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Talk:Electrical impedance/Archive 2

Very poorly written
I think this site needs a good rewrite. It readly like a bad textbook, i have the feeling much of it is basicaly a copy-edit from some crappy textbooks. Things that say "let v(t) be somecrap" is not only stuffy sounding but useless without the explanation it needs. Not to mention, variables are not explained like they are on every other wikipedia page. Variables need to be explained, I fixed up the first part of the AC steady state part like that - with variables defined in a regular readable way. The other equations should read the same way. Fresheneesz 06:26, 9 December 2005 (UTC)

Holy freaking crap and i just looked at the "see also" section... wtf waaaay too many Fresheneesz 06:29, 9 December 2005 (UTC)

 * Please don't be a dick, Fresheneesz. More importantly, please don't replace correct statements with obviously incorrect statements such as:

 * "The general formula for the sinusoidal voltage of a circuit containing resistors, capacitors, and inductors is:"...

 * In the general case, the amplitude is a function of time. For AC steady state, the amplitude must be constant with time.  Further, the objective of this part of the article is to motivate the phasor notation - not to give some general circuit equation. Please try to actually understand the purpose of a section before editing. Alfred Centauri 15:34, 9 December 2005 (UTC)

 * Sorry, now that I look over it again, after half a year, my statement was pretty freaking wrong. However, I still hold to my dick-like comments above. If this page were well written - I wouldn't have made that mistake.
 * "objective of this part of the article is to motivate the phasor notation" - I don't think this is the best way to go about this. It can be easily and very understandably said that "Phasor notation is used to simplify circuit analysis by replacing resistances, inductances, and capacitances with a common form - impedances. Impedances can be used almost exactly like resistors, which are much simpler to work with than capacitances and inductances."
 * I really do hate the "let something be something else" wording. It does indeed remind me of horrible horrible textbooks that hauted my HS and even college career. I mean... why are we letting this something be something else? Is it because its useful perhaps? Or is it arbitrary. In this case it isn't arbitrary, so "letting" something be something else is simply bad wording. Sorry if I'm being a dick. I'm going to change that wording slightly again. Fresheneesz 04:36, 26 April 2006 (UTC)

 * Simply because you hate some wording because of nightmares from high-school doesn't imply that the wording is inappropriate for an encyclopedic article (in fact, it probably implies th oppposite). Remember, this is not an introductory text book.  But heck, if it bothers you that much, change it to something you are more comfortable with.  What are you waiting for?  Alfred Centauri 02:50, 29 April 2006 (UTC)

 * Approval? Wait I did change stuff. Fresheneesz 11:24, 28 May 2006 (UTC)

 * If nothing else, the article could seriously benefit from a layman's explanation. I'd do it, but I don't know how impedence works (on the consumer level) which is what brought me here in the first place. AubreeAnn —Preceding unsigned comment added by 66.30.114.49 (talk) 02:53, 27 April 2008 (UTC)

Sinusoidal basis...
I have removed the following text from the introduction:

"Because these sinusoids comprise a basis (basis means here that we can express any function f(t) as a summation of sine functions using Fourier Series (like in vectors any vector in space can be expressed in terms of i,j,k so i,j,k are called the basis vectors)) over the space of real functions, the impedance of a linear system describes its response to any time-varying signal, not just to a sinusoid."

I have three reasons for this:

(1) The section "Circuits with General Sources" already covers this concept.

(2) This statement doesn't belong, IMHO, in the introductory paragraph.

(3) This statement is not entirely correct. The impedance (as a function of s) of a linear system describes its complex frequency domain voltage response to a time domain unit impulse of current, not a general time-varying signal.

Alfred Centauri 04:21, 21 February 2006 (UTC)

Edits
Please stop inserting the erroneous negative sign in the equation for capacitive impedance. The math, as it stood before your edits, is correct. If you do not see this, then consider the following:

 * $$\frac{1}{j}=-j\,$$

 * $$\frac{1}{j \omega C}= \frac{-j}{\omega C}\,$$

 * $$X_C = \Im(Z_C) = \frac{-1}{\omega C}\,$$

With the negative sign you inserted, $$X_C$$ becomes positive which is clearly incorrect. Alfred Centauri 00:26, 2 March 2006 (UTC)

Merge Suggestions
Disagree to both merge suggestions. Both articles should instead be candidates for deletion. The author of these two articles has been known to insert unformatted incomprehensible rubbish to the Electromagnetic induction article. Alfred Centauri 21:22, 30 March 2006 (UTC)

 * Too late, already merged. You do realize that a merge doesn't mean a copy edit - I think my merge smothely incorporates whatever "full electric impedance" may be, and disregards the crappy explanations and definitions that are already better explained in this article. Fresheneesz 05:13, 21 April 2006 (UTC)

circuit symbol
I've seen the circuit symbolization of an impedance element as both a box (like the european way to show a resistor), and I *think* i've seen it also drawn like an american resistance (i can't quite remember). In any case, we should have a picture of how to draw it on this page. Fresheneesz 04:41, 21 April 2006 (UTC)

 * [[Image:Source and load circuit Z.png]] [[Image:Source and load boxes.png]] — Omegatron 15:25, 21 April 2006 (UTC)

 * Thanks, I'll add that to the intro. Fresheneesz 04:49, 26 April 2006 (UTC)

Also, impedance elements can be shown in the following way. In australia anyway. Element L is the impedance. Nick 04:52, 12 June 2006 (UTC)

 * Really? That's an inductance everywhere I've seen it. — Omegatron 07:03, 12 June 2006 (UTC)

passive elements only?
I was wondering if impedance only works with passive elements. The only reason I asked is that "passive element" was specified in an article I merged here. Comments? Fresheneesz 05:14, 21 April 2006 (UTC)

 * Impedance is the ratio of a phasor voltage and a phasor current regardless of whether the voltage and current are associated with a passive or active element or even the same port. Consider the two-port 'z' parameters.    Alfred Centauri 14:24, 21 April 2006 (UTC)

See also should be organized
Someone should organize the see also section into subcategories. Fresheneesz 05:18, 21 April 2006 (UTC)

AC steady state
The information under the header "AC steady state" seems very far disconnected with impedances. This article is on impedances, not on phasor notation. Obviously they can be interrlated, but that section is not a good starter. We should start on what impedances are - not what voltage is. Fresheneesz 04:57, 26 April 2006 (UTC)

 * I disagree - AC steady state is crucial to the development of the impedance concept. If you don't see this, then you don't quite understand impedance.  Let's say that an RC circuit is driven by a sinusoidal source with frequency f but the circuit is not in steady state.  What is the impedance of the capacitor?
 * And, although this article is not on phasor notation, what does an impedance multiply? A time domain current?  A frequency domain current?  Or, is it a phasor current?  Is the result a time domain voltage, frequency domain voltage, or a phasor voltage?  Alfred Centauri 21:53, 26 April 2006 (UTC)

 * I may not fully understand impedances, but I know how to use them. Thats part of my point, people comming to this page may not want to know the derivation for impedances - just how to use them. So starting the page off with intricacies isn't best.
 * "what does an impedance multiply" - what do you mean by that, I don't understand that question. 68.6.112.70 03:32, 28 April 2006 (UTC)

 * I would fully agree with you - if this article were part of a textbook. If people want to learn how to use impedances, then they should try Wikibooks, don't you think?
 * An impedance multiplies a phasor current and the result is a phasor voltage. Thus, it seems reasonable (to me) to believe that in order to define an impedance, which is the ratio of voltage and current phasors, requires an introduction to phasors.  Can you describe a better way?  If you were re-write this article, how would you define impedance and then give examples of how an impedance is used without introducing phasors first?  Alfred Centauri 02:38, 29 April 2006 (UTC)

 * "If people want to learn how to use impedances, then they should try Wikibooks, don't you think?" - yes and no. If they want a step by step in depth discussion of how to use impedances, then wikibooks would be the place. But if they want a concise overview of how people use impedances, then I think wikipedia is a perfect spot for that.

 * And... don't you think that perhaps an introduction to phasors belongs in wikibooks instead of here. Or perhaps just on the page on phasors. This is one thing I would do if I were to rewrite this page - get rid of the intro to phasors, and instead link the word phasors to the article on phasors (which I think is probably done). People comming here want to know about impedances, not phasors - but if they do want to learn both, they can visit both pages. Fresheneesz 22:01, 15 May 2006 (UTC)

non-phasor impedances
I've recently learned that "impedance" is not one thing. What I mean is that the term impedance is used in more than just phasor circuit analysis. For example, in Laplace transformations, circuit elements can be transformed into impedances in the s-domain. This article (as far as I can tell) focuses only on phasor impedances, and disregards the existance of s-Domain impedances. They are very interrlated concepts, and s-domain impedances should have a place on this page. Fresheneesz 04:59, 26 April 2006 (UTC)

 * Actually, impedance is just one thing - it is the ratio of the voltage to the current when the circuit excitation is of the form:

 * $$Ae^{s_0t}\,$$

 * See the discussion under 'Rv of last edit by Wigner'. It is precisely because the complex exponential is the eigenfunction of the derivative operation that, for a capacitor or an inductor driven by a complex exponential source, the ratio of the time domain voltage to current is the same as the ratio of the frequency domain voltage to current.  Recall that, for a resistor, the resistance R is the ratio of the voltage to current in both the time domain and the frequency domain for any excitation.

 * Now, it is my opinion that the the ratio of the frequency domain voltage to current for an arbitrary excitation is not technically an impedance. On a couple of occasions, editors have added verbage to this effect and I have removed it.  The reason is that for an arbitrary excitation we still have:

 * $$V(s) = Z(s)I(s)\,$$

 * but, in the time domain, we now have:

 * $$v(t) = z(t) * i(t)\,$$

 * where * is convolution.

