Magnitude of the transfer function

[Zain]

Hi all, In one of my problems there is a question with a plot of the transfer function magnitude versus frequency. In the graph I notice that the magnitude of the transfer function is greater than 1. This is confusing as I thought that filters are not able to increase the magnitude of the output signal with respects to the input as they are only able to attenuated the amplitude or change the phase of the signals. Also to strengthen my case, as the transfer function is H(f) = Vout∠θout / Vin∠θin, shouldn't the magnitude of the transfer function always be less than or equal to 0.

<Moderator's note: Link trigger shortened: https://ccle.ucla.edu/pluginfile.php/2045528/mod_resource/content/0/HW6.pdf>

 

[FactChecker]

The short answer:
A filter with a feedback loop can accumulate certain frequencies to much higher than the input amplitude of that frequency. Which frequencies accumulate and which attenuate are determined by the phase shift of the feedback.

 

[the_emi_guy]

Transfer functions of filters can certainly be > 1.

Consider a voltage source feeding a series LC. The transfer function of current vs. freq will go to infinity at series resonance.

Active filters can be built to have gain at all frequencies, such as in an audio amp with equalizer.

 

[NascentOxygen]

Besides all of the above, a filter can include a buffer/amplifier stage in addition to the "amplitude-shaped frequency response" block, with the whole shebang still called a filter.

Even a passive filter comprising only Rs and Cs can be devised to show a voltage magnification exceeding unity over a desired frequency band.

 

[donpacino]

Have you ever listened to music through an electrical device like a car or an ipod? the signal for the music inside the device is very small, but we need to AMPLIFY the size of the signal to get loud music. small input, large output

 

[tech99]

Even a passive filter comprising only Rs and Cs can be devised to show a voltage magnification >1.00 at some frequecies.

I presume this is where the source or load are inductive. I cannot think of another case.

 

[NascentOxygen]

I presume this is where the source or load are inductive. I cannot think of another case.

No inductance involved. Assume ideal sources unless otherwise specified.

Only fractionally greater than unity gain, nevertheless, not attenuation.

 

[FactChecker]

I think this is an example of a passive "filter" whose gain at one frequency is > 1:

 

[tech99]

I think this is an example of a passive "filter" whose gain at one frequency is > 1:

Yes, but it is equivalent to an LC filter. Mass and springiness, and a distinct lack of R.

 

[FactChecker]

Yes, but it is equivalent to an LC filter. Mass and springiness, and a distinct lack of R.

I didn't have that discussion in mind. I see that you are right about the gain of an RC filter. The gain remains below 1. (see http://www.engr.sjsu.edu/wdu/ME106Spring2010/HW/HowToUseMatlabToDrawBodeDiagram.pdf )

 

[Tom.G]

Even a passive filter comprising only Rs and Cs can be devised to show a voltage magnification exceeding unity over a narrow band of frequencies.

No inductance involved. Assume ideal sources unless otherwise specified.

Only fractionally greater than unity gain, nevertheless, not attenuation.

I'm interested and curious, can you supply a concrete, real-world, example without both L & C, and with all passive elements?

 

[donpacino]

I'm interested and curious, can you supply a concrete, real-world, example without both L & C, and with all passive elements?

No. with only passive components with no energy storage devices (L and C) you are implying that you want no reactance. This means that the values the same at DC and high frequencies. I could point out voltage multipliers, but that has capacitors and diodes.

 

[tech99]

No. with only passive components with no energy storage devices (L and C) you are implying that you want no reactance. This means that the values the same at DC and high frequencies. I could point out voltage multipliers, but that has capacitors and diodes.

I think "passive" in this context means no amplifier.

 

[donpacino]

I think "passive" in this context means no amplifier.

Yes, he specified both passive and no L and C. I pointed out that it doesn't leave any elements that change with frequency, which means no resonance.

 

[FactChecker]

I think that the requirement of no L and no C would be requirement creep that is unreasonable. The next step would be no wires. ;>)

 

[Tom.G]

Yes, he specified both passive and no L and C. I pointed out that it doesn't leave any elements that change with frequency, which means no resonance.

I'm interested and curious, can you supply a concrete, real-world, example without both L & C, and with all passive elements?

Sorry for the unclear query. I'll try again. The intention of "...without L&C..." was that not both be present.
As stated:

No inductance involved. Assume ideal sources unless otherwise specified.

Only fractionally greater than unity gain, nevertheless, not attenuation.

If above unity gain is obtainable in such a case, I'm still interested in finding out how.

 

[donpacino]

ahhh ok, in that case see my above post, voltage multiplier made from diodes and caps takes in low amp ac voltage and outputs high voltage dc

 

[Tom.G]

ahhh ok, in that case see my above post, voltage multiplier made from diodes and caps takes in low amp ac voltage and outputs high voltage dc

Yup, that would do it.

However, my query was regarding this post:

Even a passive filter comprising only Rs and Cs can be devised to show a voltage magnification exceeding unity over a narrow band of frequencies.

I'm still trying to find out how that is accomplished.

 

[donpacino]

Yup, that would do it.

However, my query was regarding this post:

I'm still trying to find out how that is accomplished.

True, I didn't pick that up, I guess I lack reading comprehension! I am not sure of just R and C unfortunately

 

[NascentOxygen]

If above unity gain is obtainable in such a case, I'm still interested in finding out how.

I'm trying to cajole forum software into coughing up its secrets, but am expending much effort to no avail. From memory, the Twin Tee network can show a small voltage gain. I know it has been discussed in at least one forum thread, but I can't manage to unearth it.

