We're all familiar with typical plot (eg http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical explanations of Q. What's the analytical expression represented by that curve? [Yeah I should be able to derive it from first principles, but can't seem to do anything right today ;[ TIA
Basic question about Q of resonat RLC tank
Started by ●July 4, 2008
Reply by ●July 4, 20082008-07-04
Richard Owlett wrote:> We're all familiar with typical plot (eg > http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical > explanations of Q. > > What's the analytical expression represented by that curve? > > [Yeah I should be able to derive it from first principles, but can't > seem to do anything right today ;[It's a general definition of 3-dB bandwidth, and the curve can be anything with a peak. Single-tuned, any Q, double-tuned, any Q and coupling coefficient (maximally flat is most interesting), and others. I know you mean something particular, but if you knew how restrictive that is, you would ask differently, so I won't tell you (today.) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 4, 20082008-07-04
Richard Owlett wrote:> We're all familiar with typical plot (eg > http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical > explanations of Q. > > What's the analytical expression represented by that curve? > > [Yeah I should be able to derive it from first principles, but can't > seem to do anything right today ;[ > > TIA >I doubt that particular curve is really accurate, but a resonant, pure-bandpass tank circuit would have a transfer function A w_o/Q s H(s) = ------------------- w_o s^2 + --- s + w_o^2 Q Anything with a fairly high Q is going to have nearly the same response around resonance if it is high pass, low pass, or some strange collection of zeros (other than a notch). The big difference will be how much residual energy coming through at off-resonant frequencies. You can derive the above equation by solving the circuit below for Vout/Vin: ___ .----|___|----o-----o-------o Vout | R | | | | | /+\ --- C| Vin ( ) C --- C| L \-/ | C| | | | | | | === === === GND GND GND (created by AACircuit v1.28.6 beta 04/19/05 www.tech-chat.de) (there's an equivalent one for Iout/Iin, and another one that uses a series R,L and C, but I'm too lazy to show them just now). -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●July 4, 20082008-07-04
Jerry Avins wrote:> Richard Owlett wrote: > >> We're all familiar with typical plot (eg >> http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical >> explanations of Q. >> >> What's the analytical expression represented by that curve? >> >> [Yeah I should be able to derive it from first principles, but can't >> seem to do anything right today ;[ > > > It's a general definition of 3-dB bandwidth, and the curve can be > anything with a peak. Single-tuned, any Q, double-tuned, any Q and > coupling coefficient (maximally flat is most interesting), and others.Actually, maximally flat is least interesting (at the moment ;) I was thinking simplest case of 1 each R, L, and C as either one mesh or one node. But I had briefly considered the others you mentioned but didn't see them giving an answer of a different form *EXCEPT* for coupled case with staggered tuning. Besides I had seen the "same" plot in so many contexts (even ARRL handbooks of the 50's) that there just had to be a "common" analytical description with normalized frequency on x-axis.> > I know you mean something particular, but if you knew how restrictive > that is, you would ask differently, so I won't tell you (today.)There you go again, suggesting people think.> > Jerry
Reply by ●July 4, 20082008-07-04
Tim Wescott wrote:> Richard Owlett wrote: > >> We're all familiar with typical plot (eg >> http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical >> explanations of Q. >> >> What's the analytical expression represented by that curve? >> >> [Yeah I should be able to derive it from first principles, but can't >> seem to do anything right today ;[ >> >> TIA >> > I doubt that particular curve is really accurate, but a resonant, > pure-bandpass tank circuit would have a transfer function > > A w_o/Q s > H(s) = ------------------- > w_o > s^2 + --- s + w_o^2 > QI assume w_o is resonant freq and the numerator is {A * w_o * s}/Q That just doesn't look right, but that just means I'm having my usual problems with s-plane.> > Anything with a fairly high Q is going to have nearly the same response > around resonance if it is high pass, low pass, or some strange > collection of zeros (other than a notch). The big difference will be > how much residual energy coming through at off-resonant frequencies. > > You can derive the above equation by solving the circuit below for > Vout/Vin: > > ___ > .----|___|----o-----o-------o Vout > | R | | > | | | > /+\ --- C| > Vin ( ) C --- C| L > \-/ | C| > | | | > | | | > === === === > GND GND GND > (created by AACircuit v1.28.6 beta 04/19/05 www.tech-chat.de) > > (there's an equivalent one for Iout/Iin, and another one that uses a > series R,L and C, but I'm too lazy to show them just now). >I was trying with equiv using Vout/Iin
Reply by ●July 4, 20082008-07-04
Richard Owlett wrote: ...> There you go again, suggesting people think.Well, tou might at least think about what question you want to ask. :-) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 4, 20082008-07-04
