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Old 30th Oct 2017, 10:18 am   #1
Al (astral highway)
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Default Maths query: What is 'T' in this expression, please?

The expression is:

Vpk = sqrt(L/C) * exp(-T/4/tau) * Ipk

It's to accurately calculate current in a tank circuit, allow for the exponential delay envelope of the capacitor, and is inserted as a term into the conventional expression, so more accurate than: Vpk=sqrt(L/C)*Ipk

I've tried a few informed guesses but I'd rather be sure.

Many thanks, folks!
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Old 30th Oct 2017, 10:27 am   #2
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Default Re: Maths query: What is 'T' in this expression, please?

Temperature coefficient?
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Old 30th Oct 2017, 10:44 am   #3
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Default Re: Maths query: What is 'T' in this expression, please?

Hello, is that a guess or for sure? I haven’t encountered tempco for caps before!!
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Old 30th Oct 2017, 11:03 am   #4
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Default Re: Maths query: What is 'T' in this expression, please?

I expect that that's the expression including the exponential decay envelope in which case T will be time. A transiently energised circuit, say by closing a switch at time zero, will gradually relax to its steady-state condition. But early on in this decay the peak voltage, and current for that matter, may be higher than the steady-state values, and so may their ratio which is the expression you've shown multiplying Ipk. That ratio has its highest value at zero time (T=0) when it simplifies to sqrt(L/C).

EDIT: Just to be clear I came across the expression in the last post here http://www.edaboard.com/thread337972.html. I haven't checked that it's correct. I've just explained, as you asked, what the T stands for.

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Old 30th Oct 2017, 11:15 am   #5
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Default Re: Maths query: What is 'T' in this expression, please?

The equation is a trifle confusing since it is generally accepted that a lower case t is used to represent time, not an upper case t.
The expression exp(-T/4/tau) is also somewhat confusing and would be better written as exp(-T*tau/4), where * signifies multiply.

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Old 30th Oct 2017, 11:23 am   #6
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Default Re: Maths query: What is 'T' in this expression, please?

Quote:
Originally Posted by Skywave View Post
The equation is a trifle confusing since it is generally accepted that a lower case t is used to represent time, not an upper case t.
Hi Al, good point. I think in this case, they've popped it into upper case to avoid confusion with the Greek symbol for tau?

Quote:
Originally Posted by Skywave View Post
expression exp(-T/4/tau) is also somewhat confusing and would be better written as exp(-T*tau/4), where * signifies multiply.
Very good point there, Al.
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Old 30th Oct 2017, 11:26 am   #7
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Default Re: Maths query: What is 'T' in this expression, please?

I think it's meant to be exp(-T/(4*tau)). tau is almost certainly a time constant and the expression inside the exponential brackets needs to be dimensionless which T/tau is but T*tau isn't.

I agree about the confusing layout and the use of capital T though.

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Old 30th Oct 2017, 11:27 am   #8
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Default Re: Maths query: What is 'T' in this expression, please?

Astral Highway,

The equation appears to give a value for peak voltage in a resonant circuit since it contains L and C . Usually in such a circuit subjected to a transient or a decaying oscillation after some time, there is exponential decay, with time.

So never seeing this one before, and the fact that it's an e to the "-T" it appears to be suggestive that the T is t or time, so the equation specifies what the peak voltage will be after a time and it could be rearranged to solve for peak current. I assume tau is 2pi, but often with an equation that contains multiple terms it's hard to know what it means unless you have some text associated with it, explaining it.

Certainly it's not great if equations are just presented without some explanation of what they mean.
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Old 30th Oct 2017, 11:29 am   #9
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Default Re: Maths query: What is 'T' in this expression, please?

Quote:
Originally Posted by GrimJosef View Post
I expect that that's the expression including the exponential decay envelope ...
Ahah, I did make that clear in the OP, GJ!

Quote:
Originally Posted by GrimJosef View Post
...in which case T will be time
Right... that's a credible answer, thanks, GJ.

I can't remember where I found this expression as I wrote it down in a notebook ages ago. I'll pop it into Google to see if any other pages are returned, aside the one you reference there...

Cheers
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Last edited by Al (astral highway); 30th Oct 2017 at 11:39 am. Reason: to avoid confusion!
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Old 30th Oct 2017, 11:39 am   #10
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Default Re: Maths query: What is 'T' in this expression, please?

