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Circuit Idea/How do We Create Sinusoidal Oscillations?

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How do We Create Sinusoidal Oscillations?

(Revealing the Secret of LC Tank)

Circuit idea: Connect two heterogeneous energy storing elements to each other and charge one of them with energy.

Questions to be answered
LC tank is a legendary electrical circuit, and conceiving sinusoidal oscillations in an LC tank is a popular electrical phenomenon. It is well described in physics, electricity and electronics resources (see for example one of the best LC explanations on the web ). Browsing through these resources you will probably get to know what an LC tank is and how it operates in terms of electricity. Only, although these resources explain in detail what an LC tank does they do not say how and why it does this magic.

The point is that LC oscillation is not only a specific electrical occurrence; this is a great worldly phenomenon. There are many its manifestations in "non-electrical" world (mechanics, pneumatics, hydraulics, etc.) If you want to penetrate deeply into its nature, you will begin asking dozens of "philosophical" questions that are still unanswered. What are these questions? Here are some of them.

''Why does an LC tank consist of two "heterogeneous" accumulating elements (a capacitor and an inductor)? Why is it not assembled by two "homogeneous" elements (both capacitors or both inductors)? What is the role of these elements in the LC circuit? What do they actually do there? What are they? Why does the energy in an LC tank "have no peace"? What makes it circulate (in theory, infinitely) between the two components? What makes the accumulating elements discharge completely? Why does an LC oscillation have a sinusoidal shape? Why is it no with linear or other shape? Why does not the LC natural frequency depend on the amount of the "pushed" energy? Why is the oscillation curve symmetrical? What are the differences between LC and other (RC, relaxation, etc.) oscillations? Is there any likeness between them and what is it? What is the connection between the two-component LC tank and the one-component pendulum? What is the "capacitor" and what is the "inductor" in a pendulum?''

In this story, we will answer these questions in a manner that is quite different from the classical one. Here, we will forget temporarily the conventional "electrical" LC explanations (e.g., "...the inductor discharges through the capacitor and the field collapses..."); they will help us not so much to answer the questions above. Because we, human beings, cannot understand concrete (electrical) phenomena by concrete (electrical) explanations. In order to grasp the specific electrical ideas, we need first to reveal the most general, non-electrical ideas behind the concrete electrical phenomena.

In this exciting story we, circuit ideators and circuit philosophers, will try by ourselves to find out the truth about this great worldly phenomenon. For this purpose, we will re-invent and build step-by-step the famous LC tank, in order to grasp the basic idea benind it. Thus we will create a general, non-electrical philosophy for the specific electrical LC phenomenon.

Initial steady quantity
General sources. First of all, to do something in this world, we need a steady power source. According to the two kinds of energy - kinetic and potential, there are also two kinds of sources - a flow-like source and a pressure-like one (see the top of Fig. 1a). A flow-like source keeps up a constant flow rate and a pressure-like source keeps up a constant pressure difference of something (air, water, oil, spring tension, etc.) - Fig. 1b.

Electrical sources. Similarly, in electricity we have two kinds of sources - a current source keeping up a constant current and a voltage source keeping up a constant voltage (see the bottom of Fig. 1a).

The problem. Only, for a variety of reasons, we want to make the output quantity of the steady source begin changing periodically through time. Of course, we can take a varying source and just begin controlling it; but we want it to oscillate by itself, without our help. So, the problem is:

How do we make the steady quantity wiggle regularly through time?

Making a quantity depend on time
First and foremost, we have to make the steady quantity depend on (change through) time. At this point, it is not so important how it varies in time; it is only important that it changes in time. So, the main question is: How do we make time-dependent quantities?

Philosophy. Obviously, we have to start some process in time (but what is time?:) In order to make something change, we have to take from it or to give to it; this means to move something from one place to other... One source is not enough for this purpose since it keeps up steady its output quantity; we need two elements - source and load connected to each other, to begin transferring energy from the source to the load (and maybe, v.v.) We already have a source; it remains to choose the kind of load.

A need of accumulating. A resistor is useless for such a load since it can instantaneously change current (when voltage is instantaneously changed). A resistor cannot store energy, it cannot "remember". It is clear that we have to begin transferring and storing some quantity in an accumulating element (an integrator). Integrators can accumulate energy, matter or information; mostly they store energy. So, we can make its output quantity increase by continuously adding energy to the accumulator and to decrease it by continuously subtracting energy from the accumulator. According to the two kinds of energy - kinetic and potential, we can build two kinds of integrators - flow-to-pressure and pressure-to-flow integrator.

Flow-to-pressure integrator. In this case, we begin converting a flow-like quantity (carrying kinetic energy) into a pressure-like quantity (carrying potential energy) and storing it into an accumulating element having a property to contain (see the top of Fig. 2a). The energy moves from the flow to the pressure accumulator and the pressure-like quantity begins changing linearly through time (Fig. 2b).

Examples: moving mass pushes a spring, an accelerated pendulum raises, a child wind up a spring motor, an archer draws a bow, a flywheel puts in motion a toy car, a motor lifts a mass, an air pump blows up a tire, a water pump fills a reservoir, etc. We can even, in order to feel the process, imagine that very we are moving (in synectics, this technique for creative thinking is known as "empathy").

Note that the flow-to-pressure integrator consists of two heterogenous elements connected to each other - a source of kinetic energy and an accumulator of potential energy. If we connect two homogeneous elements to each other (a source of kinetic energy and an accumulator of kinetic energy), a big pressure appears and the process ceases momentarily; this arrangement will differentiate rather than integrate. A mechanical example: imagine what happens, if a moving car hits a car at rest.

Current-to-voltage integrator. In electricity, we have two elements that can accumulate energy - the capacitor stores potential energy and the inductor stores kinetic energy. So, we can build two kinds of electrical integrators - current-to-voltage and voltage-to-current integrator. Let's begin with the first and more popular of them.

According to the general flow-to-pressure arrangement, in a current-to-voltage integrator we begin converting a current (carrying kinetic energy) into a voltage (carrying potential energy) and storing it into a capacitor. So, the circuit includes a current source charging a capacitor (see the bottom of Fig. 2a). As a result, the voltage across the capacitor changes linearly through time (Fig. 2b). Note the current enters the positive plate of the capacitor; so, the capacitor acts here as a load.

The circuit consists of two heterogenous electrical elements connected to each other - a current source and a capacitor. If we connect two homogeneous elements to each other (a current source and an inductor), a high voltage appears across them and the process ceases momentarily; the circuit will differentiate rather than integrate.

