Lecture 23: Positive Feedback, Comparators, and Hysteresis

What this lecture covers

After two lectures on negative feedback op amp circuits, this lecture asks what happens with positive feedback instead, where a portion of the output is fed back to the noninverting input rather than the inverting one. A naive static analysis suggests the circuit behaves like an ordinary amplifier, but the lecture shows this is misleading: any small perturbation actually drives the output to one supply rail or the other. To understand why, the instructor builds a more detailed dynamic model of the op amp using an internal RC time constant and derives a first-order differential equation whose stability depends on the balance between positive and negative feedback fractions.

The second half applies this insight to build a basic comparator circuit and then improve it with positive feedback into a Schmitt-trigger-like circuit exhibiting hysteresis, which resists switching due to small noise near the input's zero crossing. After watching, you understand why positive feedback circuits are inherently unstable around an equilibrium, and how deliberately introduced hysteresis makes a comparator robust to noisy inputs.

Key ideas

• Positive vs. negative feedback: feeding back a fraction of the output to the inverting input stabilizes the circuit (negative feedback); feeding it to the noninverting input reinforces any deviation (positive feedback).
• Static analysis can mislead: a straightforward algebraic analysis of a positive-feedback op amp circuit produces a finite gain expression, vOUT = (R2/R1)*vIN, but this equilibrium is unstable and never actually observed in practice.
• Dynamic op amp model: inserting an internal RC network between the differential input and the gain stage captures how the op amp's output evolves over time, rather than assuming instantaneous response.
• Stability condition: writing the dynamics as dvO/dt + vO/T = 0, the sign of the time constant T (which depends on gamma_minus - gamma_plus, the feedback fractions to each input) determines whether the circuit is stable (decays to equilibrium), unstable (diverges to a rail), or neutrally stable (stays wherever perturbed).
• Physical analogy: stable, unstable, and neutral equilibria correspond to a ball at the bottom of a well, a ball balanced on a hilltop, and a ball on a flat tabletop, respectively.
• Basic comparator: an op amp with no feedback and an input applied to v- produces an output that slams to one supply rail or the other depending on the sign of the input, useful for converting an analog signal into a digital high/low sequence.
• Noise sensitivity of a bare comparator: when the input hovers near zero due to noise, a plain comparator's output chatters rapidly between rails.
• Hysteresis via positive feedback: adding positive feedback to the comparator makes its switching threshold depend on the circuit's current state, so the input must cross a higher threshold to switch one way and a lower threshold to switch back, eliminating chatter from small noise near the switching point.

Walkthrough

Introducing positive feedback and the naive analysis (0:00)

The instructor reviews the negative-feedback inverting amplifier pattern, then introduces the mirror-image circuit with feedback routed to the positive input instead. A quick intuitive argument shows any perturbation should drive the output toward a rail, yet a careful static analysis, replacing the op amp with its dependent-source model and solving algebraically, produces the same finite-gain expression as the negative-feedback case, vOUT = (R2/R1)*vIN. The instructor flags this contradiction as the puzzle to resolve.

Building a dynamic model of the op amp (14:28)

To understand why the static result is misleading, the instructor introduces a more detailed op amp model with an internal RC network representing the time it takes the amplifier's internal state v* to respond to changes in v+ - v-, before being multiplied by the gain A.

Deriving the stability equation (18:38)

Applying this dynamic model to a circuit with both positive feedback (R1, R2) and negative feedback (R3, R4), the instructor writes and simplifies a differential equation for the output, arriving at dvO/dt + vO/T = 0, where the time constant T depends on 1/(RC) + (A/RC)*(gamma_minus - gamma_plus), with gamma_minus and gamma_plus the feedback fractions to the inverting and noninverting inputs.

Interpreting stable, unstable, and neutral cases (27:52)

The instructor shows that when gamma_minus > gamma_plus (net negative feedback), T is positive and any perturbation decays back to equilibrium, a stable situation. When gamma_plus > gamma_minus (net positive feedback), T is negative and any perturbation grows without bound toward a rail, an unstable situation. He illustrates these cases with a ball in a well (stable), a ball on a hilltop (unstable), and a ball on a flat table (neutral).

Building a basic comparator (36:06)

With no feedback at all, an op amp with an input applied to v- acts as a comparator: the output slams to one supply rail when the input is positive and to the other when it is negative, useful for converting an analog signal into a sequence of digital ones and zeros. The instructor points out that this basic comparator chatters unpredictably when the input has noise near zero.

Adding positive feedback for hysteresis (42:15)

By feeding back a fraction of the output to the positive input, the instructor builds a comparator whose switching threshold depends on its current output state: it switches high only after the input rises above one threshold (e.g. +6 V) and switches low only after the input falls below a different threshold (e.g. -6 V). This lagging behavior, called hysteresis, is demonstrated live and shown to eliminate the chattering seen with the basic comparator when noise is present near the switching point.

Before you watch

• Watch the two previous lectures on the operational amplifier and its negative-feedback circuits, since this lecture directly contrasts positive feedback against that established pattern.
• Review first-order RC circuit dynamics (time constant, exponential decay or growth), which are reused directly to analyze the op amp's dynamic stability.
• Be comfortable with the ideal op amp's dependent-source model from the earlier op amp lecture.

Check your understanding

1. Why does a static analysis of the positive-feedback op amp circuit produce a plausible-looking gain expression that is never actually observed in practice?
2. What determines whether the dynamic model of an op amp circuit is stable, unstable, or neutrally stable, and how does this relate to the balance between feedback to the positive and negative inputs?
3. How does a basic comparator without hysteresis behave badly when its input has noise near the switching threshold, and how does adding positive feedback fix this?
4. Why does the hysteresis comparator switch at a different input threshold depending on its current output state?