 * What is this z(t)? Consider a capacitor driven by a constant current source.  The ratio of the time domain voltage to current is a function of time:

 * $$\frac{v_C(t)}{i_C(t)}=\frac{t}{C}$$

 * Would you call this an impedance? Alfred Centauri 14:03, 26 April 2006 (UTC)

I guess? Not quite sure.
 * "it is the ratio of the voltage to the current when the circuit excitation is of the form...." - I'm pretty sure that Laplace impedances don't require the function to be sinusoidal at all - especially not in that specific form. This is why they're useful over phasors.
 * "''For example, what is the impedance of a capacitor if the voltage across the capacitor is:
 * $$v_C(t) = cos(2 \pi f_1 t) + sin(2 \pi f_2 t) \,$$''"

I actually used Laplace transformations to derive the impedance of the capacitor in that case:
 * $$ Z_C(t) = \frac{1}{2 \pi f_2}+ \frac{sin(2 \pi f_2 t)}{4 \pi f_1 C} - \frac{cos(2 \pi f_1 t)}{2 \pi f_2} \ $$
 * where C is the capacitance.

"my opinion that ... is not technically an impedance" - I think it technically *is* an impedance, but perhaps you'd rather not think of it as such. Laplace impedances are still voltage over current - but are more general than phasor impedances. Fresheneesz 03:05, 27 April 2006 (UTC)

 * (1) Impedance is not a function of the instantaneous values of the voltage and current. Don't take my word for it - look it up.  Point me to a reference that says otherwise.
 * (2) IMHO, you are confusing transfer functions with impedances. The ratio of Laplace (and Fourier) voltages and currents are transfer functions.
 * (3) What you derived from the Laplace transform is not an impedance and further, it is clearly not the ratio of the time domain voltage and current associated with the capacitor which is given by:

 * $$\frac{1}{2 \pi C}\frac{cos(2 \pi f_1 t) + sin(2 \pi f_2 t)}{f_2 cos(2 \pi f_2 t) - f_1 sin(2 \pi f_1 t)}$$

 * The Laplace transform of cos(t) and sin(t) actually assume that the time signals are u(t)cos(t) and u(t)sin(t) respectively. That is, the sinusoidal sources 'turn on' at t=0.  If you use the Fourier transform (and its inverse), you should get the same answer as I get.  In any case, these time functions are not impedances.  Once again, if you can find anything in the literature that says otherwise, I would be very interested to see it.  Alfred Centauri 04:05, 27 April 2006 (UTC)

I did probably confuse transfer function for impedance in my largish function, fair enough. But in the s-domain, the laplace impedances *are* V(s)/I(s). They work like normal impedances, and are defined for more functions than sinusoidal ones. Fresheneesz 07:23, 27 April 2006 (UTC)

 * "They work like normal impedances, and are defined for more functions than sinusoidal ones" Sigh.  I think that you are not seeing the deep connection between phasors and the Laplace (or Fourier) domain.  Maybe you'll see soon that what you've said here (and have added to the Laplace transform page) is nonsense.  Maybe this will help - think of what the difference might be between an impedance function and an impedance proper.  What is the difference between 1/Cs and 1/Cs_o?  This is subtle so take your time. Alfred Centauri 13:55, 27 April 2006 (UTC)

 * Something else to consider. In words, what does the ratio V(s)/I(s) represent?  What if i(t) is the unit impulse?  Then I(s) = 1 so that the ratio V(s)/I(s) = V(s).  Now we see that this ratio is simply the Laplace transform of the voltage response to the unit impulse of current.  In other words, a transfer function.  Alfred Centauri 14:15, 27 April 2006 (UTC)

 * I've been taught that the connection between phasors and the Laplace domain is that phasors are a subset of the Laplace domain - where $$ s = j \omega $$. Replacing the "s" in Laplace impedances with "$$j \omega$$" gives the phasor impedance. Is this wrong? Fresheneesz 02:08, 28 April 2006 (UTC)

 * A phasor is the complex amplitude associated with a time function of the form:

 * $$Ae^{s_0t} \, $$

 * In physical circuits, we don't find time functions of this form since this time function is complex valued. However, we can find time functions of the form:

 * $$\Re{(Ae^{s_0t})} \,$$

 * For AC (sinusoidal) circuit analysis, we assume that the real part of s_0 is zero so that the voltages and currents are of the form:

 * $$\Re{(Ae^{j \omega_0 t})} \,$$

 * To do the analysis, we pretend that the time functions are of the form:

 * $$Ae^{j \omega_0 t} \, $$

 * and then take the real part of the solution at the end.

 * But, this doesn't diminish the fact that a phasor is the complex amplitude of a complex exponential function of time. How does this all tie in with the s-domain?

 * First, recognize that the ratio of the s-domain voltage to current associated with a circuit element is, by definition, the Laplace transform of the time domain voltage response to a time domain unit impulse of current. To see this for a capacitor, take the inverse Laplace transform of 1/Cs:

 * $$\frac{1}{C}u(t) \Leftrightarrow \frac{1}{Cs}$$

 * That is, a unit impulse of current through a capacitor produces a step of voltage scaled by 1/C.

 * Such a function in the s-domain is called a transfer function. Since this function has units of ohms, it is appropriate to call it an impedance transfer function.  What must be understood is that this function is not an impedance proper.  Now, let us evaluate this impedance transfer function for some particular value of s which we will denote as s_0.  What does this quantity represent?  It is straightforward to show that this quantity is equal to the ratio of the complex amplitudes of the time domain voltage and current functions associated with a capacitor when they are of the form:

 * $$Ae^{s_0t} \, $$

 * Try to show this mathematically - you will need to use the delta function in the s-domain - delta(s - s_0) - as the s-domain current in order to select the value 1/Cs_0 from the function 1/Cs. So, 1/Cs_0 is equal to the ratio of the voltage and current phasors when s = s_0.  This defines the impedance of the capacitor when s = s_0.

 * So, can phasors be used for more general sources? The answer is, of course, yes if the Laplace transform of the source function exists.  If we calculate the s-domain voltage across a capacitor for some general s-domain current, we must take the inverse Laplace transform in order to find the time domain voltage function, right?  But, if you consider what the process of taking the inverse transform is, you see that it is simply the 'summing up' of complex exponential time functions (with different values of s) scaled by the impedance of the capacitor at that particular value of s.  See the section in the article entitled "Circuits with general sources".  So you see, phasors are built-in to the inverse Laplace transform.  Alfred Centauri 03:54, 28 April 2006 (UTC)

 * On last thing, replacing s with jomega, in the case of the bilateral Laplace transform, simply gives the Fourier transform of the impulse response (or, in the case of unilateral transform, don't forget the implicit multiplication by u(t)). In other words, you still end up with an impedance transfer function which, when evaluated at some value of omega, gives the ratio the phasors associated with:

 * $$Ae^{j \omega_0 t} \, $$

Alfred Centauri 04:21, 28 April 2006 (UTC)

One small point I have: Impedance depends on the input wavefunction, as is shown by its dependance on ω. Thus by saying all impedances are phasor impedances, you constrain those impedances to be only sinusoidal, and lose the generality of arbitrary input waveforms. In short, the impedance depends on s, whether or not s = jω Fresheneesz 21:54, 15 May 2006 (UTC)

part 2 of discussion

 * Fresheneesz, after all I went through above, your response is like you didn't read or understand any of it! OK then, let's try a different approach using your phrasology.  Please give me an example of an input waveform that cannot be handled by phasor impedances.  Alfred Centauri 22:23, 15 May 2006 (UTC)

Alright, how bout the unit impulse. (btw, thanks for being patient). Fresheneesz 07:53, 16 May 2006 (UTC)

 * Alright then, the unit impulse. What impedance does a capacitor present to a unit impulse of current? Alfred Centauri 13:26, 16 May 2006 (UTC)

 * Well... its s-domain impedance is 1/(sC). I don't know what else to say to that. Fresheneesz 08:40, 20 May 2006 (UTC)

 * You should at least justify you claim above that the unit impulse cannot be handled by what you call phasor impedances. Also, please justify your claim  above that the impedance depends on the input waveform.  After all, what you call the s-domain impedance in your answer above does not depend on the input waveform does it?  You could describe what is the significance of 's' in the answer you gave above.  What does it mean?  How does one use it?  Why is this 's' somehow better than jω for handling the unit impulse as an input waveform?  Alfred Centauri 14:15, 21 May 2006 (UTC)

 * Well, I wouldn't think that i'd have to explain these things to you... but... phasors require a constant frequency, which doesn't work for any arbitary wavefunction. And setting omega = to 0 after time 0 gives you an impedance of infinity, which is obviously incorrect except at t=∞. And impedances depend on the input waveform because capacitors and inductors are reactive elements. At 0 frequency (for all time), a capacitor has infinite impedance - and for infinite frequency (for all time), a capacitor has 0 impedance.
 * And the impedance for the capacitor does depend on the input, however it doesn't depend on the s-domain version of the input (which makes s-domain impedance more useful). As for the significance of s, I'm not an expert in laplace tranforms, but s is a parameter that can be related to the frequency of the input. One can substitute jω for s and find the frequency response. The s-domain is "better" than phasors for handling the unit impulse, because phasors require a constant frequency and laplace transformations don't. Fresheneesz 19:51, 21 May 2006 (UTC)

 * I wanted you to explain it to me so I could better understand what is it you don't understand. What you've 'explained' to me above is incorrect and demonstrates that you don't understand impedance or the relationship between phasors and the s-domain or even the relationship between the s-domain and the time domain.  I actually thought I had expained some of this rather well in my lengthy post above.  However, either I failed to do so or you didn't bother to try to understand what I wrote.  I'll close by individually addressing some of the things you said above:

 * "phasors require a constant frequency, which doesn't work for any arbitary wavefunction." Yes, a phasor is associated with a particular frequency but that in no way implies that phasors cannot be used for arbitrary waveforms.
 * "And setting omega = to 0 after time 0 gives you an impedance of infinity, which is obviously incorrect except at t=&infin". This sentence is not even wrong - it is meaningless.
 * "And the impedance for the capacitor does depend on the input, however it doesn't depend on the s-domain version of the input (which makes s-domain impedance more useful)." Again, not even wrong.  At best, this statement can be described as confused.
 * "As for the significance of s, I'm not an expert in laplace tranforms, but s is a parameter that can be related to the frequency of the input." How can this be?  As you've already correctly stated, 1/sC is independent of the input signal.
 * "One can substitute jω for s and find the frequency response." This is the closest you came to demonstrating some understanding of this subject.  It's not quite a correct statement but it's not far off.
 * "The s-domain is "better" than phasors for handling the unit impulse, because phasors require a constant frequency and laplace transformations don't."   Yeah, you've said this before but no matter how many times you say it, the fact that it is wrong is not going to change.