Rounding the curve...
So far, we have made the steady quantity change linearly through time. Thus, we have obtained a quantity that increases continuously in time. In many cases (e.g., in ramp generators), this is a desired feature; but here we need a sinusoidal shape, not a linear one. So, the new problem is, "How do we round the linear curve?"

Then, let's scrutinize the present linear curve trying to find a remedy for converting it to a sinusoidal shaped. What does it mean? It means that after the quantity was changing linearly for a while, at this point it has already to begin changing (decreasing) its slope, to bend down, "to loose its nerve" (Fig. 3b). How do we achieve such an effect?

The general idea. So far, we have obtained a linear change by an integrator that was integrating a constant quantity. Now, we ask ourselves, "How do we make the integrator to slow down its output quantity?" The answer is: "By decreasing the input quantity." Here is the basic idea needed - to decrease the input quantity.

...by reducing the effective voltage...
Remember the imperfect RC integrator, whose linear curve is naturally rounded (exponentially shaped).

The "problem" (here, this is an advantage) of this integrating circuit (Fig. 3a) was that the capacitor was been charging by an imperfect "self-made" current source (assembled by a voltage source V and a resistor R). As a result, the voltage drop VC across the load (the capacitor) enervates the excitation voltage V; in this way, it disturbs the imperfect current source. In this arrangement, the voltage difference VR = V - VC determines the current I instead the excitation voltage V. As the current-creating voltage VR decreases through time the current decreases as well. Because of this decreasing in time current source the curve of VC begins decreasing its slope as well; this is what we want here.

...by decreasing the input current...
But here we have a perfect current source whose current does not depend on the voltage across the load (the capacitor). That means, if the load voltage increases, our current source increases its internal excitation voltage as well thus eliminating the voltage disturbance. It looks strange but here the problem is the very perfect current source:) What do we do then? How do we make it behave as an imperfect one?

Having been looking at Fig. 3, some rudiments of new idea can begin conceiving in our minds... It turns out that we can manage to round, change and finally decrease the slope of the curve by making in some way the perfect current source begin decreasing by itself the current produced when the voltage across the load increases... We can generalize this assertion:

The only way to change (decrease) the slope of the curve representing the output quantity of a perfect integrator is to begin changing (decreasing) its input quantity.

Whereas in the imperfect RC integrator the rounding is a result of decreasing the effective current-creating voltage VR, here the rounding will be a result of decreasing the whole input voltage V. Well, it is wonderful but how can we do this magic?

Of course, we can regulate the input voltage but this will require some electronic circuit. Do not remember here we "invent" the simplest "thing" that can oscillate; so, we need the simplest "trick" that can exist! What is it?

...by exhausting the input current source
It seems another new idea begins conceiving... We would like to decrease a quantity... It is produced by a power source... It has a limited reserve of energy... Eureka! If we begin taking away energy from the source, its output quantity will begin decreasing! Here are some examples: pushing the spring, the moving mass begins slowing its speed; running into an elastic barrier, we slow your run; rising, the pendulum begins slowing, etc.

The right idea. If, in a kind of way, the input current source begins weakening, depleting, exhausting its reserve of energy, the source's current will begin decreasing as well (Fig. 4a). As a result, the voltage across the capacitor will begin changing slowly and its curve will begin "loosing its nerve" (Fig. 4b). Is this source an imperfect current source as above? No, it is not; it is an exhausting but still a perfect constant current source that decreases gradually its output quantity. Here, the current decreases not because of the load voltage (as it was at the imperfect RC integrator) but because of the load consumption!

What electrical element can serve as such an exhausting perfect current source? Of course, an inductor charged with kinetic energy! It seems strange but it produces a "constant" current (for quick load changes) that gradually decreases:

A charged inductor can serve as an exhausting current source that decreases gradually its output current.

Eurika! We have obtained the arrangement needed (Fig. 4a); we have almost "invented" the LC tank:

An initially charged inductor suppplies an initially empty capacitor.

Note the current leaves the positive lead of the inductor; so, the inductor acts here as a source.

Stopping the "movement"
We are already at the top of the curve and now it's time to stop here. But how can we make the integrator's output quantity stop? Will it stop, if we stop the input quantity? No, it will not. Even if we decrease significantly the input quantity, the integrator will not cease changing its output quantity. The only way to stop the integrator's output quantity is to zero the input quantity. To convince yourself of this phenomenon imagine you fill a vessel with water: even if you lower the rate of flow to an insignificant quantity, the water level will still continue increasing. More examples: if you feed up gradually a money deposit with small change, the amount of money increases slowly (but still increases) through time; pushing the spring, the moving mass finally stops; moving pendulum stops at the top.

The "exhausting" flow source above that produces gradually decreasing flow has already helped us to change the slope of the curve and to round it. Here, at the top of the curve, it can help us now to stop the "movement". Indeed, at the maximum of the quantity, the source has completely exhausted its reserve of kinetic energy. Actually, at this instant, it is not merely a source; it is an empty source. As a result, the flow becomes zero (Fig. 5a) and the integrator stops changing its output quantity (Fig. 5b). Then, let's draw another conclusion:

The only way to stop changing the integrator's output quantity is to zero its input quantity.

In the electrical implementation (an LC tank) the inductor serves as an exhausting current source (Fig. 5a). At the top of the curve, it has fully discharged and produces no current. As a result, the voltage across the capacitor stops changing.

Why does the inductor discharge completely? We have already said that an inductor is an exhausting but still a perfect constant current source. Furthermore, an inductor is a natural, pure, monolithic current source; it is not an artificial current source that is composed of an internal voltage source and a resistor as in the case of the imperfect RC integrator. This means it can create a high enough voltage across the load to produce the current desired. In other words, the current created by the inductor does not depend on the voltage across the capacitor; this voltage is not an "opposition" to the current. As there is nothing to oppose the current produced by the inductor, no matter how small energy is left in the inductor; the current continues flowing until the inductor discharges completely.

The opposite arrangement. In order to feel the difference, compare this "discharging inductor -> charging capacitor" arrangement with the "discharging capacitor -> charging inductor" arrangement mentioned below. In the latter case, the charged capacitor is discharging and the empty inductor is charging. In this way, both the elements are finally charged with some energy. The voltage across them has some middle value - the point of the system equilibrium.

Reversing the direction
So far, we have been "moving" in one direction (initially fast, then slow) and finally we have stopped at the top of the curve. As we have been continuously "pushing" energy into the accumulator (the integrator) its output quantity has been continuously increasing in magnitude. Thus we have obtained a monotonous and quite uninteresting signal:) However, we would like to obtain a periodical quantity; so, we have here, at the top of the curve, to change, to reverse in some way the direction of the "movement". Something has to happen at this point to make the integrator's output quantity begin decreasing... What can cause this to happen?

