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Talk:Elementary algebra

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Operations
I really don't think that this section belongs in this article. The order of operations is not introduced in algebra. It is a concept introduced with basic arithmetic. It needs a mention somehow and a link for completeness but beyond that it should be prior knowledge. Anyone think I am off-base on this one?Phancy Physicist (talk) 08:18, 10 June 2010 (UTC)

Why in this section or in any section in Wikipedia would there be author attribution? "By:Ůmáīr Hafeez Měmõń'" Shouldn't this be deleted? --OneAmongBillions (talk) 13:08, 4 November 2010 (UTC)

Algebraic examples
This section is longer than the rest of the article and for some reason that bothers me. Maybe because it seems that the article is written in little bit of a "teachy" way. Which I don't think is necessarily bad but is this okay with Wikipedia's style?

Since I am talking about style the presentation-style bulleted lists are giving me pause as well. Are they OK with wikipedia standards?Phancy Physicist (talk) 08:18, 10 June 2010 (UTC)
 * I was seeking for information about the elimination method. In a typical teachy way it is explained. I would like a link or an explanation why the method works and is allowed. I guess a link to the gaussian method would suffice... but. I think it could be explained in a more cursory way here as it is about elementary algebra. A link to the gaussian method has its problems as the readers are reading about elementary algebra and that entry would confuse most people. (English is not my first language, I hope you get what I am getting at)-- Tobbe —Preceding unsigned comment added by 83.233.238.234 (talk) 14:39, 4 September 2010 (UTC)

This section seems typical of the whole article - somewhat shoddy, and highly pedantic in tone. The object of this Wiki is to identify what elementary algebra is, not how to undertake exercises in it.

The solving of equations is an overly-laboured diatribe concerning a subset of those equations whose solutions are analytically possible. What of the others, such as x^2 + ln(x) = 5 for example? — Preceding unsigned comment added by 83.158.63.7 (talk) 13:47, 13 October 2011 (UTC)

Basic algebra
"Basic algebra" redirects here but this is also a algebraic terminology: A finite-dimensional K-algebra A is basic iff every indecomposable projective module has multiplicity one in A iff all simple modules are one-dimensional. There should be a disambiguation page (and a new article for basic algebra). I could write the new article (at least a stub) but I don't know how to change the redirecting... 92.224.233.104 (talk) 20:05, 29 April 2011 (UTC)

Operations
Per the comments above, I've removed the section on Operations. I'm going to replace it with comments that are more relevant to algebra. --Iantresman (talk) 13:01, 20 July 2012 (UTC)

 * Also, can the 3.5 example for multiplication be fixed to look more like it should (3 dot 5)? hydnjo (talk) 22:07, 23 July 2012 (UTC)

 * I think I've fixed, showing both the European and American forms for $$3 \ .\ 5$$   or   $$3 \cdot 5$$. --Iantresman (talk) 08:54, 24 July 2012 (UTC)

 * OK, thanks. Oops, I just noticed that the same problem exists with the a.b example and there is a confusing comma just above the 3 . 5 that you fixed, hydnjo (talk) 11:40, 24 July 2012 (UTC)

 * Thanks for that, all done, I think. --Iantresman (talk) 12:12, 24 July 2012 (UTC)

Mixed formatting of inline math
This article mixes inline HTML and for math variables and expressions, with the latter aparently dominating. This should be made consistent by choosing either, but not both. They each have their problems, but consistency appears to be the one thing that is recommended. — Quondum☏ 18:38, 31 July 2012 (UTC)
 * Good idea, all done. Do you think that numerals should also be converted? --Iantresman (talk) 15:19, 1 August 2012 (UTC)
 * Personally, I would have said yes. However, there seems to be little consensus on these matters, and I've run into resistance relating to HTML/

/ preferences before from others. The only argument that seems to remain relatively unchallenged is consistency of presentation within a single article, but even that not always. Different preferences seem to manifest according to individual browser/preferences settings (PNG vs. MathJax, specific font choices and associated perceived clashes, the opposing issue of font choices and the perceived benefit of differentiating math from text, available installed characters for each font, etc.). So, you're on your own on this... — Quondum☏ 18:43, 1 August 2012 (UTC)
 * I'm going to go with the following rationale. If we refer to a number in an equation, then I'll use <math>, otherwise not. So if we produce an equation to represent a quantity, then no <math>. --Iantresman (talk) 19:48, 1 August 2012 (UTC)

 * Referring to point 1b of the GA-review, it might comply with the MOS guidelines as far as mathematics in concerned, but I find the current usage of inline $$$$ highly disturbing.
 * As an example, take for instance this statement:
 * ...for example, $$2x^2 + 3ab - x^2 + ab$$ is written as $$x^2 + 4ab$$, because the terms containing $$x^2$$ are added together, and, the terms containing $$ab$$ are added together. By convention, terms with the highest power (exponent), are written on the left.
 * I propose that we write this with inline HTML math —see MOS:MATH—, like for instance
 * ...for example, 2x2+3ab-x2+ab is written as x2+4ab, because the terms containing x2 are added together, and, the terms containing ab are added together. By convention, terms with the highest power (exponent), are written on the left.
 * or even better:
 * ...for example, $2x^{2}+3ab-x^{2}+ab$ is written as $x^{2}+4ab$, because the terms containing $x^{2}$ are added together, and, the terms containing $ab$ are added together. By convention, terms with the highest power (exponent), are written on the left.
 * I can't do it right now, but I'd be willing to work on this one of these days. What do you think? - DVdm (talk) 07:10, 3 August 2012 (UTC)

 * Some might take the view that since MathJax is largely working and the font, size and baseline offset problem is (or should be) largely solved by it, we should be moving towards the inline form. Try switching your preferences to MathJax to see whether you think this'll have any merit; at least it may argue against actually reformatting an article that uses inline  .  But I would not be surprised at strong objections, such as yours.  I seem to remember similar objections to use of

, so I have stopped using this template in place of HTML (and I would also like to see a tie-up between its font and that of MathJax). Straightforward HTML does seem most common in most articles, though I prefer a distinct font to distinguish math. In summary, I think that there is a general consensus not yet reached on this matter, but I am wary of kicking this debate into life again. — Quondum☏ 09:17, 3 August 2012 (UTC)

 * Up to now I had my prefs set to "Always PNG", which makes the inline math quite ugly indeed. I tried the (still experimental) MathJax on this article and I must say it looks much better. Drawback is that it takes an eternity to load in IE8. Firefox is faster. Also note that there's no caching. By the way, try opening Help:Displaying a formula with MathJax in IE8—Ouch! - DVdm (talk) 10:04, 3 August 2012 (UTC)

 * I didn't know there was a MathJax option. For now, I would stick with inline PNG math on the grounds that it is faster, and works. While it is ugly, it help differentiate the maths from the text. It will also probably be faster on mobile devices.
 * There is indeed the odd inline equation where the terms are not grouped and in order, but it might be deliberate, to demonstate grouping. If so, the text will make it clear. --Iantresman (talk) 11:55, 3 August 2012 (UTC)
 * Well, the HTML math (as in the alternatives of my example) is much faster and takes much less browser cache space and network bandwidth, than PNG, which is even more important on mobdevs. The main problems with PNG are that it presents itself differently on different browsers, and that the text falls below the lines, which is what makes it so terribly ugly for inline math. I'd definitely vote for inline-HTML and I'm prepared to do it one of these days. - DVdm (talk) 13:37, 3 August 2012 (UTC)

We can always make inline HTML look a bit like math PNG: First expression encoded as: 2x2 + 3ab - x2 + ab This works fine except where we have more complicated expressions. --Iantresman (talk) 16:27, 3 August 2012 (UTC)
 * [math png] ..for example, $$2x^2 + 3ab - x^2 + ab$$ is written as $$x^2 + 4ab$$, because..
 * [Pure HTML]..for example, 2x2 + 3ab - x2 + ab is written as x2 + 4ab, because..