Richard Owlett wrote:> Tim Wescott wrote: > >> Richard Owlett wrote: >> >>> We're all familiar with typical plot (eg >>> http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical >>> explanations of Q. >>> >>> What's the analytical expression represented by that curve? >>> >>> [Yeah I should be able to derive it from first principles, but can't >>> seem to do anything right today ;[ >>> >>> TIA >>> >> I doubt that particular curve is really accurate, but a resonant, >> pure-bandpass tank circuit would have a transfer function >> >> A w_o/Q s >> H(s) = ------------------- >> w_o >> s^2 + --- s + w_o^2 >> Q > > > I assume w_o is resonant freq and the numerator is {A * w_o * s}/QYes, that's what I meant.> That just doesn't look right, but that just means I'm having my usual > problems with s-plane.But it is, unless I'm especially cracked today. Perhaps it looks too simple?>> >> Anything with a fairly high Q is going to have nearly the same >> response around resonance if it is high pass, low pass, or some >> strange collection of zeros (other than a notch). The big difference >> will be how much residual energy coming through at off-resonant >> frequencies. >> >> You can derive the above equation by solving the circuit below for >> Vout/Vin: >> >> ___ >> .----|___|----o-----o-------o Vout >> | R | | >> | | | >> /+\ --- C| >> Vin ( ) C --- C| L >> \-/ | C| >> | | | >> | | | >> === === === >> GND GND GND >> (created by AACircuit v1.28.6 beta 04/19/05 www.tech-chat.de) >> >> (there's an equivalent one for Iout/Iin, and another one that uses a >> series R,L and C, but I'm too lazy to show them just now). >> > > I was trying with equiv using Vout/IinThen connect R from Vout to ground, and put Iin in parallel with the whole mess. Then your transfer function becomes E/I = 1/Y, and Y is just the parallel combination of admittances 1/R, 1/(Ls) and Cs: Y = 1/R + 1/(Ls) + Cs; 1 1 s/C E/I = --- = ----------------- = --------------------- Y 1/R + 1/(Ls) + Cs s^2 + s/(RC) + 1/(LC) This is equal to R at the resonant frequency (which isn't surprising because the L and C cancel each other out), and you can equate w_o and Q to the various component values (and compare them to your 1950's ARRL handbook, too -- the theory hasn't changed). -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●July 4, 20082008-07-04
Tim Wescott wrote:> Richard Owlett wrote: > [snip] >>>> >>> I doubt that particular curve is really accurate, but a resonant, >>> pure-bandpass tank circuit would have a transfer function >>> >>> A w_o/Q s >>> H(s) = ------------------- >>> w_o >>> s^2 + --- s + w_o^2 >>> Q >> >> >> >> I assume w_o is resonant freq and the numerator is {A * w_o * s}/Q > > > Yes, that's what I meant. > >> That just doesn't look right, but that just means I'm having my usual >> problems with s-plane. > > > But it is, unless I'm especially cracked today. Perhaps it looks too > simple? >I was "right". Do you mind if I decline to admit just how long it took me to realize what j**2 = -1 means relative to above ARGHH ;!
Reply by ●July 5, 20082008-07-05
Richard Owlett wrote:> We're all familiar with typical plot (eg > http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical > explanations of Q. > > What's the analytical expression represented by that curve? > > [Yeah I should be able to derive it from first principles, but can't > seem to do anything right today ;[ > > TIA >To which Jerry said: > Well, you might at least think about what question you want to ask. To which I reply: Not only did I think, but I asked EXACTLY the fully specified question I intended to ask. And Tim noted: > I doubt that particular curve is really accurate ... [snip H(s) which demonstrates Tim is correct] I got SNOOKERED! I had seen that curve so many times and in so many contexts that it gained the status of *TRUTH* ;/ The curve is always illustrated as symmetric :< And I wanted a continuous function with continuous derivatives which symetrically went to zero. I'll go play with a Gaussian function. It's symmetric and goes to zero. It just doesn't have any intuitive justification as a filter goal.
Reply by ●July 5, 20082008-07-05
Richard Owlett wrote:> Richard Owlett wrote: > >> We're all familiar with typical plot (eg >> http://en.wikipedia.org/wiki/Image:Bandwidth2.png) used in typical >> explanations of Q. >> >> What's the analytical expression represented by that curve? >> >> [Yeah I should be able to derive it from first principles, but can't >> seem to do anything right today ;[ >> >> TIA >> > > To which Jerry said: > > Well, you might at least think about what question you want to ask. > > To which I reply: > Not only did I think, but I asked EXACTLY the fully specified question I > intended to ask. > > And Tim noted: > > I doubt that particular curve is really accurate ... > [snip H(s) which demonstrates Tim is correct] > > > I got SNOOKERED! > I had seen that curve so many times and in so many contexts that it > gained the status of *TRUTH* ;/ > > The curve is always illustrated as symmetric :< > And I wanted a continuous function with continuous derivatives which > symetrically went to zero. > > I'll go play with a Gaussian function. It's symmetric and goes to zero. > It just doesn't have any intuitive justification as a filter goal.The curve is only approximately symmetric and then only if the Q is high enough. Symmetry depends on the approximation that the arithmetic mean and the geometric mean of the 3-dB points are "equal". Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