Looking at many analogies of integrating, we can conclude that the only way to change the direction of the integrator's (here, a pressure-like) output quantity is to change the direction, not the magnitude, of its input (here, a flow-like) quantity. Imagine again the water analogy: while we fill the vessel the water level increases; to make it decrease, we have to begin emptying the vessel. Or imagine the money analogy: while you feed up gradually the deposit, the amount of money increases through time; to make it decrease, you begin "emptying" the deposit. Of course, we should mention again the famous mechanical analogy: after the moving mass stops, the spring begins beating it back; after we strike into the elastic barrier, it repulses us, it acts as a kind of "reverser" that changes the direction of our movement. We are ready enough to draw two conclusions:

A flow-to-pressure integrator can reverse the flow direction.

The only way to change the direction of the integrator's output quantity is to change the direction of its input quantity.

But this integrator is not only a bare accumulator... Remember that so far a flow-like source has been exhausting slowly and charging this pressure accumulator (a flow-to-pressure integrator). So, at the end of this process, the integrator has transmuted into an opposite pressure source. Now, it is a charged with energy accumulator; it has become a power source! So, it can by itself change its output quantity by passing a current through the load. But what is the load here? The other element (the former source) acts now as a load. What kind of load is it? Remember it is an accumulutaing element that stores kinetic energy.

Pressure-to-flow integrator. So, we begin now converting a pressure-like quantity (carrying potential energy) into a flow-like quantity (carrying kinetic energy) and storing it into an accumulating element having a property of inertness (see the top of Fig. 6a). Now, the energy moves in an opposite direction - from the pressure to the flow accumulator, and the flow-like quantity should change linearly through time (Fig. 6b).

Examples: a spring (or gravitation) puts in motion a mass, a spring motor actuates a toy, a child accelerate a flywheel car, air compressed in a reservoir spins an air turbine, water stored in a lake spins a water turbine, a boulder rolls down a mountain going faster and faster, "putting the pedal to the metal" makes a constant-horsepower engine push a car faster and faster, pressurized air escaping from a balloon rocket causes it to accelerate and fly away, water rocket, "pushing off" with your foot starts a bicycle moving, etc.

Note that this arrangement includes again two heterogenous elements connected to each other - a source of potential energy and an accumulator of kinetic energy. If we connect two homogeneous elements to each other (now, a source of potential energy and an accumulator of potential energy), a big current flows and the process ceases momentarily. As above, this arrangement will differentiate rather than integrate.

Voltage-to-current integrator. According to the general pressure-to-flow arrangement, in a voltage-to-current integrator we begin converting a voltage (carrying potential energy) into a current (carrying kinetic energy) and storing it into an inductor. So, the circuit includes a voltage source charging an inductor (see the bottom of Fig. 6a). As a result, the current through the inductor should change linearly through time (Fig. 6b). Note the current leaves the positive plate of the capacitor (so, it acts here as a source) and the current enters the positive plate of the inductor (so, it acts here as a load).

The circuit consists of two heterogenous elements connected to each other - a voltage source and an inductor. If we connect two homogeneous elements to each other (a voltage source and a capacitor), a big current flows through them and the process ceases momentarily; the circuit will differentiate rather than integrate.

An exhausting voltage source As above, in order to round the curve, our source has to have a limited reserve of energy; it has to exhaust, deplete gradually. The capacitor serves exactly as such an exhausting voltage source. Only, whereas before we wanted to round the curve in the final part, here we want to round it in the initial part. Let's see what the situation is. In the beginning, the current is low and the inductor consumes a little energy. Later, the current through the inductor increases and it begins consuming more and more energy from the voltage "source". Wonderful! The curve is rounded in its initial part exactly as we wanted!

''The charged capacitor in an LC tank serves as an exhausting voltage source that decreases gradually its output voltage.

Crossing the zero level
At the end of this stage, the source has completely exhausted its reserve of potential energy. As above, at this instant, it is not a source; it is just an empty source. As a result, the pressure becomes zero (see the top of Fig. 7a) and the pressure-to-flow integrator stops increasing its output quantity (Fig. 7b). It has accumulated maximum kinetic energy at this point.

If you look again at our favorite analogies, you will see that the moving masses from all the mechanical examples above are maximum accelerated (e.g., imagine how fast we run at this point:) Here, we have to remember that these objects have an interesting property - inertness. If they were non-inert, they would stop at this point and the system would be in state of equilibrium. But they are inert; that's why, they pass by this zero point (a neutral position) and continue moving... As if there are some devilish force:) that make them continue moving beyond this point (remember Newton's laws of motion).

We can already compare the behavior of the two different accumulating elements. We have already seen how the accumulator of potential energy (the capacitor) acts as a kind of an elastic barrier that repulses the moving "mass" thus reversing the direction of the movement. Now, we can see how the accumulator of kinetic energy (the inductor), because of its inherent inertness, "passes" beyond the equilibrium point (the zero level) and continues "moving".

A little while ago, the capacitor served as an exhausting voltage source. At this point, it has fully discharged and there is no voltage across it. The current through the inductor has its maximum rate but it has stopped changing. So, at the end of the stage both the current and the energy consumption have increased to its maximum value.

The opposite arrangement. We have already managed to answer why the inductor discharges completely through the capacitor - because it is a natural true current source. Now situation is even more interesting. In order to answer why the capacitor discharges completely, let's again consider the opposite arrangement - a charged capacitor connected to an empty one. If you prefer analogies, imagine communicating vessels where the one vessel is full and the other is empty. We know the result: a big current flows in the beginning; the "opposing" voltage across the initially empty capacitor (the water level of the initially empty vessel) begins rising rapidly; as a result, it decreases and finally stops the current. At this point, the energy is distributed between the two capacitors and the system is at static equilibrium (remember Newton's third law). The initially charged capacitor has not managed to discharge completely.

Why does the capacitor discharge completely? Contrary, if we connect the charged capacitor to an initially empty inductor (the present case), the current is negligible in the beginning and the capacitor begins discharging slowly through the inductor. In the course of time, the inductor lets the current increase gradually. The increasing current discharges the capacitor more and more thus hastening its discharging; finally, the capacitor is fully discharged. Figuratively speaking, an inductor has a time-dependent behavior: in the beginning, it consumes little current but in the end it devours a large quantity of current (a capacitor has an opposite time-dependent behavior - in the beginning, it devours a maximum current but in the end it consumes little current).

At first glance, it seems the system is moving to its equilibrium; we might suppose it has managed to reach this point and the energy inhabiting the LC tank "has found a peace"...

