Transfer function challenge #3

[The Phantom]

A modern active filter building block is the GIC (generalized
immittance converter). Descriptions can be found in many textbooks
with expressions for the relevant properties, but these usually assume
the amplifier gains are infinite.

For the circuit shown, find the input impedance for finite amplifier
gains. There are two amplifiers of gain A1 and A2, as shown. There
are five impedances, Z1 through Z5. Assume the nodes are numbered 1
through 5, with node 1 being the input, node 2 between Z1 and Z2, node
3 between Z2 and Z3, etc.

Obtain a final result in the form of a fraction with the numerator
consisting of a sum of products with no fractions, and the denominator
likewise. Leave the numerator and denominator unfactored so that
comparison with other people's results is facilitated. The numerator
and denominator should look *something* like this:

Z1 Z2 Z3 A1 + Z5 A1 A2 + Z2 Z4 Z5 A2 + Z3 Z4 Z5 A1 + ...
---------------------------------------------------------------------
Z2 Z4 Z5 A2 + Z1 Z2 Z3 Z4 A1 A2 + Z2 Z3 Z4 + Z1 Z5 + A1 Z1 Z2 +...


Since you have the general expression with finite amplifier gains, you
can then substitute an expression for amplifier gain with rolloff and
see what the effect of amplifier imperfection is.

Or, let the amplifier gains go to infinity, and find the limiting
input impedance for that case.

You might use a different method to solve this problem if you have
access to a computer algebra system (CAS) such as Mathcad, than if you
do it all by hand. That would be an interesting topic for discussion:
how best to solve this kind of problem given modern computer
mathematical assistants.


Vin ----o-----------.
| |
.-. |
Z1 | | |
| | |
'-' .-----. |
.---------| | | |
| | | | |
| .-. | .-----.
| Z2 | | | \- +/
| | | | \ / A1
| '-' | V
A2 /^\ .---|---' |
/+ -\ | | |
'-----' | .-. |
| | | | | Z3 |
| | | | | |
| '----' '-' |
| |----------'
| |
| .-.
| Z4 | |
| | |
| '-'
'----------|
|
.-.
Z5 | |
| |
'-'
|
GND ---
-
created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de

 

[Winfield Hill]

The Phantom wrote...

A modern active filter building block is the GIC (generalized
immittance converter). Descriptions can be found in many textbooks
with expressions for the relevant properties, but these usually
assume the amplifier gains are infinite.
[ big snip ]
Since you have the general expression with finite amplifier gains,
you can then substitute an expression for amplifier gain with rolloff
and see what the effect of amplifier imperfection is.

Or go to Adel Sedra and Peter Brackett's 1978 book and get the answers.

 

[Jim Thompson]

[snip]

Vin ----o-----------.
| |
.-. |
Z1 | | |
| | |
'-' .-----. |
.---------| | | |
| | | | |
| .-. | .-----.
| Z2 | | | \- +/
| | | | \ / A1
| '-' | V
A2 /^\ .---|---' |
/+ -\ | | |
'-----' | .-. |
| | | | | Z3 |
| | | | | |
| '----' '-' |
| |----------'
| |
| .-.
| Z4 | |
| | |
| '-'
'----------|
|
.-.
Z5 | |
| |
'-'
|
GND ---
-
created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de

As has been discussed here (and A.B.S.E), this gyrator configuration
is non-optimum. This version I developed moves nodes slightly and
yields better performance per GBW... see my website, SED/Schematics
page.

...Jim Thompson

 

[Jim Thompson]

The Phantom wrote...

A modern active filter building block is the GIC (generalized
immittance converter). Descriptions can be found in many textbooks
with expressions for the relevant properties, but these usually
assume the amplifier gains are infinite.
[ big snip ]
Since you have the general expression with finite amplifier gains,
you can then substitute an expression for amplifier gain with rolloff
and see what the effect of amplifier imperfection is.

Or go to Adel Sedra and Peter Brackett's 1978 book and get the answers.

Anyone heard from Peter since the first hurricane? He lives in
Indialantic-by-the-Sea, Florida.

...Jim Thompson

 

[Spehro Pefhany]

The Phantom wrote...

