Question on Non-Inverting Amplifier

[Judy9911]

Homework Statement

Consider the circuit that attaches in the attachment. Calculate the Vout.

Homework Equations

Non-inverting om amp:

Gain = Vout/Vin
= 1 + (Rf+Rs)

Golden Rules:
i- = i+ =0
V- = V+

The Attempt at a Solution

Gain = Vout/Vin
= 1 + (Rf+Rs)
= 1 + (10/20)
= 6

Vx = (5/4+5) Vin
= 5/9 Vin

Vin = 72mV

KCL:

(Vx - Vin /12K) + (Vx/9K) + (Vx- Vout / 84k) = 0
Vx(1/12k + 1/9k + 1/84k) = (Vin/12k) + (Vout/84k)
[(5/9)(72*10^-3)(1/12k + 1/9k + 1/84k)] - [(72*10^-3)/12k] = (Vout/84k)
Vout = 0.189V
= 189.33 mV

I am interested to know how to find the value of Vx.
Besides, I would like to know is my final answer correct or not.
If not, please kindly point out my mistake.

Your help is very much appreciated
Thank you

 

[NascentOxygen]

Hi Judy9911!

Gain = Vout/Vin
= 1 + (Rf+Rs)
= 1 + (10/20)
= 6

Where did this formula come from? Where are the 10 and 20 data values from?

 

[NascentOxygen]

Vx = (5/4+5) Vin
= 5/9 Vin

After adding an essential set of parentheses, can you explain the derivation of this equation.

 

[Judy9911]

For the Gain = Vout/Vin
= 1 + (Rf+Rs)
= 1 + (100/20)
= 6

This is my analysis from the circuit. Sorry that I make mistakes for the equation, should be Rf = 100K ohms , while Rs = 20k ohms

Vx = (5/4+5) Vin
= 5/9 Vin

The bias current of the positive input of the op amp is 0, so the 4k ohms and 5k ohms resistors act as a voltage divider.

 

[NascentOxygen]

For the Gain = Vout/Vin
= 1 + (Rf+Rs)
= 1 + (100/20)
= 6

This is my analysis from the circuit. Sorry that I make mistakes for the equation, should be Rf = 100K ohms , while Rs = 20k ohms

Okay. It is customary to reserve ❝Vin❞ to designate the input to the circuit, and not use ❝Vin❞ for some intermediate point.

Vx = (5/4+5) Vin
= 5/9 Vin

The bias current of the positive input of the op amp is 0, so the 4k ohms and 5k ohms resistors act as a voltage divider.

Can you take a fresh look at this? Which is greater, Vx or Vin?

It would be better to designate the signal at the non-inverting input as something like v(+) rather than Vin.

 

[rude man]

You should start by giving symbols (R1, R2 etc.) to all your resistors, then write KCL or KVL & solve the simultaneous equations by Cramer's rule or whatever.

 

[paul2211]

It's not in the exact form of the non-inverting amplifier, so you cannot use that equation in this problem.

Best bet is to use nodal analysis and plug the matrix in MATLAB or something.

 

[Judy9911]

Redo this question

I try to redo this question by using the nodal analysis :

I label my diagram in detail which is attached to this post.

The following is my step to solve for Vout.

Vx= 72mV
At Node'A':

Ix + Iy - Iz = 0
(Vx-Va/12k) + (Vout - Va /84k) - (Va - Vb/4k) = 0 [Equation 1]

At Node 'B':
{ Golden rule: i+ = i- = 0}
Iz - Is = i+ = 0
Iz = Is
(Va - Vb /4k) = (Vb / 5k)
(Va/4k) = (4Vb+5Vb)/5k
5Va = 9Vb [Equation 2]

At Node 'C'
It - Iu = 0
It = Iu
(Vout - Vc)/100k = Vc/20k
Vout/100k = (5Vc + Vc)/20k
Vout = 6Vc
Vc = 1/6 Vout [Equation 3]

{ Golden rule: V+ = V-}
Therefore, Vb = Vc
Sub [Equation 3],
Vb = Vc
= (1/6) Vout [Equation 4]

Sub [Equation 4] into [Equation 2]

5Va = 9(1/6) Vout
Va = 3/10 Vout [Equation 5]

Sub [Equation 5] & [Equation 4] into [Equation 1]
(Vx/12k) - (Va/12k) + (Vout/84k) - (Va /84k) - (Va/4k) + (Vb/4k) = 0
(72m/12k) - ((3/10 Vout)/12k) + (Vout/84k) - ((3/10 Vout)/84k) - ((3/10 Vout)/4k) + (((1/6) Vout)/4k) = 0
5.04 = 42Vout
Vout = 0.12V

 

[NascentOxygen]

Looks right. You conclude with Vout/Vin = 1.667 and that's what I calculated.