 * Now that I understand the depth of you confusion, I'll be reviewing your edits much more closely. Alfred Centauri 21:13, 21 May 2006 (UTC)

 * Hmm. Instead of saying my points are wrong, most of those points say that i'm "close" or "not far off". As I've said before, i'm not an expert. Not to give you fodder for more ridicule, but in fact, I'm a student. And I know for 100% certainty that s-domain impedances are *taught as impedances*.

 * Anyways, tell me how phasors can be used for arbitrary wavefunctions, if they require a wavefunction with a constant frequency. Surely there are many many waveforms that do not have a constant frequency.

 * "This sentence is not even wrong - it is meaningless" - What I meant is that, using phasor impedances, a DC input voltage sees a capacitor as an open circuit - infinite resistance/impedance. Just because you don't understand what I write, doesn't mean its meaningless.

 * "As you've already correctly stated, 1/sC is independent of the input signal" - 1/sC is independant of the s-domain input signal. However since s depends on the time-domain input, 1/sC depends on the time-domain input. What I mean by "the impedance varies with the input" is that the resistance shown by an impedance element depends on the input signal (and of course the element itself).

 * You say my thoughts are wrong, yet you give little to no justification for those statements. Could I respectfully ask that you cut to the chase? Fresheneesz 00:12, 22 May 2006 (UTC)

 * That the s-domain transfer impedance for a capacitor or inductor is taught as an impedance I do not deny. Often, we in the EE world use technical words imprecisely but this often leads to the kind of confusion you have been led to.  But tell me, did you honestly attempt to digest the material I posted earlier?  I ask this because you have requested that I 'cut to the chase' but I wonder if there is any reason to try since you don't appear to have absorbed any of the earlier information I posted.  To work out what I perceive to be your misconceptions will require that you think carefully about what I say because much of this is subtle.

 * You ask me to tell you how phasors can be used with arbitrary input functions. Yet, I have already explained that in the article section 'General sources'.  I'm sure you are aware that virtually all the signals of interest in EE have a Fourier series or Fourier integral representation.  What comes out of the analysis integral at a particular frequency is a constant complex number.  What is this number?  It is none other than the phasor representation of that frequency component contained in the signal.  In other words, the Fourier transform of a signal is precisely the set of phasors associated with the spectral components of the signal.  Do you see?  If we take the Fourier transform of the unit impulse current signal and multiply that by the impedance transfer function for the capacitor (1/jωC), we have multiplied each phasor component of the input signal by the impedance of the capacitor at the frequency of that component.  To get the time domain voltage response, we take the inverse Fourier transform, right?  But consider what this inverse transform is.  We take the phasor voltage at a particular frequency and multiply it by a sinusoidal time function at that frequency and we do that for every frequency and then 'add' all of them up at a particular value of time.  Do you get the picture now?
 * What is the relationship between the Laplace transform and the Fourier transform? To answer that, let's first assume that our time-domain function is equal to zero for t<0.  Then, it is easy to see that if F(jω) is the Fourier transform of f(t), then F(s) [where s = σ + jω] is the Fourier transform of f(t)exp(σt).  Clearly, for the Laplace transform to exist, we must have σ < 0 - a decaying exponential.  Do you see the connection?  What is the implication of this?
 * BTW, your sentence about infinite impedance at DC is meaningless and it's not because I don't understand what you were 'trying' to say. It is because you have put together phrases that don't make any sense in the context of impedance.  Also, connect a DC current source to a capacitor and tell me if it really appears to be an open circuit.
 * You said "1/sC is independant of the s-domain input signal. However since s depends on the time-domain input, 1/sC depends on the time-domain input." On the contrary, s does not depend on the time-domain input signal.  Maybe you are trying to say something else here but I can assure you that s does not depend on the time-domain input signal.
 * I would never ridicule you for any reason other than a stubborn refusal to try to understand what I have explained here and in my earlier lengthy post. BTW, I am a student also so you can't use that excuse with me ;<)  Alfred Centauri 01:52, 22 May 2006 (UTC)

 * I did honestly try to understand what you wrote above. But lets recap what exactly we're arguing:

 * Basically, the bottom line is that you're saying 1/jωC is an impedance, but 1/sC is not. Not only that, s doesn't depend on the input signal. I know for a fact that that second assertation is not true. For example, for a sinusoidal input (where phasors apply as well), s = jω. ω is derived from the input waveform, therefore s depends on the input waveform.

 * I do see the connection between the fourier transform and the laplace tranform. And I realize that all real inputs are a superposition of sinusoidals. So ok, I was wrong about phasors not working for general waves. I'll concede there. Fresheneesz 18:29, 26 May 2006 (UTC)

 * "Basically, the bottom line is that you're saying 1/jωC is an impedance, but 1/sC is not.". Sorry, but no, that is not what I'm saying.  Here's a question for you.  What is the difference in the meaning of the variable s used in the following two equations?

 * $$v_C(t)/i_C(t) = \frac{1}{sC}$$
 * $$V_c(s)/I_c(s) = \frac{1}{sC}$$

 * I believe that if you know and understand the answer to that question, you should understand what I'm trying to say regarding impedance.

 * "Not only that, s doesn't depend on the input signal. I know for a fact that that second assertation is not true." s does not depend on the input signal.  s is the variable of the kernel of the Laplace integral transform.  It is not possible for that variable to depend on the function being transformed.  The only thing I can think of that you may be trying to say here is that the function of s that comes out of the Laplace transform depends on the time-domain signal.  If this is what you mean, then you have made a correct but trivial statement.  After all, in the theory of Laplace transforms, there is a one-to-one correspondence between the frequency and time-domain representations of a signal.  However, if this is not what your are trying to say then please give me an example where the Laplace variable s depends on the time-domain signal being Laplace transformed.  Alfred Centauri 22:15, 26 May 2006 (UTC)

 * To answer your first question, the second equation is a regular laplace imedance. I'm no too sure about the actual meaning of s in the s domain - my first thought was that it is some sort of constant that depends on the input. For the first equation to be true of a capacitor, s must be a function of time.

 * "give me an example where the Laplace variable s depends on the time-domain signal" - I already gave you an example, and you ignored it. I wrote:
 * "For example, for a sinusoidal input (where phasors apply as well), s = jω. ω is derived from the input waveform, therefore s depends on the input waveform." Fresheneesz 01:29, 28 May 2006 (UTC)

 * That you don't know the actual meaning of s in the s domain and that you believe that s must a function of time in the first equation explains helps me understand why you believe that a sinusoidal signal demonstrates that s depends on the input waveform.

part 3

 * The simple fact is that s is a complex parameter that is not a function of time nor does it depend on the input function. You will hopefully see all this when you understand what the actual meaning of s is.  Good luck in your studies.  Alfred Centauri 02:23, 28 May 2006 (UTC)

 * Thanks for the luck, but you ignored my example for the second time. Am I wrong that s depends on ω for a sinusoidal input? Fresheneesz 11:20, 28 May 2006 (UTC)

 * Here's what I asked for: "...please give me an example where the Laplace variable s depends on the time-domain signal being Laplace transformed."  You have confirmed twice that the example you wish to give is a 'sinusoidal input'.  Alright then, let the time domain capacitor current be sinusoidal:

 * $$i_C(t) = A \cos{\omega t} + B \sin{\omega t}\,$$

 * where t is an independent real variable and ω is a given real parameter

 * The Laplace transform of this sinusoidal input current is then given by:

 * $$I_c(s) = A \frac{s}{s^2 + \omega ^2} + B \frac{\omega}{s^2 + \omega ^2}$$

 * where s is an independent complex variable.

 * Now, will you please explain to me why you insist that the independent complex variable s in the later equation above depends in any way on the former equation or for that matter, on ω? Alfred Centauri 12:35, 28 May 2006 (UTC)

 * Well, I don't understand exactly why this is, but replacing s with j ω does give you regular phasors. This is correct, right? Could you explain to me why replacing s with j ω is done, because I don't understand it fully. However, my argument hinges on the fact that ω is part of the input waveform, and if s depends on ω then it also depends on the input. Fresheneesz 20:02, 28 May 2006 (UTC)

 * Recall an earlier comment where I wrote...

 * "What is the difference in the meaning of the variable s used in the following two equations?

 * $$v_C(t)/i_C(t) = \frac{1}{sC}$$
 * $$V_c(s)/I_c(s) = \frac{1}{sC}$$

 * I believe that if you know and understand the answer to that question, you should understand what I'm trying to say regarding impedance."

 * The answer is that in the first equation, s is a parameter while in the second equation, s is the independent variable. Your argument is based on not drawing a distinction between the two.  The fact that the same notation is used here isn't helping you see the distinction.  That's why I usually distinguish between the two by using a subscript 0 for the parameter so my previous equations would now look like:

 * $$v_C(t)/i_C(t) = \frac{1}{s_0C}$$
 * $$V_c(s)/I_c(s) = \frac{1}{sC}$$

 * and

 * $$i_C(t) = A \cos{\omega_0 t} + B \sin{\omega_0 t}\,$$

 * where t is an independent real variable and $$\omega_0$$ is a given real parameter

 * The Laplace transform of this sinusoidal input current is then given by:

 * $$I_c(s) = A \frac{s}{s^2 + \omega_0 ^2} + B \frac{\omega_0}{s^2 + \omega_0 ^2}$$

 * where s is an independent complex variable.

 * Now, if you do the replacement s = jω, you get the following:

 * $$I_c(j \omega) = A \frac{j \omega}{- \omega^2 + \omega_0 ^2} + B \frac{\omega_0}{- \omega^2 + \omega_0 ^2}$$

 * See the difference? In the last equation, ω is the independent variable while $$\omega_0$$ is a number - a parameter.  The Bode plot of this current would have log(ω) on the horizontal axis while $$\omega_0$$ would appear as a point on this axis.  Get the picture?  Alfred Centauri 22:03, 28 May 2006 (UTC)

 * Update: to address your question of why replace s with $$j \omega$$:  recall that s is a complex variable with a real part and an imaginary part.  This is often written mathematically as:

 * $$s = \sigma + j \omega\,$$

 * So, σ is the real part of s and ω is the imaginary part of s. If we choose to set σ to zero, the (bi-lateral) Laplace transform is identical to the Fourier transform.  However, in the case of $$s_0$$ (a parameter in a time domain equation), if we set $$\sigma_0$$ equal to zero, we get the familiar 'phasor' form, e.g.,

 * $$\frac{1}{s_0C} = \frac{1}{j\omega_0C} when \sigma_0 = 0$$

 * Alfred Centauri 01:26, 29 May 2006 (UTC)

 * Ok I sorta think I see what you've been saying. Does part of it involve that the input waveform depends on ω and so does s, but s doesn't depend on the input?

