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# suggestions for circuit design- RF Cafe Forums

Because of the high maintenance needed to monitor and filter spammers from the RF Cafe Forums, I decided that it would be best to just archive the pages to make all the good information posted in the past available for review. It is unfortunate that the scumbags of the world ruin an otherwise useful venue for people wanting to exchanged useful ideas and views. It seems that the more formal social media like Facebook pretty much dominate this kind of venue anymore anyway, so if you would like to post something on RF Cafe's Facebook page, please do.

Below are all of the forum threads, including all the responses to the original posts.

jaya000
Post subject: suggestions for circuit design Posted: Tue Feb 20, 2007 8:10 am

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
To start learning RF i planned for one small signal 2 stage antenna preamplifier. Frequency range 88-108 Mhz.Mhz. Accordingly i hv done the following according to the bias condition required. .
I hv selected input transistor with following S parameters.

f S11 S21 S12 S22
GHz MAG ANG MAG ANG MAG ANG MAG ANG
0.10 0.71 -49.5 18.61 139 0.023 65.8 0.834 -22

Feeding this data into a software vierpol (two port in English) but in old version. Then i got following Zin, Zout. .

For input/output impedance BJT @100 Mhz. Calculated on Vierpol in Elekta
Polar Rect
Input impedance Zi = 102.100 -65.3 42.622 -92.668
Output Impedence Zo = 233.22 -64 102.237 -209.617

The output transistor with following S parameters.

FRE S11 S21 S12 S22 K MAG1
(MHz) MAG ANG MAG ANG MAG ANG MAG ANG (dB)
100 0. .45 -78 26.73 125 0.01 69 0.68 -31 0.77 34.3

For Input output impedance BJT @100 Mhz. Calculated on Vierpol in Elekta

Polar Rect
Input Impedance Zin = 58.490 -47.8 39.289 -43.33
Output Impedance Zout = 148.82 -52.5 90.596 -118.067

I got it in polar but also coverted into rectangular.
All this is with 50 ohm input output.

I am newbie in RF. Is this ok till now?
If this is ok then i will like to know from RF experts/Gurus here that which type of circuit i must use for input/output & interstage matching? Circuit input output impedance will be 75 ohms.

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fred47
Post subject: Suggestions for circuit designPosted: Tue Feb 20, 2007 1:05 pm

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Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi!

Two words: Smith Chart.

The Smith Chart is a graphical representation of the s-parameters, with a grid representing impedances relative to the system reference impedance (usually 50 Ohms, but in your case, I'd use 75 Ohms). You can still buy paper versions, but you can download a program called QuickSmith and let your computer do all the boring work. It's free, and a nice piece of work.

Good Luck!

Fred

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jaya000
Post subject: Posted: Tue Feb 20, 2007 10:54 pm

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
Thanks fred47,
As suggested by u, I hv downloaded the Quicksmith. Now just guide me how to get started.
Should i devide the zin zout by 50?
But then What to do for impedance of quicksmith chart? Should it be also normalised to 1?
I am newbie so needs help to start with. I hv not seen any document about smith chart matching with 75 ohms.

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mhtplsh
Post subject: Posted: Wed Feb 21, 2007 1:20 pm

Captain

Joined: Wed Feb 21, 2007 12:50 pm
Posts: 18
Fred47,
I am stucked here.
The noise parameter window asks for the following parameter.

G0 ( Mag)
G0 ( Deg)
Rn (Ohms)
Tell me how to get or calculate this.

If i set the smith chart to 75 ohms do i need to change the s parameters & zin Zout to 75 ohm as asked in some other post here.

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fred47
Post subject: QuickSmithPosted: Wed Feb 21, 2007 1:59 pm

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
jaya000:
Good observation - if you were using paper, you'd need to normalize by dividing by the system impedance (50 or 75 Ohms). For QuickSmith, you set the system impedance - the Center or "Prime" point - and use ordinary impedances.

Just enter the values - QuickSmith assumes 50+j0 Ohms for the Center "Prime" point unless you tell it differently with the "toolbox" (which you get to by clicking the wrench icon at the top of the chart page).

