Basic Op Amp closed-loop gain question

This roughly matches figure 24 for LF357

Now use the information in this video at about 3:15

"Christopher Howard" snipped-for-privacy@librehacker.com wrote in message news: snipped-for-privacy@librehacker.com...

These days it's much easier to do all this in LTSpice.

Just search for a model of LF357 and get it from here

Not too far off Figure 27

Version 4.1 SHEET 1 880 680 WIRE 224 -64 160 -64 WIRE 368 -64 288 -64 WIRE 160 32 160 -64 WIRE 224 32 160 32 WIRE 368 32 368 -64 WIRE 368 32 304 32 WIRE -304 128 -304 80 WIRE -256 128 -304 128 WIRE -256 144 -256 128 WIRE -64 144 -96 144 WIRE 32 144 -64 144 WIRE 160 144 160 32 WIRE 160 144 112 144 WIRE 368 160 368 32 WIRE 400 160 368 160 WIRE 336 176 336 128 WIRE 336 176 304 176 WIRE -304 192 -304 128 WIRE 224 192 176 192 WIRE 336 208 304 208 WIRE 160 224 160 144 WIRE 224 224 160 224 WIRE 176 240 176 192 WIRE 368 240 368 160 WIRE 368 240 304 240 WIRE -96 256 -96 144 WIRE 336 288 336 208 WIRE -96 384 -96 336 FLAG -256 144 0 FLAG -304 0 +15 FLAG -304 272 -15 FLAG 336 128 +15 FLAG 336 288 -15 FLAG 400 160 Vout FLAG -64 144 Vin FLAG 176 240 0 FLAG -96 384 0 SYMBOL voltage -304 -16 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V1 SYMATTR Value 15V SYMBOL voltage -304 176 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V2 SYMATTR Value 15V SYMBOL voltage -96 240 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V3 SYMATTR Value AC 1 SYMBOL res 128 128 R90 WINDOW 0 0 56 VBottom 2 WINDOW 3 32 56 VTop 2 SYMATTR InstName R1 SYMATTR Value 4 SYMBOL res 320 16 R90 WINDOW 0 0 56 VBottom 2 WINDOW 3 32 56 VTop 2 SYMATTR InstName R2 SYMATTR Value 2k SYMBOL cap 288 -80 R90 WINDOW 0 0 32 VBottom 2 WINDOW 3 32 32 VTop 2 SYMATTR InstName C1 SYMATTR Value 2pF SYMBOL LF357_NS 256 208 R0 SYMATTR InstName U2 TEXT -464 384 Left 2 !.ac dec 1000 1E6 100E6

Am 06.08.2025 um 05:42 schrieb Edward Rawde:

Am 06.08.2025 um 05:42 schrieb Edward Rawde:

This assumes that the LTSpice op amp models are sufficiently reliable.

For most purposes they are reliable enough, but there are always corner cases where a subtle effect that wasn't worth modelling produces a misleading simulation.

Jim Thompson claimed that pretty much all the op amp Spice models were behavioral models, and that he could devise more accurate behavioral models than most application engineers. He was touting for trade, but nobody disagreed with him.

Yes, especially considering real-world nonlinearities.

Spice can handle non-linear equations. As Jim Thompson used to point out, the op amp models usually available for use in spice tend to be behavioral models, rather than transistor level models, and there's no guarantee that either kind of model is going to completely capture the real world behavour of your op amp, even one that has been around as long as the LF357. It was first introduced in the early 1970's when I was working for Kent Instruments in Luton, England, and we used quite a few of them - though more of the LF356 which was stable at unity gain.

National Semiconductor's Linear Brief LB-42 by Robert A. Pease tells you how to use an LF357 as a fast (50V/us) unity gain buffer/follower.

john larkin snipped-for-privacy@glen--canyon.com wrote: > On Tue, 5 Aug 2025 23:42:58 -0400, "Edward Rawde" > snipped-for-privacy@invalid.invalid wrote: >

The theory of linear, time-invariant systems is so general and so powerful, and provides such an abundance of useful insight, that I for one would be crippled without it.

For a beginner to give up so easily is a definite pistol + shoehorn move.

To the OP: The closed-loop gain isn’t hard to derive—strictly high school sophomore/junior stuff. Simple algebra, solving simultaneous equations by substitution, and plotting using complex numbers.

First, the output voltage equals the open-loop gain times the voltage difference between the plus (noninverting) and minus (inverting) inputs:

V_out = A_Vol *(V_inP - V _inM).

Above some frequency in the low audio, A_Vol rolls off, but we’ll carry it along explicitly so you can estimate its effects.