 * Also, to bring this conversation back to its original point, I think that there should be at least a small note on the name "laplace impedances" and what that means as opposed to the impedances on this page. What do you think? Fresheneesz 19:45, 29 May 2006 (UTC)

 * Let me try again. You are confusing the ω that is the imaginary part of s with the frequency parameter of the time domain signal.  These are not the same variables!  Maybe this will help you to 'see' this:

 * $$v(t) = V_p cos(\alpha t)\,$$
 * $$V(s) = V_p \frac{s}{s^2 + \alpha^2}$$
 * $$V(j\omega) =V(s)|_{\sigma = 0}= V_p \frac{j\omega}{-\omega^2 + \alpha^2}$$

 * s and ω do not depend on the input v(t) in any way whatsoever. You can specify α to be any real number and I can specify σ & ω to be any real numbers without regard to your choice of α.  The only thing that depends on both of our selections is the value of V(s) for a given α, σ, & ω.  Alfred Centauri 13:11, 30 May 2006 (UTC)

Sorry guys to interrupt. I've finished UK A-Levels (lol) including Physics, and have heard of impedance before. I believe it has something to do with resistance, and from what I've heard/read so far, it has something to do with resistance in circuits in which alternating current passes. However, this page is very confusing. I checked Wikibooks and found nothing on this subject. Could someone possibly describe qualitatively what impedance is, and whether I'm right in saying how it is related to resistance (above)? I am interested in what it is.

In addition, I think I found a website a while ago which explained impedance simply. However, (typical of me) I forgot what it said and forgot what the address is. If I find it, can I put a link to it in the External Links section of the main article? Thanks in advance. --FSHero 10:10, 31 July 2006 (UTC)

Thankyou, thankyou! I too turned to this page to re-inform myself of my understanding of the term (my EE courses were 35 years ago). This article is apparently addressed to people who can argue over it, not to those seeking understanding. --Kcrossle 15:23, 31 August 2006 (UTC)

lol! Unfortunate... FSHero 14:55, 18 September 2006 (UTC)

Needs a phasor Diagram
This article needs a phasor diagram, to show geometrically what the impendance is. RZ heretic 00:55, 1 November 2006 (UTC)

Shockingly bad
With all respects to the editors who are certainly trying hard, there is much in this article which is so poorly stated and imprecise that it is misleading. (By the way, this seems to be true not just of this article but also of others in the EE domain). I would not send a student to this site to use it as a reference because of the errors. Every undergraduate electrical engineering textbook treats this subject with better explanations and without the errors. This article is full of redundancies and contradictions, its too hard to go through this line by line to correct it all. Please, editors, look at an EE circuits textbook for the appropriate definitions, try to not dumb down the concepts so casually and *please* leave out the bizarre generalizations.

 * I do agree that this article badly needs rewriting, which I plan to do shortly. Moreover, the last edits are really bad. There are not such things as "full", "effective" or "reactive" impedances. There is just "impedance" which is a complex number with a real part and an imaginary part. LPFR 13:26, 5 November 2006 (UTC)

 * As a casual reader, I also groaned I saw those silly phrases. My suggestion to the editor LPFR above is to simply blow it all away and start over, if they will let you.

 * You clearly are ignorant of how Wikipedia works. First, if you see something that is wrong then fix it - 'armchair' editors are not welcome here.  Second, blowing an article away is simply bad form here and will almost certainly be reverted.  Third, and most importantly, are you really so naive as to think that a complete rewrite of this article will fix it?  Don't you understand that every Tom, Dick, and Harry that has just skimmed a copy of some generic "Introduction to Electronics" book will edit the rewritten article until it looks pretty much the same as it does now?  Are you prepared to monitor the "Electrical Impedance" article from now on and revert any edits that don't meet your standards?  If so, wouldn't that make you part of the 'they' you derisively referred to?  Alfred Centauri 22:44, 6 November 2006 (UTC)

 * I'm afraid that User:Alfred Centauri is right. I stop any further collaboration with WP. It is worthless to write anything if any low-level knowledge and high-ego person will "correct" your writings. I think that WP is doomed to represent the "mean-street-knowledge" level. I was puzzled reading the scientific level of serious wikipedians (as project members). You seldom or never found University teachers or retired academic members. The best you found are engineers or thesis students. Of course, you do not need to be old or have academic titles to know things, but it helps. I now understand why there are so few university teachers and experienced people. Last edits in this article washed out my last wishes to be useful. -- LPFR 09:27, 17 November 2006 (UTC)

Removed sketch
I removed this sketch, under the heading "Four Quadrant Operation":

--Heron 20:56, 8 February 2007 (UTC)

Possible changes
This article contains a lot of useful information, however I feel that anyone who is not already reasonably familiar with the concept of impedance is likely to leave non the wiser.

I am a grad student in experimental physics and recently wrote a short paper on impedance for the benefit of undergrad project students in our lab and thought it might be nice to put a version of it online.

I have created an initial outline that you can find here.

I realise that a lot of people have put a lot of work into this article over the years and I don't want to step on peoples toes; my main concern is to make clear from the beginning how impedance differs from resistance, particularly the inclusion of the voltage-current phase relationship.

I would appreciate comments from any and all interested parties on my talk page.

--DJIndica 23:51, 26 May 2007 (UTC)

 * See my comments above in the section titled "Shockingly Bad". To summarize, feel free to make wholesale changes if you like; be bold.  I did the same thing a couple of years ago and it is time for this article to be 'reset' again.  Just keep in mind that this article is subject to lots of well-meaning but ill-informed edits.  Alfred Centauri 01:54, 27 May 2007 (UTC)

 * The edit on DJIndica's page looks like a pretty big improvement in understandability. I would guess quite a bit of information is left out of that subpage that's in the current one, but the new edit looks good enough to replace the current one, as long as the left out information is considered. Fresheneesz 19:23, 27 May 2007 (UTC)

I have made the changes to the article, I have removed quite a lot; I hope people will consider carefully before reinstating sections. Where I have removed things it is because I do not think it helps explain the concept of impedance, I have generally included links to more in depth articles on the removed topics.

Comments welcome!--DJIndica 03:41, 3 June 2007 (UTC)

Resistance and Ohm's Law
Ohm's law could be viewed as the definition of a fixed resistor. In that case, impedance "generalizes" Ohm's law. If Ohm's law is a behavioral relationship then it says things like v(t)/i(t) = R for any ideal voltage or current source, not just sinusoidal AC. Because it is called a "law," I have always thought Ohm's law referred to a behavioural relationship, otherwise it would be called Ohm's definition. But maybe I am confused. gaussmarkov 08:54, 26 June 2007 (PDT)

 * IMHO, Ohm's law is a law in the context of physics. However, in the context of (ideal) circuit theory, a resistor circuit element is defined as that circuit element where v(t) = R i(t).  That is, Ohm's law defines the ideal resistor circuit element in the context of circuit theory.  Alfred Centauri 16:46, 26 June 2007 (UTC)

 * That makes sense. Thank you for responding.  Perhaps the article should have some sort of statement that it is about ideal circuit theory?  I hope it's alright if I press my question a little: Ohm's law as a definition of the ideal fixed resistor places no restrictions on v(t) or i(t) other than their proportionality.  They do not have to be proportional sinusoidal AC.  As I understand this article, it says that the definition of impedance requires sinusoidal AC.  How can impedance be a generalization if it makes an additional restriction?  All the best, gaussmarkov 19:58, 26 June 2007 (UTC)

 * Yes, I see your point and it is a good one. Consider the following:  if the source in an ideal circuit has the form of a complex exponential and the circuit is in steady state, the voltage and current relationship for any resistor, inductor, or capacitor in the circuit has the form of Ohm's law:  v(t) = Z i(t).  In such a circuit, the inductor and capacitor have an imaginary valued Z while the resistor has a real Z.  This is the generalization referred to in the article - the application of an Ohm's law like relationship to inductors and capacitors when certain conditions hold.  What do you suggest for a different word or phrase to use here that would improve the accuracy of the article?  Alfred Centauri 23:41, 26 June 2007 (UTC)

 * Great. I tried to be bold and took your question as an invitation to write the wording.  I also offered a slight re-organization of the introductory paragraph.  I am hopeful, of course, that this is helpful. gaussmarkov 06:04, 27 June 2007 (UTC)

 * I'm not convinced that impedance is analogous to resistance.  The concept of impedance is clearly some type of extension of the concept of resistance since resistance is a component (the real part) of impedance.  Please reconsider your edit.  Alfred Centauri 21:56, 27 June 2007 (UTC)

 * For lack of any objection, I've changed 'analagous' to 'an extension of'. Alfred Centauri 02:15, 30 June 2007 (UTC)

 * I like the use of extends to rather than analog of, I have used the same language in the Ohm's law section. I'm not happy with "Often, impedance is represented formally as a complex quantity"; for the general case impedance is a complex quantity, the article makes this clear by discussing its real and imaginary parts (and magnitude and phase).  This was changed very recently by gaussmarkov, I would like to change it back but would be interested to hear any opposing arguments.--DJIndica 15:14, 1 July 2007 (UTC)

 * The change to extends to is fine with me. I think it is a good choice.

 * I do not think that impedance is a complex quantity. I think it is merely a two-dimensional concept. Commonly, we refer to its magnitude and its phase.  These are not inherently related to complex representations, though that is formally (or algebraically) convenient.  Alternatively, we refer to its resistive and reactive dimensions.  Again, there is no necessary connection with the complex plane.  To make this point another way, all of the derivations can be carried out without complex algebra.