 * Yes, but look at the source of your spanned Pure HTML. It's pretty hard to code:
 * (1) math PNG or MathJax:
 * $$2x^2 + 3ab - x^2 + ab$$ -- with source $$2x^2 + 3ab - x^2 + ab$$
 * (2) Your pure HTML (with the i.m.o. awkward to maintain span tags):
 * 2x2 + 3ab - x2 + ab -- with source 2x2 + 3ab - x2 + ab
 * (3) Almost like your pure HTML with template:Math which uses size 165% in stead of your less-standardish 150%:
 * $2x^{2} + 3ab - x^{2} + ab$ -- with source $$
 * (4) My proposed HTML with template:Math:
 * $2x^{2}+3ab-x^{2}+ab$ -- with source $2x^{2}+3ab-x^{2}+ab$
 * When you take any standard math book, I think the obvious way to go is the fourth, and i.m.o. it's not that hard to code and maintain. - DVdm (talk) 17:36, 3 August 2012 (UTC)

 * This is risking starting a debate when others pitch in. However, considering that (a) I (and evidently others) find MathJax and PNG are rather slow and download size is greater, and (b) presentation seems to be best IMO (I've seen contention on this, due to browser font choices) with the

template for simple formulae (especially inline), (c) these shortcomings of have not yet been and might not be corrected for some time despite some progress, and (d) the template approach should still allow centralised font choices to match MathJax if need be, my personal support is for option 4. In fact, for the simple stand-alone formulae such as in an article like this one, I would possibly support

or $$ there too. It'll cope with fractions (e.g. $a + b 3$) and surds (e.g. $\sqrt{x^{m}$) a bit clumsily, but that is the worst I see. But we don't have a really good solution. — Quondum☏ 22:39, 3 August 2012 (UTC)
 * I would not touch the standalone math. I think that should remain in TeX, as in most other articles (e.g. Escape velocity). I.m.o. it's just the inline math that really would benefit from getting htmled. That way the standalone math stands out and can be chosen by the reader (i.e. PNG or Jax) and the text becomes homogeneous. - DVdm (talk) 09:21, 4 August 2012 (UTC)

Update. It looks like the article failed the GA-review, so I will not touch the inline math for now. - DVdm (talk) 16:44, 11 August 2012 (UTC)

 * It seems that there is no recommended style guide for inline equations, which seems like a surprising omission considering its importance. I note one earlier discussion on "Inline math notation", but it is inconclusive. I've suggested re-opening the discussion. --Iantresman (talk) 17:46, 13 August 2012 (UTC)

 * Good idea. By the way, I have now activated MathJax (again). I admit that it makes all math —that is, inline and standalone— look significantly better. Major drawback is that some articles take even more significant ages to load :-( - DVdm (talk) 18:03, 13 August 2012 (UTC)

The introduction
"US algebra"?

"The pushing of algebra from high school, where it has traditionally been taught, to elementary school has met with some controversy. Whereas students 30 years ago memorized multiplication tables in math class, students today, driven by the new Common Core Standards, are being introduced to variables as early as 6th grade."

So, look at that: if we're introducing elementary algebra, we should rely on overall accounts, or, when we refer to a national account, we'd better treat it as an example, an instance, not leaving such a passage without any allocation as if the subject of the article is certainly national by default. M? Lincoln Josh (talk) 12:25, 19 January 2013 (UTC)

 * The first question is what is "elementary algebra"? I do not know any generally accepted definition, and reading this article, I agree that the only definition that one may extract from it is "the part of algebra that is taught in elementary school". Thus a better title would be "Algebra in elementary school". But, IMO, the main question is not if it is "US-only" or not (although this question is relevant). The problem, shared with most mathematics articles of elementary level, is that it fails the main objectives of an encyclopedia: an encyclopedia is aimed to provide information that the reader does not has at hand. In other word, in case of students, an encyclopedia should provide insights that are not in textbooks (WP:NOTTEXTBOOK). Here are some examples of points where the article is wrong:
 * Variables: They represent not only numbers, but they may represent any mathematical object. They "describe" (3 times in the article) nothing, but "allow or help to describe"
 * Quadratic equation: Only the formula is needed here, the various ways to derive it appear or should appear in quadratic equation. It must be mentioned with links that formulas exist for degree 3 and 4 but not for higher degrees
 * Evaluating (and simplifying) expressions: It must be said that the process may be and has been automatized (computer algebra).
 * Systems of linear equations: a single example is useful. The description of the various elementary methods should be left to system of linear equations and/or simultaneous equations. On the other hand it is fundamental to mention that the methods extend to any number of equations and unknowns, and that there are computer programs that solve them efficiently. Even for small kids, it may be interesting that weather forecasting needs to solve huge such systems.
 * This list is far to be complete, but I hope that it will help to understand what is wrong in this article. D.Lazard (talk) 15:56, 19 January 2013 (UTC)

Variables
I'm not sure what do you mean "hat describes what". For the letters, I used the term "designate". Lincoln Josh (talk) 13:30, 17 February 2013 (UTC)

 * I suppose that you are talking about section Variables. I have edited it and hope that this solves your problem. D.Lazard (talk) 16:13, 17 February 2013 (UTC)

Equation balance analogy animations?
If on the off-chance these happen to help: feel free to take or leave. Thanks. M∧Ŝc2ħεИτlk 22:34, 18 May 2013 (UTC)

Conflict with process-oriented nature of subject
Although introduction to the foundations for the techniques in axiom is important, making that the article does not reflect how elementary algebra is taught. Rather, what is being taught is a system of techniques - substitution, descriptions in terms of variables or indeterminates, and solution of an equation or isolation of a quantity. So the treatment should seek first to explain the processes, what's possible because they work, and how they integrate with one another (to constitute elementary algebra), and then to axiomatize them in order to explain why they work, or that they work in other situations where those axioms hold. ᛭ LokiClock (talk) 10:59, 13 February 2014 (UTC)
 * As Wikipedia is not text book, and this article is not about pedagogy, but about mathematics, this does not matter if it does not reflect how elementary algebra is taught. On the other hand, it is important for an encyclopedia to provide to the reader a wider view on the subject than that he can find in his/her textbook. I cannot find any axiomatization in this article, and axiomatization is not relevant here: elementary algebra was widely used by mathematicians and students a long time before the introduction of axiomatization in mathematics (at the end of 19th century). I do not understand why you are talking about axiomatization. I agree that the description of the elementary methods is much too detailed here: we can describe them briefly and refer to the corresponding articles for details. On the other hand the described problems should be put in their context, and they are not. For example it should be said (and this is not said) that the linear equations are always solvable, as well that quadratic ones, and that the other equations in one variables can be solved only in very specific case (in practice, only in examples that are devised for being solvable by students). Similarly, for linear systems it should be said that the presented methods may be generalized to any number of equations and variables, and that these methods can be automatized into algorithms and computer programs. Another information of encyclopedic nature, which is lacking, is the fact that the rules for transforming and simplify expressions may be and have been automatized in computer algebra systems. All kinds know about computers. It is very strange that the mathematic teachers are reluctant to explain what is used in computer and what the computers can do in the area of their courses. Wikipedia must provide this information, because it is of encyclopedic nature. D.Lazard (talk) 20:26, 13 February 2014 (UTC)

Exponential and logarithmic equations?
The article contains a section Elementary algebra. While this may well be something that is taught in high school algebra courses, is it really part of what is called algebra? I think algebra in the elementary sense only includes the operations addition, subtraction, multiplication, division, and taking n-th roots. Loraof (talk) 21:19, 19 January 2017 (UTC)

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Hatnotes
Right now the article is headed by four hatnotes, dated between August 2012 and January 2013. I propose that I remove three of them:

 * Needs reorganization — the current organization looks fine to me.

