The way an op amp goes about doing its magic ???

[jalbers@bsu.edu]

I have a few questions about feedback. For discussion consider an
inverting op amp configuration with R2 feedback @ 4K ohms and R1
(connected to V-) @ 1K at 1K ohms. The open loop gain to be 200,000 .

Assume that Vin is at 2 volts, according to my calculations using Vout
= (V+ - V-)*200,000 and V- = (Vin-Vout)*R2/(R1+R2)+Vout, V- should be
around .0003999 volts and Vout should be around -7.9980005 volts.

My question is in what fashion does Vout get to -7.9980005? I know it
happens but it seems like magic. I have looked at the schematic
diagram of a 741 for example. It will probably take decades before I
would be able to understand something like that. Is it instantaneous
or does the op amp spend some time hunting for this value? If it
hunts, does it swing low to the negative rail and move towards the
upper rail or do something else?

I did some experimenting using excel trying to iterate between Vout =
(V+ - V-)*200,000 and V- = (Vin-Vout)*R2/(R1+R2)+Vout to see if the
thing will somehow converge. Here are some examples:


Assume that the op amp guessed the output was -8.1 making V- = -.02
(Voltage Divider), but this would mean that Vout should be 400
(amplifying the difference between V+ and V- , ....

Vout V-
-8.1 -0.02
400 81.6
-1632000 -326398.4
6527968000 1305593602
-2.61119E+13 -5.22237E+12
1.04447E+17 2.08895E+16
-4.1779E+20 -8.3558E+19


Assume that the op am guessed the output was -7.9

Vout V-
-7.9 0.02
-400 -78.4
1568000 313601.6
-6272032000 -1254406398
2.50881E+13 5.01763E+12
-1.00353E+17 -2.00705E+16
4.0141E+20 8.0282E+19
-1.60564E+24 -3.21128E+23
6.42256E+27 1.28451E+27
-2.56902E+31 -5.13805E+30
1.02761E+35 2.05522E+34

Vout for both of these does not seem to converge on anything??? I
don't think that clipping off Vout to stay within the positive and
negative rails would help.

How does an op amp do its magic?

Any help would be greatly appreciated. Thanks

 

[DJ Delorie]

For R1=1k and R2=4k inverting...

V- = Vin + (Vout-Vin)/5

V- = (Vin*4 + Vout)/5
= Vin*4/5 + Vout/5

The op-amp will try to make V- = 0, resulting in:

Vout/5 = -Vin*4/5
Vout = -Vin*4

Starting from "off", though, the instant you apply +2v to Vin, this
happens:

Vout = 0
Vin = +2
V- = (2*4+0)/5 = 1.6v

The op amp will try to do this:

Vout = (V+ - V-) * 200,000

I.e. Vout = -320,000 volts.

HOWEVER, it cannot do this instantaneously! The output voltage will
gradually (perhaps over a few nano- or microseconds, depending on the
op amp) change from its old value (0v) to the new one (-320,000).
I.e. it will start going negative. In this example, let's use 1V/nS
as the fastest the output can change.

AS IT CHANGES, the V- value changes also. Which changes the op amp's
desired Vout. This is how "feedback" happens. Let's say a nanosecond
later, Vout is at -1v. V- is now (2*4+(-1))/5 or 1.4v, and the op amp
wants Vout to be -280,000 volts. Continuing on the 1v/nS trend:

nS Vin V- Vout(desired) Vout(actual)

-1 0 0.0 0 0
0 2 1.6 -320,000 0
1 2 1.4 -280,000 -1
2 2 1.2 -240,000 -2
3 2 1.0 -200,000 -3
4 2 0.8 -160,000 -4
5 2 0.6 -120,000 -5
6 2 0.4 -80,000 -6
7 2 0.2 -40,000 -7
8 2 ~0.0 -8 -8 (approx)

Now, if there's any stray capacitance or inductance in your circuit
(likely), the output value may overshoot, just like any other signal.
If it overshoots, the op amp will reduce its output accordingly, etc,
until it stabilizes. Hopefully the overshoot will be tiny and
stabilize quickly.

The open loop gain tells you how close to ideal the op amp is. For an
open loop gain of 100, for example, the V- would have to be 0.08 to
get a Vout of -8.0. You'd need a slightly more positive Vout to do
this, so the actual Vout is more like -7.62:

Vout = - V- * 100
V- = Vin*4/5 + Vout/5
Vin = 2

V- = -Vout / 100
V- = 1.6 + Vout/5

-Vout/100 = 1.6 + Vout/5
-Vout = 160 + Vout * 20
-Vout * 21 = 160
Vout = - 7.62
V- = 0.076

For a gain of 200,000, the result is Vout = 7.9998 and V- = 0.00004.

 

[John Larkin]

On Wed, 6 Aug 2008 11:32:43 -0700 (PDT), "jalbers@bsu.edu"
<jalbers@bsu.edu> wrote:

I have a few questions about feedback. For discussion consider an
inverting op amp configuration with R2 feedback @ 4K ohms and R1
(connected to V-) @ 1K at 1K ohms. The open loop gain to be 200,000 .