A modern active filter building block is the GIC (generalized
immittance converter). Descriptions can be found in many textbooks
with expressions for the relevant properties, but these usually
assume the amplifier gains are infinite.
[ big snip ]
Since you have the general expression with finite amplifier gains,
you can then substitute an expression for amplifier gain with rolloff
and see what the effect of amplifier imperfection is.

Or go to Adel Sedra and Peter Brackett's 1978 book and get the answers.

Prof. Sedra liked it so much that it's featured on the cover..

Best regards,
Spehro Pefhany

 

[xray]

Anyone heard from Peter since the first hurricane? He lives in
Indialantic-by-the-Sea, Florida.

FWIW My cousin does too and they escaped without any signinfcant damage
(leeward side). Hopefully Peter was as fortunate. Oh, it would have been
the 2nd one that was challenging in Indiatlantic.

 

[The Phantom]

[snip]

Vin ----o-----------.
| |
.-. |
Z1 | | |
| | |
'-' .-----. |
.---------| | | |
| | | | |
| .-. | .-----.
| Z2 | | | \- +/
| | | | \ / A1
| '-' | V
A2 /^\ .---|---' |
/+ -\ | | |
'-----' | .-. |
| | | | | Z3 |
| | | | | |
| '----' '-' |
| |----------'
| |
| .-.
| Z4 | |
| | |
| '-'
'----------|
|
.-.
Z5 | |
| |
'-'
|
GND ---
-
created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de

As has been discussed here (and A.B.S.E), this gyrator configuration
is non-optimum.

I picked it because it has a pleasing symmetry, not because of any
performance advantage or disadvantage. That would be an interesting
topic for another thread. This one's purpose is, as you said:

"...I'd like to have the young-bucks here actually do the grunt work
and get proficient."

This version I developed moves nodes slightly and

 

[Dr. Neutron]

The Phantom wrote...

A modern active filter building block is the GIC (generalized
immittance converter). Descriptions can be found in many textbooks
with expressions for the relevant properties, but these usually
assume the amplifier gains are infinite.
[ big snip ]
Since you have the general expression with finite amplifier gains,
you can then substitute an expression for amplifier gain with rolloff
and see what the effect of amplifier imperfection is.

Or go to Adel Sedra and Peter Brackett's 1978 book and get the answers.

That wouldn't be playing the game. Kind of like the physics test question that says,
"You are standing on the street in front of a tall building with a mercury barometer. How
can you determine the height of the building?"

The street-wise kid says, "I'd find the building super and tell him, 'I'll give you
this excellent mercury barometer if you'll tell me the height of the building'".

(Then there's the fellow who says, "I'd go to the top of the building and drop the
barometer off and time how long it takes to hit the sidewalk. From that I can determine
the height of the building".)

Look it up only to verify your own work.

 

[Ban]

[snip]

Vin ----o-----------.
| |
.-. |
Z1 | | |
| | |
'-' .-----. |
.---------| | | |
| | | | |
| .-. | .-----.
| Z2 | | | \- +/
| | | | \ / A1
| '-' | V
A2 /^\ .---|---' |
/+ -\ | | |
'-----' | .-. |
| | | | | Z3 |
| | | | | |
| '----' '-' |
| |----------'
| |
| .-.
| Z4 | |
| | |
| '-'
'----------|
|
.-.
Z5 | |
| |
'-'
|
GND ---
-
created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de

As has been discussed here (and A.B.S.E), this gyrator configuration
is non-optimum.

I picked it because it has a pleasing symmetry, not because of any
performance advantage or disadvantage. That would be an interesting
topic for another thread. This one's purpose is, as you said:

"...I'd like to have the young-bucks here actually do the grunt work
and get proficient."

This version I developed moves nodes slightly and

yields better performance per GBW... see my website, SED/Schematics
page.

...Jim Thompson

I remember a famous audio A/D converter manufacturer showing reconstruction
filters in his data sheets with this gyrator based circuit. But they have
magically disappeared and were replaced with simple Sallen-Key type
lowpasses. The idea of Cauer (elliptic) filters doesn't go very well with
high-end audio, and anyway with oversampling have become useless.
I couldn't get in my simulation programm a stable output with this gyrator,
when simulating a "super-capacitor" with Z1 and Z5 being capacitors. Bad
alternating bursts were showing up, now I understand why AD thew that
structure out.
There are better ways to form BiQuad filters, I use a 5opamp approach, which
lets me directly and independently implement the coefficients or a 3 opamp
state variable filter, which has 3 resistors to programm "f" and "Q" for
numerator and denominator. There are also solutions with 1 and 2 opamps, but
they do not separate the coefficients and are difficult to adjust.