I've posted conversion details (50 to 75 Ohms) in another thread in this subject. But the impact of that is minimized by a program like QuickSmith.

mhtplsh: I'd be a genius if I could adequately summarize low-noise amplifier design in a short post. I'd suggest Guillermo Gonzales' book on microwave amplifier design. I'll see if I can come up with a quick procedure, but I'm not sure if I've got the time.

Good Luck!
Fred

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jaya000
Post subject: Posted: Thu Feb 22, 2007 2:16 am

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
Thanks Fred47,
Cannot find ur post of 50 to 75 ohm conversion.
Then on the web i got the 50 ohm to 75 ohm conversion .xls sheet. I can fill my 50 ohm S parameter, input output impedance & can get 75 ohm conversions. Elekta software can give direct 75 ohm input output impedance from 50 ohm S parameters. I will just confirm it today.
I can set quicksmith to 75 ohms for input matching circuit design.
Further there is Zin & Rx(L) on smithchart. I think Zin to be set to 75 ohms & Rx(L) will be input impedance of the device in 75 ohms.
will this be ok to proceed.

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mhtplsh
Post subject: Posted: Thu Feb 22, 2007 2:39 am

Captain

Joined: Wed Feb 21, 2007 12:50 pm
Posts: 18
Fred47 sir,
I got the info about the above mentioned parameters from a book.
The manufacturer NEC hv given noise figure for 500 Mhz minimum. I want to design premp for VHF. Hence cannot do anything about NF input in Quicksmith. I am thinking of practical thoughtful solution.
So to give some input NF in quicksmith with careful study of the datasheet, I hv come to conclusion that i can take noise figure of 1.25db for 500 mhz at 10V 7 ma & use it for 100 Mhz at 6.5V 13.5 ma.
There might some minor difference.

Further I hv checked the another transistor package with same chip for 200 Mhz & noted the noise figure. It is around 1.10db @ 200 mhz to 1.20db @ 500 Mhz at low 4.5V, 3ma.

But for practical purpose will this be ok?

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fred47
Post subject: ckt designPosted: Thu Feb 22, 2007 1:32 pm

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi mhtplsh!

Your NF assumptions look safe to me - even a bit conservative.

As a side note, at one time I needed a very low noise figure in the low-MHz range - the best I could do was with a [b]microwave[\b] transistor, suitably rolled off so it didn't oscillate. (And yes, it was an NEC transistor!)

Good Luck!
Fred

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fred47
Post subject: 50-to-75 OhmPosted: Thu Feb 22, 2007 1:34 pm

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi jaya000!

Here's a copy of the post you couldn't find:

Hi!
To get 75 Ohm s-parameters from 50 Ohm parameters, convert to impedance (for example, by plotting on a Smith Chart normalized to 50 Ohms, or by formula), then convert to 75 Ohm s-parameters (for example, by plotting on a Smith Chart normalized to 75 Ohms, or again by formula).

A reminder about the Smith Chart: it's normalized to a value of unity for the center point ("prime center", etc.). That means that every impedance you plot is divided by the actual impedance. Philip Smith did that so that you'd always be able to use the same chart. Straight impedances, like those you get from a data sheet, aren't normalized, so if they give a value like "10+j5 Ohms" you can normalize and plot like always.

That means that all the techniques that you've learned for 50 Ohms transfer directly to 75 Ohms.

Formula: the impedance matrix can be calculated from the scattering (s) matrix by this equation:

Z = (1+S)(1-S)^-1

and the scattering (s) matrix from the impedance matrix by

S=(Z-1)*(Z+1)^-1

A reminder that if S has eigenvalues of +1 or -1, Z doesn't exist. (This isn't usually a problem, but just in case...)

