Talk:Electrical impedance/Archive 1

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11:25, 28 May 2006 (UTC)

changed.. bits

I changed two bits saying, essentially, "with constant voltage, capacitors in the circuit prevent current from flowing", since the capacitor could be in parallel with a resistor, for instance, and then a current would flow. Also, my statement that impedance in DC circuits is redundant is false: DC doesn't mean constant voltage, and even if the voltage is normally constant, the behavior during switch on/off is analyzed using the impedance. AxelBoldt 17:36 Feb 23, 2003 (UTC)

Nominal impedance

I am not sure what the nominal impedance article is supposed to show. I tried to update it a little, but couldn't really get it into a workable form without being too heavy on physics/math. Simplifying it too much will just give people ideas of impedance that are wrong, and inhibit further understanding. We should get a regular old guitar player to read through it and mark it up with questions they have.

I'm thinking maybe it should just be absorbed into this article, and the first section or so of this article should be a layman's introduction. Opinions? - Omegatron 21:48, Sep 13, 2004 (UTC)

One thing that confuses me

i got confused while trying to write the nominal impedance article (teaching something is the best way of learning it yourself). maybe this should be posted in the thevenin's theorem article instead.

you can create a filter out of passive components, say a 5th order chebyshev bandpass filter (if that's a possible filter). anyway, if you plotted the frequency response of this filter with a resistive load, the graph goes all over the place and makes lots of squigglies. and yet you can collapse the entire circuit down into a single voltage divider with a voltage source, R+jX source impedance, and R+jX load impedance? That seems like more complication than can fit into just those two terms. maybe I don't grasp something. - Omegatron 01:29, Sep 14, 2004 (UTC)

I just took a brief look, but I think the confusion arises because you may have forgotten that the R+jX term only applies at one frequency. That is this is another simplification. The filter is commonly specified in the frequency domain (at steady state - another simplification) and the impedance will, in general, be a function of frequency. By the way, for some filters the transient response is important too. --AJim 02:57, 14 Sep 2004 (UTC)

Hmm. For a load that is simply a capacitor, the impedance would be

$Z=R+jX=0+j{-1 \over 2\pi fC}={-j \over 2\pi fC}$

Right? So the R + jX is not an approximation at a particular frequency, it contains the magnitude of resistance and phase change for all frequencies (because of the f term). And somehow any combination of inductors capacitors and resistors can apparently be collapsed down into a single R + jX term (which seems suspicious to me). Perhaps during the thevenin conversion the voltage source magnitude is the part that is only valid at a specific frequency?

I'm pretty sure transient (step) response, impulse response, and frequency response are just different ways of looking at the same thing. - Omegatron 03:43, Sep 14, 2004 (UTC)

In Thevenin's theorem, the equivalent impedance is only half the story. There is also the equivalent voltage source, which can be arbitrarily complex. I believe you will find that that is where the complexity goes when you do the reduction. Gazpacho 03:57, 14 Sep 2004 (UTC)

Aha! So is it true that everything can be collapsed into a single R+jX? (I mean one for source and one for load.) - Omegatron 04:40, Sep 14, 2004 (UTC)

Yes. The input can produce a complex signal at the load, but the coupling between that signal and the load is linear. Gazpacho 05:35, 14 Sep 2004 (UTC)

---

Removed from article:

I quote from the book "Understanding Telephone Electronics" by John L Fike Ph.D, PE. Adj Professor of Electrical Engineering, Southern Methodist University, Staff Consultant , Texas Instruments Learning Center. and George .E. Friend, Consultant, Telecommunications, Dallas, Texas and Staff Consultant, Texas Instruments Learning Center. Chapter 1, page 15.

"The amount of echo delay depends upon the distance from the transmitter to the point of reflection. The effect of the delay on the talker may be barely noticable to very irritating, to downright confusing. Echo also affects the listener on the far end but to a lesser degree. Echos are caused by mismatches in transmission line impedances which usually occur at the hybrid interface between a two wire line and a 4 wire transmission system. The effect of echo is reduced by inserting a loss in the lines." (Italics and bolding all mine). I rest my case, but an admission of error and an apology from Omegatron would be nice.

Rewrite in progress...