 

[Jim Thompson]

On Thu, 09 Sep 2004 18:12:26 -0700, Jim Thompson


I picked it because it has a pleasing symmetry, not because of any
performance advantage or disadvantage. That would be an interesting
topic for another thread. This one's purpose is, as you said:

"...I'd like to have the young-bucks here actually do the grunt work
and get proficient."

[snip]

Unfortunately it rarely happens :-(

...Jim Thompson

 

[Roy McCammon]

A modern active filter building block is the GIC (generalized
immittance converter). Descriptions can be found in many textbooks
with expressions for the relevant properties, but these usually assume
the amplifier gains are infinite.

For the circuit shown, find the input impedance for finite amplifier
gains. There are two amplifiers of gain A1 and A2, as shown. There
are five impedances, Z1 through Z5. Assume the nodes are numbered 1
through 5, with node 1 being the input, node 2 between Z1 and Z2, node
3 between Z2 and Z3, etc.

Obtain a final result in the form of a fraction with the numerator
consisting of a sum of products with no fractions, and the denominator
likewise. Leave the numerator and denominator unfactored so that
comparison with other people's results is facilitated. The numerator
and denominator should look *something* like this:

Z1 Z2 Z3 A1 + Z5 A1 A2 + Z2 Z4 Z5 A2 + Z3 Z4 Z5 A1 + ...
---------------------------------------------------------------------
Z2 Z4 Z5 A2 + Z1 Z2 Z3 Z4 A1 A2 + Z2 Z3 Z4 + Z1 Z5 + A1 Z1 Z2 +...


Since you have the general expression with finite amplifier gains, you
can then substitute an expression for amplifier gain with rolloff and
see what the effect of amplifier imperfection is.

Or, let the amplifier gains go to infinity, and find the limiting
input impedance for that case.

You might use a different method to solve this problem if you have
access to a computer algebra system (CAS) such as Mathcad, than if you
do it all by hand. That would be an interesting topic for discussion:
how best to solve this kind of problem given modern computer
mathematical assistants.


Vin ----o-----------.
| |
Z1 |
| .-----. |
V2 .---------| | | |
| | | .-----.
| Z2 | \- +/
| | | \ / A1
| | | V
A2 /^\ .---|---' V3 |
/+ -\ | | |
'-----' | Z3 |
| '----' | |
| |----------' V4
| |
| Z4
| |
'----------| V5
|
Z5
|
GND

Easy enough.
V5 = V3 = Vin
I5 = V5/Z5
I4 = I5
V4 = V5+Z4*I4
I3 = (V3-V4)/Z3
I2 = I3
V2 = V3 + Z2*I2
I1 = (Vin - V2)/Z1
Zin = Vin/I1

 

[Jim Thompson]

[snip]

Easy enough.
V5 = V3 = Vin

This is only true if the amplifier gains are infinite. You lose points for not
following directions. As I said above:
"For the circuit shown, find the input impedance for finite amplifier gains."

Not as easy.

But this does raise an interesting additional question. If the amplifier gains are
finite, what additional condition(s) would be required so that V5=V3, or so that V3=Vin.
Would it be possible at all?

[snip]

Not only are the gains finite, but they have, over most of their
useful frequency range, a phase shift of 90°.

...Jim Thompson

 

[The Phantom]

Easy enough.
V5 = V3 = Vin

This is only true if the amplifier gains are infinite. You lose points for not
following directions. As I said above:
"For the circuit shown, find the input impedance for finite amplifier gains."

Not as easy.

But this does raise an interesting additional question. If the amplifier gains are
finite, what additional condition(s) would be required so that V5=V3, or so that V3=Vin.
Would it be possible at all?

I5 = V5/Z5
I4 = I5
V4 = V5+Z4*I4
I3 = (V3-V4)/Z3
I2 = I3
V2 = V3 + Z2*I2
I1 = (Vin - V2)/Z1
Zin = Vin/I1

Again, as I said above:
Obtain a final result in the form of a fraction with the numerator consisting of a sum of
products with no fractions, and the denominator likewise.