Creating the second half of the sinusoid
So far, we was considering in details how to create the first half (the first and the second quarter) of the sinusoid curve. The common sense suggests that we can create in the same manner the second half. Then, let's try it.

The third quarter

At the end of the second quarter and at the beginning of the third quarter, the situation is the same as it was at the first quarter. From one side, the inductor is fully charged with kinetic energy and it has transmuted from an integrator into a current source again (Fig. 8a). From the other side, the empty capacitor is connected to this "current source". What does the inductor begin doing then? Of course, it will start charging the capacitor and begin exhausting gradually. The only difference is the direction of the current that creates across the capacitor a negative voltage. As a result, the curve begins falling (in the beginning - rapidly, then - slowly) and finally stops at the bottom. It is rounded in the final part as we want.

The fourth quarter

Similarly, in the beginning of the final fourth quarter, the situation is the same as it was at the second quarter. From one side, the capacitor is fully charged with potential energy and it has transmuted from an integrator into a voltage source again (Fig. 9a). From the other side, the empty inductor is connected to this "voltage source". What does the capacitor begin doing then? Of course, it will start charging the inductor and begin exhausting gradually. The only difference is the negative voltage across the capacitor that creates a current with an opposite direction. As a result, the curve stays under the zero level but begins rising (in the beginning - slowly, then - rapidly) and finally crosses the zero level (Fig. 9b). It is rounded at the initial part as we want.

A generalization
At the end of this incredible story, let's generalize what we have managed to do.

Basic structure

In the beginning, we have assembled a system of two connected to each other heterogeneous elements that can process two different kinds of energy: they can convert energy (acting as true energy converters), store energy (acting as accumulators) and supply loads by energy (acting as sources). The one of them can convert potential to kinetic energy, store the kinetic energy and transfer gradually this kinetic energy to the load keeping up a gradually decreasing flow-like quantity; the other element can convert kinetic to potential energy, store this potential energy and transfer gradually this potential energy to the load keeping up a gradually decreasing pressure-like quantity. The first element has a property of inertness while the second one has a property of strain; so, the first element can "pass" beyond the equilibrium point and continue moving while the second one can reverse the direction of the "movement". Each of these reversible elements can act as a load as well as an gradually exhausting source. These kinds of rechargeable sources are fitted perfectly to each other - when the one element acts as a flow-output source, the other one acts as a flow-input load and v.v. - when the other element acts as a pressure-output source, the first one acts as a pressure-input load. As a result, they can exchange energy gradually in time between each other.

Electrical circuit
In electricity, we have implemented this arrangement by connecting an inductor and a capacitor to each other; thus we have obtained the famous circuit of an LC tank. The inductor can convert voltage to current, store the kinetic energy of moving electrons in the magnetic field and transfer gradually this kinetic energy to the load keeping up a gradually decreasing current; the capacitor can convert current to voltage, store this potential energy in the electrostatic field and transfer gradually this potential energy to the load keeping up a gradually decreasing voltage. Simply speaking, the inductor acts as a rechargable current source and the capacitor acts as a rechargable voltage source.

Operation
In the beginning (see point a on Fig. 11), there is no energy in the LC tank; it is at rest. Then, we charge the inductor connecting momentarily a current source (with the same success we might charge initially the capacitor connecting momentarily a voltage source and this is the more usual case in electronic oscillators). After we disconnect the source a wonder begins...

Phase 1. The inductor (acting as a positive current source) discharges gradually through the capacitor (acting as an accumulating load). The energy moves from left to right; the voltage increases above the zero level. At the top (point b), the capacitor becomes a positive voltage source. Phase 2. The capacitor (acting as a positive voltage source) discharges gradually through the inductor (acting as an accumulating load). The energy moves from right to left; the voltage decreases toward the zero level. At the end (point c), the inductor becomes a negative current source. Phase 3. The inductor (acting as a negative current source) discharges gradually through the capacitor (acting as an accumulating load). The energy moves from left to right; the voltage decreases below the zero level. At the bottom (point d), the capacitor becomes a negative voltage source. Phase 4. The capacitor (acting as a negative voltage source) discharges gradually through the inductor (acting as an accumulating load). The energy moves from right to left; the voltage increases toward the zero level. At the end (point e), the inductor becomes a positive current source.

Assembling. If there are no losses in the circuit, the four phases will repeat continuosly one after another (1 - 2 - 3 - 4 - 1 - etc.) thus forming the whole LC sinusoidal oscillation (Fig. 11). The LC tank keeps the initially injected energy intact but this energy is not at rest, it "has no peace". The energy inhabitting the LC tank moves continuously to and fro between the two elements and changes alternatively from one to other form. Accordingly, the electrical quantities (current and voltage) representing the corresponding kinds of energy (kinetic and potential) change in а sinusoidal manner; so, we can use them as outputs.

Natural frequency
Energy independence. As we have seen, after we inject energy into an LC tank it begins producing a sinusoidal oscillation with a natural period. If we inject more energy, the amplitudes of the voltage and the current increase but they change more quickly and v.v.; as a result, the natural period remains constant. The reason of this "energy-independent time behavior" is the variable consumption: when we increase the initially injected energy, the consumption increases as well and v.v. In the case when we charge the inductor with more energy, the current passing through the inductor will reach a bigger magnitude and then, when we disconnect the "pushing" current source, this bigger current will determine a bigger consumption; as a result, the time remains the same. In the other case, when we charge the capacitor with more energy and then disconnect the "pushing" voltage source, the inductor will begin gradually consuming a bigger current; as a result, again the time remains the same (we can observe a similar "time constant" effect in an RC circuit). So, we can make a conclusion: the natural frequency of an LC oscillation does not depend on the amount of the "pushed" energy (on the magnitudes of the voltage and current).

LC dependence. Contrary, if we change the values of the "storage properties" (inductance L and capacity C) of the accumulating elements (inductor and capacitor), the natural frequency changes as well. We can easily realize by intuition that the bigger the value of the storage property is, the bigger the duration of the corresponding quarter period is (e.g., if the capacitor is "bigger", the inductor needs more time to charge it).

Symmetry. But why the oscillation remains symmetrical (TPhase1 = TPhase2 = TPhase3 = TPhase4), even if we change the value only of one storage property (the inductance L or the capacity C)? The answer is simple. During an every half of a period, each of the two elements changes two times its "role": during the first quarter it acts as a load and during the second quarter it acts as a source. In this way, its storage property defines the duration of both the neighboring quarters; changing the property, we change simultaneously both the quarter durations. As a result, the curve remains symmetrical when we change either the inductance L or the capacity C or both of them.