It looks like this discussion (in Italian) is about the voltage gain of a Twin Tee network (modified).

http://www.electroyou.it/forum/view...start=40&sid=4339d3073d6aa35532301ccf1c14f3a8

Using this type of network, it's feasible to create an oscillator around an emitter follower. The emitter follower, as you know, having a gain less than unity, requires the RC feedback network to exhibit a gain slightly greater than unity at some frequency, for oscillation.

 

[The Electrician]

I'm trying to cajole forum software into coughing up its secrets, but am expending much effort to no avail. From memory, the Twin Tee network can show a small voltage gain. I know it has been discussed in at least one forum thread, but I can't manage to unearth it.

It looks like this discussion (in Italian) is about the voltage gain of a Twin Tee network (modified). http://www.electroyou.it/forum/view...start=40&sid=4339d3073d6aa35532301ccf1c14f3a8

Using this type of network, it's feasible to create an oscillator around an emitter follower. The emitter follower, as you know, having a gain less than unity, requires the RC feedback network to exhibit a gain slightly greater than unity at some frequency, for oscillation.

You may be looking for this thread: https://www.physicsforums.com/threads/rc-networks.235927/

Here's a link to the IRE paper: http://ieeexplore.ieee.org/document/4050644/

I also found this: https://electronics.stackexchange.c...-network-with-voltage-gain-greater-than-unity

 

[NascentOxygen]

You may be looking for this thread: https://www.physicsforums.com/threads/rc-networks.235927/

Indeed, that is almost certainly the thread I had in mind, from 2008. Have you archived the missing Figure, by any chance?

 

[the_emi_guy]

I simulated the circuit that is in the electronics.stackexchange link with LTspice. Sure enough, the gain is > 1 for all frequencies below ~2.2KHz.
The gain peaks to 1.15 at 1.1KHz.

Pretty awesome, I never thought about getting voltage gain out of an RC network.

 

[NascentOxygen]

I simulated the circuit that is in the electronics.stackexchange link with LTspice. Sure enough, the gain is > 1 for all frequencies below ~2.2KHz.
The gain peaks to 1.15 at 1.1KHz.

Not ALL lower frequencies, surely? Are you able to attach a graph?

 

[The Electrician]

Indeed, that is almost certainly the thread I had in mind, from 2008. Have you archived the missing Figure, by any chance?

I did not archive it, but somewhere I have a copy of the Epstein paper.

 

[the_emi_guy]

Graph attached.
Voltage gain is unity right at DC of course, but then moves north of unity until it crosses over at 2.2KHz
.

 

[Tom.G]

Well, just goes to prove that this is the place to learn something 'new'! The original paper was published in 1951. Here is a link that claims to supply it if you give them your e-mail and agree to be tracked, sold, etc.
https://www.researchgate.net/publication/3471098_Synthesis_of_Passive_RC_Networks_with_Gains_Greater_than_Unity

Here's one that claims a version with a gain of 1.47.
https://forum.allaboutcircuits.com/threads/rc-networks.2741/page-3

And one that claims it is impossible, until you get to the last three posts.
https://groups.google.com/forum/?hl=en#!topic/sci.electronics.design/mQqD52y6IXE

Thanks @NascentOxygen, for bringing this up/pointing it out.

 

[The Electrician]

Well, just goes to prove that this is the place to learn something 'new'! The original paper was published in 1951. Here is a link that claims to supply it if you give them your e-mail and agree to be tracked, sold, etc.
https://www.researchgate.net/publication/3471098_Synthesis_of_Passive_RC_Networks_with_Gains_Greater_than_Unity

Here's one that claims a version with a gain of 1.47.
https://forum.allaboutcircuits.com/threads/rc-networks.2741/page-3

And one that claims it is impossible, until you get to the last three posts.
https://groups.google.com/forum/?hl=en#!topic/sci.electronics.design/mQqD52y6IXE

Thanks @NascentOxygen, for bringing this up/pointing it out.

Here's another paper to get. The maximum gain depends on the order of the network, and apparently Fialkow/Gerst derive this: http://ieeexplore.ieee.org/document/4051325/

 

[NascentOxygen]

Graph attached.
Voltage gain is unity right at DC of course, but then moves north of unity until it crosses over at 2.2KHz

Very good. (I was distractedly thinking of the modified Twin Tee with its bandpass response.)

The effect of a load on the voltage enhancement is another consideration in the design of these.

An arrangement resembling a series pair of modified Bridged-Tee networks gives bandpass response with peak gain ×1.12 and zero phase.

An RC Circuit Giving Over-Unity Gain. Conrad L Longmire. TeleTech April 1947 pp 40-41,112. http://www.americanradiohistory.com/Archive-Tele-Tech/Tele-Tech-1947-04.pdf

 

[The Electrician]

Very good. (I was distractedly thinking of the modified Twin Tee with its bandpass response.)

The effect of a load on the voltage enhancement is another consideration in the design of these.

An arrangement resembling a pair of modified Bridged-Tee networks gives bandpass response with gain ×1.12 and zero phase. An RC Circuit Giving Over-Unity Gain. Conrad L Longmire. TeleTech April 1947 pp 40-41,112. http://www.americanradiohistory.com/Archive-Tele-Tech/Tele-Tech-1947-04.pdf

How did you find this? What search terms did you use? I've seen interesting references to that old magazine, Tele-Tech, but I would never have expected that such relatively obscure articles would be indexed.

 

[NascentOxygen]

How did you find this? What search terms did you use? I've seen interesting references to that old magazine, Tele-Tech, but I would never have expected that such relatively obscure articles would be indexed.

The article by Longmire was cited by authors in one of the google hits. If I couldn't have free access to the article, maybe I could read from its bibliography. This here: http://ieeexplore.ieee.org/document/4051325/references

So I simply needed to find an archive of the Tele-Tech publication.