OK, an example (strictly made up, doesn't correspond to any likely part!):
S11 = 0.5+1.0j S12 = 0.1 + 0j
S21 = 3.0 - 0.5j S22 = 1.0 + 0.1j

The normalized Z matrix is
-1.0 + 0.5j -0.5 + 0j
-15 + 2.5j -3.5 + 5.0j

and the 50-Ohm-denormalized Z matrix is
-50 + 25j -25.+0j
-750 + 125j -175 + 250j

Now, normalize to 75 Ohms by dividing by 75. The 75-Ohm-normalized Z matrix is
-0.66667 + 0.33333j -0.33333 + 0.00000j
-10.00000 + 1.66667j -2.33333 + 3.33333j

Finally, convert back to the scattering (s) matrix:
0.16578 + 1.15508j 0.12834 + 0.03209j
4.01070 + 0.32086j 1.09626 + 0.16043j

I got the answers with the help of the GPL program Octave - makes life a lot easier than using a calculator!

Good Luck!
Fred

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jaya000
Post subject: Posted: Thu Feb 22, 2007 10:11 pm

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
To learn i decided to first design with 50 ohm input output.
Opened Quicksmith & selected amplifier design.

I inserted following S parameters with polar input selected
f S11 S21 S12 S22
! GHz MAG ANG MAG ANG MAG ANG MAG ANG
0.10 0.71 -49.5 18.61 139 0.023 65.8 0.834 -22

Got K = 0.783 Delta = 0.0426
Potentially unstable transistor

Then entered noise parameters
NF GO ANGLE Rn
1.15 .18 126 7.5
In the literature it is given as Rn/50 = .15
So I multiplied it with 50. Hence entered 7.5 is this ok?

Set chart parameter to 50 ohms.
Then on the circle menu selected stability circles.
Got the message : For both Source plane & Load plane : The stability region is outside the circle

Then selected available gain circle.
It shows that MAG is 29.078 but i selected 20db gain as transistor was unstable K =0.783
Then selected the noise circle as 1.5 db. as the 1.2 db circle was too small.

Then selected Source impedance & Load impedeance from transfer
I got the following readings :
GammaS 1.056 <49.289
Zs = -4.086+j70.525
Ga = 20db NF 1.5db
GammaL = 1.014<49.281
ZL = -2.015+j108.975

Then i clicked back button.
Guide me now how to go further.
My goal is to design 88-108 mHz input filter with mimimum .1db or less ripple in the required

bandwidth.

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IR
Post subject: Posted: Fri Feb 23, 2007 3:29 am

Joined: Mon Jun 27, 2005 2:02 pm
Posts: 373
Location: Germany
As a general procedure you should design your matching to satisfy stability, Ga and NF requirements.

When you display on the Smith Chart the Gain, Noise circles then at some areas these circles are intersecting together. These intersection points will satisfy all of the required parameters. Thes point on the Smith Chart are the required impedances. Of course that you have to assure that these impedances are also at the region stable to avoid oscillations!

Your matching networks should convert 50 ohm to these impedances...

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fred47
Post subject: Amplifier designPosted: Fri Feb 23, 2007 12:58 pm

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Thanks IR - that's a good terse statement!

Hi jaya000,

0.1 dB gain flatness seems rather tight for the broadcast band, where antenna matching is hardly that good.

There's a lot unmentioned that you need to think about:
1. What does the response need to be outside the 88-108 MHz band? Do you need a tight bandpass response to eliminate interference, or does it not matter at all?

2. What kind of return loss do you need at the input and the output? (That is, how accurate a 50 or 75 Ohm impedance do you need?)

In general, you can't have both good filtering and good impedance matching over a broad frequency range.

Do you have a good university library available to you? (In other words, would it do any good to give you the names of some books on the subject of amplifier design & impedance matching, or would that be unhelpful?)

Good Luck,
Fred

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jaya000
Post subject: Posted: Fri Feb 23, 2007 3:45 pm

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
Thanks fred47 & IR,
I can get the book i want due to my good friend circle.
Even one friend ( not a RF man ) told me that he can arrange his company computer with serenade 8 on weekends if i want to learn.
So just guide me, i will do it with all my energy & resources.
I will try to look into the suggestion of IR.
I got stuckup at the point as mentioned above. I am not able to move ahead.

