"Stretching" an inductor

[Joel Kolstad]

Does anyone know of a simple network that has, say, more than twice the
reactance at 2*f0 as it does at f0? Essentially I'm after a reactance of
2*pi*f*(something super-linear).

You can readily convince yourself that going the other way is no problem... if
you put a small capacitor in series with an inductor, the slope of the
reactance is lowered (since the capacitor's reactance is dropping with
frequency), hence making the reactance at 2*f0 less than twice what the
circuit has at f0.

---Joel

 

[Jim Thompson]

Does anyone know of a simple network that has, say, more than twice the
reactance at 2*f0 as it does at f0? Essentially I'm after a reactance of
2*pi*f*(something super-linear).

You can readily convince yourself that going the other way is no problem... if
you put a small capacitor in series with an inductor, the slope of the
reactance is lowered (since the capacitor's reactance is dropping with
frequency), hence making the reactance at 2*f0 less than twice what the
circuit has at f0.

---Joel

Huh? Won't a parallel tank do what you want? Although you're saying
reactance, rather than "real".

...Jim Thompson

 

[John Popelish]

Does anyone know of a simple network that has, say, more than twice the
reactance at 2*f0 as it does at f0? Essentially I'm after a reactance of
2*pi*f*(something super-linear).

You can readily convince yourself that going the other way is no problem... if
you put a small capacitor in series with an inductor, the slope of the
reactance is lowered (since the capacitor's reactance is dropping with
frequency), hence making the reactance at 2*f0 less than twice what the
circuit has at f0.

There may be some version of a generalized impedance converter that
produces a resistive impedance proportional to frequency squared. I
have seen some that produce a negative resistance inversely
proportional to frequency squared, i.e. -1/D*w^2, where D is a
constant based on resistor and capacitor values in the circuit.

 

[Tom Bruhns]

Huh?? If I put 100nH in series with 30pF, at 100MHz the net reactance
is +j9.78 ohms; at 200MHz, it's +j99.14 ohms. Last time I checked, 99
is more than twice 9.8, not less. In general, if you double any
frequency above the resonance, you get a reactance more than twice as
high, though the effect is the strongest for starting frequencies near
the resonance.

Cheers,
Tom

 

[TuT]

There may be some version of a generalized impedance converter that
produces a resistive impedance proportional to frequency squared. I have
seen some that produce a negative resistance inversely proportional to
frequency squared, i.e. -1/D*w^2, where D is a constant based on resistor
and capacitor values in the circuit.

That's an FDNR, (Frequency Dependant Negative Resistor), and can be realised
with the basic four-op-amp gyrator circuit using two capacitors and three
resistors. You can get a negative resistance that either increases or
decreases as the square of the frequency, depending where you put the
capacitors, but I've never found a way of doing anything useful with this
circuit.

You can make a frequency dependant voltage divider with an FNDR and a
resistor, but the circuit goes through a phase inversion, the point of
inflection being at the frequency where -R = R

 

[TuT]

That's an FDNR, (Frequency Dependant Negative Resistor), and can be
realised with the basic four-op-amp gyrator circuit . . .

[snip]

Make that "two-op-amp gyrator circuit".

 

[Tom Bruhns]

You wrote, ". . . but I've never found a way of doing anything useful
with this circuit."

If as you wrote in the paragraph after that, you put it in series with
a resistor, you get a circuit that behaves much like a series resonant
circuit. Not only does it go through a phase inversion, but it goes
through zero resistance. Similarly if you put it in parallel with a
resistor, you get something that behaves much like a parallel resonant
circuit: where the gyrator negative resistance equals the parallel
positive resistance, the net resistance goes to infinity. If you think
of C-R-L circuits (or better, 1/sC, R, sL) as having components that
contribute in quadrature, with 90 degrees going from C to R and R to L,
you see that you can have the same effect with three components that
behave as R, sL and s^2Gyrator; you've just multiplied everything by s.
Or you can have a {1/s^2Gyrator, 1/sC, R} set. And you can expand
your filtering horizons by having a set {1/s^2Gyrator, 1/sC, R, sL,
s^2Gyrator} . . . that lets you implement higher order filters in
simpler topologies (if only the gyrator were a simple passive two-lead
part!)

Gyrators don't seem to be used very often, but I have seen them used to
(presumably) keep the op amp out of the direct signal path, in an
attempt to have it contribute less distortion in the passband.