Final conclusion
LC tank is the simplest electrical circuit that can generate oscillations with a perfect sinusoidal shape. This unique circuit consists only of two elements but it can do all that is needed to produce sinusoidal oscillations somewhat by itself. It is an almost ideal circuit that resemble a perpetual motion machine:) - we have only to inject initial energy and then it continues acting by itself.


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.NET Development Foundation/Annexes

Annexes

Annexes
The annexes are a series of articles on specific subjects that are referenced by the main text but were not integrated in it.

Generic Types
This page is a discussion on the need to have generic types in the context of collections.

Building custom attributes
This page is a tutorial on the building of a custom attribute that will be accessed thru the reflexion functionality.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: .NET Development Foundation/Annexes


text:
Talk:Arimaa/Overview

Possible contradiction in last paragraph: Is the patent granted or pending? --67.60.92.8 (talk) 14:22, 30 December 2007 (UTC)

Citations needed
I would like to include the links about the Arimaa online festival. I need the link to the Arimaa matches, the strategy workshop, and to the interview with Omar Syed.

title: Talk:Arimaa/Overview


text:
XQuery/DBpedia with SPARQL and Simile Timeline - Album Chronology

In this example, the SIMILE project's Timeline JavaScript is used to display the chronology of the albums released by a selected artist or group. As with the football teams in the previous example, the data is extracted from DBpedia via a SPARQL query.

This example has three components:
 * an XQuery script which creates the HTML page, linking to the Timeline JavaScript
 * an XQuery script to query DBpedia for albums for a selected group, assign a year of release to each album and generate a set of events in the form expected by the TimeLine script.
 * an XQuery script to provide an index of groups in a given category

HTML Page script
The script accepts one parameter, the name of the artist or group. The JavaScript required to customize the Timeline is defined in-line in a CDATA section. The onLoad function accepts two parameters, the name of the artist and a date to provide the initial focus for the timeline. The event stream is provided by the call to the XQuery script group2tl.xq, passing the group name.

The JavaScript interface here is minimal.

declare option exist:serialize "method=xhtml media-type=text/html";

let $group:= request:get-parameter("group","Eagles") return   <![CDATA[ function onLoad(group,start) { var theme = Timeline.ClassicTheme.create; theme.event.label.width = 400; // px theme.event.bubble.width = 300; theme.event.bubble.height = 300;

var eventSource1 = new Timeline.DefaultEventSource;

var bandInfo = [ Timeline.createBandInfo({       eventSource:    eventSource1,        theme:          theme,        date:           start,        width:          "100%",         intervalUnit:   Timeline.DateTime.YEAR,         intervalPixels: 45    }),

]; Timeline.create(document.getElementById("my-timeline"), bandInfo); Timeline.loadXML("group2tl.xq?group="+group, function(xml, url) { eventSource1.loadXML(xml, url); });

} ]]>    {$group} Albums

TimeLine events group2tl.xq
This script has a similar structure to the football team script. A prototype SPARQL query is edited to change the default name to the supplied name, the query sent to the DBpedia SPARQL service and the resultant SPARQL XML result converted to XML tuples.

Note - the group name needs an additional encoding because SPARQL requires URIs to be uri encoded and group names may contain characters such as ( and ) which require encoding. Hence Queen_(band) must appear in the URI in SPARQL as Queen_%28band%29. In addition the whole query is uri encoded which allows this encoding to be recovered when the whole query is decoded by the SPARQL service. (Whew!)

As with the Football clubs, there may be multiple tuples for the same album and the best (at present just the first non null) set of data needs to be extracted from these tuples. The year of release is hacked out of the multitude of different date formats which may appear and each album represented in the format expected by Timeline, with the pop-up contents serialised. This is limited to the album cover if available and links to DBpedia and Wikipedia.

declare namespace r = "http://www.w3.org/2005/sparql-results#";

declare variable $query := " PREFIX p:  PREFIX rdf:  SELECT * WHERE {      ?album p:artist  .            ?album rdf:type .     OPTIONAL {?album p:cover ?cover}.     OPTIONAL {?album p:name ?name}.     OPTIONAL {?album p:released ?dateofrelease}.   } ";

declare function local:execute-sparql($query as xs:string) { let $sparql := concat("http://dbpedia.org/sparql?format=xml&default-graph-uri=http://dbpedia.org&query=",                   encode-for-uri($query)                  ) return doc($sparql) };

declare function local:sparql-to-tuples($rdfxml ) { for $result in $rdfxml//r:result return { for $binding in $result/r:binding return if ($binding/r:uri) then  element {$binding/@name}  { attribute type {"uri"} , string($binding/r:uri) }                    else   element {$binding/@name}  { attribute type {$binding/@datatype}, string($binding/r:literal) }            } };

declare function local:clean($text) { let $text:= util:unescape-uri($text,"UTF-8") let $text := replace($text,"http://dbpedia.org/resource/","") let $text := replace($text,"\(.*\)","") let $text := replace($text,"_"," ") return $text };

declare function local:year-from-date($d) { let $d := replace($d,"[^0-9\-]","") let $dp := tokenize($d,"-") let $year := $dp[1] return if ($year castable as xs:integer and string-length($year)=4) then $year else }; let $group := request:get-parameter ("group","The_Allman_Brothers_Band") let $groupx := replace($group," ","_") let $queryx := replace($query,"The_Allman_Brothers_Band",encode-for-uri($group)) let $result := local:execute-sparql($queryx) let $tuples := local:sparql-to-tuples($result) return

Execution

 * Pink Floyd
 * Leonard Cohen

Group Index
This script queries DBpedia for the resources which belong to a specified category, for example Rock_and_Roll_Hall_of_Fame_inductees. A table of group names in alphabetical order provides links to the Timeline view using the script above, and to an HTML table view of the discography.

declare namespace r = "http://www.w3.org/2005/sparql-results#";

declare option exist:serialize "method=xhtml media-type=text/html"; declare variable $query := " PREFIX skos:  PREFIX p:  SELECT * WHERE {      ?group  skos:subject .   } "; declare function local:execute-sparql($query as xs:string) { let $sparql := concat("http://dbpedia.org/sparql?format=xml&default-graph-uri=http://dbpedia.org&query=",                  escape-uri($query,true)                   ) return doc($sparql) };

declare function local:clean($text) { let $text:= util:unescape-uri($text,"UTF-8") let $text := replace($text,"\(.*\)","") let $text := replace($text,"_"," ") return $text };

let $category := request:get-parameter("category","Rock_and_Roll_Hall_of_Fame_inductees") let $queryx := replace($query,"Rock_and_Roll_Hall_of_Fame_inductees",$category) let $result := local:execute-sparql($queryx) return {local:clean($category)}

Index of Rock and Roll Groups and Artists


text:
Technical Theatre/Lighting/Ellipsoidal

An Ellipsoidal reflector spotlight or Profile Spotlight (abbreviated to Ellipsiodal or ERS) is a light with a strong, well defined beam, and is very versatile. Leko and Source Four are two of the most common brand names of these lights.

