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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:

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90-minute plan

1. Warm-up (x min):
   1. ….
2. Core concepts & derivations (x min):
   1. …
3. Practice (x min): …
4. Wrap-up (x min): Summary box; common pitfalls checklist.

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 $u_{\rm n}$. They result in the differential voltage $u_{\rm d} = u_{\rm p} - u_{\rm n}$.

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 ($R_{\rm I}=0$, $R_{\rm O}\rightarrow \infty$, $A_{\rm D}\rightarrow \infty$) 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

See Bang–bang_control

De-Noise

Analog-to-Digital Converter (ADC)

Common pitfalls

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Exercises

Worked examples

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Embedded resources