 * May be confusing or unclear — I don’t think that’s a general problem with the article; if there are any specific confusing passages, someone could put confusing tags there.

 * Doesn’t give a worldwide view — while it’s probably true that elementary algebra is taught differently in different countries, this article is not about what is taught and how; it’s about the topic of elementary algebra itself, which is a topic that is independent of location.

I think these three hatnotes should be removed. I think it’s legitimate to retain the other hatnote, saying the article contains excessive intricate detail. I don’t have an opinion about that, but I can see that someone might reasonably think so.

Loraof (talk) 17:34, 19 February 2018 (UTC)

 * I think you can get rid of all of them. Besides the worldwide view tag, the others were all placed at the same time by a now banned user (probable sockpuppet). While there are a couple of places where there is too much detail, I don't think that it is an overall problem. The worldwide tag comes from Hong Kong by a still active user. If he wishes to expand on this point he still can, but I agree with your statement about this. --Bill Cherowitzo (talk) 19:47, 19 February 2018 (UTC)

Absolute value, inequalities
Absolute value is not mentioned here; is there any reason for that? It seems like an important topic to me. Also it might be useful to have an example of solving an inequality, and mention that the sign has to flip when multiplying by a negative number. —Lights and freedom (talk ~ contribs) 20:03, 6 October 2021 (UTC)

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The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 11:52, 23 August 2022 (UTC)
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title: Talk:Elementary algebra


text:
Electronic oscillator

An electronic oscillator is an electronic circuit that produces a periodic, oscillating or alternating current (AC) signal, usually a sine wave, square wave or a triangle wave,  powered by a direct current (DC) source. Oscillators are found in many electronic devices, such as radio receivers, television sets, radio and television broadcast transmitters, computers, computer peripherals, cellphones, radar, and many other devices.

Oscillators are often characterized by the frequency of their output signal:
 * A low-frequency oscillator (LFO) is an oscillator that generates a frequency below approximately 20 Hz. This term is typically used in the field of audio synthesizers, to distinguish it from an audio frequency oscillator.
 * An audio oscillator produces frequencies in the audio range, 20 Hz to 20 kHz.
 * A radio frequency (RF) oscillator produces signals above the audio range, more generally in the range of 100 kHz to 100 GHz.

There are two general types of electronic oscillators: the linear or harmonic oscillator, and the nonlinear or relaxation oscillator. The two types are fundamentally different in how oscillation is produced, as well as in the characteristic type of output signal that is generated.

The most-common linear oscillator in use is the crystal oscillator, in which the output frequency is controlled by a piezo-electric resonator consisting of a vibrating quartz crystal. Crystal oscillators are ubiquitous in modern electronics, being the source for the clock signal in computers and digital watches, as well as a source for the signals generated in radio transmitters and receivers. As a crystal oscillator’s “native” output waveform is sinusoidal, a signal-conditioning circuit may be used to convert the output to other waveform types, such as the square wave typically utilized in computer clock circuits.

Harmonic oscillators

Linear or harmonic oscillators generate a sinusoidal (or nearly-sinusoidal) signal. There are two types:

Feedback oscillator
The most common form of linear oscillator is an electronic amplifier such as a transistor or operational amplifier connected in a feedback loop with its output fed back into its input through a frequency selective electronic filter to provide positive feedback. When the power supply to the amplifier is switched on initially, electronic noise in the circuit provides a non-zero signal to get oscillations started. The noise travels around the loop and is amplified and filtered until very quickly it converges on a sine wave at a single frequency.

Feedback oscillator circuits can be classified according to the type of frequency selective filter they use in the feedback loop:

 * In an RC oscillator circuit, the filter is a network of resistors and capacitors. RC oscillators are mostly used to generate lower frequencies, for example in the audio range.  Common types of RC oscillator circuits are the phase shift oscillator and the Wien bridge oscillator. LR oscillators, using inductor and resistor filters also exist, however they are much less common due to the required size of an inductor to achieve a value appropriate for use at lower frequencies.

 * In an LC oscillator circuit, the filter is a tuned circuit (often called a tank circuit) consisting of an inductor (L) and capacitor (C) connected together, which acts as a resonator.  Charge flows back and forth between the capacitor's plates through the inductor, so the tuned circuit can store electrical energy oscillating at its resonant frequency. The amplifier adds power to compensate for resistive energy losses in the circuit and supplies the power for the output signal. LC oscillators are often used at radio frequencies, when a tunable frequency source is necessary, such as in signal generators, tunable radio transmitters and the local oscillators in radio receivers.  Typical LC oscillator circuits are the Hartley, Colpitts and Clapp circuits.
 * In a crystal oscillator circuit the filter is a piezoelectric crystal (commonly a quartz crystal).  The crystal mechanically vibrates as a resonator, and its frequency of vibration determines the oscillation frequency. Crystals have a very high Q-factor and also better temperature stability than tuned circuits, so crystal oscillators have much better frequency stability than LC or RC oscillators. Crystal oscillators are the most common type of linear oscillator, used to stabilize the frequency of most radio transmitters, and to generate the clock signal in computers and quartz clocks. Crystal oscillators often use the same circuits as LC oscillators, with the crystal replacing the tuned circuit; the Pierce oscillator circuit is also commonly used. Quartz crystals are generally limited to frequencies of 30 MHz or below.  Other types of resonators, dielectric resonators and surface acoustic wave (SAW) devices, are used to control higher frequency oscillators, up into the microwave range.  For example, SAW oscillators are used to generate the radio signal in cell phones.

Negative-resistance oscillator
In addition to the feedback oscillators described above, which use two-port amplifying active elements such as transistors and operational amplifiers, linear oscillators can also be built using one-port (two terminal) devices with negative resistance, such as magnetron tubes, tunnel diodes, IMPATT diodes and Gunn diodes. Negative-resistance oscillators are usually used at high frequencies in the microwave range and above, since at these frequencies feedback oscillators perform poorly due to excessive phase shift in the feedback path.

In negative-resistance oscillators, a resonant circuit, such as an LC circuit, crystal, or cavity resonator, is connected across a device with negative differential resistance, and a DC bias voltage is applied to supply energy. A resonant circuit by itself is "almost" an oscillator; it can store energy in the form of electronic oscillations if excited, but because it has electrical resistance and other losses the oscillations are damped and decay to zero. The negative resistance of the active device cancels the (positive) internal loss resistance in the resonator, in effect creating a resonator with no damping, which generates spontaneous continuous oscillations at its resonant frequency.

The negative-resistance oscillator model is not limited to one-port devices like diodes; feedback oscillator circuits with two-port amplifying devices such as transistors and tubes also have negative resistance. At high frequencies, three terminal devices such as transistors and FETs are also used in negative resistance oscillators. At high frequencies these devices do not need a feedback loop, but with certain loads applied to one port can become unstable at the other port and show negative resistance due to internal feedback. The negative resistance port is connected to a tuned circuit or resonant cavity, causing them to oscillate. High-frequency oscillators in general are designed using negative-resistance techniques.

List of harmonic oscillator circuits
Some of the many harmonic oscillator circuits are listed below:

 * Armstrong oscillator, a.k.a. Meissner oscillator
 * Hartley oscillator
 * Colpitts oscillator
 * Clapp oscillator
 * Seiler oscillator
 * Vackář oscillator
 * Pierce oscillator
 * Tri-tet oscillator
 * Cathode follower oscillator
 * Wien bridge oscillator
 * Phase-shift oscillator
 * Cross-coupled oscillator
 * Dynatron oscillator
 * Opto-electronic oscillator
 * Robinson oscillator

Relaxation oscillator

A nonlinear or relaxation oscillator produces a non-sinusoidal output, such as a square, sawtooth or triangle wave. It consists of an energy-storing element (a capacitor or, more rarely, an inductor) and a nonlinear switching device (a latch, Schmitt trigger, or negative-resistance element) connected in a feedback loop. The switching device periodically charges the storage element with energy and when its voltage or current reaches a threshold discharges it again, thus causing abrupt changes in the output waveform.