The gain circle & noise circle r not intersecting anywhere. The noise cicle is small & nearly in the center. The gain circle is almost at outside of smith chart.
Filter ripple is not important for me. But able to design with ur guidance is important. The amplifier is to boost fm signals for my car.
waiting for ur new suggestions on how to proceed further with Quicksmith chart.

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jaya000
Post subject: Posted: Sat Feb 24, 2007 4:55 am

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
Then entered noise parameters
NF GO ANGLE Rn
1.15 .18 126 7.5
In the literature it is given as Rn/50 = .15
So I multiplied it with 50. Hence entered 7.5 is this ok?

I admitted Rn= .15, then the noise circle became big & intersected the gain circle.

Now let me know what u would hv proceeded further to design 88~108 mhz.

This will help me to go further from where i got stucked.

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fred47
Post subject: amplifier designPosted: Tue Feb 27, 2007 12:50 am

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi jaya000,
Two very popular books on amplifier design:

Gonzalez, Guillermo,Microwave Transistor Amplifiers: Analysis & Design, 2nd edition.

Abrie, Peter,The Design of Microwave Amplifiers and Oscillators.

BTW, I don't get the same numbers you got - K = 0.325 was what I got, and Delta = 0.692; NFmin came out with 2.9 dB NF, not 1.15. I don't know what the difference is.

You might want to find a transistor that's not quite so "hot" - that's a lot of gain available!

Good Luck,
Fred

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jaya000
Post subject: Posted: Tue Feb 27, 2007 2:33 pm

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
Respected Fred47 :
Thank u very much for ur co-operation & help till now.

U said :
"BTW, I don't get the same numbers you got - K = 0.325 was what I got, and Delta = 0.692; NFmin came out with 2.9 dB NF, not 1.15. I don't know what the difference is. "

Yes u r right. I hv one typing decible mistake. I hv corrected & got the same K = 0.325 and Delta = 0.692.
But i hv NF data from the manufacturer as below.

Noise parameter
Nf =1.15 GO .18 ang. 126 Rn/50 = .15

Hence i did the following to go further.

The Maximum Gain available is 29.080 but the stability facor K is very poor .325
Hence decided for 16 db gain so that i will finally get 13~14 db.
The noise circle with 1.2db is inside the gain circle. Then increased the noise upto 1.33db so that noise circle intersects gain circle.
Here i got the following :

GammaS = 1.026 < 132.591 Zs = -0.758+j21.949

GammaL = 1.555 <10.463 Zl = -267.655+j335.299

Now i want to design one matching Bandpass filter for 88~108 Mhz. The ripple can be .25db to .5db at the input. For output i will prefer to use 3 pole low pass filter or just impedance matching LC circuit.
Guide me how to proceed further?
I bought the Gonzalez, Guillermo,Microwave Transistor Amplifiers: Analysis & Design, 2nd edition. book. It is quite expensive. it costed me US\$ 95.00.

If this is ok, then guide me for final stage of the BP filter design help.

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fred47
Post subject: Amplifier designPosted: Wed Feb 28, 2007 3:41 am

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi jaya000,

OOPS!!! STOP RIGHT THERE!!! IT'S NOT WORTH GOING ON!

A negative real part of the input or output impedance means you almost certainly have an oscillator, not an amplifier.

You wrote:
GammaS = 1.026 < 132.591 Zs = -0.758+j21.949
-------------------------------------------/\
GammaL = 1.555 <10.463 Zl = -267.655+j335.299
----------------------------------------/\

The arrows point to the negative real impedances you report.

I'd seriously suggest a different transistor!

Good Luck!
Fred

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jaya000
Post subject: Posted: Wed Feb 28, 2007 11:32 pm

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
Thanks fred47.
I got it now. It is due to u i learned a lot.
For me this will be a new start.
I will now restart with some other transistor as suggested by u.
with regards
Jaya000

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jaya000
Post subject: Posted: Thu Mar 01, 2007 2:31 am

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
I hv seen this transistor used in Japanese vhf boosters. Hence i once more played with quicksmith. Due to help from fred47 i came to know the problem. Hence i increased the NF circle & got the following readings.

