Block 23 — Comparator Circuits

Learning objectives

Preparation at Home

Well, again

• read through the present chapter and write down anything you did not understand.
• Also here, there are some clips for more clarification under 'Embedded resources' (check the text above/below, sometimes only part of the clip is interesting).

For checking your understanding please do the following exercises:

• …

90-minute plan

1. Warm-up (5–10 min):
   1. Recall: op-amp with negative feedback vs. no feedback.
   2. Live demo or simulation: sweep \(u_{\rm p}\) across \(u_{\rm m}\) and observe comparator switching.
2. Core concepts (45–50 min):
   1. Comparator basics: inputs, differential voltage \(u_{\rm d}=u_{\rm p}-u_{\rm m}\), saturation behavior.
   2. Output stages: open-collector vs. push-pull; role of pull-up resistor.
   3. Noise problem at the switching point.
   4. Non-inverting Schmitt trigger:
      1. positive feedback
      2. hysteresis
      3. derivation and interpretation of \(U_{\rm sh,u}\) and \(U_{\rm sh,l}\).
3. Applications (15–20 min):
   1. Bang-bang control
   2. De-noising / signal conditioning
   3. Comparator as basic ADC element
4. Wrap-up (5 min):
   1. Key takeaways
   2. Typical mistakes and outlook to further applications

Conceptual overview

Core content

Comparator

Up to now we focussed on operational amplifier, which is only usable in a closed-loop setup. However, it also as a „special brother“, the comparator.
The differences form the comparator in contrast to the operational amplifier are:

1. It is only used in positive feedback. It should never be used in negative feedback.
2. It is optimized for fast switching
3. It only outputs in saturation, which means it only has two possible outputs, see details below.

The symbol is related to the op-amps triangular shape - often the exact same symbol is used.

We again have two inputs: The non-inverting input $u_{\rm p}$ and the inverting input $u_{\rm m}$. They result in the differential voltage $u_{\rm d} = u_{\rm p} - u_{\rm m}$.

So, but what is the output, now? For this, it helps to have a look onto the simulation below.

There are two types of comparators:

1. comparators with open-collector output:
   This type outputs the minimum value, when the non-inverted input is bigger than the inverted one.
   Otherwise, the output is high-ohmic or undefined.
   This is sometimes shown by a diamond shape ◇ on the output.
   For these type, a pull-up resistor is needed to have a readable output in case of $u_{\rm d}>0$.
   $$u_{\rm O,OC}= \biggl\{ \begin{array}{l} &&\text{undefined} &&\text{for} &&u_{\rm d}>0 \\ &&U_{\rm sat, min} &&\text{for} &&u_{\rm d}<0 \end{array}$$
2. comparators with push-pull output:
   This type outputs the minimum value, when the non-inverted input is bigger than the inverted one.
   Otherwise, it outputs the maximum value.
   $$u_{\rm O, PP}= \biggl\{ \begin{array}{l} &&U_{\rm sat, max} &&\text{for} &&u_{\rm d}>0\\ &&U_{\rm sat, min} &&\text{for} &&u_{\rm d}<0 \end{array}$$

Similar to the operational amplifier, the situation $u_{\rm d}=0$ is important.
This time, $u_{\rm d}=0$ is not automatically reached, but it is the „turning point“ for changing the output value.

The values of the output voltages $U_{\rm sat, min}$ (and $U_{\rm sat, max}$, when defined) are given by the voltage supply of the comparator,
In the first simulation they are set unipolar to $U_{\rm sat, min}=0 ~\rm V$ and $U_{\rm sat, max}=5 ~\rm V$.

Non-inverting Schmitt Trigger

Based on the comparator, we can try to setup a „op-amp like“ circuitry. However, we have to take care, that we use a positive feedback.
The most important circuit is similar to the inverting amplifier, but with positive feedback is it the non-inverting Schmitt trigger.

The golden rules ($R_{\rm I}\rightarrow \infty$, $R_{\rm O}=0$, $A_{\rm D}\rightarrow \infty$) also apply here.

Therefore, the currents through the resistors $R_1$ and $R_2$ are the same: $i_1 = i_2$ (given, that $R_{\rm O}\rightarrow \infty$).

$$ u_{\rm D}=0 \quad \rightarrow \quad u_{\rm O} \text{ changes its state} $$

At the „turning point“ with $u_{\rm D}=0$, the input and output voltages are equal to the voltages over the resistances.
However, the signs have to be considered (when $u_{\rm O}$ is positive, $u_{\rm i}$ has to be negative for $u_{\rm D}=0$): $$ u_1 = - u_{\rm I} \\ u_2 = u_{\rm O} $$

Then, the currents $i_1$ and $i_2$ are given by $$ i_1 = - {{u_{\rm I}}\over{R_1}} \\ i_2 = {{u_{\rm O}}\over{R_2}} $$

And therefore, this „turning point“ is given by $$ u_{\rm I} = - {{R_1}\over{R_2}} \cdot u_{\rm O} $$

These „turning points“ are called threshold.
The upper threshold $U_{\rm sh,u}$ and the lower threshhold $U_{\rm sh,l}$ are given by $$ \boxed{ U_{\rm sh,u} = + {{R_1}\over{R_2}} \cdot u_{\rm O} \\ U_{\rm sh,l} = - {{R_1}\over{R_2}} \cdot u_{\rm O} } $$

The shown „switching effect“ is called hysteresis.
The curve is called hysteresis loop and shows the switching at the upper and lower threshold.