 * I see that there is also some confusion about these matters remaining in the article. In section titled Complex voltage and current, someone states "we must bear in mind that these are real quantities and take only the real part at the end of a calculation." Actually, we typically want the magnitude and the phase, neither of which is the real part. gaussmarkov 23:01, 5 September 2007 (UTC)

 * "I do not think that impedance is a complex quantity". Impedance is a complex quantity by definition as it is the ratio of the voltage and current phasors regardless of your argument to the contrary.  All that matters here are authoritative references of which there are plenty that dispute your assertion.  BTW, taking the real part of the complex exponential form of the voltages and currents does give the magnitude and phase as part of the  real time dependent function which is, in the end, precisely the form of the actual voltages and currents.  Perhaps you were thinking that the section implied taking the real part of the phasors instead?  Alfred Centauri 00:48, 6 September 2007 (UTC)

 * Hmm. We may not be able to get the authorities to agree on the definition of impedance.  What do you consider to be an authority?  I have not seen any citations to authorities here.  The references at the end of the article surely don't qualify.  I will try to find time to go to the library tomorrow.

 * Perhaps it would be helpful to distinguish between impedance and complex impedance, as many authors do? You would then be referring to the latter.  Impedance without modifiers is variously defined, sometimes as complex, sometimes as the ratio of the amplitudes of sinusoidal voltage and current, and sometimes as the ratio of appropriately phase-shifted voltage and current.  In the last two cases, impedance is only the magnitude of complex impedance.  But I think that I have also seen it defined as a vector (not specifically complex) so that phase is included.  There is plenty of physics course material online that bears out what I am saying.  See for example, http://www.physclips.unsw.edu.au/jw/AC.html#impedance.  Or the external link at the end of the article, which refers to the complex impedance method.

 * If impedance is indeed a complex quantity, then we will have to change its units. The ohm is not a complex unit of measure.  Stating, as the article does, that the SI unit of impedance is the ohm suggests that impedance refers only to the magnitude as just described.

 * On the second point, my answer to your question is yes, I was thinking of the phasors because that is what the text is discussing. Let's take the text apart.  The complete thought is "In order to simplify calculations, the voltage and current are commonly represented as complex quantities denoted as $\scriptstyle{\tilde{V}}$ and $\scriptstyle{\tilde{I}}$; we must bear in mind that these are real quantities and take only the real part at the end of a calculation.|undefined"  I agree with the first sentence, because it says voltage and current are represented as complex quantities.  This is similar to what I wrote previously about impedance.  Removing represented would be, in my view, a mistake.

 * Now, as I read it, the antecedent for the word these in the second sentence is complex quantities. I expect most readers will read the real part (mentioned next) of a complex quantity as its real component.  That real part, as I said before, is neither magnitude nor phase, though these are both real as well.  All the best, gaussmarkov 07:21, 6 September 2007 (UTC)

 * "The proportionality constants between the complex amplitudes of the voltage and the current in an inductor or capacitor play a similar role to that of the resistance in a resistor. These quantities are called the impedances (of which resistance is a subclass)" (Su, Kendall, 1978, Fundamentals of Circuits, Electronics, and Signal Analysis).

 * Since this article is about electrical impedance, it follows that EE textbooks (such as the one I cite above) and EE reference books are authoritative references here and not technical level electronics and physics textbooks.

 * As to your second point, please note that the section in question gives the voltage and current as complex valued functions of time. These are not phasors!  Phasors are discussed after this section you give above.  Don't forget, the use of phasors to solve AC circuits is, as I have stated before on this talk page, in fact justified by representing (pretending) that the voltages and currents in the circuit are complex valued functions of time, solving, and then taking the real part at the end.  Then, we find that we only need to keep up with the complex amplitude (the phasors) of the complex valued time functions while solving the circuit.  But, at the end, if we want the actual time varying functions back, we must first multiply the phasor solution by $$e^{j\omega_0 t}$$ (giving us back the complex valued time function) and then take the real part of that to get the actual real valued time dependent voltage or current.

 * Based on your question about this, I have changed "complex quantity" to "complex valued functions of time" Alfred Centauri 14:51, 6 September 2007 (UTC)

 * BTW, in this textbook which I have taught from, the terms complex impedance and impedance are used interchangeably. That is, impedance is not used to mean the magnitude of the complex impedance in this textbook.  —Preceding unsigned comment added by Alfred Centauri (talk • contribs) 15:15, 6 September 2007 (UTC)

 * Indeed, I confused phasors with voltage and current as functions of time. Thank you for straightening me out.

 * I spent some time in the Engineering Library this morning and found some books that described impedance in noncomplex ways. See for example, Jones, Laughton, & Say, Electrical Engineer's Reference Book.  As I read him, Hambly (which you mention) clearly starts with complex impedance and then shortens it to impedance.  I was surprised to read that you view EE textbooks as the only authorities.  At the bottom of this page this article is classified under Physics articles with comments | Unassessed physics articles.  Perhaps we could apply the WP:NPOV and broaden the range of authorities?  It did not occur to me that electrical implied EE and I am confident that a lot of readers are not coming from or looking for an EE treatment.

 * I made a few edits that I hope are at least constructive, if not acceptable. Cheers, gaussmarkov 22:21, 6 June 2007 (UTC)

 * BTW, I meant to add that the majority of EE textbooks that I examined certainly talk exclusively about impedance as complex. I want to acknowledge this.  In my rewording of the introductory paragraph, I replace the phrase "in general" with "often" to reflect this.  Perhaps I should have used "usually" -- this seems reasonable also if others prefer it. All the best, gaussmarkov 22:32, 6 June 2007 (UTC)

 * "Perhaps we could apply the WP:NPOV and broaden the range of authorities?" I don't follow you.  If, for example, someone adds something here from a reference of the level of, e.g., "Electronics for Dummies" which clearly contradicts or distorts (for the sake of simplicity) the information in one or more widely cited EE textbooks or reference books, removing that material is not even remotely a POV issue.  FWIW, I encourage and have in fact advocated for technical level treatment, e.g., non-complex treatment of impedance in electrical/electronics articles as long as it is not misleading or outright incorrect.  Also, keep in mind that many EE students do in fact refer to Wikipedia so, IMHO, it is essential that EE level material is included in these articles.  Cheers, Alfred Centauri 22:29, 6 September 2007 (UTC)

 * I had in mind your exclusion: "EE reference books are authoritative references here and not technical level electronics and physics textbooks." I think it is reasonable to include the electronics and physics textbooks as authoritative in an article on electrical impedance.  -- gaussmarkov 05:44, 7 June 2007 (UTC)

 * "it is reasonable to include the electronics and physics textbooks as authoritative in an article on electrical impedance". Of course it's reasonable and I don't think that I said that they should be excluded.  I think you've misintepreted what I meant.  There is a hiearchy of authority when it comes to reference materials, right?  So, if an editor puts in material that is questionable and supports it with a reference whose intended audience is the layperson, technician, or non-EE and then another editor removes or corrects the material and supports the edit with a widely used EE reference, which reference is authoritative in this case?  —Preceding unsigned comment added by Alfred Centauri (talk • contribs) 12:54, 7 September 2007 (UTC)

Sorry to break into this discussion like this, but I thought I would take it upon myself to improve the section in question by expanding upon the idea of "taking the real part" with a more-detailed description of why this works. Please pare it down if I've gone overboard here! Oli Filth 15:14, 8 September 2007 (UTC)

Etymology section
While I am sympathetic with the point made in the edit summary associated with the addition of this section ("who came up with the term doesn't need to be in the first sentence"); the creation of a full section seems like overkill unless there are other points of interest in the etymology of the word impedance. A mention of Oliver Heavyside does not need to be in the first sentence, but given his importance in the development of this and related concepts, I feel he is worthy of a mention and the opening section is probably the best place for it. I will revert shortly unless there are good arguments made for keeping it, or suggestions for additional etymological content.--DJIndica 21:56, 12 September 2007 (UTC)

 * Yes, a new section is probably overkill, but can you try something different like making it the second sentence. No need to state everything in one sentence. —Pengo 02:42, 13 September 2007 (UTC)

== WikiProject Electronics quality and importance assessment ==

It has been noted by a number of contributors that this page can vary significantly in quality over time due to the large number of edits, some of which are ill informed and detract from the information content and/or readability, and can add false or misleading information. It seems to me that it would be a good idea to rate the quality and importance of the article using the WikiProject Electronics scales.

My thinking is that this will in the first instance hopefully put it on the radar of more people with genuine knowledge and interest in this area. In the long run, if the article reaches the point where it is assessed as a good article then it may encourage more thoughtful edits in the future. I have added the banner at the top of this page and taken the liberty of adding a 'high importance' tag; I don't think it is controversial to state that impedance is an important topic in electronics.

It would be good to see comments here, suggesting people's thoughts on the current quality of the article along with suggestions for improvements, perhaps based on good and featured articles on related topics.--DJIndica 23:02, 19 October 2007 (UTC)

Electromagnetism template
This article is one of only a couple of those listed in the electromagnetism template that does not include the template. There are definitely arguments against including it (not least that it is rather long and would require moving the images around), but I think it is probably worth having. I have tried playing around with the positioning of the first image in combination with the template; an example of one possibe arrangement can be found here. --DJIndica 16:57, 22 October 2007 (UTC)

Merger proposal
It is proposed to merge Impedance of different devices (derivations) into this article. The article says little or nothing that is not already said here and said better and in places is just plain wrong. Besides which it is practically orphaned.  Sp in ni ng  Spark  18:42, 14 June 2008 (UTC)

 * I have created a version of this article including the derivations at User:DJIndica/Sandbox2. I'd appreciate any comments.--DJIndica (talk) 12:29, 18 July 2008 (UTC)

Question
Vp stands for peak voltage, right? I'm referring to the part about capacitors in the Device Examples section of the article. If not please explicitly state what Vp is, since I don't think it is something that can be easily inferred from the article. 96.60.80.133 (talk) 03:09, 13 November 2008 (UTC)

I would also like to add that I found this article very helpful. From reading some of the discussion here I understand that some of you guys are extremely intelligent and I thank you for all your contributions. 96.60.80.133 (talk) 03:38, 13 November 2008 (UTC)

 * Vp is the voltage amplitude, which is indeed the peak voltage given that there is no DC offset.--DJIndica (talk) 19:01, 13 November 2008 (UTC)

title: Talk:Electrical impedance/Archive 2


text:
Laestrygonians

In Greek mythology, the Laestrygonians or Laestrygones  (Λαιστρυγόνες) were a tribe of man-eating giants. They were said to have sprung from Laestrygon, son of Poseidon.