Quote:
Originally Posted by Argus25 View Post
Astral Highway,

The equation appears to give a value for peak voltage in a resonant circuit since it contains L and C . Usually in such a circuit subjected to a transient or a decaying oscillation after some time, there is exponential decay, with time.

So never seeing this one before, and the fact that it's an e to the "-T" it appears to be suggestive that the T is t or time, so the equation specifies what the peak voltage will be after a time...
Interesting, and again, yes, I think you're on to something here. I'd like current at 90 degrees of a cycle and tracking one cycle, so now I can work this out. Thank you Argus...


Quote:
Originally Posted by Argus25 View Post
assume tau is 2pi, but often with an equation that contains multiple terms it's hard to know what it means unless you have some text associated with it, explaining it.
I think having had a look at the other options, Tau might just be RC?

I agree that it's a poor equation that arrives with no explanation. I normally meticulously record references in my workbook but in this case I have nothing!!!
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Old 30th Oct 2017, 11:48 am   #11
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Default Re: Maths query: What is 'T' in this expression, please?

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Originally Posted by astral highway View Post
Ahah, I did make that clear in the OP, GJ!
On my screen the OP says delay, not decay.

Quote:
Right... that's a credible answer, thanks, GJ. Only what time?
Is it the period of one cycle?
T is a variable, so the expression can be used for whatever time you choose. What it seems to be saying is that as time goes on the ratio Vpk/Ipk gets smaller and smaller, tending to zero at very large times. This seems unphysical to me. But as I say I haven't checked whether the expression is correct or whether it only applies for a particular circuit configuration or excitation. All I've done is suggest that T is time I'm afraid.

EDIT I've just read more of the thread I linked to and it seems that that is the formula for the decaying oscillations in a tank circuit after it has been disconnected at time T=0 from its power source. In that case tau may be determined by the amount of loading on the circuit.

Cheers,

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Last edited by GrimJosef; 30th Oct 2017 at 11:54 am.
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Old 30th Oct 2017, 11:56 am   #12
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Default Re: Maths query: What is 'T' in this expression, please?

Traditionally, tau is used for time constant which for some circuits is RC. For an LC resonator it is related to Q (something like Q/(2 pi f) ?) - which might still work out to be RC for series R. Why there is a factor of 4 in the exponential is unclear, as I would expect 1.

The decay will be of the resonance, not the capacitor alone. Ipk would be the initial peak current, while Vpk is the decaying peak - so a confusing way of writing it.

Given the problems with this formula, you might be best to start again in finding out how resonances decay rather than using something which could turn out to be unreliable.
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Old 30th Oct 2017, 12:03 pm   #13
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Default Re: Maths query: What is 'T' in this expression, please?

On the Tau thing, it was proposed in the past that a symbol should be used to replace 2.pi, because it seemed to crop up everywhere in physics equations rather than just Pi. So it would have been better to have a single symbol representing 2.pi.

But of course, as with any symbol, it could mean anything, because in some text it could say be tau = RC instead. RC has units of time which makes sense for your equation. So perhaps that is why in the version of the equation you have they used a T for time rather than t, but it's hard to know for sure.
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Old 30th Oct 2017, 12:22 pm   #14
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Default Re: Maths query: What is 'T' in this expression, please?

Astral Highway,

You mentioned you wanted the current at 90 degrees of a cycle , presumably 90 degrees into the cycle. If you have an LCR resonant circuit subject to a transient (either a fixed voltage applied from zero or a step change in voltage) then the exponentials are important as there is energy loss or gained in the circuit and the oscillations decay, assuming the damping is low, and some new equilibrium is reached. So this requires a certain equation.

However if you are dealing with a steady state situation, like in some power oscillator, where the energy losses are being replaced (to sustain oscillation) then you need a different equation which will not contain the exponential decay.

So one other thing that helps is trying to figure out which specific equation applies to the specific set of circuit circumstances you are dealing with.

What is the arrangement of the resonant circuit you are dealing with and is it in a steady state or subject to transients ?

Sorry I can't paste in a link, but Google "using tau to replace 2 Pi" makes interesting reading.

Last edited by Argus25; 30th Oct 2017 at 12:28 pm.
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Old 30th Oct 2017, 1:12 pm   #15
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Default Re: Maths query: What is 'T' in this expression, please?