Square-wave relaxation oscillators are used to provide the clock signal for sequential logic circuits such as timers and counters, although crystal oscillators are often preferred for their greater stability. Triangle-wave or sawtooth oscillators are used in the timebase circuits that generate the horizontal deflection signals for cathode ray tubes in analogue oscilloscopes and television sets. They are also used in voltage-controlled oscillators (VCOs), inverters and switching power supplies, dual-slope analog to digital converters (ADCs), and in function generators to generate square and triangle waves for testing equipment. In general, relaxation oscillators are used at lower frequencies and have poorer frequency stability than linear oscillators.

Ring oscillators are built of a ring of active delay stages. Generally the ring has an odd number of inverting stages, so that there is no single stable state for the internal ring voltages. Instead, a single transition propagates endlessly around the ring.

Some of the more common relaxation oscillator circuits are listed below:

 * Multivibrator
 * Pearson–Anson oscillator
 * Ring oscillator
 * Delay-line oscillator
 * Royer oscillator

Voltage-controlled oscillator (VCO)
An oscillator can be designed so that the oscillation frequency can be varied over some range by an input voltage or current. These voltage controlled oscillators are widely used in phase-locked loops, in which the oscillator's frequency can be locked to the frequency of another oscillator. These are ubiquitous in modern communications circuits, used in filters, modulators, demodulators, and forming the basis of frequency synthesizer circuits which are used to tune radios and televisions.

Radio frequency VCOs are usually made by adding a varactor diode to the tuned circuit or resonator in an oscillator circuit. Changing the DC voltage across the varactor changes its capacitance, which changes the resonant frequency of the tuned circuit. Voltage controlled relaxation oscillators can be constructed by charging and discharging the energy storage capacitor with a voltage controlled current source. Increasing the input voltage increases the rate of charging the capacitor, decreasing the time between switching events.

Theory of feedback oscillators
A feedback oscillator circuit consists of two parts connected in a feedback loop; an amplifier $$A$$ and an electronic filter $$\beta(j\omega)$$. The filter's purpose is to limit the frequencies that can pass through the loop so the circuit only oscillates at the desired frequency. Since the filter and wires in the circuit have resistance they consume energy and the amplitude of the signal drops as it passes through the filter. The amplifier is needed to increase the amplitude of the signal to compensate for the energy lost in the other parts of the circuit, so the loop will oscillate, as well as supply energy to the load attached to the output.

Frequency of oscillation - the Barkhausen criterion
To determine the frequency(s) $$\omega_0\;=\;2\pi f_0$$ at which a feedback oscillator circuit will oscillate, the feedback loop is thought of as broken at some point (see diagrams) to give an input and output port. A sine wave is applied to the input $$v_i(t) = V_ie^{j\omega t}$$ and the amplitude and phase of the sine wave after going through the loop $$v_o = V_o e^{j(\omega t + \phi)}$$ is calculated
 * $$v_o = A v_f\,$$     and      $$v_f = \beta(j\omega) v_i \,$$      so      $$v_o = A\beta(j\omega) v_i\,$$

Since in the complete circuit $$v_o$$ is connected to $$v_i$$, for oscillations to exist
 * $$v_o(t) = v_i(t)$$

The ratio of output to input of the loop, $${v_o \over v_i} = A\beta(j\omega)$$, is called the loop gain. So the condition for oscillation is that the loop gain must be one
 * $$A\beta(j\omega_0) = 1\,$$

Since $$A\beta(j\omega) $$ is a complex number with two parts, a magnitude and an angle, the above equation actually consists of two conditions:
 * The magnitude of the gain (amplification) around the loop at ω0 must be unity
 * $$|A||\beta(j\omega_0)| = 1\, \qquad\qquad\qquad\qquad\qquad\qquad \text{(1)} $$
 * so that after a trip around the loop the sine wave is the same amplitude. It can be seen intuitively that if the loop gain were greater than one, the amplitude of the sinusoidal signal would increase as it travels around the loop, resulting in a sine wave that grows exponentially with time, without bound. If the loop gain were less than one, the signal would decrease around the loop, resulting in an exponentially decaying sine wave that decreases to zero.
 * The sine wave at the end of the loop must be in phase with the wave at the beginning of the loop. Since the sine wave is periodic and repeats every 2π radians, this means that the phase shift around the loop at the oscillation frequency ω0  must be zero or a multiple of 2π radians (360°)
 * $$\angle A + \angle \beta = 2 \pi n \qquad n \in 0, 1, 2... \, \qquad\qquad \text{(2)}$$

Equations (1) and (2) are called the Barkhausen stability criterion. It is a necessary but not a sufficient criterion for oscillation, so there are some circuits which satisfy these equations that will not oscillate. An equivalent condition often used instead of the Barkhausen condition is that the circuit's closed loop transfer function (the circuit's complex impedance at its output) have a pair of poles on the imaginary axis.

In general, the phase shift of the feedback network increases with increasing frequency so there are only a few discrete frequencies (often only one) which satisfy the second equation. If the amplifier gain $$A$$ is high enough that the loop gain is unity (or greater, see Startup section) at one of these frequencies, the circuit will oscillate at that frequency. Many amplifiers such as common-emitter transistor circuits are "inverting", meaning that their output voltage decreases when their input increases. In these the amplifier provides 180° phase shift, so the circuit will oscillate at the frequency at which the feedback network provides the other 180° phase shift.

At frequencies well below the poles of the amplifying device, the amplifier will act as a pure gain $$A$$, but if the oscillation frequency $$\omega_0$$ is near the amplifier's cutoff frequency $$\omega_C$$, within $$0.1\omega_C$$, the active device can no longer be considered a 'pure gain', and it will contribute some phase shift to the loop.

An alternate mathematical stability test sometimes used instead of the Barkhausen criterion is the Nyquist stability criterion. This has a wider applicability than the Barkhausen, so it can identify some of the circuits which pass the Barkhausen criterion but do not oscillate.

Frequency stability
Temperature changes, aging, and manufacturing tolerances will cause component values to "drift" away from their designed values. Changes in frequency determining components such as the tank circuit in LC oscillators will cause the oscillation frequency to change, so for a constant frequency these components must have stable values. How stable the oscillator's frequency is to other changes in the circuit, such as changes in values of other components, gain of the amplifier, the load impedance, or the supply voltage, is mainly dependent on the Q factor ("quality factor") of the feedback filter. Since the amplitude of the output is constant due to the nonlinearity of the amplifier (see Startup section below), changes in component values cause changes in the phase shift $$\phi\;=\;\angle A\beta(j\omega)$$ of the feedback loop. Since oscillation can only occur at frequencies where the phase shift is a multiple of 360°, $$\phi\;=\;360n^\circ$$, shifts in component values cause the oscillation frequency $$\omega_0$$ to change to bring the loop phase back to 360n°. The amount of frequency change $$\Delta \omega$$ caused by a given phase change $$\Delta \phi$$ depends on the slope of the loop phase curve at $$\omega_0$$, which is determined by the $$Q $$
 * $${d\phi \over d\omega}\Bigg|_{\omega_0} = -{2Q \over \omega_0}\,$$     so       $$\Delta \omega = -{\omega_0 \over 2Q}\Delta \phi \,$$

RC oscillators have the equivalent of a very low $$Q$$, so the phase changes very slowly with frequency, therefore a given phase change will cause a large change in the frequency. In contrast, LC oscillators have tank circuits with high $$Q$$ (~102). This means the phase shift of the feedback network increases rapidly with frequency near the resonant frequency of the tank circuit. So a large change in phase causes only a small change in frequency. Therefore the circuit's oscillation frequency is very close to the natural resonant frequency of the tuned circuit, and doesn't depend much on other components in the circuit. The quartz crystal resonators used in crystal oscillators have even higher $$Q$$ (104 to 106) and their frequency is very stable and independent of other circuit components.