I have begun a rewrite of this article for the following reasons:

(1) The crucial concept of AC steady state has been left out (2) The development of the concept of impedance is, as a result, problematic (3) Some info is just plain wrong as in the case of the sections 'constant signals', 'fixed frequency signals', and 'variable frequency signals'

To start, I have rewritten the introduction. My next task is to develop the concept of impedance by introducing the notion of AC steady state to further reinforce the analogy with Ohm's law. This section will replace the sections on 'constant signals' through 'fixed frequency signals'. I will then extend this idea to case of a discrete and then continuous distribution of frequencies. This section will replace 'variable frequency signals'. Alfred Centauri 23:53, 21 August 2005 (UTC)

Update: I have made the first pass at the rewrite of this article. I see that Omegatron is watching over my shoulder (many thanks!). I considered adding a section on complex power but had second thoughts - that subject is probably covered elsewhere and if not, I can add that in a separate article. Alfred Centauri 17:01, 22 August 2005 (UTC)

Are you 130.207.237.254? — Omegatron 18:31, August 22, 2005 (UTC)

Dang! I guess I am. I must have timed out whilst composing an edit and didn't notice on the subsequent edits. Sorry about that. I'll pay more attention next time.

Dang again - I didn't add the sig Alfred Centauri 21:56, 24 August 2005 (UTC)

Rv of last edit by Witger

Witger - I reverted your changes for the reason that in any derivation of electrical impedance, the first step is to assume that either the voltage or current is sinusoidal. Additionally, electrical impedance is defined as the ratio of a phasor voltage to a phasor current. A non-sinusoidal quantity does not have a phasor representation. If you have evidence to the contrary, please link to or post it below. Thanks. Alfred Centauri 20:21, 26 August 2005 (UTC)

After reflecting on the above, I believe that I have made too strong a statement. In fact, one can use the full blown complex exponential to derive a quantity called impedance that is more general than what is discussed in this article. However, this is more of theoretical interest than practical interest. Not that there is anything wrong with that!

First, one has to assume that the circuit is in steady state and is driven by a complex exponential source, e.g.:

$v_{S}(t)=V_{0}e^{st}\,$

where s is complex:

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

It follows that the voltage and currents in the circuit are of the same form. If the voltage across a capacitor is of this form then the ratio of the time domain capacitor voltage to current is:

${\frac {v_{C}(t)}{i_{C}(t)}}={\frac {1}{sC}}\,$

That is, the impedance of the capacitor is a complex quantity as opposed to a pure imaginary quantity as is the case with sinusoidal excitation.

The catch is this: no physical voltage or current can be represented as a complex exponential for two reasons:

(1) no imaginary time domain voltages or currents exist

(2) no real exponential voltages or currents exist - a real exponential is unbounded.

It should be further noted that impedance cannot be defined for the sum of complex exponentials or for that matter the sum of sinusoids. For example, what is the impedance of a capacitor if the voltage across the capacitor is:

$v_{C}(t)=cos(2\pi f_{1}t)+sin(2\pi f_{2}t)\,$

where $f_{1}<>f_{2}$. Alfred Centauri 23:40, 26 August 2005 (UTC)

You've just given me a flashback!

That is how I was taught, actually. Haven't seen it in a long time. Probably confused me more than anything, and I had to unlearn it in order to learn something that I would actually use, but it is being taught.  :-) — Omegatron 02:32, August 27, 2005 (UTC)

When I first studied electronics at a technical school, AC circuit analysis was introduced using phasors. Later, when I first studied electrical engineering at a university, AC circuit analysis was introduced using the complex exponential. I remember thinking "what the heck is that thing?" Later still, when I was teaching an EE class to non-EE majors, the text introduced AC circuit analysis with phasors. I'm of the opinion that phasor anaylsis should be used initially with the complex exponential stuff saved for later signal processing classes. Alfred Centauri 11:50, 27 August 2005 (UTC)

notation

Now that I've had it pointed out to me, I'm a stickler for notation:

1. $R={\frac {v_{R}\left(t\right)}{i_{R}\left(t\right)}}$
2. $i_{C}(t)=C{\frac {dv_{C}(t)}{dt}}$
3. $v_{C}(t)=V_{p}sin\left(2\pi ft\right)$
4. ${\frac {dv_{C}(t)}{dt}}=2\pi fV_{p}cos\left(2\pi ft\right)$
5. $I_{c}=j2\pi fCV_{c}\,$
6. $Z_{\mathrm {capacitor} }={\frac {V_{c}}{I_{c}}}={\frac {1}{j2\pi fC}}$
7. $v_{L}(t)=L{\frac {di_{L}(t)}{dt}}$
8. $Z_{\mathrm {inductor} }=j2\pi fL\,$

Is $i_{C}$ the current of the capacitor, or the current of the capacitance?  :-) Does C here mean capacitance or could it be equally notated by $i_{\mathrm {cap} }$? Likewise with $v_{L}$, etc.