Gamma S = 0.987 < -172.904
Zs = 0.335+j-3.100

Ga = 16.00db
Fi = 1.45db

Gamma L = 0.953 < 6.701
ZL = 299.09+j732.347

Now this r positive. Suggest me now how to go ahead.

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fred47
Post subject: circuit design suggestionsPosted: Fri Mar 02, 2007 12:18 am

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi jaya000!

The next step is to do the impedance matching.

QuickSmith makes that relatively easy - it will transfer your calculated information to the Smith Chart. Then you observe where you are with respect to the prime center, and arrange to get there by moving along the reactance and susceptance (1/X) circles, depending on whether you are adding a capacitor or inductor in series or in parallel.

You need to do this for both input and output.

Then you check to see if your design meets your bandwidth requirements. Again, QuickSmith provides the "Sweep" function so you can look at S21 (the "forward gain" - the spec of most interest to most people.)

If you're lucky, it's OK. If not, then you're up for either using a pre-canned program such as Eagleware Genesis, AWR Microwave Office, or Agilent HFSS, etc. Those are generally pretty pricy - you might want to see if you can get access to one via a friend or a professor.

The alternative is to learn how to do broadband - and that's a whole 'nother book, my friend. (Specifically, Carlin's Wideband Circuit Design <grin> ). I do seriously hope that you're lucky!

Fred

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jaya000
Post subject: Posted: Sat Mar 03, 2007 11:13 pm

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
thanks fred47.
I got different idea to calculate input/output filter.
just let me tell u about input filter.
As the s parameters r 50 ohms, and i want 75 ohm input
I will calculate 75ohm to 50 ohm bandpass filter first by software.
then i will calculate simple LC filter for 5o ohms to Zs = 0.335+j-3.100 as impedance matching.
Then combine this two circuits.
This way i will hv control over frequency response.

Waiting for ur guidance & opinion,

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fred47
Post subject: Amplifier inputPosted: Sun Mar 04, 2007 1:12 am

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi jaya000!

Your proposal sounds good - but it really doesn't work that way.

Think about how a filter works for a moment.

The energy at the frequencies you wants goes through, the energy at the frequencies you don't want doesn't go through. What happens to that energy?

The short answer is that it gets reflected. The longer answer is that a filter is (usually) made up of only reactive parts - coils and capacitors - and not dissipative parts like resistors. So the energy can't go through the filter, can't disappear in the filter - it must be reflected.

For energy to be reflected, the impedance must change - if there were no change in impedance, the energy would just continue on forward. The specification that tells you this is called "Return Loss", and for a filter, outside of the passband, that's close to zero dB - all the energy hitting the filter is reflected.

For this to happen, the filter must be properly terminated - and that usually means "with the characteristic impedance". In your case, it almost doesn't matter whether that's 75 or 50 Ohms.

Your "simple LC filter" to convert from 50 ohms to 0.335 - j3.1 (did you really mean 50 x 0.33f - j3.1 ?) will not look like a 50 Ohm resistor, regardless - so it will dramatically affect the performance of your filter.

It's really all ONE problem, which you can't "divide and conquer". That's why Agilent, Eagleware, Advanced Wave Research, etc. can charge so much for their computer programs - some problems are just really HARD.

You can sometimes get OK results from "cut-and-try" with a simulator, such as Linear Technology's free SwitcherCAD/LTSpice. Of course, no simulator gives perfect results at those frequencies, due to unmodeled aspects of the components. But it helps get you in the right ballpark, at any rate.

Good Luck!
Fred

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jaya000
Post subject: Posted: Sun Mar 04, 2007 7:31 am

Captain

Joined: Tue Feb 20, 2007 5:43 am
Posts: 14
Respected FRED47,
Thanks for ur nice answer. It is painful to know that u hv got cold. Hence reply when u r ok.

U said :
For this to happen, the filter must be properly terminated - and that usually means "with the characteristic impedance". In your case, it almost doesn't matter whether that's 75 or 50 Ohms.

Your "simple LC filter" to convert from 50 ohms to 0.335 - j3.1 (did you really mean 50 x 0.33f - j3.1 ?) will not look like a 50 Ohm resistor, regardless - so it will dramatically affect the performance of your filter.

