Comparator Hysteresis Calculator: Theory & Equations

Introduction to the Comparator Hysteresis Calculator

The Comparator with Hysteresis Calculator is an essential electronic design tool for electrical engineers. It allows you to calculate the precise resistor values required to set the High Threshold Voltage ($V_{th}$) and Low Threshold Voltage ($V_{tl}$) for a comparator circuit.

By determining the correct Reference Voltage ($V_{ref}$) and the resistor ratio ($R_2/R_1$), you can design stable, noise-immune circuits. Below, we explore the fundamental theory behind comparators, the importance of hysteresis, and the mathematical formulas used in this calculator.

What is a Comparator?

A Comparator is an integrated circuit (IC) that compares two input voltages and outputs a digital signal indicating which is larger.

• Function: It acts as a one-bit Analog-to-Digital Converter (ADC).
• Operation:
  • If $V_{IN} > V_{REF}$, the output swings to the positive supply rail (Logic 1).
  • If $V_{IN} < V_{REF}$, the output swings to the negative supply rail (Logic 0).

Comparators are widely used in level detection, window detectors, and Analog-to-Digital converters. Common comparator ICs include the LM339, LM393, and TL331.

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What is Hysteresis? (The Schmitt Trigger)

The Problem: Noise and "Chatter"

In a standard comparator, if the input signal moves slowly or contains noise, the output may flicker rapidly (bounce) between high and low states when the input voltage is near the reference threshold. This oscillation is often called "chatter."

The Solution: Hysteresis

Hysteresis eliminates this noise by introducing two distinct threshold voltages instead of one:

1. High Threshold ($V_{th}$): The voltage the input must rise above to switch the output Low.
2. Low Threshold ($V_{tl}$): The voltage the input must fall below to switch the output High.

By adding a feedback resistor to the circuit (Positive Feedback), the comparator "remembers" its current state and resists changing until the input significantly crosses the new threshold. This configuration is often called a Schmitt Trigger.

How the Circuit Works

An inverting comparator with hysteresis uses a positive feedback network.

• Resistors $R_1$ and $R_2$ establish a reference voltage at the non-inverting input (+).
• Resistor $R_h$ (Feedback) connects the output to the non-inverting input.

When the output is High, the feedback resistor pulls the voltage at the non-inverting input slightly higher (creating $V_{th}$). When the output is Low, the feedback resistor pulls the non-inverting input slightly lower (creating $V_{tl}$). This separation creates the "Hysteresis Window."

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Comparator Hysteresis Formulas

The calculator uses the following equations to determine the relationship between the threshold voltages and the resistor network for an Inverting Comparator with Hysteresis.

1. Calculate Resistor Ratio ($R$)

First, we define the ratio of the feedback resistors:

$R = \frac{R_2}{R_1}$

The ratio is determined by the difference between the supply voltages and the desired hysteresis window width:

$R = \frac{V_P - V_N}{V_{th} - V_{tl}}$

2. Calculate Threshold Voltages

Once the ratio and reference voltage are known, the thresholds are defined as:

High Threshold Voltage ($V_{th}$):

$V_{th} = V_{ref} \cdot \left(\frac{1+R}{R}\right) - V_n \cdot \left(\frac{1}{R}\right)$

Low Threshold Voltage ($V_{tl}$):

$V_{tl} = V_{ref} \cdot \left(\frac{1+R}{R}\right) - V_p \cdot \left(\frac{1}{R}\right)$

3. Calculate Reference Voltage ($V_{ref}$)

$V_{ref} = V_n + (V_{th} - V_n) \cdot \frac{R}{1+R}$

Where:

• $V_{th}$: High Threshold Voltage
• $V_{tl}$: Low Threshold Voltage
• $V_{ref}$: Internal Reference Voltage applied to the resistor divider
• $V_p$: Positive Supply Voltage ($V_{CC}$)
• $V_n$: Negative Supply Voltage ($V_{EE}$ or Ground)

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Op-Amp vs. Dedicated Comparator

Can you use a standard Operational Amplifier (Op-Amp) like an LM741 or LM358 as a comparator?

• Yes, but with limitations. Op-amps are designed for linear operation and negative feedback. When used as a comparator (open-loop), they are slow to recover from saturation and have slow slew rates.
• Dedicated Comparators (like the LM393) are designed for open-loop operation. They have very fast switching times and open-collector outputs, making them ideal for driving digital logic circuits.

Applications

• Noise Suppression: Cleaning up noisy sensor signals.
• Debouncing: Smoothing out mechanical switch signals.
• Relaxation Oscillators: Creating square waves.
• Zero-Crossing Detectors: Detecting when an AC signal crosses zero volts.