Should it be $V_{c}$ or $V_{C}$? And so on... — Omegatron 03:32, August 28, 2005 (UTC)

This is such a good question! In most commercial schematics, you will see both a ref des and value such as "R100" and "47" so that we would say: "the resistor R100 has a resistance of 47 ohms". Ohm's law for this resistor looks like:

${\frac {v_{R100}}{i_{R100}}}=47\Omega \,$

Now, consider a simple circuit as presented in most textbooks where you'll probably see a resistor labeled with R. I suppose this is to be interpreted as 'the resistor R has a resistance of R ohms'. Ohm's law for this resistor looks like:

${\frac {v_{R}}{i_{R}}}=R\Omega \,$

That is, R is used in this equation in two different ways - as a label for a device and as a label for the resistance of the resistor. Rarely does this notation cause confusion yet I wonder if there is a better notation that is not too cumbersome.

I suppose that I should use $Z_{C}$ instead of $Z_{capacitor}$ but I wanted to emphasize that this is a general result. Maybe that's not such a good idea.

Regarding the uppercase versus lower case subscripts. I'm using the convention that a phasor quantity has an upper case letter with a lower case subscript. Perhaps a short section or a link to a short article on notation is in order? Alfred Centauri 13:56, 28 August 2005 (UTC)

What I meant for the first part was the difference between italics for variables and roman font for descriptions. So $v_{\mathrm {R} }$ would mean "the voltage of the resistor", while $v_{R}$ would mean, "the voltage of the resistance variable". I think the first is correct in this case. R is just a shorthand for resistor.

When I made Image:Source and load circuit.png, it originally had "source" and "load" written out [1]. I've since decided that it's not worth it, because the equations become super large with all those words written out. — Omegatron 15:22, August 28, 2005 (UTC)

distinction between phase and angle

I have reverted changes by Witger in the magnitude and phase portion of the article for the following reasons:

Phase is the unit magnitude complex number that is multiplied by the modulus (a real number) of the complex number:

$Z=\sigma +j\omega =({\sqrt {\sigma ^{2}+\omega ^{2}}})e^{j\phi }$

The angle or phase angle or argument is the real number given by the inverse tangent of the ratio of the imaginary part to the real part:

$\phi =tan^{-1}{\frac {\omega }{\sigma }}\,$

That is, the phase is:

$cos(\phi )+jsin(\phi )\,$

Alfred Centauri 17:48, 28 August 2005 (UTC)

Sorry Alfred on the formula for phase. Now I see what you mean by :$tan^{-1}\,$ .

Anyway, there seems to be some confusion between cotangent and arcustangent involved:

I think we agree on the following:

$\tan(phi)={\omega }/{\sigma }\,$

Then, if I am not mistaken, the more conventional notation should be:

$\phi =atg{\frac {\omega }{\sigma }}\,$

Note: Instead of $atg\,$ you might prefer also $arctan\,$ .

OK?

Regards,

Witger.

I've seen arctan and $tan^{-1}$ used with about equal frequency but I think this may be the first time I've seen atg. I vote for arctan if you think $tan^{-1}$ is too obscure. Alfred Centauri 12:05, 29 August 2005 (UTC)

I've only seen arctan used in computer programming and other places where superscripts are not appropriate or possible. Nearly all math texts I've seen use $tan^{-1}$, but the two are equivalent, and I don't see any problem with which one is used, as long as you don't flip back and forth in sequences of related equations. --ssd 15:42, 10 December 2005 (UTC)

interpretation of phase angles

I think it would be helpful to have some qualitative interpretation of the phase angles, e.g., the current leads the voltage in a capacitive circuit, etc. Like a lot of WP articles on scientific topics, this one could be improved by doing everything that can be done with easy math first, and only introducing harder math toward the end of the article. It could also use circuit diagrams, graphs, etc.--bcrowell

Hooray for that attitude! Someone has to write it, though.  :-) — Omegatron 13:38, 16 September 2005 (UTC)

Small note, unhelpful sentence or two

On the part in AC steady state: "For DC circuit analysis, each voltage and current is represented by a constant real number. Thus, it is reasonable to suppose that the rules developed for DC circuit analysis can be used for AC circuit analysis by using complex numbers instead of real numbers."

I don't think it would be "reasonable" to just assume that. Not to mention, why doesn't it go into *why* the rules for DC circuits be used for AC circuits. Needs clarification. Fresheneesz 06:21, 9 December 2005 (UTC)

The very next section - Definition of Impedance - explains the 'why'. The intent of the statement that you object to is to motivate the notion that we define impedance in order to generalize Ohm's law (and what follows from Ohm's law) to AC circuits in AC steady state. Alfred Centauri 16:04, 9 December 2005 (UTC)

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