Cheers,
Tom

 

[Joel Kolstad]

Huh?? If I put 100nH in series with 30pF, at 100MHz the net reactance
is +j9.78 ohms; at 200MHz, it's +j99.14 ohms. Last time I checked, 99
is more than twice 9.8, not less.

Thanks Tom, that'll certainly do it. When I said "small capactior," in my
case I was thinking "below resonance" -- as you say, when you're above
resonance the reverse is true. I should have thought it through some more
before posting.

---Joel

 

[Joel Kolstad]

Hmm... in fact... ignore that bit about above vs. below resonance too -- turns
out I dropped a minus sign in my original calculations. :-(

 

[Don Bowey]

You wrote, ". . . but I've never found a way of doing anything useful
with this circuit."

If as you wrote in the paragraph after that, you put it in series with
a resistor, you get a circuit that behaves much like a series resonant
circuit. Not only does it go through a phase inversion, but it goes
through zero resistance. Similarly if you put it in parallel with a
resistor, you get something that behaves much like a parallel resonant
circuit: where the gyrator negative resistance equals the parallel
positive resistance, the net resistance goes to infinity. If you think
of C-R-L circuits (or better, 1/sC, R, sL) as having components that
contribute in quadrature, with 90 degrees going from C to R and R to L,
you see that you can have the same effect with three components that
behave as R, sL and s^2Gyrator; you've just multiplied everything by s.
Or you can have a {1/s^2Gyrator, 1/sC, R} set. And you can expand
your filtering horizons by having a set {1/s^2Gyrator, 1/sC, R, sL,
s^2Gyrator} . . . that lets you implement higher order filters in
simpler topologies (if only the gyrator were a simple passive two-lead
part!)

Gyrators don't seem to be used very often, but I have seen them used to
(presumably) keep the op amp out of the direct signal path, in an
attempt to have it contribute less distortion in the passband.

Cheers,
Tom

That's an FDNR, (Frequency Dependant Negative Resistor), and can be realised
with the basic four-op-amp gyrator circuit using two capacitors and three
resistors. You can get a negative resistance that either increases or
decreases as the square of the frequency, depending where you put the
capacitors, but I've never found a way of doing anything useful with this
circuit.

You can make a frequency dependant voltage divider with an FNDR and a
resistor, but the circuit goes through a phase inversion, the point of
inflection being at the frequency where -R = R

Check out the Sept 14 EDN Magazine, "design ideas" Brick-wall lowpass audio
filter for a practical circuit which uses three gyrators.

I don't care much for EDN, but occasionally a contributor sends them
something really good. This is one.

Don

 

[Jim Thompson]

You wrote, ". . . but I've never found a way of doing anything useful
with this circuit."

If as you wrote in the paragraph after that, you put it in series with
a resistor, you get a circuit that behaves much like a series resonant
circuit. Not only does it go through a phase inversion, but it goes
through zero resistance. Similarly if you put it in parallel with a
resistor, you get something that behaves much like a parallel resonant
circuit: where the gyrator negative resistance equals the parallel
positive resistance, the net resistance goes to infinity. If you think
of C-R-L circuits (or better, 1/sC, R, sL) as having components that
contribute in quadrature, with 90 degrees going from C to R and R to L,
you see that you can have the same effect with three components that
behave as R, sL and s^2Gyrator; you've just multiplied everything by s.
Or you can have a {1/s^2Gyrator, 1/sC, R} set. And you can expand
your filtering horizons by having a set {1/s^2Gyrator, 1/sC, R, sL,
s^2Gyrator} . . . that lets you implement higher order filters in
simpler topologies (if only the gyrator were a simple passive two-lead
part!)

Gyrators don't seem to be used very often, but I have seen them used to
(presumably) keep the op amp out of the direct signal path, in an
attempt to have it contribute less distortion in the passband.

Cheers,
Tom



Check out the Sept 14 EDN Magazine, "design ideas" Brick-wall lowpass audio
filter for a practical circuit which uses three gyrators.

I don't care much for EDN, but occasionally a contributor sends them
something really good. This is one.

Don

Cute. But my favorite filter book, Herrero & Willoner, has techniques
where you simply enter pass and stop bands and go directly to Laplace,
thence to active filter, gyrator or otherwise.

...Jim Thompson