Commonly the light includes:
 * A gel holder
 * A gobo holder
 * Several shutters to shape the beam.
 * An adjustable barrel, allowing a lighting designer to change the size and focus of the light thrown by the unit by changing the distance between the mirror and the smooth, plano-convex lens or lenses

There are several advantages of the ERS that make it one of the most common lights in a theatre. Its Ellipsoidal lens means only a 500w or 750w bulb is needed, instead of a 1000w bulb, while retaining the same brightness. Some ERSes, notably the Source Four, also include a dichroic reflector, so the beam is much cooler than other lights. Dichroic reflectors only reflect light from the visible spectrum, allowing infrared (heat) radiation to pass out the back of the light. This prolongs the life of gels and gobos.

Ellipsoidals can have either a fixed field angle, normally from 5° to 50°, or a variable field angle, commonly called a "Zoom". ETC, who manufacture the Source 4, produce a variety of attachments, commonly called "barrels", to change the field angle of the Source 4.

Uses
Ellipsoidals can be used for colour washes, with an attachment to change the field angle, but this job is better suited to PAR cans, Fresnels and Floodlights. Ellipsoidals are good for gobo projection, and highlighting actors or scenery on stage. They can also be used to god-spot actors by placing them at a sharp (5° - 10°) angle. This gives the effect of divine intervention, or, if used with a green gel, alien abduction.

title: Technical Theatre/Lighting/Ellipsoidal


text:
Talk:Circuit Idea/How do We Create Sinusoidal Oscillations?

A dedication
I dedicate this incredible story to my former student [mailto:Kirillgenev@abv.bg Cyril] who has inspired me to start the series of stories about this great (circuit) phenomenon.

The story that I have told here is the most incredible happening in my teacher's life. I have been teaching circuitry for more than 20 years. I have been having good, responsive, clever, curious, capable and even talented students that have been helping and inspiring me in my pursuit of revealing the truth about circuits through these years. But, as a rule, my best students loose interest in the subject after passing the examination. I have been receiving sporadically emails from some of them at that when they need some help.

That's why, imagine how much I was amazed when Cyril whom I was teaching basic circuitry two years ago, came on one of my consultation hours and showed willingness for discussing circuit phenomena. In the beginning, I supposed that Cyril has just to pass some examination or he has failed to take an examination or he has to make a project... But imagine how much I was amazed when I knew that Cyril wanted just to understand circuits! He brought out a few textbooks, showed to me obscure circuit diagrams of various oscillators and told me that he had not understood anything from these explanations. He just wanted to know why, when and how oscillations arise, where currents flow, why the inductor and capacitor of an LC tank discharge completely, what make energy circulate between the two components of the LC circuit, what transistors and op-amps do in oscillating circuits, how an RC oscillator oscillates although it does not actually contain any oscillating circuit, etc. Then I started reading the vague texts and, as a result, I realized that I also do not understand them. I knew but I didn't realize the basic ideas behind these great phenomena.

It happened half a month ago; since that time I have no peace. I stopped writing temporarily the introductory methodological stories of Circuit idea and began thinking about oscillation phenomena. During my consultation hours Cyril and I discuss these famous circuits. Usually, our "brainstorming sessions" last two hours. I record them on a solid-state recorder; then I write down them on a paper and finally I use the valuable thoughts in creating circuit stories. So, I would like to dedicate this exciting introductory story about the philosophy of oscillations in this world to Cyril who has inspired me to start and then to keep this series of stories. Circuit-fantasist (talk) 17:27, 30 December 2007 (UTC)

In the beginning
Will you believe me how difficult it was for me to start this page? How do I begin writing a new story about something so simple, old, and legendary? Because it is easy to explain complex things in this world but it is so difficult to explain the simplest things. Everybody knows how an LC tank operates in terms of electricity and nobody understands why it operates just in this way. Thousands of writers have explained the concrete electrical LC circuit in thousands of sources down the years. Don't you think it's time finally to extract the general idea from the bare 2-component circuit, to reveal the philosophy behind it?

The challenge
I realize that it is a great challenge to begin writing a story about such a great phenomenon and to pretend to say something new about the topic. Those who venture to start such an adventure have to be prepared for bearing (mostly negative) reactions of others. People do not like someone to express own new ideas about old things even, if they are more than obvious. This upsets their mental equilibrium and they react to this "intervention" trying to redress the balance. Contrary, people adopt willingly, use and benefit from nobody's, conventional, approved and "reputable" ideas. I have known this truth from my experience since I have been bearing the (negative) reactions of people around me through my life...

But, in spite of all, I begin writing this incredible story about the great, not only an electrical but in the first place, a worldly phenomenon. Circuit-fantasist (talk) 10:18, 12 January 2008 (UTC)

Succession problems
Maybe, the main problem of this page is that it is based on a few preceding pages, which are not yet written. These pages are dedicated to the behavior of accumulating elements (capacitors and inductors) and to relaxation oscillations; they are included in the contents:

Stories about capacitor's behavior:

Is a Capacitor a Voltage Source? (comparison with a "real" voltage source) Making a Capacitor Integrate (building a resistive-capacitive integrator) Making a Capacitor Differentiate (building an "incorrect" capacitive and a "correct" capacitive-resistive differentiator) Discharging a Capacitor through a Capacitor (transferring instantly energy between voltage "sources")

Stories about inductor's behavior:

Is an Inductor a Current Source? (comparison with a "real" current source) Making an Inductor Integrate (building an inductive-resistive integrator) Making an Inductor Differentiate (building an "incorrect" inductive differentiator) Discharging an Inductor through an Inductor (a conflict between "incorrect" connected current sources)

Stories about relaxation oscillations:

How to Make the Voltage across a Capacitor Wiggle (creating a philosophy of the relaxation oscillator by comparing with the LC one)

Of course, it would be better to write first these simpler pages before to start the more complex page about LC tank. But what do I do when I was so eager to reveal the secret of the legendary LC tank? I have been longing for that for years; so, I have yielded to temptation:) That's why I have decided to start this page and, in order to compensate the absence of the pages above, to insert explanatory texts about capacitor's and inductor's behavior where it is needed. Later, when we write the missing pages, we might move or copy these texts to the appropriate places. Circuit-fantasist (talk) 17:48, 15 January 2008 (UTC)

What the talk page to contain
I suggest to shape this talk page as a second, enlarged and more colorful written version of the main page that to serve as a counterpart of the basic story. Here we - wikibookeans, circuit ideators, circuit philosophers and just curious visitors, might discuss more freely the subject. In the discussion page, we will comment what and how to say while in the main page we will just say it.