This means u can't combine two filters.

My combine means i was supposed use the bandpass filter from the 75 ohm side. Then at 50 ohm side i would hv connected ( putting in series) it to another filter with one side as 50 ohms & another side to Zs 0.335+j3.100 with smithchart impedance selected for 50 ohms as mentioned below. This figures r from quicksmith

Gamma S = 0.987 < -172.904
Zs = 0.335+j-3.100

But as there is no termination between the two filters, so it will not work.

U said : u can sometimes get OK results from "cut-and-try" with a simulator

Then how about design & assemble some filter with active circuit & try & play with sweep generator. My friend has got one.

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nubbage
Post subject: Posted: Mon Mar 05, 2007 5:29 am

General

Joined: Fri Feb 17, 2006 12:07 pm
Posts: 218
Location: London UK
Hi All contributers to this excellent topic.
I have sat on the side reading progress. I had no experience to contribute to active ie amplifier, circuits.
Now the attention has moved to the filter elements, it is more in my territory.
It is tempting to use a rapid roll-off filter for applications like this, so an ellyptic function or a Chebychev type might be selected.
However, not only the impedance varies rapidly at the band edge, but the plot of phase vs frequency oscillates rapidly. The end effect of the latter is cross-modulation between channels, and if you have an application where there are many strong carriers being amplified with some near the band edges, the inter-modulation can be severe.
For this reason a slower roll-off Butterworth filter would be better.
One supplementary question I would pose for IR or Fred is :
Is there a clean, well-behaved way of reducing gain and improving stability margin with a wide-band amplifier using negative feedback, perhaps overall from output back to input? Intuitively, this might lead to oscillation outside the band where the relative phase of S21 cannot be certain, and hence goes from negative feedback to positive.

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fred47
Post subject: circuit stuffPosted: Mon Mar 05, 2007 2:05 pm

General

Joined: Wed Feb 22, 2006 3:51 pm
Posts: 104
Hi nubbage,

I may be showing my age, but amplifier neutralization (and even unilateralization) has been around for a long time. But it's mostly a relatively narrow-band technique.

I think negative feedback is great - gain is cheap, and negative feedback buys lower distortion, more stable characteristics, and some other benefits as well, but there are a few "interesting points":

1. Broadband circuits can't really use reactive components for the feedback - the variation in impedance makes truly broadband operation unreasonable. That leaves resistive feedback - and for a low-noise amplifier, that's not good. The resistor raises the noise figure. (There is an exception - see below)

2. For narrow-band circuits, the traditional oscillation avoidance technique of limiting the range over which the amplifier has greater than unity gain works well. Traditionally, this has been done with parallel-tuned circuit "tanks" as the amplifier output load.

3. The MMIC gain block amplifiers (like the Minicircuits MAR-1) have resistive negative feedback built in. If I had to create an amplifier quickly, I might well use those instead of discretes.

4. There exists a set of techniques called "noiseless feedback", which use either transformers or directional couplers. This is a fascinating but problem-filled area - parasitics are an amazingly difficult problem. I have designs in production which use this approach - but there was a lot of work to get there.

Nubbage, I don't quite follow your statement "the plot of phase vs frequency oscillates rapidly. The end effect of the latter is cross-modulation between channels, and if you have an application where there are many strong carriers being amplified with some near the band edges, the inter-modulation can be severe. "

I was under the impression that intermodulation resulted only from non-linear effects - and phase shift "distortion" is a linear effect. ("Linear" in this case having the operational meaning that multiplying the input waveform by a constant doesn't change the output waveform in any way other than its amplitude).

You might be thinking on a system level, where the amplifier following the filter creates the IM - but I'm not sure how to interpret your statement. You might also be thinking about FM demodulation, where the "capture effect" comes into play. FM demodulation, of course, is also a nonlinear operation.

(The first part of your statement, about the rapid change in phase shift for elliptic-function (Cauer) and Chebyshev filters is, of course, undeniably true, and it's absolutely true that ignoring that aspect can get you into trouble.)

Thanks!
Fred

Posted  11/12/2012
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