmr-10">Ω</span>). Resistors are called <span
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class="ecbx-1000">linear components </span>because
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their voltage-current response is linear i.e. an increase in voltage or current causes a linear increase
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in the other. <span
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class="ecbx-1000">Inductors </span>are governed by:
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<div class="math-display" >
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<img
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src="oscillators1x.png" alt=" di 1 ∫
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V = Ldt, i = L V dt
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" class="math-display" ></div>
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<!--l. 39--><p class="nopar" > where <span
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class="cmmi-10">L </span>is inductance, measured in Henries (H). Note that this means that the voltage across an
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inductor responds to a change in current. If a current is constant, then the voltage vanishes. But if
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we change the current, a voltage is generated across the inductor. That means if we send an
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<span
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class="ecti-1000">alternating current </span>which is always changing through the inductor, then we will get a
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voltage across the inductor. This is unusual because an inductor is essentially just a
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short-circuit and yet when a changing current passes through it, it will have a voltage like a
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resistive element! You could say that inductors <span
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class="ecti-1000">resist a change in current </span>in this sense of
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resistance.
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<!--l. 49--><p class="indent" > Finally, <span
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class="ecbx-1000">Capacitors </span>are governed by the equation:
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<div class="math-display" >
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<img
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src="oscillators2x.png" alt="i = C dv,
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dt
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" class="math-display" ></div>
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<!--l. 50--><p class="nopar" > where <span
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class="cmmi-10">C </span>is the capacitance of the component, measured in Farads (F). Here, the capacitor’s
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behavior is similar but the relationship is sort of reversed or flipped, if you will. Now, if we have a
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constant voltage across the capacitor then no current will flow. This makes sense because a
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capacitor is essentially made from two conductive plates seperated from one another by a
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non-conductive material. This is effectively a break in the circuit, as indicated by the standard
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electrical symbol for a capacitor. But if we change the voltage across the capacitor, it starts to
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conduct! changing the voltage somehow forces current to flow between the plates! It’s no mystery,
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this is due to some complicated rules of physics known generally as <span
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class="ecti-1000">electrodynamics</span>, but that’s for
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another time.
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<!--l. 59--><p class="indent" > Okay. We can connect these three equations mathematically by utilizing <span
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class="ecbx-1000">Kirchoff’s Laws of</span>
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<span
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class="ecbx-1000">Voltage and Current</span>. This sounds a little complicated but it relies on some straightforward
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principles. The core idea is that any current which enters a wire junction (usually called a <span
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class="ecti-1000">node</span>)
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must exit the junction in some way. For many reasons, another logical rule that follows from this is
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that the voltages across each component in a loop must sum to zero. If this were not true, it would
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result in charge pooling somewhere in the wire, which is almost always impossible. Here’s the
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laws:
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<ul class="itemize1">
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<li class="itemize"><span
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class="ecbx-1000">KIRCHOFF’S CURRENT LAW: </span>All current entering a node must exit i.e. the
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sum of currents entering/leaving a node must always be <span
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class="ecti-1000">zero.</span>
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</li>
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<li class="itemize"><span
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class="ecbx-1000">KIRCHOFF’S VOLTAGE LAW: </span>The sum of component voltages over any loop
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of wire must be zero.</li></ul>
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<!--l. 71--><p class="indent" > If the current entering each node must also be leaving it, in our RLC circuit this means the
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current through each node is the same. This is because the circuit is a closed loop! If the current
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were not the same across each node, then it would have to pool somewhere or escape into thin air.
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We can express this mathematically:
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<div class="math-display" >
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<img
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src="oscillators3x.png" alt="iR + iL + iC = 0.
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" class="math-display" ></div>
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<!--l. 74--><p class="nopar" > We know the equations for the current through a resistor and capacitor:
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<div class="math-display" >
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<img
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src="oscillators4x.png" alt="v dv
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R-+ iL + C dt = 0.
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" class="math-display" ></div>
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<!--l. 76--><p class="nopar" > We know that voltage across an inductor is proportional to the change in current.
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If we integrate with respect to time, we can get a figure for the current through an
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inductor:
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<div class="math-display" >
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<img
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src="oscillators5x.png" alt=" ∫
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iL =-1 vdt
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L
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" class="math-display" ></div>
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<!--l. 79--><p class="nopar" >
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<div class="math-display" >
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<img
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src="oscillators6x.png" alt=" ∫
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v-+ 1- v dt+ Cdv = 0.
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R L dt
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" class="math-display" ></div>
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<!--l. 80--><p class="nopar" > Now we just differentiate:
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<div class="math-display" >
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<img
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src="oscillators7x.png" alt=" d2v 1 dv 1
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C --2 + ----+ --v = 0,
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dt R dt L
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" class="math-display" ></div>
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<!--l. 82--><p class="nopar" > or
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<div class="math-display" >
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<img
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src="oscillators8x.png" alt="av′′+ bv′ + cv = 0; a = C, b = 1-, c = 1-.