According to Thucydides (6.2.1.) and Polybius (1.2.9) the Laestrygones inhabited southeast Sicily. The name is akin to that of the Lestriconi, a branch of the Corsi people of the northeast coast of Sardinia (now Gallura).

Mythology
Odysseus, the main character of Homer's Odyssey, visited them during his journey back home to Ithaca. The giants ate many of Odysseus's men and destroyed eleven of his twelve ships by launching rocks from high cliffs. Odysseus's ship was not destroyed because it was hidden in a cove near shore. Everyone on Odysseus's ship survived the incident. "His soldiers, with a dozen ships, arrive at 'the rocky stronghold of Lamos: Telepylus, the city of the Laestrygonians'."

Lamos is not mentioned again, perhaps being understood as the founder of the city or the name of the island on which the city is situated. In this land, a man who could do without sleep could earn double wages; once as a herdsman of cattle and another as a shepherd, as they worked by night as they did by day. The ships entered a harbor surrounded by steep cliffs, with a single entrance between two headlands. The captains took their ships inside and made them fast close to one another, where it was dead calm.

Odysseus kept his own ship outside the harbor, moored to a rock. He climbed a high rock to reconnoiter, but could see nothing but some smoke rising from the ground. He sent two of his company and an attendant to investigate the inhabitants. The men followed a road and eventually met a young woman on her way to the Fountain of Artakia to fetch some water, who said she was a daughter of Antiphates or Antiphatus, the king, and directed them to his house. However, when they got there they found a gigantic woman, the wife of Antiphates who promptly called her husband, who immediately left the assembly of the people and upon arrival snatched up one of the men, killing and eating him on the spot. The other two men ran away, but Antiphates raised an outcry, so that they were pursued by thousands of Laestrygonians, who are either giants or very large men and women. They threw vast rocks from the cliffs, smashing the ships, and speared the men like fish. Odysseus made his escape with his single ship because it was not trapped in the harbor; the rest of his company was lost. The surviving crew went next to Aeaea, the island of Circe.

According to historian Angelo Paratico the Laestrygonians were the result of a legend originated by the sight by Greek sailors of the Giants of Mont'e Prama, recently excavated in Sardinia.

Later Greeks believed that the Laestrygonians, as well as the Cyclopes, had once inhabited Sicily.

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

title: Laestrygonians


text:
Toga

The toga, a distinctive garment of ancient Rome, was a roughly semicircular cloth, between 12 and 20 ft in length, draped over the shoulders and around the body. It was usually woven from white wool, and was worn over a tunic. In Roman historical tradition, it is said to have been the favored dress of Romulus, Rome's founder; it was also thought to have originally been worn by both sexes, and by the citizen-military. As Roman women gradually adopted the stola, the toga was recognized as formal wear for male Roman citizens. Women found guilty of adultery and women engaged in prostitution might have provided the main exceptions to this rule.

The type of toga worn reflected a citizen's rank in the civil hierarchy. Various laws and customs restricted its use to citizens, who were required to wear it for public festivals and civic duties.

From its probable beginnings as a simple, practical work-garment, the toga became more voluminous, complex, and costly, increasingly unsuited to anything but formal and ceremonial use. It was and is considered ancient Rome's "national costume"; as such, it had great symbolic value; however even among Romans, it was hard to put on, uncomfortable and challenging to wear correctly, and never truly popular. When circumstances allowed, those otherwise entitled or obliged to wear it opted for more comfortable, casual garments. It gradually fell out of use, firstly among citizens of the lower class, then those of the middle class. Eventually, it was worn only by the highest classes for ceremonial occasions.

Varieties
The toga was an approximately semi-circular woollen cloth, usually white, worn draped over the left shoulder and around the body: the word "toga" probably derives from tegere, to cover. It was considered formal wear and was generally reserved for citizens. The Romans considered it unique to themselves, thus their poetic description by Virgil and Martial as the gens togata ('toga-wearing race'). There were many kinds of toga, each reserved by custom to a particular usage or social class.
 * Toga virilis ("toga of manhood") also known as toga alba or toga pura: A plain white toga, worn on formal occasions by adult male commoners, and by senators not having a curule magistracy. It represented adult male citizenship and its attendant rights, freedoms and responsibilities; traditionally given at a father's discretion to his son during the feast of Liberalia, to mark the onset of puberty and legal "coming of age", at around 14 years of age or more.
 * Toga praetexta: a white toga with a broad purple stripe on its border, worn over a tunic with two broad, vertical purple stripes. It was formal costume for:
 * Curule magistrates in their official functions, and traditionally, the Kings of Rome.
 * Freeborn boys, and some freeborn girls, before they came of age. It marked their protection by law from sexual predation and immoral or immodest influence. A praetexta was thought effective against malignant magic, as were a boy's bulla, and a girl's lunula.
 * Some priesthoods, including the Pontifices, Tresviri Epulones, the augurs, and the Arval brothers.
 * Toga candida: "Bright toga"; a toga rubbed with chalk to a dazzling white, worn by candidates (from Latin candida, "pure white") for public office. Thus Persius speaks of a cretata ambitio, "chalked ambition". Toga candida is the etymological source of the word candidate.
 * Toga pulla: a "dark toga" was supposed to be worn by mourners at elite funerals. A toga praetexta was also acceptable as mourning wear, if turned inside out to conceal its stripe; so was a plain toga pura. Wearing a toga pulla at the feast that ended mourning was irreligious, ignorant, or plain bad manners. Cicero makes a distinction between the toga pulla and an ordinary toga deliberately "dirtied" by its wearer as a legitimate mark of protest or supplication.
 * Toga picta ("painted toga"): Dyed solid purple, decorated with imagery in gold thread, and worn over a similarly decorated tunica palmata; used by generals in their triumphs. During the Empire, it was worn by consuls and emperors. Over time, it became increasingly elaborate, and was combined with elements of the consular trabea.
 * Trabea, associated with citizens of equestrian rank; thus their description as trabeati in some contemporary Roman literature. It may have been a shorter form of toga, or a cloak, wrap or sash worn over a toga. It was white with some form of decoration. In the later Imperial era, trabea refers to elaborate forms of consular dress. Some later Roman and post-Roman sources describe it as solid purple or red, either identifying or confusing it with the dress worn by the ancient Roman kings (also used to clothe images of the gods) or reflecting changes in the trabea itself. More certainly, equites wore an angusticlavia, a tunic with narrow, vertical purple stripes, at least one of which would have been visible when worn with a toga or trabea, whatever its form.
 * Laena, a long, heavy cloak worn by Flamen priesthoods, fastened at the shoulder with a brooch. A lost work by Suetonius describes it as a toga made "duplex" (doubled by folding over upon itself).

As "national dress"
The toga's most distinguishing feature was its semi-circular shape, which sets it apart from other cloaks of antiquity like the Greek himation or pallium. To Rothe, the rounded form suggests an origin in the very similar, semi-circular Etruscan tebenna. Norma Goldman believes that the earliest forms of all these garments would have been simple, rectangular lengths of cloth that served as both body-wrap and blanket for peasants, shepherds and itinerant herdsmen. Roman historians believed that Rome's legendary founder and first king, the erstwhile shepherd Romulus, had worn a toga as his clothing of choice; the purple-bordered toga praetexta was supposedly used by Etruscan magistrates, and introduced to Rome by her third king, Tullus Hostilius.

In the wider context of classical Greco-Roman fashion, the Greek enkyklon (ἔγκυκλον, "circular [garment]") was perhaps similar in shape to the Roman toga, but never acquired the same significance as a distinctive mark of citizenship. The 2nd-century diviner Artemidorus Daldianus in his Oneirocritica derived the toga's form and name from the Greek tebennos (τήβεννος), supposedly an Arcadian garment invented by and named after Temenus. Emilio Peruzzi claims that the toga was brought to Italy from Mycenaean Greece, its name based on Mycenaean Greek te-pa, referring to a heavy woollen garment or fabric.

In civil life
Roman society was strongly hierarchical, stratified and competitive. Landowning aristocrats occupied most seats in the senate and held the most senior magistracies. Magistrates were elected by their peers and "the people"; in Roman constitutional theory, they ruled by consent. In practice, they were a mutually competitive oligarchy, reserving the greatest power, wealth and prestige for their class. The commoners who made up the vast majority of the Roman electorate had limited influence on politics, unless barracking or voting en masse, or through representation by their tribunes. The Equites (sometimes loosely translated as "knights") occupied a broadly mobile, mid-position between the lower senatorial and upper commoner class. Despite often extreme disparities of wealth and rank between the citizen classes, the toga identified them as a singular and exclusive civic body.

Togas were relatively uniform in pattern and style but varied significantly in the quality and quantity of their fabric, and the marks of higher rank or office. The highest-status toga, the solidly purple, gold-embroidered toga picta could be worn only at particular ceremonies by the highest-ranking magistrates. Tyrian purple was supposedly reserved for the toga picta, the border of the toga praetexta, and elements of the priestly dress worn by the inviolate Vestal Virgins. It was colour-fast, extremely expensive and the "most talked-about colour in Greco-Roman antiquity". Romans categorised it as a blood-red hue, which sanctified its wearer. The purple-bordered praetexta worn by freeborn youths acknowledged their vulnerability and sanctity in law. Once a boy came of age (usually at puberty) he adopted the plain white toga virilis; this meant that he was free to set up his own household, marry, and vote. Young girls who wore the praetexta on formal occasions put it aside at menarche or marriage, and adopted the stola. Even the whiteness of the toga virilis was subject to class distinction. Senatorial versions were expensively laundered to an exceptional, snowy white; those of lower ranking citizens were a duller shade, more cheaply laundered.

Citizenship carried specific privileges, rights and responsibilities. The formula togatorum ("list of toga-wearers") listed the various military obligations that Rome's Italian allies were required to supply to Rome in times of war. Togati, "those who wear the toga", is not precisely equivalent to "Roman citizens", and may mean more broadly "Romanized". In Roman territories, the toga was explicitly forbidden to non-citizens; to foreigners, freedmen, and slaves; to Roman exiles; and to men of "infamous" career or shameful reputation; an individual's status should be discernable at a glance. A freedman or foreigner might pose as a togate citizen, or a common citizen as an equestrian; such pretenders were sometimes ferreted out in the census. Formal seating arrangements in public theatres and circuses reflected the dominance of Rome's togate elect. Senators sat at the very front, equites behind them, common citizens behind equites; and so on, through the non-togate mass of freedmen, foreigners, and slaves. Imposters were sometimes detected and evicted from the equestrian seats.