Tunability
The frequency of RC and LC oscillators can be tuned over a wide range by using variable components in the filter. A microwave cavity can be tuned mechanically by moving one of the walls. In contrast, a quartz crystal is a mechanical resonator whose resonant frequency is mainly determined by its dimensions, so a crystal oscillator's frequency is only adjustable over a very narrow range, a tiny fraction of one percent. It's frequency can be changed slightly by using a trimmer capacitor in series or parallel with the crystal.

Startup and amplitude of oscillation
The Barkhausen criterion above, eqs. (1) and (2), merely gives the frequencies at which steady-state oscillation is possible, but says nothing about the amplitude of the oscillation, whether the amplitude is stable, or whether the circuit will start oscillating when the power is turned on. For a practical oscillator two additional requirements are necessary:
 * In order for oscillations to start up in the circuit from zero, the circuit must have "excess gain"; the loop gain for small signals must be greater than one at its oscillation frequency
 * $$|A\beta(j\omega_0)| > 1\,$$

A typical rule of thumb is to make the small signal loop gain at the oscillation frequency 2 or 3. When the power is turned on, oscillation is started by the power turn-on transient or random electronic noise present in the circuit. Noise guarantees that the circuit will not remain "balanced" precisely at its unstable DC equilibrium point (Q point) indefinitely. Due to the narrow passband of the filter, the response of the circuit to a noise pulse will be sinusoidal, it will excite a small sine wave of voltage in the loop. Since for small signals the loop gain is greater than one, the amplitude of the sine wave increases exponentially.
 * For stable operation, the feedback loop must include a nonlinear component which reduces the gain back to unity as the amplitude increases to its operating value.

During startup, while the amplitude of the oscillation is small, the circuit is approximately linear, so the analysis used in the Barkhausen criterion is applicable. When the amplitude becomes large enough that the amplifier becomes nonlinear, technically the frequency domain analysis used in normal amplifier circuits is no longer applicable, so the "gain" of the circuit is undefined. However the filter attenuates the harmonic components produced by the nonlinearity of the amplifier, so the fundamental frequency component $$\sin \omega_0 t$$ mainly determines the loop gain (this is the "harmonic balance" analysis technique for nonlinear circuits).

The sine wave cannot grow indefinitely; in all real oscillators some nonlinear process in the circuit limits its amplitude, reducing the gain as the amplitude increases, resulting in stable operation at some constant amplitude. In most oscillators this nonlinearity is simply the saturation (limiting or clipping) of the amplifying device, the transistor, vacuum tube or op-amp. The maximum voltage swing of the amplifier's output is limited by the DC voltage provided by its power supply. Another possibility is that the output may be limited by the amplifier slew rate.

As the amplitude of the output nears the power supply voltage rails, the amplifier begins to saturate on the peaks (top and bottom) of the sine wave, flattening or "clipping" the peaks. Since the output of the amplifier can no longer increase with increasing input, further increases in amplitude cause the equivalent gain of the amplifier and thus the loop gain to decrease. The amplitude of the sine wave, and the resulting clipping, continues to grow until the loop gain is reduced to unity, $$|A\beta(j\omega_0)|\;=\;1\,$$, satisfying the Barkhausen criterion, at which point the amplitude levels off and steady state operation is achieved, with the output a slightly distorted sine wave with peak amplitude determined by the supply voltage. This is a stable equilibrium; if the amplitude of the sine wave increases for some reason, increased clipping of the output causes the loop gain $$|A\beta(j\omega_0)|$$ to drop below one temporarily, reducing the sine wave's amplitude back to its unity-gain value. Similarly if the amplitude of the wave decreases, the decreased clipping will cause the loop gain to increase above one, increasing the amplitude.

The amount of harmonic distortion in the output is dependent on how much excess loop gain the circuit has: An exception to the above are high Q oscillator circuits such as crystal oscillators; the narrow bandwidth of the crystal removes the harmonics from the output, producing a 'pure' sinusoidal wave with almost no distortion even with large loop gains.
 * If the small signal loop gain is made close to one, just slightly greater, the output waveform will have minimum distortion, and the frequency will be most stable and independent of supply voltage and load impedance. However, the oscillator may be slow starting up, and a small decrease in gain due to a variation in component values may prevent it from oscillating.
 * If the small signal loop gain is made significantly greater than one, the oscillator starts up faster, but more severe clipping of the sine wave occurs, and thus the resulting distortion of the output waveform increases. The oscillation frequency becomes more dependent on the supply voltage and current drawn by the load.

Design procedure
Since oscillators depend on nonlinearity for their operation, the usual linear frequency domain circuit analysis techniques used for amplifiers based on the Laplace transform, such as root locus and gain and phase plots (Bode plots), cannot capture their full behavior. To determine startup and transient behavior and calculate the detailed shape of the output waveform, electronic circuit simulation computer programs like SPICE are used. A typical design procedure for oscillator circuits is to use linear techniques such as the Barkhausen stability criterion or Nyquist stability criterion to design the circuit, then simulate the circuit on computer to make sure it starts up reliably and to determine the nonlinear aspects of operation such as harmonic distortion. Component values are tweaked until the simulation results are satisfactory. The distorted oscillations of real-world (nonlinear) oscillators are called limit cycles and are studied in nonlinear control theory.

Amplitude-stabilized oscillators
In applications where a 'pure' very low distortion sine wave is needed, such as precision signal generators, a nonlinear component is often used in the feedback loop that provides a 'slow' gain reduction with amplitude. This stabilizes the loop gain at an amplitude below the saturation level of the amplifier, so it does not saturate and "clip" the sine wave. Resistor-diode networks and FETs are often used for the nonlinear element. An older design uses a thermistor or an ordinary incandescent light bulb; both provide a resistance that increases with temperature as the current through them increases.

As the amplitude of the signal current through them increases during oscillator startup, the increasing resistance of these devices reduces the loop gain. The essential characteristic of all these circuits is that the nonlinear gain-control circuit must have a long time constant, much longer than a single period of the oscillation. Therefore over a single cycle they act as virtually linear elements, and so introduce very little distortion. The operation of these circuits is somewhat analogous to an automatic gain control (AGC) circuit in a radio receiver. The Wein bridge oscillator is a widely used circuit in which this type of gain stabilization is used.

Frequency limitations
At high frequencies it becomes difficult to physically implement feedback oscillators because of shortcomings of the components. Since at high frequencies the tank circuit has very small capacitance and inductance, parasitic capacitance and parasitic inductance of component leads and PCB traces become significant. These may create unwanted feedback paths between the output and input of the active device, creating instability and oscillations at unwanted frequencies (parasitic oscillation). Parasitic feedback paths inside the active device itself, such as the interelectrode capacitance between output and input, make the device unstable. The input impedance of the active device falls with frequency, so it may load the feedback network. As a result, stable feedback oscillators are difficult to build for frequencies above 500 MHz, and negative resistance oscillators are usually used for frequencies above this.

