Another inductor simulator

The Phantom a écrit :

R'3 has the wrong sign, plus the components values comes from a second order taylor expansion of the real TF, giving a totally bogus impedance magnitude at high frequency. The phase is surprisingly accurate though (0.5° WC for 10R,10R,0.1R and 100u). Another interesting point is that the approximation quality decreases while increasing the Taylor expansion order.

notice the minus signs:

---------[Ra]---+------. | | -Rb -Ca | |

----------------+------'

with Rb=R1.R2/R3 Ra=R1+R2+Rb Ca=C.R1.R2/(R3^2) which in fact is a strange funny way of making an inductor.

on:

(Shakes magic 8 ball) Answer: No

on:

Doesn't qualify. That's *less* than an opinion. :-)

Furthermore, if the answer was no, then a correct equivalent circuit was to be provided. Did the magic 8 ball give one?

"The Phantom" schrieb im Newsbeitrag news: snipped-for-privacy@4ax.com...

Hello Phantom,

You made not the best equivalent circuit. Place a resistor Rp in parallel to the inductor+resistor and remove R3'.

IN --|Rp|-- GND IN --|L1|--|Rs|--GND

L=R1*R2*C Rs=R1+R2 Rp=R1*R2/Rser

(Rser is the series resistance of the capacitor or an external series resistor)

You will be surprised how good it now matches the active circuit's impedance over frequency. I checked my calculations with LTspice of course.

Best regards, Helmut

I don't get the same driving point impedance from this as from the op amp circuit.

But can you come with an equivalent circuit that doesn't use negative capacitors, etc.? Find one that uses only positive components.

Please understand that it is not *my* equivalent circuit; it is Siegfried Linkwitz's circuit.

Linkwitz's two equivalent circuits are both less than optimum.

It is possible to do better. Can you find a better equivalent?

on:

I only shook it once. You only get one answer per shake.

"The Phantom" schrieb im Newsbeitrag news: snipped-for-privacy@4ax.com...

Hello Phatom,

You can tell a lot, but I don't believe it. In this case you can't beat my solution as presented with the same number of devices. I doubt that you have simulated with my circuit.

Best regards, Helmut

Hellmut,

I have simulated your circuit, but it isn't possible to verify the correctness of the equivalent circuit that way. I will explain..

The equivalent circuit you have posted is very, very close to the optimum. It is so close that if you plot its impedance (using typical values of R1, R2, R3 and C, say for an audio equalizer) versus frequency over the audio band, and at the same time plot the impedance of the optimum equivalent circuit, you won't be able to see any difference without zooming in on the plot quite a bit.

It is necessary to analyze the driving point impedances algebaically to see the difference.

The driving point impedance of the op amp circuit given at:

(assuming an ideal op amp. I have assumed this in both of these threads and Linkwitz has obviously made the same assumption) is:

s C R1 R2 + s C R1 R3 + s C R2 R3 + R1 + R2 Zin = -------------------------------------------------------- s C R3 + 1

The driving point impedance of the equivalent circuit you posted is:

s C R1^2 R2^2 + R1 R2^2 + R1^2 R2 Zin = --------------------------------------------- s R1 R2 R3 + R1 R2 + R1 R3 + R2 R3

Now, if you make a small change to the circuit you posted, namely, don't put Rp in parallel with the series combination of L and Rs as you did, but rather put Rp in parallel only with the inductance element, then you will get this driving point impedance, which is *exactly* the same as the op amp circuit:

s C R1 R2 + s C R1 R3 + s C R2 R3 + R1 + R2 Zin = -------------------------------------------------------- s C R3 + 1

Now we have an equivalent circuit which has *exactly* the same driving point impedance as the op amp circuit. But if you plot the impedances (without zooming on the plot) in Spice using typical values for R1, R2, R3 and C such as would be used for an audio application, you won't be able to tell the difference. Only an algebraic analysis can show the difference.

Your circuit gives a result very close to the optimum, and for practical purposes it would be quite usable, but it isn't *exactly* the same as the op amp circuit. That's what I meant when I said it is possible to do better.

It's certainly possible that I have made a mistake in all this, so would you please work it out for yourself and let me know if you agree?

And, for the next point of discussion I intended for this thread, can you figure out a way to make the parallel branch, Rp, vanish?

"The Phantom" schrieb im Newsbeitrag news: snipped-for-privacy@4ax.com...

Hello Phantom,

Thanks for your feedback. I tried with Rp parallel to L and indeeed it's about 2.5 times more accurate, but not totally equivalent to the active circuit in my simulations.

The gain of the amplifier has to be G=1+Rser/R2 instead of G=1. Now the input impedance can increase towards inifinity as with an inductor without Rp. Rser is the series resistance/resistor of the capacitor C.

Best regards, Helmut

Since the algebra shows that such a configuration should be exactly equivalent, what would be the reason that simulation shows a difference?

Yes, this is correct, and if Rser is small, then the gain doesn't have to be increased much. Notice that if R3 in the .gif (what you have called Rser) becomes zero, then Rp vanishes (becomes infinite) also.

The inductor simulator in the other thread I posted doesn't have this property. Have you also examined that simulation?

Hello Phantom,

I rounded the nunbers used in the simulation too much. Instead of the exact value 1.99712H for the inductor, I had used 2H. Now with the precise numbers, the simulaton of all three circuits (Laplace, active, equ. L) fully agree.

Yes, I had simulated it. I now checked it again and found a nice property. OK, it was more empiracilly.

If the capacitor has no series resistance(Rserc=0): Rp=R2-R1

If the capacitor has some series resistance (Rserc>0): Rp=(R2-R1)*(1-Rserc/R1)

Thanks for the discussion of these circuits.

Best regards, Helmut

Helmut,

Sorry about misspelling your name in another post. :-(

I'm going to copy this last part over to that other thread since it applies to that simulator.