Applications

Bang-Bang Control

In the shown simulation, Bang–bang_control is realized with a comparator including hysteresis. and a simple first-order plant (RC network).

The circuit can be interpreted as follows:

• The comparator with positive feedback (via $R_1$ and $R_2$) forms a Schmitt trigger with an upper threshold $U_{\rm sh,u}$ and a lower threshold $U_{\rm sh,l}$.
• The output of the comparator switches only between its two saturation values ($U_{\rm sat,max}$ and $U_{\rm sat,min}$), which is characteristic of bang-bang behavior.
• The resistor–capacitor combination ($R$, $C$) represents a controlled system (plant) with inertia: the capacitor voltage changes only gradually.

The operating principle is:

• If the output voltage $u_{\rm O}$ is high, the capacitor is charged through $R$, causing the feedback signal to increase.
• As soon as the capacitor voltage reaches the upper threshold $U_{m sh,u}$, the comparator switches abruptly to its lower saturation level.
• The capacitor now discharges (or charges in the opposite direction), until the voltage reaches the lower threshold $U_{\rm sh,l}$.
• At this point, the comparator switches back to the high saturation level.

As a result, the system continuously oscillates between the two thresholds. The comparator output is a two-level (on/off) signal, while the capacitor voltage varies smoothly between $U_{\rm sh,l}$ and $U_{\rm sh,u}$.

This example illustrates key properties of bang-bang control:

• the actuator (comparator output) has only two states,
• the controlled variable is kept within a band defined by the hysteresis,
• the switching frequency depends on the system dynamics (here the $RC$ time constant) and the hysteresis width.

Such control principles appear in thermostats, relaxation oscillators, power electronics, and simple closed-loop controllers where simplicity and robustness are more important than exact regulation.

De-Noise

Real analog signals are often corrupted by noise.
When such a signal is fed directly into a comparator, small noise amplitudes around the threshold can cause rapid switching of the output (chatter).

The Schmitt trigger solves this problem by its two distinct thresholds \(U_{\rm sh,u}\) and \(U_{\rm sh,l}\).
As long as the input signal remains between these two values, the output state does not change.

This makes comparators with hysteresis ideal for:

• cleaning up slowly varying or noisy sensor signals,
• debouncing mechanical switches,
• converting noisy analog waveforms into clean digital signals.

Analog-to-Digital Converter (ADC)

At its core, every analog-to-digital converter contains at least one comparator.

A comparator performs a binary decision:
Is the input voltage larger or smaller than a given reference?

In the simplest case (1-bit ADC):

• one comparator compares \(u_{\rm I}\) with a reference voltage \(U_{\rm ref}\),
• the output represents a single digital bit.

More complex ADCs like the flash ADC (shown in the simulation below) use multiple comparators or reuse one comparator repeatedly with different reference values.

Thus, understanding comparator behavior is fundamental for understanding how analog information is converted into digital form.

Common pitfalls

• Treating a comparator like a linear op-amp: assuming the output follows a linear gain law \(u_{\rm O}=A_{\rm D}\,u_{\rm d}\). In reality, the output almost always saturates at \(U_{\rm sat,min}\) or \(U_{\rm sat,max}\).
• Using negative-feedback intuition: expecting the circuit to automatically enforce \(u_{\rm d}=0\). Without negative feedback, \(u_{\rm d}=0\) is only the *switching boundary*, not an operating point.
• Mixing up inputs: confusing the non-inverting input \(u_{\rm p}\) and inverting input \(u_{\rm m}\), which leads to predicting the wrong output polarity.
• Ignoring the output stage type:
  1. forgetting that an open-collector comparator cannot actively drive a high level,
  2. omitting the required pull-up resistor, resulting in an undefined/high-ohmic output.
• Forgetting saturation limits: assuming ideal logic levels, while real comparators are limited by their supply voltages \(U_{\rm sat,min}\), \(U_{\rm sat,max}\).
• No hysteresis for noisy signals: using a plain comparator where a Schmitt trigger is required, leading to output chatter when \(u_{\rm I}\) fluctuates near the threshold.
• Sign errors in Schmitt-trigger thresholds: losing track of the sign of \(u_{\rm O}\) when deriving or applying

\[ U_{\rm sh,u}=+\frac{R_1}{R_2}u_{\rm O}, \qquad U_{\rm sh,l}=-\frac{R_1}{R_2}u_{\rm O}. \]

Exercises

Conceptual checks

1. Explain in one or two sentences why a comparator is normally operated without negative feedback.
2. What information about the input signal does the comparator output represent when \(u_{\rm O}\) is in saturation?
3. Why is \(u_{\rm d}=0\) a special point for a comparator, even though it is not a stable operating point?

Exercises

Embedded resources