History
The first practical oscillators were based on electric arcs, which were used for lighting in the 19th century. The current through an arc light is unstable due to its negative resistance, and often breaks into spontaneous oscillations, causing the arc to make hissing, humming or howling sounds which had been noticed by Humphry Davy in 1821, Benjamin Silliman in 1822, Auguste Arthur de la Rive in 1846, and David Edward Hughes in 1878. Ernst Lecher in 1888 showed that the current through an electric arc could be oscillatory.

An oscillator was built by Elihu Thomson in 1892 by placing an LC tuned circuit in parallel with an electric arc and included a magnetic blowout. Independently, in the same year, George Francis FitzGerald realized that if the damping resistance in a resonant circuit could be made zero or negative, the circuit would produce oscillations, and, unsuccessfully, tried to build a negative resistance oscillator with a dynamo, what would now be called a parametric oscillator. The arc oscillator was rediscovered and popularized by William Duddell in 1900. Duddell, a student at London Technical College, was investigating the hissing arc effect. He attached an LC circuit (tuned circuit) to the electrodes of an arc lamp, and the negative resistance of the arc excited oscillation in the tuned circuit. Some of the energy was radiated as sound waves by the arc, producing a musical tone. Duddell demonstrated his oscillator before the London Institute of Electrical Engineers by sequentially connecting different tuned circuits across the arc to play the national anthem "God Save the Queen". Duddell's "singing arc" did not generate frequencies above the audio range. In 1902 Danish physicists Valdemar Poulsen and P. O. Pederson were able to increase the frequency produced into the radio range by operating the arc in a hydrogen atmosphere with a magnetic field, inventing the Poulsen arc radio transmitter, the first continuous wave radio transmitter, which was used through the 1920s.

The vacuum-tube feedback oscillator was invented around 1912, when it was discovered that feedback ("regeneration") in the recently invented audion (triode) vacuum tube could produce oscillations. At least six researchers independently made this discovery, although not all of them can be said to have a role in the invention of the oscillator. In the summer of 1912, Edwin Armstrong observed oscillations in audion radio receiver circuits and went on to use positive feedback in his invention of the regenerative receiver. Austrian Alexander Meissner independently discovered positive feedback and invented oscillators in March 1913. Irving Langmuir at General Electric observed feedback in 1913. Fritz Lowenstein may have preceded the others with a crude oscillator in late 1911. In Britain, H. J. Round patented amplifying and oscillating circuits in 1913. In August 1912, Lee De Forest, the inventor of the audion, had also observed oscillations in his amplifiers, but he didn't understand the significance and tried to eliminate it until he read Armstrong's patents in 1914, which he promptly challenged. Armstrong and De Forest fought a protracted legal battle over the rights to the "regenerative" oscillator circuit which has been called "the most complicated patent litigation in the history of radio". De Forest ultimately won before the Supreme Court in 1934 on technical grounds, but most sources regard Armstrong's claim as the stronger one.

The first and most widely used relaxation oscillator circuit, the astable multivibrator, was invented in 1917 by French engineers Henri Abraham and Eugene Bloch. They called their cross-coupled, dual-vacuum-tube circuit a multivibrateur, because the square-wave signal it produced was rich in harmonics, compared to the sinusoidal signal of other vacuum-tube oscillators.

Vacuum-tube feedback oscillators became the basis of radio transmission by 1920. However, the triode vacuum tube oscillator performed poorly above 300 MHz because of interelectrode capacitance. To reach higher frequencies, new "transit time" (velocity modulation) vacuum tubes were developed, in which electrons traveled in "bunches" through the tube. The first of these was the Barkhausen–Kurz oscillator (1920), the first tube to produce power in the UHF range. The most important and widely used were the klystron (R. and S. Varian, 1937) and the cavity magnetron (J. Randall and H. Boot, 1940).

Mathematical conditions for feedback oscillations, now called the Barkhausen criterion, were derived by Heinrich Georg Barkhausen in 1921. The first analysis of a nonlinear electronic oscillator model, the Van der Pol oscillator, was done by Balthasar van der Pol in 1927. He showed that the stability of the oscillations (limit cycles) in actual oscillators was due to the nonlinearity of the amplifying device. He originated the term "relaxation oscillation" and was first to distinguish between linear and relaxation oscillators. Further advances in mathematical analysis of oscillation were made by Hendrik Wade Bode and Harry Nyquist in the 1930s. In 1969 Kaneyuki Kurokawa derived necessary and sufficient conditions for oscillation in negative-resistance circuits, which form the basis of modern microwave oscillator design.

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

title: Electronic oscillator


text:
Societas Europaea

A societas Europaea (, ; "European society" or "company"; plural: societates Europaeae; abbr. SE) is a public company registered in accordance with the corporate law of the European Union (EU), introduced in 2004 with the Council Regulation on the Statute for a European Company. Such a company may more easily transfer to or merge with companies in other member states.

As of April 2018, more than 3,000 registrations have been reported, including the following nine components (18%) of the Euro Stoxx 50 stock market index of leading eurozone companies (excluding the SE designation): Airbus, Allianz, BASF, E.ON, Fresenius, LVMH Moët Hennessy Louis Vuitton (and its subsidiary Dior), SAP, Schneider Electric and Unibail-Rodamco-Westfield.

National law continues to supplement the basic rules in the Regulation on formation and mergers. The European Company Regulation is complemented by an Employee Involvement Directive which manages the rules for participation by employees on the company's board of directors. There is also a statute allowing European Cooperative Societies.

Formation
The statute provides five ways of forming a European limited company:
 * 1) By merger of national companies from different member states
 * 2) By creation of a European limited company as a parent company in a holding structure of joint stock companies and limited liability companies from different member states
 * 3) By the creation of a joint venture between companies (or other entities) in different member states
 * 4) By the creation of an SE subsidiary of a national company
 * 5) By the conversion of a national company into an SE

Formation by merger is available only to public limited companies from different member states. Formation of an SE holding company is available to public and private limited companies with their registered offices in different member states or having subsidiaries or branches in member states other than that of their registered office. Formation of a joint subsidiary is available under the same circumstances to any legal entities governed by public or private law.

Minimum capital
The SE must have a minimum subscribed capital of €120,000 as per article 4(2) of the directive, subject to the provision that where a member state requires a larger capital for companies exercising certain types of activities, the same requirement will also apply to an SE with its registered office in that member state (article 4(3)).

Registered office
The registered office of the SE designated in the statutes must be the place where it has its central administration, that is to say its true centre of operations. The SE may transfer its registered office within the European Economic Area without dissolving the company in one member state in order to form a new one in another member state; however, such a transfer is subject to the provisions of 8 which require, inter alia, the drawing up of a transfer proposal, a report justifying the legal and economic aspects of the transfer and the issuing, by the competent authority in the member state in which the SE is registered, of a certificate attesting to the completion of the required acts and formalities.

Laws applicable
The order of precedence of the laws applicable to the SE is clarified.

Registration and liquidation
The registration and completion of the liquidation of an SE must be disclosed for information purposes in the Official Journal of the European Communities. Every SE must be registered in the state where it has its registered office, in a register designated by the law of that state.

Statutes
The statutes of the SE must provide as governing bodies the annual general meeting of shareholders and either a management board and a supervisory board (two-tier system) or an administrative board (single-tier system). Under the two-tier system the SE is managed by a management board. The member or members of the management board have the power to represent the company in dealings with third parties and in legal proceedings. They are appointed and removed by the supervisory board. No person may be a member of both the management board and the supervisory board of the same company at the same time. But the supervisory board may appoint one of its members to exercise the functions of a member of the management board in the event of absence through holidays. During such a period the function of the person concerned as a member of the supervisory board shall be suspended. Under the single-tier system, the SE is managed by an administrative board. The member or members of the administrative board have the power to represent the company in dealings with third parties and in legal proceedings. Under the single-tier system the administrative board may delegate the power of management to one or more of its members.