Assume that Vin is at 2 volts, according to my calculations using Vout
= (V+ - V-)*200,000 and V- = (Vin-Vout)*R2/(R1+R2)+Vout, V- should be
around .0003999 volts and Vout should be around -7.9980005 volts.

My question is in what fashion does Vout get to -7.9980005? I know it
happens but it seems like magic. I have looked at the schematic
diagram of a 741 for example. It will probably take decades before I
would be able to understand something like that. Is it instantaneous
or does the op amp spend some time hunting for this value? If it
hunts, does it swing low to the negative rail and move towards the
upper rail or do something else?

Opamps are designed to be stable and approach the equilibrium point
smoothly and monotonically. At one or more places inside the opamp,
there are capacitors that on-purpose slow down the response to make it
so. Two things generally happen:

For big output steps, the output slew-rate limits, at some number of
volts per microsecond; it can't move any faster than that.

For small output steps, the forward gain behaves like a 1st order RC
lowpass filter. When you close a feedback loop around that, it moves
and settles smoothly. It's common for the break frequency of this
open-loop lowpass response to be low, 10 Hz or some such.

The actual step response can be deduced using control theory. But
intuitively, just assume that the opamp responds slowly to an input
error signal.

Some opamps may overshoot a little, especially very fast ones.

Try adding a little capacitance from the inverting node to ground.
That will mess up the carefully-planned transient response and make
the step response overshoot, ring, or even oscillate. The extra
capacitance turns the system from a single-delay thing (mostly stable)
to a 2-delay loop, less stable.

I did some experimenting using excel trying to iterate between Vout =
(V+ - V-)*200,000 and V- = (Vin-Vout)*R2/(R1+R2)+Vout to see if the
thing will somehow converge. Here are some examples:

It's hard to model accurately with Excel. LT Spice is a better tool.

John

 

[Eeyore]

"jalbers@bsu.edu" wrote:

I have a few questions about feedback. For discussion consider an
inverting op amp configuration with R2 feedback @ 4K ohms and R1
(connected to V-) @ 1K at 1K ohms. The open loop gain to be 200,000 .

Open loop gains are approximate, average, not accurate values.

Assume that Vin is at 2 volts, according to my calculations using Vout
= (V+ - V-)*200,000 and V- = (Vin-Vout)*R2/(R1+R2)+Vout, V- should be
around .0003999 volts and Vout should be around -7.9980005 volts.

My question is in what fashion does Vout get to -7.9980005?

It's called negative feedack and works with all amplifiers. That's what
you need to study up on.

Graham

 

[John Fields]

On Wed, 6 Aug 2008 11:32:43 -0700 (PDT), "jalbers@bsu.edu"
<jalbers@bsu.edu> wrote:

How does an op amp do its magic?

consider a voltage divider: (View in Courier)


E1
|
R1
|
+--E2
|
R2
|
GND


The voltage at E1 can be described by:


E2(R1+R2)
E1 = -----------
R2

Next, assume that we know the values of R1, R2, the desired value of
E2, (as shown below) and that we need to apply a voltage At E1 which
will cause E2 to be what we want.

E1
|
[10k]
|
+--1V
|
[1k]
|
GND

In that case:

E2(R1+R2) 1V (1e4R + 1e3R)
E1 = ----------- = ------------------ = 11V
R2 1e3R

Next, assume we have a magic box which supplies E1 to the divider and
that we can connect to the voltage divider like this:


+--------+--E1
| |
[10k] +--+--+
| | |
+-----|- +|--+---E2
| +-----+ |
[1k] [GEN]
| |
GND GND

its job being to make the voltage on the - terminal be precisely the
voltage on the + terminal, which we supply using a DC source and apply
at E2.

Let's say that E2 is 0V. Then,


E2(R1+R2) 0V (1e4R + 1e3R)
E1 = ----------- = ------------------ = 0V
R2 1e3R


Now, lets raise E2 to 0.1V. Then:


0.1V (1e4R + 1e3R)
E1 = -------------------- = 1.1V
1e3R

If we tabulate E1 for E2s from 0V to 1V 1n 0.1V increments, well
arrive at:

E2 E1
V V
------+------
0.1 1.1
0.2 2.2
0.3 3.3
0.4 4.4
0.5 5.5
0.6 6.6
0.7 7.7
0.8 8.8
0.9 9.9
1.0 11.0

Note that in every instance E1 is 11 times higher than E2, so we can
say the circuit has a gain of 11.

Handily, if we need more or less gain we can get it merely by changing
the ratio of R1:R2. In this case (which is that of a non-inverting op
amp) the voltage gain is described as being:

R1
Av = ---- + 1
R2

So, the opamp's "magic" lies in its being able to sense when the + and
- terminals are at different voltages and then to drive its output in
the direction required to make them equal.

JF