Magnitude Response Calculator - Calculator Academy

Enter the frequency, resonant frequency, and quality factor into the calculator to determine the magnitude response in decibels (dB).

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Magnitude Response Formula

This calculator evaluates the magnitude response of a normalized resonant system at a specific frequency. It shows how strongly the system responds at the measured frequency compared with its peak response at resonance.

The linear magnitude ratio used by the calculator is:

\left|H(f)\right| = \frac{1}{\sqrt{1 + Q^2\left(\frac{f}{f_r} - \frac{f_r}{f}\right)^2}}

To express that response in decibels, use:

MR = 20\log_{10}\left(\left|H(f)\right|\right)

Combining both steps gives the full calculator equation:

MR = 20\log_{10}\left(\frac{1}{\sqrt{1 + Q^2\left(\frac{f}{f_r} - \frac{f_r}{f}\right)^2}}\right)

Variable Definitions

MR: Magnitude response in decibels (dB).
f: The frequency being evaluated.
f_r: The resonant frequency of the system.
Q: The quality factor, which controls how narrow or broad the resonance is. Higher values mean sharper selectivity.

How to Interpret the Result

0 dB means the response is at the normalized peak.
Negative dB values mean the frequency is attenuated relative to the peak.
As the measured frequency moves farther away from resonance, the response becomes more negative.
A larger Q produces a steeper, narrower response around resonance.
A smaller Q produces a broader response with less aggressive roll-off.

At resonance, the response reaches its maximum in this model:

f = f_r \Rightarrow MR = 0 \text{ dB}

This means the calculator is especially useful for understanding tuned circuits, band-pass behavior, filter selectivity, and other systems where the response peaks near one frequency.

How to Calculate Magnitude Response

Enter the measured frequency f.
Enter the resonant frequency f_r.
Enter the quality factor Q.
Apply the frequency-ratio term inside the square root.
Convert the resulting linear magnitude to decibels using the base-10 logarithm.

If you already know the result in decibels and want the corresponding linear amplitude ratio, use:

\left|H(f)\right| = 10^{MR/20}

Useful Resonance Relationships

The half-power condition corresponds to approximately -3 dB. For this response model, that occurs when:

\left|\frac{f}{f_r} - \frac{f_r}{f}\right| = \frac{1}{Q}

For a high-Q resonant system, the bandwidth is often approximated by:

B \approx \frac{f_r}{Q}

This approximation is most useful when the resonance is fairly sharp and the system behaves like a narrow band-pass response.

Example

If the measured frequency is 500 Hz, the resonant frequency is 1000 Hz, and the quality factor is 5, the magnitude response is:

MR = 20\log_{10}\left(\frac{1}{\sqrt{1 + 5^2\left(\frac{500}{1000} - \frac{1000}{500}\right)^2}}\right) \approx -17.58 \text{ dB}

A result of -17.58 dB indicates the system response at 500 Hz is much smaller than the peak response at resonance.

Solving for Quality Factor

If the calculator is being used to find Q instead of magnitude response, the equation can be rearranged to:

Q = \frac{\sqrt{10^{-MR/10} - 1}}{\left|\frac{f}{f_r} - \frac{f_r}{f}\right|}

Because this model is normalized to a peak of 0 dB, valid magnitude-response inputs are typically 0 dB or less.

Input Tips

Use the same unit for both f and f_r. Hz, kHz, and MHz all work as long as they match.
Q is unitless and should be entered as a positive value.
Frequency values should be greater than zero.
Very high-Q systems can change rapidly with small shifts in frequency, so precise inputs matter.
If your result seems too large in magnitude, check for mixed units such as Hz for one field and kHz for the other.

Why Magnitude Response Matters

Magnitude response is one of the clearest ways to understand how a system treats different frequencies. Engineers and students use it to evaluate:

filter sharpness and selectivity,
resonant amplification behavior,
off-resonance attenuation,
approximate bandwidth, and
how sensitive a design is to frequency variation.

When combined with resonant frequency and quality factor, magnitude response gives a compact picture of how a system behaves across its operating range.