The following operations require the authorization of the supervisory board or the deliberation of the administrative board:
 * any investment project requiring an amount more than the percentage of subscribed capital
 * the conclusion of supply and performance contracts where the total turnover provided for therein is more than the percentage of turnover for the previous financial year
 * the raising or granting of loans, the issue of debt securities and the assumption of liabilities of a third party or suretyship for a third party where the total money value in each case is more than the percentage of subscribed capital
 * the setting-up, acquisition, disposal or closing down of undertakings, businesses or parts of businesses where the purchase price or disposal proceeds account for more than the percentage of subscribed capital
 * the percentage referred to above is to be determined by the statutes of the SE. It may not be less than 5% nor more than 25%.

Annual accounts
The SE must draw up annual accounts comprising the balance sheet, the profit and loss account, and the notes to the accounts, and an annual report giving a fair view of the company's business and of its position; consolidated accounts may also be required.

Taxation
In tax matters, the SE is treated the same as any other multinational, i.e., it is subject to the tax regime of the national legislation applicable to the company and its subsidiaries. SEs are subject to taxes and charges in all member states where their administrative centres are situated.

Winding-up
Winding-up, liquidation, insolvency, and suspension of payments are in large measure to be governed by national law. When an SE transfers its registered office outside the Community, or in any other manner no longer complies with requirements of article 7, the member state must take appropriate measures to ensure compliance or take necessary measures to ensure that the SE is liquidated.

Status of the legislation and implementation
Council Regulation (EC) No 2157/2001 of 8 October 2001 on the Statute for a European company (SE).

Council Directive 2001/86/EC of 8 October 2001 supplementing the Statute for a European company with regard to the involvement of employees.

See also: Europa's collection of press releases, regulations, directives and FAQs on the European Company Statute.

United Kingdom
Following the withdrawal of the UK from the European Union, any SE registered in the United Kingdom converted to a United Kingdom Societas and UK Societas replaced SE in its name. UK Societas retain many of the elements from the SE framework but importantly without the ability to transfer their registered office outside of the UK.

Employee participation
The regulation is complemented by the Council Directive supplementing the Statute for a European Company with regard to the involvement of employees (informally "Council Directive on Employee Participation"), adopted 8 October 2001. The directive establishes rules on worker involvement in the management of the SE.

EU member states differ in the degree of worker involvement in corporate management. In Germany, most large corporations are required to allow employees to elect a certain percentage of seats on the supervisory board. Other member states have no such requirement, and furthermore in these states such practices are largely unknown and considered a threat to the rights of management.

These differing traditions of worker involvement have held back the adoption of the statute for over a decade. States without worker involvement provisions were afraid that the SE might lead to having such provisions being imposed on their companies; and states with those provisions were afraid they might lead to those provisions being circumvented.

A compromise, contained in the directive, was worked out as follows: worker involvement provisions in the SE will be decided upon by negotiations between employees and management before the creation of the SE. If agreement cannot be reached, provisions contained in the directive will apply. The directive provides for worker involvement in the SE if a minimum percentage of employees from the entities coming together to form the SE enjoyed worker involvement provisions. The directive permits member states to not implement these default worker involvement provisions in their national law, but then an SE cannot be created in that member state if the provisions in the directive would apply and negotiations between workers and management are unsuccessful.

Definition
Definition of employee participation: it does not mean participation in day-to-day decisions, which are a matter for the management, but participation in the supervision and strategic development of the company.

Participation

 * If the two parties do not reach a satisfactory arrangement, a set of standard principles set out in the annex to the directive becomes applicable.
 * Several models of participation are possible: firstly, a model in which the employees form part of the supervisory board or of the administrative board, as the case may be; secondly, a model in which the employees are represented by a separate body; and finally, other models to be agreed between the management or administrative boards of the founder companies and the employees or their representatives in those companies, the level of information and consultation being the same as in the case of the second model. The general meeting may not approve the formation of an SE unless one of the models of participation defined in the directive has been chosen.
 * The employees' representatives must be provided with such financial and material resources and other facilities as enable them to perform their duties properly.
 * With regard to a European company formed through a merger, the standard principles relating to worker participation will apply where at least 25% of the employees had the right to participate in decisions before the merger. Here a political agreement proved impossible until the Nice summit in December 2000. The compromise adopted by the heads of state or governments allowed a member state not to apply the directive to SEs formed from a merger, in which case the SE could not be registered in the member state in question unless an agreement had been concluded between the management and employees, or that no SE employee had the right of participation before the formation of the SE.

Employment contracts and pensions
Employment contracts and pensions are not covered by the directive. With regard to occupational pension schemes, the SE is covered by the provisions laid down in the proposal for a directive on institutions for occupational schemes, presented by the Commission in October 2000, in particular in connection with the possibility of introducing a single pension scheme for all their employees in the European Union.

Development
Two approaches have been attempted to solve the problems cited above. One approach is to harmonize the company law of the member states. This approach has had some successes, but after thirty years only limited progress has been made. It is difficult to harmonize widely different regulatory systems, especially when they reflect different national attitudes to issues such as worker involvement in the management of the company.

The other approach is to construct a whole new system of EU company law, that co-exists with the individual company laws of the member states. Companies would have the choice of operating either under national regulations or under the EU-wide system. However, this approach has been only somewhat more effective than the harmonization approach: while states are not as concerned about having foreign traditions of corporate governance imposed on their companies, which the harmonization approach could well entail; they also wish to ensure that the EU-wide system would be palatable to the traditions of their national companies, so that they will not be put at a disadvantage compared to the other member states.

The European Company Statute represents a step in this direction, albeit a limited one. While it establishes some common EU rules on the SE, these rules are incomplete, and the holes in the rules are to be filled in using the law of the member state in which the SE is registered. This has been due to the difficulties of agreeing on common European rules on these issues.

Registrations
As of 11 April 2018, 3,015 registrations have been made. In terms of registrations, the Czech Republic is vastly overrepresented, accounting for 79% of all Societates Europaeae as of December 2015. 9 of the 50 constituents of the Euro Stoxx 50 stock market index of leading eurozone companies are as of December 2015 Societates Europaeae. On 8 October 2004, Dutch company MPIT Structured Financial Services is Europe's first SE company.

Annual registrations by member state are presented in the following chart:

Registrations of new societates are to be published in the Official Journal of the European Union. There is no official union-wide register of societates, as they are registered in the nation in which their corporate seats are located. worker-participation.eu does however maintain a database of current and planned registrations. Examples of companies include: {{Legend|#ADD8E6|Components of the Euro Stoxx 50 stock market index of leading eurozone companies}}

title: Societas Europaea


text:
Electronic mixer

An electronic mixer is a device that combines two or more electrical or electronic signals into one or two composite output signals. There are two basic circuits that both use the term mixer, but they are very different types of circuits: additive mixers and multiplicative mixers. Additive mixers are also known as analog adders to distinguish from the related digital adder circuits.

Simple additive mixers use Kirchhoff's circuit laws to add the currents of two or more signals together, and this terminology ("mixer") is only used in the realm of audio electronics where audio mixers are used to add together audio signals such as voice signals, music signals, and sound effects.

Multiplicative mixers multiply together two time-varying input signals instantaneously (instant-by-instant). If the two input signals are both sinusoids of specified frequencies f1 and f2, then the output of the mixer will contain two new sinusoids that have the sum f1 + f2 frequency and the difference frequency absolute value |f1 - f2|.

Any nonlinear electronic block driven by two signals with frequencies f1 and f2 would generate intermodulation (mixing) products. A multiplier (which is a nonlinear device) will generate ideally only the sum and difference frequencies, whereas an arbitrary nonlinear block will also generate signals at 2·f1-3·f2, etc. Therefore, normal nonlinear amplifiers or just single diodes have been used as mixers, instead of a more complex multiplier. A multiplier usually has the advantage of rejecting – at least partly – undesired higher-order intermodulations and larger conversion gain.