 

[The Phantom]

[snip]

Vin ----o-----------.
| |
.-. |
Z1 | | |
| | |
'-' .-----. |
.---------| | | |
| | | | |
| .-. | .-----.
| Z2 | | | \- +/
| | | | \ / A1
| '-' | V
A2 /^\ .---|---' |
/+ -\ | | |
'-----' | .-. |
| | | | | Z3 |
| | | | | |
| '----' '-' |
| |----------'
| |
| .-.
| Z4 | |
| | |
| '-'
'----------|
|
.-.
Z5 | |
| |
'-'
|
GND ---
-
created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de

As has been discussed here (and A.B.S.E), this gyrator configuration
is non-optimum. This version I developed moves nodes slightly and
yields better performance per GBW.

How did you model the limited GBW of the amps? A single pole
rolloff? Can you post the model and the numbers for the parameters of
the rolloff?

.. see my website, SED/Schematics

 

[Jim Thompson]

On Thu, 09 Sep 2004 18:12:26 -0700, Jim Thompson
[snip]

As has been discussed here (and A.B.S.E), this gyrator configuration
is non-optimum. This version I developed moves nodes slightly and
yields better performance per GBW.

How did you model the limited GBW of the amps? A single pole
rolloff? Can you post the model and the numbers for the parameters of
the rolloff?

.. see my website, SED/Schematics

page.

See "Op-Amp-Config.zip" on the Subcircuits & Symbols page of my
website. It is modeled just like a real op-amp: slewed pole-splitting
capacitor plus an excess phase all-pass to emulate the delay in the
devices.

...Jim Thompson

 

[Winfield Hill]

Ban wrote...

I remember a famous audio A/D converter manufacturer showing reconstruction
filters in his data sheets with this gyrator based circuit. But they have
magically disappeared and were replaced with simple Sallen-Key type
lowpasses. The idea of Cauer (elliptic) filters doesn't go very well with
high-end audio, and anyway with oversampling have become useless.
I couldn't get in my simulation programm a stable output with this gyrator,
when simulating a "super-capacitor" with Z1 and Z5 being capacitors.

You have to have proper models for the components (capacitor esr, opamp
series output inductance etc.), and moreover, mature complete designs
add a single stabilizing part for robustness.

Bad alternating bursts were showing up, now I understand why AD thew
that structure out.

Very good, please enlighten us.

 

[Roy McCammon]

Very good, please enlighten us.

I've got several 10's of thousnads of notch filters based
on that topology out in the field for 20+ years. My first
prototype oscilated, but I redid it with lower impedences
and it worked fine ever sense. Though, I have had a very
good analog designer tell me that he had had problems with
that topology.

 

[Winfield Hill]

Roy McCammon wrote...

I've got several 10's of thousnads of notch filters based
on that topology out in the field for 20+ years. My first
prototype oscillated, but I redid it with lower impedences
and it worked fine ever sense.

Excellent, I'd really like to have a go at the rejected
version, do you have any around or can you give us the
exact circuit and part values?

 

[Roy McCammon]

Roy McCammon wrote...



Excellent, I'd really like to have a go at the rejected
version, do you have any around or can you give us the
exact circuit and part values?

the best I can do is show the working version and try to guess at the
old values.

 

[Rich Grise]

The Phantom wrote...

A modern active filter building block is the GIC (generalized
immittance converter). Descriptions can be found in many textbooks
with expressions for the relevant properties, but these usually
assume the amplifier gains are infinite.
[ big snip ]
Since you have the general expression with finite amplifier gains,
you can then substitute an expression for amplifier gain with rolloff
and see what the effect of amplifier imperfection is.

Or go to Adel Sedra and Peter Brackett's 1978 book and get the answers.

That wouldn't be playing the game. Kind of like the physics test
question that says,
"You are standing on the street in front of a tall building with a mercury
barometer. How can you determine the height of the building?"

The street-wise kid says, "I'd find the building super and tell him,
'I'll give you
this excellent mercury barometer if you'll tell me the height of the
building'".

(Then there's the fellow who says, "I'd go to the top of the building
and drop the
barometer off and time how long it takes to hit the sidewalk. From that I
can determine the height of the building".)

Look it up only to verify your own work.

You left out, measure your own shadow and the shadow of the building at
the same time, and do similar triangles. Use the barometer to figure out
where to go have a pina colada. %-}

Cheers!
Rich