 

[Judy9911]

Thanks for giving me the guideline.

 

[rude man]

I got the same.

 

[The Electrician]

The solution to these sorts of problems can be solved in a very systematic way with linear algebra:

I happened to notice that this circuit exhibits a peculiar behavior. Try analyzing the circuit with an opamp gain of 7 instead of the ideal gain of ∞.

It is peculiar enough that I'm going to start a new thread in the Electrical Engineering forum.

 

[rude man]

The solution to these sorts of problems can be solved in a very systematic way with linear algebra:

I happened to notice that this circuit exhibits a peculiar behavior. Try analyzing the circuit with an opamp gain of 7 instead of the ideal gain of ∞.

It is peculiar enough that I'm going to start a new thread in the Electrical Engineering forum.

Op amp gain of 7: output = 0.1020V.
What is so peculiar about this? Just change 1 equation, and add 1 more.

 

[The Electrician]

Op amp gain of 7: output = 0.1020V.
What is so peculiar about this? Just change 1 equation, and add 1 more.

I get .0582 volts. Anybody else?

But. actually I meant to explore something else:

Sorry. I spent so much time playing with a small change that I forgot to mention it. I should have said to reverse the v+ and v- terminals (or let the opamp gain be -7 rather than +7).

In the original circuit, exchange the + and - inputs of the opamp but otherwise let the gain still be ∞. If an analysis is performed using the "Golden rules" nothing changes because the golden rule is still just v+ = v-; no distinction is made between the v+ and v- terminals in the "Golden rule". The gain A of the opamp isn't even considered.

But, try this. Obtain a general expression for Vout/Vin in terms of the resistor values and A for the case where the + and - terminals are as shown in the OP's schematic, and also obtain the expression for Vout/Vin with the + and - terminals reversed.

Now take the limit as A→∞ for both cases and note that Vout/Vin is 5/3 in both cases. You get the same transfer function regardless of the orientation of the + and - inputs to the opamp.

 

[rude man]

The 'golden rule' does not apply to just any op amp circuit. It only applies if the amplifier output is not saturated and if the gain is infinite (if gain = 7 the + and - inputs are different, for another example). By reversing the + and - inputs to the op amp in this circuit I'd bet the amp goes into saturation and V+ will be very different from V-. But I'm too lazy to do the computations, and I haven't used my PSPICE in years.

 

[The Electrician]

The "Golden rule" apparently applies to the OP's circuit, doesn't it?

Look at the OP's solution in post #8. Which equations would be different if the + and - inputs of the opamp were reversed?

The OP made the assumptions that are usually invoked to justify the use of the "Golden rules", namely that A = ∞ (this one was unstated) and i+ = i- =0.

How is the OP or NascentOxygen or rude man or anyone else to know if the opamp output is saturated as a result of solving these equations? What further mathematical tests (this excludes building the circuit or using a simulator) could determine if the opamp output is saturated?

 

[rude man]

The "Golden rule" apparently applies to the OP's circuit, doesn't it?

How is the OP or NascentOxygen or rude man or anyone else to know if the opamp output is saturated as a result of solving these equations? What further mathematical tests (this excludes building the circuit or using a simulator) could determine if the opamp output is saturated?

That's a very good question. And the procedure is as follows:

1. assume the amp is unsaturated and let V+ = V-. Solve the equations. Then, if the output is within the amp's bounds (e.g. +/-10V for a +/-15V-powered amp), the assumption was warranted and the result is OK.

3. If the output computes to saturation, V+ ≠ V- and V+ and V- now become separate voltages. The output is then defined (fixed) as + or - 10V and the equations solved for all the independent nodes. This procedure works for any condition where the output is saturated. Of course, this circuit would typically be useless.

Textbook questions seldom try to trick the student into giving him/her a saturated amplifier, so most often the assumption of V+ = V- is warranted, and the output computes to a non-saturated value.

BTW the same can be said for exceeding the common-mode voltage limits, which is why when the system of equations are solved, all nodes should be computed, not just the output. Some op amps have very limited common-mode voltage limits, others can handle almost to the + and - power supply rails. (The 'common-mode voltage' is the actual voltage at each of the two op amp inputs referred to ground, not to each other. So you can have V+ = V- = 12V which looks good at first blush (inputs equal) but the 10V common-mode voltage would probably exceed the max limit & the amp would most probably saturate.)

Try it with this circuit (switch the op amp inputs) & see!

 

[rude man]

I should add that the above assumes "fully-compensated" op amps, so any dc gain is OK, even very low gains. Also, it assumes resistors only. Op amp stability is an advanced concept. Analog IC suppliers like Analog Devices Inc. have excellent free publications if interested.