Frankly speaking, I have the feeling that the discussion page will become more interesting than the main one. That's why, I am divided in my decision of what to develop; this evening I decided to enrich the talk page with some wisdom:) Circuit-fantasist (talk) 19:36, 15 January 2008 (UTC)

What the secret of LC oscillations is
Maybe, we have first to say here what the truth about LC oscillation is and then to discuss how to present this wisdom to visitors (these are different things). Well, let's begin with the first.

The initial point: A steady quantity
At the starting point we need a steady quantity. Where to get it from? As usual in this world, we need energy to create a quantity. We can get this energy from some power source. So, the first thing what we need (in this world) is a power source.

The problem: How to make it oscillate?
Then, we ask yourself, "How do we make the steady quantity oscillate?" But what does mean "to oscillate"? Simply speaking, to oscillate means just to make the quantity increase and decrease consecutively and regularly. How do we do this magic?

Preparing a steady quantity to be changed
It seems it's too difficult to make the very power source oscillate (power sources keep up constant their output quantities). Then, what do we do? We can act like in many situations of routine: instead to change the whole source energy we get some amount of energy from the source and store it in some "place" with the purpose to change it. As there are two kinds of energy - kinetic and potential, we can store energy in two kinds of energy accumulators (integrators). Let's do it; we have got the initial steady quantity. Examples: if we have stored a kinetic energy in an inductor, we obtain a constant current; if we have stored a potential energy in a capacitor, we obtain a constant voltage.

Changing the accumulated quantity
Now, we can begin doing the "veritable work" - to increase and decrase the steady quantity, in order to make it oscillate. How can we make it increase? Of course, by adding continuosly energy to the accumulator. And how can we make it decrease? Contrary, by subtracting continuosly energy from the accumulator. In both the cases we move, transport, transfer energy from one place to other - from the source to the accumulator and v.v.

Relaxation oscillations
(...We might explain what an LC oscillation is, if we say what it is not. Here, we might say that an LC oscillation is not a relaxation one. Only, we would explain what a "relaxation oscillation" is. This phenomenon is no less vague and mystic than the LC phenomenon (I have never met good resources about it). Now, we have opportunity to reveal the secret behind relaxation phenomenon by comparing it with LC one. After that, we may move these thoughts to the future story about relaxation oscillations...)

The basic idea. According to the powerful idea above, we might first build the simpler and more intuitive relaxation oscillator by using three elements: a flow (current) source, an accumulator of potential energy (a capacitor) and a load (a resistor). Such an arrangement operates in two stages as follows: in the first stage, we connect the current source to the accumulator and begin charging it; in the second stage, we connect the accumulator to the load (turn on the switch SW) and discharge it. Let's repeat it in other words: in a period of a relaxation oscillation, we first draw a kind of energy (usually, potential) from the flow source to fill the accumulator, then "suck dry" the energy from the accumulator and just "throw out" it to the environment. Thus, in the first stage of every period the source has to restore the energy in the accumulator.

Features. Note that the relaxation arrangement needs only one accumulator. It lies on the way of energy and serves as a buffer that interrupts temporarily the flow (only, the average flow remains constant). In this way, it intermits, tears to "pieces" the energy flow. It is interesting fact that a continuous current enters the accumulator while an intermittent current leaves it; so, we might say that a relaxation oscillator "discretisizes", "portens", "doses" the unceasing input flow of energy. At the same time, the accumulator's contents increases and decreases periodically. The shape of the curve is not sinusoidal because at the points of switching (at the peaks) the "flow-creating" quantity has a maximum magnitude. Note also that increase and decrease can have different durations. Examples of this possibility (see the analogies below) are a photoflash (slow charging, fast discharging), a toilet tank, a husband that fills slowly and continuously a family money deposit while his wife empties rapidly and regularly it:)

What does "relaxation oscillation" mean? I have been asking myself many times what the word "relaxation" means in this context. What does "relax" here? Maybe, the capacitor? If yes, when does the capacitor relax? Maybe, when we charge it (in this case, "relaxation" would mean restoring, retrieving, recreating the content of the accumulating element). Or maybe, the capacitor relaxes (returns to its initial "zero energy" state of equilibrium) when we discharge it...

Relaxation analogies. We can see many manifestations of this phenomenon in nature, routine, engineering and finally, in electronics. For example, I have a remembrance from my childhood how sometimes faulty toilet tanks began working intermittently (filling -> emptying -> filling -> emptying... up to infinity:) Similarly, imagine you were foolish enough to make a common money deposit with your wife:) Then, you begin working hard to feed up continuously the deposit. As a result, the amount of money increases continuously through time. Only, at some point your wife decides that's enough and draw out all the money accumulated. You begin working again... and she empties the deposit again... and this repeats again... and again... and again...:) Circuit-fantasist (talk) 20:06, 18 January 2008 (UTC)

LC oscillations

The basic idea. Contrary, in order to create an ideal LC oscillation (assume an LC tank without losses), we charge the accumulator only once in the beginning (by turning on the switch SW in the figure on the right). Then, we "suck dry" the energy from the accumulator but do not "throw it away"; instead, we convert it into an opposite kind (kinetic to potential or potential to kinetic) and store the converted energy in another accumulator. We move, transfer, convey the energy from the one accumulator to the other. After, we use this energy to restore the energy of the first accumulator instead to "suck" energy from the source (we move back the energy). By the way, maybe is more correctly to say "energy moves" rather than "we move the energy". But why the energy moves by itself? What makes it move? What makes it move to and fro without end? Don't you think there is something mystic in this endless "ping-pong" movement?

Implementation. For this purpose, the LC arrangement consists of two heterogenious (regarding to the two kinds of energy) accumulating elements. Each of them can either provide energy (when it serves as a source) or store the energy (when it serves as a load). When acts as a source, it is a weakening, depleting, exhausting source because the other (accumulating) element "sucks out" the source's energy. The energy moves from the source to the accumulator and the source's output quantity decreases.

Similarities

 * 1) Both the oscillating circuits contain (at least one) accumulating element that acts either as a source or as an integrator.

