Bode Resonator Phase Difference Theoretical Value Calculator

This calculator computes the theoretical phase difference of a Bode resonator system based on input parameters such as resonant frequency, damping ratio, and frequency of interest.

Input Parameters

Calculation Results

Calculation Formula

Phase Difference (φ) = arctan(2ζω/ω₀ - ω²/ω₀²)

Where:
φ = Phase difference in radians
ζ = Damping ratio
ω = Operating frequency
ω₀ = Resonant frequency

Bode Resonator Phase Difference Theoretical Value Calculator Usage Guide

Learn how to use the Bode Resonator Phase Difference Theoretical Value Calculator and its working principles

How to Use This Calculator

  1. Enter the resonant frequency (ω₀) in radians per second. This is the natural frequency at which the resonator oscillates.
  2. Enter the damping ratio (ζ). This value represents how much energy is dissipated in each oscillation, typically between 0 (undamped) and 1 (critically damped).
  3. Enter the operating frequency (ω) at which you want to calculate the phase difference, also in radians per second.
  4. Click the "Calculate" button to compute the theoretical phase difference.
  5. The results will be displayed in both radians and degrees.

Understanding the Formula

The phase difference (φ) between the input and output of a damped resonator is calculated using the formula:

φ = arctan(2ζω/ω₀ - ω²/ω₀²)

This formula shows how the phase difference depends on the damping ratio and the ratio of operating frequency to resonant frequency. At resonance (ω = ω₀), the phase difference is 90 degrees (π/2 radians) for a lightly damped system (ζ < 1).

Practical Applications

This calculator is useful in control systems engineering, particularly when designing and analyzing filter circuits, oscillators, and vibration isolation systems. The phase difference is a critical parameter in determining system stability and response characteristics.

Example

For a resonator with a resonant frequency of 10 rad/s, a damping ratio of 0.1, and operating at 12 rad/s:

  • Input values: ω₀ = 10, ζ = 0.1, ω = 12
  • Calculated phase difference: 0.615 radians (35.36 degrees)