Various anecdotes reflect the toga's symbolic value. In Livy's history of Rome, the patrician hero Lucius Quinctius Cincinnatus, retired from public life and clad (presumably) in tunic or loincloth, is ploughing his field when emissaries of the Senate arrive, and ask him to put on his toga. His wife fetches it and he puts it on. Then he is told that he has been appointed dictator. He promptly heads for Rome. Donning the toga transforms Cincinnatus from rustic, sweaty ploughman – though a gentleman nevertheless, of impeccable stock and reputation – into Rome's leading politician, eager to serve his country; a top-quality Roman. Rome's abundant public and private statuary reinforced the notion that all Rome's great men wore togas, and must always have done so.

Work and leisure

Traditionalists idealised Rome's urban and rustic citizenry as descendants of a hardy, virtuous, toga-clad peasantry, but the toga's bulk and complex drapery made it entirely impractical for manual work or physically active leisure. The toga was heavy, "unwieldy, excessively hot, easily stained, and hard to launder". It was best suited to stately processions, public debate and oratory, sitting in the theatre or circus, and displaying oneself before one's peers and inferiors while "ostentatiously doing nothing".

Every male Roman citizen was entitled to wear some kind of toga – Martial refers to a lesser citizen's "small toga" and a poor man's "little toga" (both togula), but the poorest probably had to make do with a shabby, patched-up toga, if he bothered at all. Conversely, the costly, full-length toga seems to have been a rather awkward mark of distinction when worn by "the wrong sort". The poet Horace writes "of a rich ex-slave 'parading from end to end of the Sacred Way in a toga three yards long' to show off his new status and wealth."

In the early 2nd century AD, the satirist Juvenal claimed that "in a great part of Italy, no-one wears the toga, except in death"; in Martial's rural idyll there is "never a lawsuit, the toga is scarce, the mind at ease". Most citizens who owned a toga would have cherished it as a costly material object, and worn it when they must for special occasions. Family, friendships and alliances, and the gainful pursuit of wealth through business and trade would have been their major preoccupations, not the otium (cultured leisure) claimed as a right by the elite. Rank, reputation and Romanitas were paramount, even in death, so almost invariably, a male citizen's memorial image showed him clad in his toga. He wore it at his funeral, and it probably served as his shroud.

Despite the overwhelming quantity of Roman togate portraits at every social level, and in every imaginable circumstance, at most times Rome's thoroughfares would have been crowded with citizens and non-citizens in a variety of colourful garments, with few togas in evidence. Only a higher-class Roman, a magistrate, would have had lictors to clear his way, and even then, wearing a toga was a challenge. The toga's apparent natural simplicity and "elegant, flowing lines" were the result of diligent practice and cultivation; to avoid an embarrassing disarrangement of its folds, its wearer had to walk with measured, stately gait, yet with virile purpose and energy. If he moved too slowly, he might seem aimless, "sluggish of mind" - or, worst of all, "womanly". Vout (1996) suggests that the toga's most challenging qualities as garment fitted the Romans' view of themselves and their civilization. Like the empire itself, the peace that the toga came to represent had been earned through the extraordinary and unremitting collective efforts of its citizens, who could therefore claim "the time and dignity to dress in such a way".

Patronage and salutationes

Patronage was a cornerstone of Roman politics, business and social relationships. A good patron offered advancement, security, honour, wealth, government contracts and other business opportunities to his client, who might be further down in the social or economic scale, or more rarely, his equal or superior. A good client canvassed political support for his patron, or his patron's nominee; he advanced his patron's interests using his own business, family and personal connections. Freedmen with an aptitude for business could become extremely wealthy; but to negotiate citizenship for themselves, or more likely their sons, they had to find a patron prepared to commend them. Clients seeking patronage had to attend the patron's early-morning formal salutatio ("greeting session"), held in the semi-public, grand reception room (atrium) of his family house (domus). Citizen-clients were expected to wear the toga appropriate to their status, and to wear it correctly and smartly or risk affront to their host.

Martial and his friend Juvenal suffered the system as clients for years, and found the whole business demeaning. A client had to be at his patron's beck and call, to perform whatever "togate works" were required; and the patron might even expect to be addressed as "domine" (lord, or master); a citizen-client of the equestrian class, superior to all lesser mortals by virtue of rank and costume, might thus approach the shameful condition of dependent servitude. For a client whose patron was another's client, the potential for shame was still worse. Even as a satirical analogy, the equation of togate client and slave would have shocked those who cherished the toga as a symbol of personal dignity and auctoritas – a meaning underlined during the Saturnalia festival, when the toga was "very consciously put aside", in a ritualised, strictly limited inversion of the master-slave relationship.

Patrons were few, and most had to compete with their peers to attract the best, most useful clients. Clients were many, and those of least interest to the patron had to scrabble for notice among the "togate horde" (turbae togatae). One in a dirty or patched toga would likely be subject to ridicule; or he might, if sufficiently dogged and persistent, secure a pittance of cash, or perhaps a dinner. When the patron left his house to conduct his business of the day at the law courts, forum or wherever else, escorted (if a magistrate) by his togate lictors, his clients must form his retinue. Each togate client represented a potential vote: to impress his peers and inferiors, and stay ahead in the game, a patron should have as many high-quality clients as possible; or at least, he should seem to. Martial has one patron hire a herd (grex) of fake clients in togas, then pawn his ring to pay for his evening meal.

The emperor Marcus Aurelius, rather than wear the "dress to which his rank entitled him" at his own salutationes, chose to wear a plain white citizen's toga instead; an act of modesty for any patron, unlike Caligula, who wore a triumphal toga picta or any other garment he chose, according to whim; or Nero, who caused considerable offence when he received visiting senators while dressed in a tunic embroidered with flowers, topped off with a muslin neckerchief.

Oratory

In oratory, the toga came into its own. Quintilian's Institutio Oratoria (circa 95 AD) offers advice on how best to plead cases at Rome's law-courts, before the watching multitude's informed and critical eye. Effective pleading was a calculated artistic performance, but must seem utterly natural. First impressions counted; the lawyer must present himself as a Roman should: "virile and splendid" in his toga, with statuesque posture and "natural good looks". He should be well groomed – but not too well; no primping of the hair, jewellery or any other "feminine" perversions of a Roman man's proper appearance. Quintilian gives precise instructions on the correct use of the toga – its cut, style, and the arrangements of its folds. Its fabric could be old-style rough wool, or new and smoother if preferred – but definitely not silk. The orator's movements should be dignified, and to the point; he should move only as he must, to address a particular person, a particular section of the audience. He should employ to good effect that subtle "language of the hands" for which Roman oratory was famed; no extravagant gestures, no wiggling of the shoulders, no moving "like a dancer".

To a great extent, the toga itself determined the orator's style of delivery: "we should not cover the shoulder and the whole of the throat, otherwise our dress will be unduly narrowed and will lose the impressive effect produced by breadth at the chest. The left arm should only be raised so far as to form a right angle at the elbow, while the edge of the toga should fall in equal lengths on either side." If, on the other hand, the "toga falls down at the beginning of our speech, or when we have only proceeded but a little way, the failure to replace it is a sign of indifference, or sloth, or sheer ignorance of the way in which clothes should be worn". By the time he had presented his case, the orator was likely to be hot and sweaty; but even this could be employed to good effect.

In public morals
Roman moralists "placed an ideological premium on the simple and the frugal". Aulus Gellius claimed that the earliest Romans, famously tough, virile and dignified, had worn togas with no undergarment; not even a skimpy tunic. Towards the end of the Republic, the arch-conservative Cato the Younger favoured the shorter, ancient Republican type of toga; it was dark and "scanty" (exigua), and Cato wore it without tunic or shoes; all this would have been recognised as an expression of his moral probity. Die-hard Roman traditionalists deplored an ever-increasing Roman appetite for ostentation, "un-Roman" comfort and luxuries, and sartorial offences such as Celtic trousers, brightly coloured Syrian robes and cloaks. The manly toga itself could signify corruption, if worn too loosely, or worn over a long-sleeved, "effeminate" tunic, or woven too fine and thin, near transparent. Appian's history of Rome finds its strife-torn Late Republic tottering at the edge of chaos; most seem to dress as they like, not as they ought: "For now the Roman people are much mixed with foreigners, there is equal citizenship for freedmen, and slaves dress like their masters. With the exception of the Senators, free citizens and slaves wear the same costume." The Augustan Principate brought peace, and declared its intent as the restoration of true Republican order, morality and tradition.

Augustus was determined to bring back "the traditional style" (the toga). He ordered that any theatre-goer in dark (or coloured or dirty) clothing be sent to the back seats, traditionally reserved for those who had no toga; ordinary or common women, freedmen, low-class foreigners and slaves. He reserved the most honourable seats, front of house, for senators and equites; this was how it had always been, before the chaos of the civil wars; or rather, how it was supposed to have been. Infuriated by the sight of a darkly clad throng of men at a public meeting, he sarcastically quoted Virgil at them, "Romanos, rerum dominos, gentemque togatam " ("Romans, lords of the world and the toga-wearing people"), then ordered that in future, the aediles ban anyone not wearing the toga from the Forum and its environs – Rome's "civic heart". Augustus's reign saw the introduction of the toga rasa, an ordinary toga whose rough fibres were teased from the woven nap, then shaved back to a smoother, more comfortable finish. By Pliny's day (circa 70 AD) this was probably standard among the elite. Pliny also describes a glossy, smooth, lightweight but dense fabric woven from poppy-stem fibres and flax, in use from at least the time of the Punic Wars. Though probably appropriate for a "summer toga", it was criticised for its improper luxuriance.

