Tutorial: Toward the Ideal Op-Amp

The questions below are due on Monday October 28, 2024; 10:00:00 PM.
 
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In this problem, we'll explore the transition from our Voltage-Controlled Voltage Source model of an op-amp to the "ideal" op-amp assumption. We'll explore these relationships through a couple of example circuits.

1) Buffer

To start, let's consider the following circuit, known as a "buffer" or a "voltage follower," where both the input voltage v_i and the output voltage v_o are measured relative to the same reference point:

One way to analyze this circuit would be to replace the op-amp with a voltage-controlled voltage source like we've seen in class:

Modeling the op-amp in this way, what is the value of v_o, in terms of v_i and k?

v_o =~

Op-amps are designed with the intention of not only behaving like a VCVS in this way, but also having a large value of k. We'll often idealize op-amps for purposes of analysis by taking the limit as k\to\infty.

In this model, as we let k\to\infty, what is the value of v_o?

\displaystyle{\lim_{k\to\infty}v_o} =~

And, as we let k\to\infty, what is the value of v_- (the voltage at the op-amp's - input terminal)?

\displaystyle{\lim_{k\to\infty}v_-} =~

2) Non-inverting Amplifier

Now let's take a look at another circuit before we talk about the general trends here: the non-inverting amplifier.

As before, we can start our analysis by replacing the op-amp symbol with our voltage-controlled voltage source model:

Modeling the op-amp in this way, what is the value of v_o, in terms of v_i, k, R_1, and R_2?

v_o =~

Op-amps are designed with the intention of not only behaving like a VCVS in this way, but also having a large value of k. We'll often idealize op-amps for purposes of analysis by taking the limit as k\to\infty.

In this model, as we let k\to\infty, what is the value of v_o?

\displaystyle{\lim_{k\to\infty}v_o} =~

And, as we let k\to\infty, what is the value of v_-?

\displaystyle{\lim_{k\to\infty}v_-} =~

3) Inverting Amplifier

Now let's take a look at one last circuit before we talk about the general trends here: the inverting amplifier.

As before, we can start our analysis by replacing the op-amp symbol with our voltage-controlled voltage source model:

Modeling the op-amp in this way, what is the value of v_o, in terms of v_i, k, R_1, and R_2?

v_o =~

Op-amps are designed with the intention of not only behaving like a VCVS in this way, but also having a large value of k. We'll often idealize op-amps for purposes of analysis by taking the limit as k\to\infty.

In this model, as we let k\to\infty, what is the value of v_o?

\displaystyle{\lim_{k\to\infty}v_o} =~

And, as we let k\to\infty, what is the value of v_-?

\displaystyle{\lim_{k\to\infty}v_-} =~

4) Consequences

This may be a surprising result at first! We've seen three different circuits here, and for both, we saw that even though they produced different outputs, in the limit as k\to\infty, they all had the property that v_- = v_+!

It turns out that this consequence generalizes completely. For any circuit connected in negative feedback (i.e., where the output affects the voltage at the - input but not the + input), taking this limit will always have the effect of driving the voltages v_+ and v_- to be equal.

For purposes of analysis, then, it turns out that we can arrive at the same result much more simply by not using the VCVS model for the op-amp at all, but rather applying this consequence a priori. Specifically, we can analyze the circuit under the following assumptions, without replacing the op-amp with the VCVS:

  • no current flows into or out of either of the input terminals of the op-amp; and
  • the voltages at the two input terminals are equal.

For our purposes in 6.200, we will almost always start from the ideal op-amp assumption, only falling back to other more complex op-amp models if the need arises.

Check Yourself 1:

Try repeating the analysis of all three of these circuits using the ideal op-amp assumption, solving for v_o in each case. You should arrive at the same results you got after taking the k\to\infty limit in the VCVS model, but hopefully with substantially less tedium!