Additive mixers
Additive mixers add two or more signals, giving out a composite signal that contains the frequency components of each of the source signals. The simplest additive mixers are resistor networks, and thus purely passive, while more complex matrix mixers employ active components such as buffer amplifiers for impedance matching and better isolation.

Multiplicative mixers
An ideal multiplicative mixer produces an output signal equal to the product of the two input signals. In communications, a multiplicative mixer is often used together with an oscillator to modulate signal frequencies. A multiplicative mixer can be coupled with a filter to either up-convert or down-convert an input signal frequency, but they are more commonly used to down-convert to a lower frequency to allow for simpler filter designs, as done in superheterodyne receivers. In many typical circuits, the single output signal actually contains multiple waveforms, namely those at the sum and difference of the two input frequencies and harmonic waveforms. The output signal may be obtained by removing the other signal components with a filter.í

Mathematical treatment
The received signal can be represented as

 * $$E_\mathrm{sig} \cos(\omega_\mathrm{sig}t+\varphi)\,$$

and that of the local oscillator can be represented as

 * $$E_\mathrm{LO} \cos(\omega_\mathrm{LO}t).\,$$

For simplicity, assume that the output I of the detector is proportional to the square of the amplitude:
 * $$I\propto \left( E_\mathrm{sig}\cos(\omega_\mathrm{sig}t+\varphi) + E_\mathrm{LO}\cos(\omega_\mathrm{LO}t) \right)^2$$

 * $$ =\frac{E_\mathrm{sig}^2}{2}\left( 1+\cos(2\omega_\mathrm{sig}t+2\varphi) \right)$$

 * $$ + \frac{E_\mathrm{LO}^2}{2}(1+\cos(2\omega_\mathrm{LO}t)) $$

 * $$ + E_\mathrm{sig}E_\mathrm{LO} \left[

\cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi) + \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi) \right] $$
 * $$ =\underbrace{\frac{E_\mathrm{sig}^2+E_\mathrm{LO}^2}{2}}_{constant\;component}+\underbrace{\frac{E_\mathrm{sig}^2}{2}\cos(2\omega_\mathrm{sig}t+2\varphi) + \frac{E_\mathrm{LO}^2}{2}\cos(2\omega_\mathrm{LO}t) + E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi)}_{high\;frequency\;component}$$

 * $$ + \underbrace{E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi)}_{beat\;component}.

$$

The output has high frequency ($$2\omega_\mathrm{sig}$$, $$2\omega_\mathrm{LO}$$ and $$\omega_\mathrm{sig}+\omega_\mathrm{LO}$$) and constant components. In heterodyne detection, the high frequency components and usually the constant components are filtered out, leaving the intermediate (beat) frequency at $$\omega_\mathrm{sig}-\omega_\mathrm{LO}$$. The amplitude of this last component is proportional to the amplitude of the signal radiation. With appropriate signal analysis the phase of the signal can be recovered as well.

If $$\omega_\mathrm{LO}$$ is equal to $$\omega_\mathrm{sig} $$ then the beat component is a recovered version of the original signal, with the amplitude equal to the product of $$ E_\mathrm{sig} $$ and $$E_\mathrm{LO} $$; that is, the received signal is amplified by mixing with the local oscillator. This is the basis for a Direct conversion receiver.

Implementations
Multiplicative mixers have been implemented in many ways. The most popular are Gilbert cell mixers, diode mixers, diode ring mixers (ring modulation) and switching mixers. Diode mixers take advantage of the non-linearity of diode devices to produce the desired multiplication in the squared term. They are very inefficient as most of the power output is in other unwanted terms which need filtering out. Inexpensive AM radios still use diode mixers.

Electronic mixers are usually made with transistors and/or diodes arranged in a balanced circuit or even a double-balanced circuit. They are readily manufactured as monolithic integrated circuits or hybrid integrated circuits. They are designed for a wide variety of frequency ranges, and they are mass-produced to tight tolerances by the hundreds of thousands, making them relatively cheap.

Double-balanced mixers are very widely used in microwave communications, satellite communications, ultrahigh frequency (UHF) communications transmitters, radio receivers, and radar systems.

Gilbert cell mixers are an arrangement of transistors that multiplies the two signals.

Switching mixers use arrays of field-effect transistors or vacuum tubes. These are used as electronic switches, to alternate the signal direction. They are controlled by the signal being mixed. They are especially popular with digitally controlled radios. Switching mixers pass more power and usually insert less distortion than Gilbert cell mixers.

Lydmikser ミキサー Mieszacz

title: Electronic mixer


text:
Eubulides

Eubulides (Εὑβουλίδης; fl. 4th century BCE) of Miletus was a philosopher of the Megarian school who is famous for his paradoxes.

Life
According to Diogenes Laërtius, Eubulides was a pupil of Euclid of Megara, the founder of the Megarian school. He was a contemporary of Aristotle, against whom he wrote with great bitterness. He taught logic to Demosthenes, and he is also said to have taught Apollonius Cronus, the teacher of Diodorus Cronus, and the historian Euphantus.

Paradoxes of Eubulides
Eubulides is most famous for inventing the forms of seven famous paradoxes, some of which, however, are also ascribed to Diodorus Cronus: Alpha ignored the man approaching him and treated him as a stranger. The man was his father. Did Alpha ignore his own father and treat him as a stranger?
 * 1) The Liar (pseudomenos) paradox: A man says: "What I am saying now is a lie." If the statement is true, then he is lying, even though the statement is true. If the statement is a lie, then he is not actually lying, even though the statement is a lie. Thus, if the speaker is lying, he tells the truth, and vice versa.
 * 2) The Masked Man (enkekalymmenos) paradox: "Do you know this masked man?" "No." "But he is your father. So – do you not know your own father?"
 * 3) The Electra (Elektra) paradox: Electra doesn't know that the man approaching her is her brother, Orestes.  Electra knows her brother.  Does Electra know the man who is approaching?
 * 4) The Overlooked Man (dialanthanôn) paradox:
 * 1) The Heap (sôritês) paradox: A single grain of sand is certainly not a heap.  Nor is the addition of a single grain of sand enough to transform a non-heap into a heap: when we have a collection of grains of sand that is not a heap, then adding but one single grain will not create a heap. And yet we know that at some point we will have a heap.
 * 2) The Bald Man (phalakros) paradox: A man with a full head of hair is obviously not bald.  Now the removal of a single hair will not turn a non-bald man into a bald one.  And yet it is obvious that a continuation of that process must eventually result in baldness.
 * 3) The Horns (keratinês) paradox: What you have not lost, you have. But you have not lost horns. Therefore, you have horns.

The first paradox (the Liar) is probably the most famous, and is similar to the famous paradox of Epimenides the Cretan. The second, third and fourth paradoxes are variants of a single paradox and relate to the problem of what it means to "know" something and the identity of objects involved in an affirmation (compare the masked-man fallacy). The fifth and sixth paradoxes are also a single paradox and is usually thought to relate to the vagueness of language. The final paradox, the horns, is a paradox related to presupposition.

Legacy
These paradoxes were very well known in ancient times, some are alluded to by Eubulides' contemporary Aristotle and even partially by Plato. Chrysippus, the Stoic philosopher wrote about the paradoxes developed by Eubulides and characterized the Horns paradox as an intractable problem (aporoi logoi). Aulus Gellius mentions how the discussion of such paradoxes was considered (for him) after-dinner entertainment at the Saturnalia, but Seneca, on the other hand, considered them a waste of time: "Not to know them does no harm, and mastering them does no good."

title: Eubulides