 

[rude man]

The "Golden rule" apparently applies to the OP's circuit, doesn't it?

Yes, but not to yours if you reverse the op amp inputs.

Look at the OP's solution in post #8. Which equations would be different if the + and - inputs of the op amp were reversed?

I did not check the OP's equations, but the answer to that question should be obvious if you did. They would most definitely be different and probably compute to a saturated output.

 

[The Electrician]

That's a very good question. And the procedure is as follows:

1. assume the amp is unsaturated and let V+ = V-. Solve the equations. Then, if the output is within the amp's bounds (e.g. +/-10V for a +/-15V-powered amp), the assumption was warranted and the result is OK.

OK. Solving the OP's problem equations with the + and - inputs as originally specified, the output is .12 volts; not saturated.

Now reverse the + and - inputs and solve again (using the golden rule method, not a simulator, as I specified when I asked the question). The output is once again .12 volts. This procedure says that the opamp doesn't saturate with either orientation of the + and - inputs. Is this true?

 

[The Electrician]

Look at the OP's solution in post #8. Which equations would be different if the + and - inputs of the opamp were reversed?

I did not check the OP's equations, but the answer to that question should be obvious if you did. They would most definitely be different and probably compute to a saturated output.

They aren't different, and therein lies the problem with the golden rule method.

 

[rude man]

They aren't different, and therein lies the problem with the golden rule method.

Alas, I must disagree.

 

[NascentOxygen]

Now reverse the + and - inputs and solve again (using the golden rule method, not a simulator, as I specified when I asked the question). The output is once again .12 volts.

Can you show your working?

 

[The Electrician]

Can you show your working?

The working is just what you see in post #8. In particular this part:

{ Golden rule: V+ = V-}
Therefore, Vb = Vc
Sub [Equation 3],
Vb = Vc
= (1/6) Vout [Equation 4]

Nowhere else are V+ and V- mentioned, and it doesn't matter whether V+ = Vb, or V+ = Vc, so the final result is independent of the orientation of V+ and V-.

If an analysis is performed keeping A finite, the result for the OP's orientation of V+ and V- is:

Vout/Vin = 35A/(21A + 156)

If V+ and V- are reversed, the result is:

Vout/Vin = 35A/(21A - 156)

Notice that if A→∞ the limiting value is 35/21 = 5/3 for both cases, the same result obtained with the golden rule method in both cases. Part of the golden rule derives from A→∞, so this is not surprising, but the golden rule solution provides no indication that saturation may occur.

 

[NascentOxygen]

I get .0582 volts. Anybody else?

Same here.

But. actually I meant to explore something else:

Sorry. I spent so much time playing with a small change that I forgot to mention it. I should have said to reverse the v+ and v- terminals (or let the opamp gain be -7 rather than +7).

I see.

But, try this. Obtain a general expression for Vout/Vin in terms of the resistor values and A for the case where the + and - terminals are as shown in the OP's schematic, and also obtain the expression for Vout/Vin with the + and - terminals reversed.

Now take the limit as A→∞ for both cases and note that Vout/Vin is 5/3 in both cases. You get the same transfer function regardless of the orientation of the + and - inputs to the opamp.

Interesting. For A→∞ the gain is +1⅔ in both cases.

But with A=7 we have ^v_out/_vin = 0.8086

and, reversed with A=─7, we have ^v_out/_vin = ─27.2

... that's doubly interesting.

 

[rude man]

OK, Electrician, time for me to eat crow.

You were right, the equations are the same. And what I wrote in post #17 was trash.

I forgot about a little thing called 'stability':

An op amp circuit consisting of resistors only will obey the 'golden rule' if and only if the net feedback factor is negative. Always providing the amp is 'fully compensated', not saturated by an excessively large input and the op amp gain is infinite (and the offset voltage and current are zero, blah blah ...).

The feedback factor, usually written as β, is the fraction of the op amp's output fed back to its input. β is a positive number if the feedback is negative.

You can have both positive and negative feedback in a circuit and the golden rule still applies, providing the net feedback is negative. I'm attaching a sketch as a simple example.

Your circuit in fact does have both positive (via the 84K resistor) and negative (via the 100K) feedback. I will compute β_- and β₊ and the net β = β_- - β₊ in a while. I suggest you try this also. Here β₊ stands for the positive-feedback factor.

If you reverse the op amp inputs, β_- becomes β₊, β₊ becomes β_-, and the circuit is unstable and the 'golden rule' does not apply.

So sorry about this. You are to be congratulated for sticking to your guns and insisting on knowing what's going on!