Differences

 * 1) A relaxation oscillator consists of only one accumulating element while an LC oscillator consists of two accumulating elements.
 * 2) The relaxation accumulator is either flow or pressure like; the LC accumulators are heterogenious (the one is flow-like and the other is pressure-like).
 * 3) A relaxation oscillator stores only one kind of energy (usually potential) in the accumulator while an LC oscillator stores two opposite kinds of energy (kinetic and potential) in the two accumulators.
 * 4) In a relaxation oscillator we "throw away" the energy while in an LC oscillator we treasure it temporarily in an additional accumulator with the purpose of future usage. That's why, LC oscillators are more economical than relaxation ones.
 * 5) Resonance phenomenon does not exist in a relaxation circuit; it can be observed only in an LC tank.
 * 6) In a relaxation oscillator the energy moves only in one direction (source -> accumulator -> load) while in an LC oscillator the energy changes periodically its direction (it circulates between the two elements).
 * 7) The shape of a relaxation oscillation is peaked, angular while the shape of an LC oscillation is rounded (sinusoidal). The reason of that is that at the peaks the source of a relaxation oscillator changes its output quantity with the maximum magnitude while the "source" of an LC oscillator (charged accumulator) does not change its output quantity.
 * 8) The shape of the relaxation oscillation can be asymmetrical (the increase and the decrease can have different durations) while the shape of the LC oscillation is precisely symmetrical.

(to be continued...) Circuit-fantasist (talk) 08:01, 17 January 2008 (UTC)

How to reveal the truth about LC oscillations
If we assume that we have already revealed the secret of LC oscillations, now we have to decide how to present this wisdom to visitors (one thing is to understand somethning but the other thing is to explain it in the most credible, interesting and attractive way). For this purpose, we have to create a "scenario" of this story.

Starting and maintaining oscillation
In every kind of oscillator (including LC and relaxation oscillators), there must be at least 2 different components that "remember" 2 "instantaneous values": at any point in the oscillaton, there has to be at least the output voltage value, and also something else that remembers if we are going "up" or "down".
 * It sounds very philosophical. I need time to assimilate this wisdom:) Circuit-fantasist (talk) 08:57, 19 January 2008 (UTC)

Since resistors can instantaneously change current (when voltage is instantaneously changed), they are useless for "remembering".
 * I like this thought; I have inserted it into the story. Circuit-fantasist (talk) 08:57, 19 January 2008 (UTC)

Components that can be used for remembering:
 * capacitors
 * inductors
 * positive-feedback amplifiers (flip-flops, Schmitt trigger inverters, etc.)

 * David, I have finally finished this incredible story about LC oscillations and now I can reply with pleasure to your assertion. It sounds very fresh and original; as I can see, it is a sort of "informational viewpoint" at the topic about oscillators. IMO, it fits very well relaxation oscillators but it is not so suitable for explaining LC oscillators (regarding to RC oscillators, they are wrapped in mystery). An "energetical viewpoint" would fit LC oscillators quite better than the "energetical one". So, I suggest including this assertion into the page about relaxation oscillations. Well, let's exchange some thoughts about the topic.

 * First at all, it goes without saying that any oscillator has to contain a "remembering element", which represents the current state of the oscillator and which continuously changes its contents. From this viewpoint, we can think of an oscillator as a sequential circuit - e.g., a counter that increments or decrements continuously its contents (a "digital analogy"). Oscillators discussed here are analog devices; so, their "remembering elements" are analog as well. The capacitor is the simplest analog memory cell (as far as I know, the John Atanasoff's computer was based on a capacitor memory).
 * Then, we need an input source that charges gradually through time the memory element. In a case of a capacitor memory, we need a current source.
 * After, we need a comparator that compares the current magnitude with the maximum (if we are moving up) or minimum one (if we are moving down). For this purpose, we may use either two separate comparators or a comparator with hysteresis (the so called Schmitt trigger).
 * Finally, as you have said, we need a second memory element that "remembers" the comparator's condition (a comparator with hysteresis serves both as a comparing and a "remembering" element). Only, this element is a digital (discrete) memory that "remembers" only one bit.

 * Following the "scenario" above we will actually build a relaxation oscillator that will produce a triangle-shaped oscillation. If we try to apply this procedure to an LC oscillator, IMO we will obtain quite artificial and unnatural explanation. In an LC oscillator, we obtain "remembering", comparing and even "rounding" somewhat by itself. IMO, we have to look at LC oscillator rather from an "energetical viewpoint" than from this "informational viewpoint".

 * Circuit-fantasist (talk) 17:13, 31 January 2008 (UTC)

In a real LC oscillator, there also is something else that regulates how energy from the power supply is fed into the rest of the circuit -- it "pushes" the rest of the circuit.

Some people use a complicated circuit (op amps, PTR resistors, etc.) that very gently puts in just enough energy to compensate for the parasitic resistance of the LC, making as "pure" a sine wave as possible. There are a variety of less complicated circuits that are perhaps easier to understand, but generate a waveform that is more distorted from the "ideal" sine wave.

Would it help or make things more confusing to discuss an example "pusher" circuit?

--DavidCary (talk) 22:51, 18 January 2008 (UTC)

 * David, you have actually started the discussion about sustaining oscillation in an LC tank ("pushing" energy into an LC tank). I have started the talk page of this story and copied this discussion there. I suggest to continue discussing about "pushing" circuits on the new talk page.

 * Circuit-fantasist (talk) 17:04, 20 January 2008 (UTC)


text:
User:Shamino

See my Wikipedia user page for my bio information.

title: User:Shamino


text:
Haskell/Solutions/Laziness

 * 1) Because   needs to evaluate   to convert it to a.
 * 2)   is stricter: it must walk through the whole list to discover its length, even if it does not need to examine the single values.

Instead, to evaluate, when   evaluates the passed list it obtains  , where thunk_0 represents the result of. Thus, you can verify that  (actually,   is equivalent to  ) while   fails. Also note that, if  is a valid   list (which does not contain   among its elements), all following expressions fail to evaluate: g (1:undefined) => length undefined => undefined

whereas the following evaluate successfully: g (1:undefined:ls) => length (undefined:ls) => 1 + length ls g (undefined:ls) => length ls

The difference between  and  may appear confusing, but is correct, because in the first case   replaces a list (in particular, the tail of the cons cell passed to  ), while in the second case   replaces a list value (which is not needed).

title: Haskell/Solutions/Laziness


text:
Technical Theatre/Lighting/Floodlights

Floodlights are basic theatrical lighting instruments, consisting of primarily of a reflector box and a lamp, usually attached to a yoke to allow the instrument to be hung. They are often used in the theater for color washes, or left uncolored for use as work lights. They offer practically no control of the beam or focus. Gels placed over a floodlight tend to fade quickly because floodlights generate a great deal of heat. However, floodlights tend to be among the least expensive lighting instruments due to their simple construction and few complicated parts.They are also used in football games,cricket games,netball games and many more.

title: Technical Theatre/Lighting/Floodlights