Women
Some Romans believed that in earlier times, both genders and all classes had worn the toga. Radicke (2002) claims that this belief goes back to a Late Antique scholiast misreading of earlier Roman writings. Women could also be citizens, but by the mid-to-late Republican era, respectable women were stolatae (stola-wearing), expected to embody and display an appropriate set of female virtues: Vout cites pudicitia and fides as examples. Women's adoption of the stola may have paralleled the increasing identification of the toga with citizen men, but this seems to have been a far from straightforward process. An equestrian statue, described by Pliny the Elder as "ancient", showed the early Republican heroine Cloelia on horseback, wearing a toga. The unmarried daughters of respectable, reasonably well-off citizens sometimes wore the toga praetexta until puberty or marriage, when they adopted the stola, which they wore over a full-length, usually long-sleeved tunic.

Higher-class female prostitutes (meretrices) and women divorced for adultery were denied the stola. Meretrices might have been expected or perhaps compelled, at least in public, to wear the "female toga" (toga muliebris). This use of the toga appears unique; all others categorised as "infamous and disreputable" were explicitly forbidden to wear it. In this context, modern sources understand the toga – or perhaps merely the description of particular women as togata – as an instrument of inversion and realignment; a respectable (thus stola-clad) woman should be demure, sexually passive, modest and obedient, morally impeccable. The archetypical meretrix of Roman literature dresses gaudily and provocatively. Edwards (1997) describes her as "antithetical to the Roman male citizen". An adulterous matron betrayed her family and reputation; and if found guilty, and divorced, the law forbade her remarriage to a Roman citizen. In the public gaze, she was aligned with the meretrix. When worn by a woman in this later era, the toga would have been a "blatant display" of her "exclusion from the respectable Roman hierarchy". However, the view that a convicted adulteress (moecha damnata) actually wore a toga in public has been challenged; Radicke believes that the only prostitutes who could be made to wear particular items of clothing were unfree, compelled by their owners or pimps to wear the relatively shorter, "skimpy", less costly toga exigua, more revealing, easily opened and thus convenient to their profession.

Roman military

Until the so-called "Marian reforms" of the Late Republic, the lower ranks of Rome's military forces were "farmer-soldiers", a militia of citizen smallholders conscripted for the duration of hostilities, expected to provide their own arms and armour. Citizens of higher status served in senior military posts as a foundation for their progress to high civil office (see cursus honorum). The Romans believed that in Rome's earliest days, its military had gone to war in togas, hitching them up and back for action by using what became known as the "Gabine cinch". In 206 BC, Scipio Africanus was sent 1,200 togas and 12,000 tunics for his operations in North Africa. As part of a peace settlement of 205 BC, two formerly rebellious Spanish tribes provided Roman troops with togas and heavy cloaks. In the Macedonian campaign of 169 BC, the army was sent 6,000 togas and 30,000 tunics. From at least the mid-Republic on, the military reserved their togas for formal leisure and religious festivals; the tunic and sagum (heavy rectangular cloak held on the shoulder with a brooch) were used or preferred for active duty.

Late republican practice and legal reform allowed the creation of standing armies, and opened a military career to any Roman citizen or freedman of good reputation. A soldier who showed the requisite "disciplined ferocity" in battle and was held in esteem by his peers and superiors could be promoted to higher rank: a plebeian could achieve equestrian status. Non-citizens and foreign-born auxiliaries given honourable discharge were usually granted citizenship, land or stipend, the right to wear the toga, and an obligation to the patron who had granted these honours; usually their senior officer. A dishonourable discharge meant infamia. Colonies of retired veterans were scattered throughout the Empire. In literary stereotype, civilians are routinely bullied by burly soldiers, inclined to throw their weight around.

Though soldiers were citizens, Cicero typifies the former as "sagum wearing" and the latter as "togati". He employs the phrase cedant arma togae ("let arms yield to the toga"), meaning "may peace replace war", or "may military power yield to civilian power", in the context of his own uneasy alliance with Pompey. He intended it as metonym, linking his own "power to command" as consul (imperator togatus) with Pompey's as general (imperator armatus); but it was interpreted as a request to step down. Cicero, having lost Pompey's ever-wavering support, was driven to exile. In reality, arms rarely yielded to civilian power. During the early Roman Imperial era, members of the Praetorian Guard (the emperor's personal guard as "First Citizen", and a military force under his personal command), concealed their weapons under white, civilian-style togas when on duty in the city, offering the reassuring illusion that they represented a traditional Republican, civilian authority, rather than the military arm of an Imperial autocracy.

In religion
Citizens attending Rome's frequent religious festivals and associated games were expected to wear the toga. The toga praetexta was the normal garb for most Roman priesthoods, which tended to be the preserve of high status citizens. When offering sacrifice, libation and prayer, and when performing augury, the officiant priest covered his head with a fold of his toga, drawn up from the back: the ritual was thus performed capite velato (with covered head). This was believed a distinctively Roman form, in contrast to Etruscan, Greek and other foreign practices. The Etruscans seem to have sacrificed bareheaded (capite aperto). In Rome, the so-called ritus graecus ("Greek rite") was used for deities believed Greek in origin or character; the officiant, even if a Roman citizen, wore Greek-style robes with wreathed or bare head, not the toga. It has been argued that the Roman expression of piety capite velato influenced Paul's prohibition against Christian men praying with covered heads: "Any man who prays or prophesies with his head covered dishonors his head."

An officiant capite velato who needed free use of both hands to perform ritual—as while plowing the sulcus primigenius undertaken at the founding of new colonies—could employ the "Gabine cinch" or "robe" (cinctus Gabinus) or "rite" (ritus Gabinus) which tied the toga back. This style, later said to have been part of Etruscan priestly dress, was associated by the Romans with their early wars with nearby Gabii and was thus used during Roman declarations of war.

Materials
The traditional toga was made of wool, which was thought to possess powers to avert misfortune and the evil eye; the toga praetexta (used by magistrates, priests and freeborn youths) was always woollen. Wool-working was thought a highly respectable occupation for Roman women. A traditional, high-status mater familias demonstrated her industry and frugality by placing wool-baskets, spindles and looms in the household's semi-public reception area, the atrium. Augustus was particularly proud that his wife and daughter had set the best possible example to other Roman women by, allegedly, spinning and weaving his clothing.

Hand-woven cloth was slow and costly to produce, and compared to simpler forms of clothing, the toga used an extravagant amount of it. To minimise waste, the smaller, old-style forms of toga may have been woven as a single, seamless, selvedged piece; the later, larger versions may have been made from several pieces sewn together; size seems to have counted for a lot. More cloth signified greater wealth and usually, though not invariably, higher rank. The purple-red border of the toga praetexta was woven onto the toga using a process known as "tablet weaving"; such applied borders are a feature of Etruscan dress.

Modern sources broadly agree that if made from a single piece of fabric, the toga of a high status Roman in the late Republic would have required a piece approximately 12 ft in length; in the Imperial era, around 18 ft, a third more than its predecessor, and in the late Imperial era around 8 ft wide and up to 18 - 20 ft in length for the most complex, pleated forms.

Features and styles
The toga was draped, rather than fastened, around the body, and was held in position by the weight and friction of its fabric. Supposedly, no pins or brooches were employed. The more voluminous and complex the style, the more assistance would have been required to achieve the desired effect. In classical statuary, draped togas consistently show certain features and folds, identified and named in contemporary literature.

The sinus (literally, a bay or inlet) appears in the Imperial era as a loose over-fold, slung from beneath the left arm, downwards across the chest, then upwards to the right shoulder. Early examples were slender, but later forms were much fuller; the loop hangs at knee-length, suspended there by draping over the crook of the right arm.

The umbo (literally "knob") was a pouch of the toga's fabric pulled out over the balteus (the diagonal section of the toga across the chest) in imperial-era forms of the toga. Its added weight and friction would have helped (though not very effectively) secure the toga's fabric onto the left shoulder. As the toga developed, the umbo grew in size.

The most complex togas appear on high-quality portrait busts and imperial reliefs of the mid-to-late Empire, probably reserved for emperors and the highest civil officials. The so-called "banded" or "stacked" toga (Latinised as toga contabulata) appeared in the late 2nd century AD and was distinguished by its broad, smooth, slab-like panels or swathes of pleated material, more or less correspondent with umbo, sinus and balteus, or applied over the same. On statuary, one swathe of fabric rises from low between the legs, and is laid over the left shoulder; another more or less follows the upper edge of the sinus; yet another follows the lower edge of a more-or-less vestigial balteus then descends to the upper shin. As in other forms, the sinus itself is hung over the crook of the right arm. If its full-length representations are accurate, it would have severely constrained its wearer's movements. Dressing in a toga contabulata would have taken some time, and specialist assistance. When not in use, it required careful storage in some form of press or hanger to keep it in shape. Such inconvenient features of the later toga are confirmed by Tertullian, who preferred the pallium. High-status (consular or senatorial) images from the late 4th century show a further ornate variation, known as the "Broad Eastern Toga"; it hung to the mid-calf, was heavily embroidered, and was worn over two pallium-style undergarments, one of which had full length sleeves. Its sinus was draped over the left arm.

Decline

In the long term, the toga saw both a gradual transformation and decline, punctuated by attempts to retain it as an essential feature of true Romanitas. It was never a popular garment; in the late 1st century, Tacitus could disparage the urban plebs as a vulgus tunicatus ("tunic-wearing crowd"). Hadrian issued an edict compelling equites and senators to wear the toga in public; the edict did not mention commoners. The extension of citizenship, from around 6 million citizens under Augustus to between 40 and 60 million under the "universal citizenship" of Caracalla's Constitutio Antoniniana (212 AD), probably further reduced whatever distinctive value the toga still held for commoners, and accelerated its abandonment among their class. Meanwhile, the office-holding aristocracy adopted ever more elaborate, complex, costly and impractical forms of toga.

The toga nevertheless remained the formal costume of the Roman senatorial elite. A law issued by co-emperors Gratian, Valentinian II and Theodosius I in 382 AD (Codex Theodosianus 14.10.1) states that while senators in the city of Rome may wear the paenula in daily life, they must wear the toga when attending their official duties. Failure to do so would result in the senator being stripped of rank and authority, and of the right to enter the Curia Julia. Byzantine Greek art and portraiture show the highest functionaries of court, church and state in magnificently wrought, extravagantly exclusive court dress and priestly robes; some at least are thought to be versions of the Imperial toga. In the West, the kings and aristocrats of new European kingdoms styled their dress after that of late military generals rather than the senatorial order, and the toga thus did not survive the end of centralized Roman governance.

title: Toga