Second, the input voltage comes from the feedback network. (We’ll do the generic case first, then specialize to inverting and noninverting.) The two inputs are treated differently,

Vout = A_VOL * (V_inP - V_inM) (1)

Vout * Ri + V_sM * Rf V_inM = -------------------- (2) Ri + Rf

Substituting (2) into (1) and collecting terms,

Vout = A_VOL V_inP - A_VOL (Vout R1 + V_sM Rf) ------------------ (3) Ri + Rf

Vout(1 + A_VOL Ri/(Ri+Rf)) = A_VOL (v_inP - V_sm Rf / (Ri + Rf) (5)

Dividing both sides by A_VOL, we get the main result

• Vout(1/A_VOL + Ri/(Ri+Rf)) = V_inP - V_sm Rf/(Ri + Rf)* (6)

1. High gain case (1/A_VOL -> 0)

Vout = V_inP * (Ri + Rf) / Ri - V_sm Rf / Ri (7)

Vout = V_inP * (Rf / Ri + 1) - V_sm Rf / Ri (8)

The stage has an inverting gain of -Rf/Ri and an inverting gain of

1. Noninverting, V_sm = 0

V_inP VinP (1+Rf/Ri) Vout = ---------------------- = -------------------- (9) (1/A_VOL + Ri/(Ri+Rf)) 1 + (1+Rf/Ri)/A_VOL

1. Inverting, V_inP = 0

-V_sm - V_sm (Rf/Ri) Vout = ---------------------- = -------------------- (10) (1/A_VOL + Ri/(Ri+Rf)) 1 + (1+Rf/Ri)/A_VOL

There's a gain correction factor in both cases. Because of its importance in stability analysis, we define the _loop gain_ A_VL as

A_VL = A_VOL / (1 + Rf / Ri), (11)

so the denominators of (9) and (10) become

This is very nearly 1.0 when the loop gain is high, but if we try to get too much gain, it won't quite get to Rf/Ri. For instance, if A_VOL=

Keeping the loop gain in mind is one of the keys to making good feedback amplifiers, because it makes it easy to estimate how accurate things will be as you go through the design.

The other important thing about expression (12) is that it goes to zero if A_VL = -1 at some frequency, which causes (9) and (10) to blow up there. This singularity is one expression of the oscillation (Barkhausen) condition.

Oscillation doesn't require this condition to be met perfectly--it'll still oscillate if the phase shift is bigger than 180 at the unity-gain cross--but the actual conditions require a bit more math, and of course once the oscillation starts the whole thing is completely nonlinear.

Suffice it to say that if phase of A_VL gets close to +-180 degrees at the frequency where |A_VL| is unity, all sorts of trouble results.

The difference between the actual phase of A_VL and -180 degrees is called the _phase margin_ phi_M. You'll get good settling behavior if phi_M is between 90 and 60 degrees. Between 60 and 45, you start to get a lot of overshoot, and below 45 the transient response will be enough to scare small children.

There is one small subtlety that I've swept under the rug, namely that voltage is a relative term, so just as V_inP is referenced to V_inM, V_out is actually referenced to some poorly-defined voltage somewhere between the amplifier's supply rails. All that is down at DC, though, where A_VOL is very high, so we can almost always ignore it safely.

Cheers

Phil Hobbs

I'm not able to view these, due to blocking proprietary JavaScript, and browser incompatibilities. Would you recommend a similar video on youtube? I can download youtube videos with yt-dlp.

Look at the post from Phil Hobbs and let me know if you understand it.

Shouldn't (2) be

Vout * Ri + V_sM * Ri V_inM = -------------------- (2) Ri + Rf

Or did I miss or misunderstand something?

A couple of concepts are basic and intuitive, like Av = G / (1 + GH), and the idea that closed-loop gain is flakey without open-loop gain to spare. People should understand that much. But when it gets down to predicting the numbers of real circuits, Spice is a lot easier than nonlinear+complex equations.

Real frequency and transient response are nasty to crank out by hand.

As Mike E says, the real value of Spice is to train your instincts.

It’s a voltage divider, which computes the weighted average of Vout and V_sM. The weights are different because of the different resistance values.

Cheers

Phil Hobbs

We’ve been having this argument for 20 years now, and will hopefully be going for many more. ;)

You do seem to have a higher opinion of the spherical cows today than usual—ISTR that you use a lot of UniversalOpAmp2s, and Digikey is fresh out of those. ;)

Coincidentally, as I write, I’m doing some curve fits of very high-Q induction coils for VLF. They don’t follow the theoretical Lorentzian curve very well at all—the series/parallel impedance transformation ratio is about 2E6, corresponding to a Q of 1400ish, but f0/FWHM is only a tenth of that.

All the Q formulas are supposed to agree at high Q. Mysterious.

Cheers

Phil Hobbs

Ok the current through Ri is (V_inM - V_sM)/Ri and through Rf it is (Vout - V_inM)/Rf

If you equate those you do indeed get your equation 2 so now I'm trying to figure out what caused me to think there was an issue earlier.

I resemble that remark. ;)

Cheers

Phil Hobbs

Just to check my sanity I got the same result by first finding the current in both resistors as (Vout - V_sM)/(Rf+Ri) Then use that to find the drop across Ri and equate it to V_inM - Vs_M My error was trying to take a mental shortcut which didn't work.

If you've got trainable instincts. People with a taste for self-delusion have a high opinion of their capacity to guess right. Mike Engelhart wasn't going to alienate a lot of his customers by being too honest.

High Q induction coils tend to be bank wound to minimise parallel capacitance. On occasion I've tried to model them as pairs, or triads or quads of linked coils in series, but it never made much difference to the results. If I'd been being hyper-picky, it might have.