

Suggestions for Circuit Design RF Cafe Forums

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
88108 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
General
Joined: Wed Feb 22,
2006 3:51 pm Posts: 104 Hi!
Two words: Smith Chart.
The Smith Chart is a graphical representation of the sparameters,
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 hv downloaded & tried Quicksmith. It is nice software.
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 lownoise 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 lowMHz 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:
50to75 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 sparameters 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 sparameters (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)(1S)^1
and the scattering (s) matrix from
the impedance matrix by
S=(Z1)*(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 50Ohmdenormalized Z matrix is 50 + 25j 25.+0j
750 + 125j 175 + 250j
Now, normalize to 75 Ohms by dividing
by 75. The 75Ohmnormalized 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. I tried to design in quicksmith with
all 50 ohm parameters. 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 88108 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
Site Admin
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 88108 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. waiting for reply,
Top
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
Top
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 cooperation & 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+j3.100
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 precanned 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+j3.100 as
impedance matching. Then combine this two circuits. This
way i will hv control over frequency response.
Waiting for
ur guidance & opinion,
Top
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 "cutandtry" 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+j3.100
But as there is no termination between
the two filters, so it will not work.
U said : u can sometimes
get OK results from "cutandtry" 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 rolloff 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 crossmodulation between
channels, and if you have an application where there are many strong
carriers being amplified with some near the band edges, the intermodulation
can be severe. For this reason a slower rolloff Butterworth
filter would be better. One supplementary question I would pose
for IR or Fred is : Is there a clean, wellbehaved way of reducing
gain and improving stability margin with a wideband 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 narrowband 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 lownoise amplifier, that's
not good. The resistor raises the noise figure. (There is an exception
 see below)
2. For narrowband 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 paralleltuned circuit "tanks" as the amplifier
output load.
3. The MMIC gain block amplifiers (like the
Minicircuits MAR1) 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 problemfilled 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
crossmodulation between channels, and if you have an application
where there are many strong carriers being amplified with some near
the band edges, the intermodulation can be severe. "
I
was under the impression that intermodulation resulted only from
nonlinear 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 ellipticfunction (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