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R L
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" class="math-display" ></div>
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<!--l. 83--><p class="nopar" > I would like to state here that the variable <span
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class="cmmi-10">v </span>is a function dependent on time, so it should be
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written as <span
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class="cmmi-10">v</span><span
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class="cmr-10">(</span><span
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class="cmmi-10">t</span><span
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class="cmr-10">) </span>for clarity. We prefer to be unclear here, because it is less cluttered to write
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equations that way. Just keep this in mind.
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<!--l. 87--><p class="indent" > What we have stumbled upon here is a truth that I find quite exciting but it has been the
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nightmare of many underclassmen electrical engineers forced to learn this before finishing their
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math coursework (who could I possibly be talking about here?): This circuit is governed by a
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<span
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class="ecbx-1000">homogenous second-order ordinary differential equation! </span>Unfortunately we cannot go over
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the basics of ODE’s here. Like many stressed undergraduate engineers before you, you
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will have to take my word as gospel. A homogenous second-order ODE is basically
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an equation where the function (<span
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class="cmmi-10">v </span>in this case) is not defined directly in terms of an
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independent variable like time or <span
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class="cmmi-10">x </span>or space etc, but in terms of its own differentials. This
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equation is <span
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class="ecti-1000">second-order </span>because the function is defined in terms of its second differential.
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If you are very clever, you may already be thinking of how one might solve such an
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equation to find <span
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class="cmmi-10">v</span><span
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class="cmr-10">(</span><span
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class="cmmi-10">t</span><span
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class="cmr-10">) </span>in terms of <span
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class="cmmi-10">t </span>alone and not in terms of its differentials. If you are
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even cleverer, you may be thinking about the classic function <span
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class="cmmi-10">f</span><span
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class="cmr-10">(</span><span
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class="cmmi-10">t</span><span
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class="cmr-10">) = </span><span
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class="cmmi-10">e</span><sup><span
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class="cmmi-7">x</span></sup><span
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class="cmmi-10">, </span>because its
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differential/integral is itself. If you are some kind of genius, you may even be considering also the
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trigonometric functions (sine, cosine, not so much tangent here), because they have a similar
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property. This will put you on the right track. Going forward, you will find that the
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equations that govern these circuits involve many natural exponentials and sine waves
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because of this property. Most ordinary differential equations are solved using complicated
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techniques that involve relating differentials with natural exponentials and waves. I
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cannot get into specifics here but if you are interested, I would keep this property in
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mind.
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<!--l. 105--><p class="indent" > OKAY. Back to the equation. Our equation can be represented by something called its
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<span
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class="ecti-1000">characteristic equation:</span>
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<div class="math-display" >
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<img
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src="oscillators9x.png" alt="ar2 + br+ c = 0.
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" class="math-display" ></div>
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<!--l. 106--><p class="nopar" > This is just a representation of the equation that represents the order of each differential by a
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power of the variable <span
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class="ecti-1000">r</span>. If we solve it like a polynomial we get the quadratic formula:
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<div class="math-display" >
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<img
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src="oscillators10x.png" alt=" √ -2-----
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r1,2 = - b±--b---4ac.
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2a
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" class="math-display" ></div>
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<!--l. 109--><p class="nopar" > You might realize that it is possible for the root of our characteristic equation to be complex
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(<span
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class="cmmi-10">r </span><span
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class="cmr-10">= </span><span
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class="cmmi-10">λ </span><span
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class="cmr-10">+ </span><span
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class="cmmi-10">μi</span>). When this is the case, our solution is of the form:
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<div class="math-display" >
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<img
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src="oscillators11x.png" alt="v(t) = C1eλtcos(μt)+ C2eλtsin(μt),
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" class="math-display" ></div>
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<!--l. 112--><p class="nopar" > where <span
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class="cmmi-10">C</span><sub><span
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class="cmr-7">1</span></sub><span
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class="cmmi-10">,C</span><sub><span
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class="cmr-7">2</span></sub> are constants determined by the <span
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class="ecti-1000">initial conditions </span>of the system. Once all
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the math is done, we will take a break to build some intuition for all this nonsense.
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The important part is that the reason this is a solution has to do with the fact that
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the differentials of exponentials and sine waves are equal to themselves or related to
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themselves.