Using tau to replace 2 pi would be very confusing, as tau is already traditionally used for time constant. Other multiples of pi also crop up in maths and physics, so nothing special about 2 pi - that just occurs when circle circumference is being considered, either explicitly or implicitly. For example, area of a circle is pi r^2; simpler than tau/2 r^2. When doing some research on the Class E power amplifier a constant which kept cropping up was pi^2 +4; I think that is simpler than tau^2/4 +4.
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Old 30th Oct 2017, 2:18 pm   #16
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Default Re: Maths query: What is 'T' in this expression, please?

Argus25,

Thank you. I’m out with just my phone so unable to reformat the quote function, sorry. But to answer your question, the circuit is indeed a power oscillator under construction - I want to model every node before I build this (fourth !) iteration of it.

So yes, steady state. On the other hand, in CW mode (likely normally only for testing or brief exhibit) a transient response does arise at the instant tne half waved doubled caps in the power supply dump into the tank circuit at a higher voltage than after a few cycles of operation. This is because the valve will always be heated to a steady state temperature before power is applied to it .
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Old 30th Oct 2017, 11:01 pm   #17
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Default Re: Maths query: What is 'T' in this expression, please?

G8HQP Dave,
Yes I'm inclined to agree about the use of tau, but it's interesting the large number of authors advocating it for 2pi, I think I read the Scientific American article on it, which is why when I saw tau in the equation I thought it might have been that. Most of the equations I have and use keep the RC just as that as either can be a variable.

Astral highway,
I'm pretty sure I have the equation which will give the current at any time after a transient is applied to a resonant circuit. I'm away from home for a week and I can't attach it. So I will send it when I get back, if it has not been posted by someone on this thread by then. It's not the kind of thing you find in most Electronic text books, typically vintage electrical engineering texts instead .

Last edited by Argus25; 30th Oct 2017 at 11:07 pm. Reason: Spelling
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Old 31st Oct 2017, 12:39 am   #18
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Quote:
Originally Posted by Argus25 View Post
Yes I'm inclined to agree about the use of tau, but it's interesting the large number of authors advocating it for 2pi, I think I read the Scientific American article on it, which is why when I saw tau in the equation I thought it might have been that. Most of the equations I have and use keep the RC just as that as either can be a variable.
It has only ever been very infrequently that I have met the Greek letter tau in equations that relate to electrical phenomena. So there is a school of thought that is advocating it as a replacement for 2.pi, eh? I can see no logical justification for that substitution whatsoever; where is the difficulty or inappropriateness of simply writing 2.pi? Electrical equations are often complicated quite enough: why make them even more obscure? Goodness knows, the simple expression 2.pi has been around long enough to have earned some sort of respect, no? The replacement of "2.pi" by "tau" seems to be obfuscation simply for the sake of it: ridiculous.

Well, that's my opinion . . . . for what it's worth (which is probably not a lot! )
However, the following link gives the whole story . . .

https://www.scientificamerican.com/a...asier-than-pi/

And to my mind, there is a good deal of nonsense in that! For example, at one point the article states:
"At its heart, pi refers to a semicircle",
which anyone who has studied the very basics of the geometry of a circle knows that that is patent rubbish: by definition, pi is the ratio of a circle's circumference to its diameter, as in C = pi*d, where C = circumference & d = diameter.

Al.

Last edited by Skywave; 31st Oct 2017 at 12:57 am.
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Old 31st Oct 2017, 7:27 am   #19
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Default Re: Maths query: What is 'T' in this expression, please?

2*pi*frequency is often shortened to lower-case omega.

But that has the justification that little omega is 'angular frequency' in radians/second instead of frequency in Hetrz.

I have come across tau as timeconstant before, but more usually it's just left as CR. But in this case there are no CR timeconstant components. It is the lossiness of a resonator and tau is a function of resonant frequency and Q.

Q has many definitions and one of them is the reciprocal of the fraction of resonator energy lost per radian of time at the resonant frequency.

The multiple definitions of Q makes it a handy translator allowing you to hop between different views of the operation of a circuit.

David
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Old 31st Oct 2017, 12:50 pm   #20
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2*pi*frequency is often shortened to lower-case omega.
But that has the justification that little omega is 'angular frequency' in radians/second instead of frequency in Hertz.
Indeed: I did contemplate saying that in my last post, as per above.

Substituting lower-case omega for 2.pi.f is a sensible substitution, since f is a variable, where as replacing 2.pi by tau isn't, since 2.pi is a constant. So, if we choose to shorten 2.pi.f when using tau to get the 'equivalent' of 2.pi.f, we get 'tau.f': unquestionably ridiculous!

Al.
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