What is the best approach to find the minimum value of the feedback resistor that will cause this op-amp to saturate?

Question

I am presented with a non-inverting amplifier; Vg = 5 V for this problem. I need to find the minimum feedback resistor value that will cause the op-amp to saturate.

I did the following: set Vout to 10 V (because of the non-inverting amplifier circuit equation) to then equal (30+x)/30 * 5 which gives me a 42 kΩ resistor.

I also tried doing a voltage divider: (30+x)/30 * (5(45/75)) but this gave me a higher value. I know the resistor value is supposed to be 35 kΩ but I don't know how to approach this to advance toward the answer.

Answer 1

I would,

• Calculate (1). \$ V_+ = 5 \frac {45}{30+45} = 3 \ \text V\$.
• Set (2) = (1). \$ V_- = 3\ \text V\$
• You know (3), \$0\ \text V\$, and (4), \$10\ \text V\$.
• Calculate (5).
  $$ V_- = 10 \frac {15}{15 + x} = 3 $$ $$ 3(15 + x) = 10 \times 15 $$ $$ 45 + 3x = 150 $$ $$ 3x = 105 $$ $$ x = 35 \ \text k\Omega$$

Answer 2

If by "best approach" you mean reasonably general (in terms of variables right until the last step), and reasonably rigorous, then here's my approach. I'd divide the operation into two stages:

^{simulate this circuit – Schematic created using CircuitLab}

Stage 1 is a simple resistor potential divider, with the following relationship between \$V_G\$ and \$V_H\$:

$$ V_H = V_G\frac{R_2}{R_1+R_2} $$

Stage 2 is a classic non-inverting amplifier. Op-amp OA1's output may be considered a voltage source, meaning that the load R5 at its output can be disregarded. I'm of course assuming that it won't draw more current than the op-amp can source, and that any current it does draw will not diminish the output swing of ±10V.

The relationship between stage 2 input \$V_H\$ and output \$V_{OUT}\$ is:

$$ V_{OUT} = V_H\left(1 + \frac{R_3}{R_4}\right) $$

We combine those two equations into one, to obtain a general equation relating \$V_{OUT}\$ and \$V_G\$. Here I'll do that by substituting \$V_H\$ from the first equation into the second:

$$ V_{OUT} = V_G\frac{R_2}{R_1+R_2}\left(1 + \frac{R_3}{R_4}\right) $$

There's only one unknown here: \$R_3\$. All other resistances are known, and input and output conditions are:

$$ \begin{aligned} V_G = +5V \\ \\ V_{OUT} = +10V \\ \\ \end{aligned} $$

I'm setting output \$V_{OUT}\$ to its maximum possible value, with the understanding that this represents saturation, and a further increase in \$V_G\$ will not result in \$V_{OUT}\$ increasing beyond that.

Now it's just a question of plugging in all known values and solving for R3.

Answer 3

\$V_G\$ is 5 volts hence, there is 3 volts at the non-inverting input. This is due to the 30 kΩ resistor and the 45 kΩ resistor acting as a potential divider: -

The gain of a generalized non-inverting op-amp circuit is: -

So, you know Vout (10 volts) and you know Vin (3 volts). You also know that R2 is fixed at 15 kΩ. Let me know if you are still having problems doing the final step.