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<!--l. 117--><p class="indent" > Now we can start getting specific with our constants. If we assume our system begins
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with the components already charged (meaning we have allowed a current source to
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keep a constant voltage across the components for enough time for the capacitor to
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be charged to the positive voltage <span
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class="cmmi-10">V</span> <sub><span
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class="cmr-7">+</span></sub>), then we can say <span
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class="cmmi-10">v</span><sub><span
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class="cmr-7">0</span></sub> <span
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class="cmr-10">= </span><span
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class="cmmi-10">V</span> <sub><span
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class="cmr-7">+</span></sub><span
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class="cmmi-10">. </span>If it has sat for a
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long long time, then there will be no change in voltage, so also we can say <span
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class="cmmi-10">v</span><span
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class="cmsy-10">′</span><sub><span
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class="cmr-7">0</span></sub> <span
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class="cmr-10">= 0</span><span
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class="cmmi-10">.</span>
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So:
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<div class="math-display" >
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<img
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src="oscillators12x.png" alt="v(0) = V+ = C1, v′(0) = 0 = λC1 + μC2,
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" class="math-display" ></div>
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<!--l. 121--><p class="nopar" >
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<div class="math-display" >
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<img
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src="oscillators13x.png" alt="C1 = V+, C2 = - λV+.
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μ
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" class="math-display" ></div>
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<!--l. 122--><p class="nopar" >
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<!--l. 124--><p class="indent" > We can keep defining constants and it will quickly cause your mind to cloud over. The point
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here is that if we charge up this circuit and then let it run, the voltage and current will oscillate
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back and forth. I have included below a graph of a potential oscillation. The frequency of this
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oscillation is determined by the constant <span
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class="cmmi-10">μ</span>, which is equal to the inverse of the root of
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the product of inductance and capacitance (<span
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class="cmmi-10">μ </span><span
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class="cmr-10">=</span> <img
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src="oscillators14x.png" alt="--1-
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√LC--" class="frac" align="middle">). In electrical engineering, it is
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usually called the <span
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class="ecbx-1000">angular frequency </span>and is more commonly denoted with the greek
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character <span
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class="cmmi-10">ω</span>. We found the constants <span
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class="cmmi-10">C</span><sub><span
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class="cmr-7">1</span></sub><span
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class="cmmi-10">,C</span><sub><span
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class="cmr-7">2</span></sub> to show that they are dependent on the initial
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voltage we charge the circuit to, as well as the properties of the three components. We
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have shown mathematically that the behavior of the circuit under certain parameters
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(such that <span
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class="cmmi-10">b</span><sup><span
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class="cmr-7">2</span></sup> <span
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class="cmmi-10">< </span><span
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class="cmr-10">4</span><span
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class="cmmi-10">ac</span>) will be oscillatory in nature. But why does the circuit oscillate
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sometimes?
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<div class="center"
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>
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<!--l. 137--><p class="noindent" >
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<!--l. 138--><p class="noindent" ><img
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src="./img//natural-RLC.png" alt="PIC"
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width="21" height="21" ></div>
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<!--l. 141--><p class="indent" > Well, recall that an inductor generates a voltage once the current through it changes, and that
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a capacitor begins conducting current once the voltage across it changes. When we charge the
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capacitor to some voltage and then close the circuit, the voltage across the capacitor suddenly
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becomes the voltage across the resistor as well. When there is a voltage across a resistor which is
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connected in a loop, then a current must flow. Conversely, the moment the circuit is closed it
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forces the voltage to drop because a current must flow through the resistor. If the capacitor
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were alone in series with the resistor, then it would simply discharge to a voltage of
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0.
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<!--l. 148--><p class="indent" > As the capacitor pushes current through the inductor, the inductor begins to respond. Initially
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it acts as a short, but as the current through it changes, it begins to generate a voltage. That
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voltage causes the inductor to push current into the capacitor again, charging it. Then once the
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inductor has discharged its stored energy, the capacitor is recharged and it begins to conduct
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again. This continues until all of the electrical energy is dissipated through the resistor (and
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realistically also through the resistances in the capacitor and inductor) as heat until there is none
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left. Kind of cool!
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<!--l. 154--><p class="indent" > If we tuned the capacitance and inductance properly, we could get one of these circuits to
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oscillate at an audible frequency. This would not be an ideal circuit for music making, and there
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are two big reasons for this:
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<ul class="itemize1">
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<li class="itemize">The circuit will only oscillate for a short period after it is triggered, preventing us from
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ever using it to play any sustained note.
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</li>
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<li class="itemize">YOU CAN’T TUNE THE FREQUENCY! You would need a variable inductor or
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transformer, and those solutions quickly become impractical. Unless you like plucky
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drone music, you’re out of luck here.</li></ul>
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<!--l. 164--><p class="indent" > I’m sure there are many other kinds of passive oscillating circuits, but I think we have done
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enough here. Next we will consider <span
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class="ecti-1000">active oscillators.</span>
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<!--l. 169--><p class="noindent" >
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<h4 class="subsectionHead"><span class="titlemark">1.2 </span> <a
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id="x1-30001.2"></a>Active Oscillators</h4>
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