Designing an Analog Circuit to Calculate √(x² + y²)

A Comprehensive Guide to Building Accurate Vector Magnitude Calculators in Analog Electronics

Key Takeaways

• Comprehensive Stages: The circuit design involves squaring input signals, summing them, and extracting the square root through specialized techniques.
• Component Selection: High-precision analog multipliers and operational amplifiers are critical for accuracy and performance.
• Implementation Methods: Multiple approaches exist, including log-antilog circuits, MOSFET-based configurations, and feedback-loop systems, each with distinct advantages.

Introduction

Calculating the magnitude of a vector defined by two components is a fundamental operation in various electronic and signal processing applications. Specifically, producing an output equivalent to √(x² + y²) involves implementing the Pythagorean theorem in an analog circuit. This guide explores detailed methodologies to design such a circuit, ensuring high accuracy and reliability.

Circuit Design Overview

1. Squaring the Input Signals

The first step involves squaring both input signals, x and y. This can be achieved using analog multipliers, which multiply the input by itself to produce x² and y² respectively.

a. Analog Multipliers

Integrated circuits like the AD633 or MPY534 are commonly used analog multipliers that can square input signals. By connecting the same input to both inputs of the multiplier, the output will be the square of the input signal.

b. Circuit Implementation

Each input signal is fed into dedicated multiplier circuits:

• x² Circuit: Connect input x to both X1 and X2 inputs of the multiplier to obtain x².
• y² Circuit: Similarly, connect input y to both X1 and X2 inputs to obtain y².

2. Summing the Squared Signals

After obtaining x² and y², the next step is to sum these values to form (x² + y²).

a. Summing Amplifier Configuration

An operational amplifier (op-amp) can be configured as a summing amplifier. Using equal resistor values for each input ensures that both squared signals are added with appropriate scaling.

b. Circuit Design

Connect the outputs of the x² and y² multipliers to the input resistors of the summing amplifier. The op-amp will produce an output voltage proportional to the sum of the squared inputs:

V_sum = x² + y²

3. Square Root Extraction

The final stage involves extracting the square root of the summed signal to produce the desired output.

a. Log-Antilog Circuit

This method utilizes logarithmic and exponential amplifiers to compute the square root.

1. Logarithmic Amplifier: Converts V_sum to its natural logarithm.
2. Scaling Stage: An op-amp scales the logarithmic output by 0.5.
3. Exponential Amplifier: Converts the scaled log value back to the linear domain, effectively computing the square root.

b. MOSFET-Based Square Root Circuit

This approach leverages MOSFETs to directly compute the square root without logarithmic conversion:

1. Squaring Circuits: Similar to the previous stages, x² and y² are generated.
2. MOSFET Configuration: Utilize MOSFETs operating in specific regions to enforce the square-root relationship.
3. Output: The circuit outputs V_out = √(x² + y²).

c. Feedback-Loop-Based Circuit

Involves a feedback mechanism where the output is regulated to satisfy the square-root condition:

1. Summing Integrator: Combines x² + y² with a feedback term proportional to V_out².
2. Multiplier: Generates the feedback term V_out².
3. Low-Pass Filter: Ensures stability by filtering high-frequency components.
4. Operation: The feedback loop maintains V_out = √(x² + y²).

Detailed Implementation Steps

Stage 1: Squaring the Inputs

Utilize precision analog multipliers to generate the squares of the input signals:

• x² Generation: Input x is connected to both inputs of the multiplier to produce x².
• y² Generation: Similarly, input y is connected to both inputs of another multiplier to produce y².

Stage 2: Summing the Squared Signals

An op-amp in a summing configuration adds the squared signals:

• The summed output V_sum equals x² + y².
• Proper scaling ensures that any inherent scaling factors of the multipliers are accounted for.

Stage 3: Extracting the Square Root

Several methods can be employed to extract the square root of V_sum:

a. Log-Antilog Method

This method transforms the multiplicative relationship into additive terms:

• The logarithmic amplifier computes ln(V_sum).
• An op-amp scales this value by 0.5.
• The exponential amplifier then computes e^(0.5 * ln(V_sum)) = √(V_sum).

b. MOSFET-Based Approach

Using MOSFETs in specific regions, the circuit directly outputs the square root:

• Operational transistors enforce the relationship V_out = √(V_sum).
• Gate voltages are adjusted to maintain proportionality and stability.

c. Feedback-Loop System

A feedback mechanism ensures the output satisfies the square root condition:

• The summing integrator receives V_sum and the feedback term V_out².
• The multiplier generates V_out², which is fed back into the integrator.
• Low-pass filtering keeps the system stable and responsive.

Component Selection and Precision

Analog Multipliers

Choosing high-precision analog multipliers is crucial. Chips like the AD633 offer reliable performance with features such as low offset and high linearity, essential for accurate squaring operations.

Operational Amplifiers

Op-amps used in summing and scaling stages must have low noise, high bandwidth, and precise gain characteristics to maintain signal integrity and accuracy.

Temperature Compensation

Both analog multipliers and op-amps can exhibit performance variations with temperature changes. Implementing temperature compensation techniques ensures consistent circuit behavior across varying environments.

Implementation Methods Comparison

Practical Considerations

Scaling and Calibration

Analog multipliers like the AD633 introduce scaling factors (e.g., V_out = (X1 * X2)/10). It's essential to calibrate the summing and square root stages to account for these scale factors, ensuring the final output accurately represents √(x² + y²).

Dynamic Range

The circuit must accommodate the expected range of input signals. Implementing adjustable gain stages can help manage varying input amplitudes, preventing saturation or signal clipping.

DC Offsets

Since logarithmic amplifiers require positive inputs, adding a DC offset to V_sum may be necessary. Post square-root extraction, this offset should be removed to maintain the integrity of the output signal.

Conclusion

Designing an analog circuit to compute √(x² + y²) involves a multi-stage process of squaring inputs, summing them, and extracting the square root. Various implementation methods, including log-antilog circuits, MOSFET-based approaches, and feedback-loop systems, offer different advantages depending on accuracy requirements, component availability, and application contexts. Precision in component selection and careful calibration are imperative to achieving reliable and accurate outputs. By thoroughly understanding and executing each stage, one can develop a robust analog circuit capable of accurately processing two input signals to produce the desired vector magnitude.

References

medium.com

Analog Circuit that Solves Pythagorean Theorem

electronics.stackexchange.com

Extracting the Square Root of a Voltage