What is the difference between a Bode plot and a Nyquist plot?

A Bode plot shows magnitude and phase separately against frequency (two graphs). A Nyquist plot combines them into a single polar plot showing real vs. imaginary parts of the frequency response as a parametric curve. Bode is better for design; Nyquist is better for stability analysis of complex systems.

What does -20 dB/decade mean?

It means the magnitude decreases by a factor of 10 (20 dB) for every factor-of-10 increase in frequency (one decade). This is the slope contributed by a single real pole. Two poles give −40 dB/decade, three give −60 dB/decade, and so on.

How do I create a Bode plot online?

Enter your transfer function into the LAPLACE Calculator at www.lapcalc.com. It will compute and display both magnitude and phase plots with exact values, corner frequencies, and stability margins marked automatically.

Bode Plot

Q: How do you plot a Bode plot step by step?

Factor the transfer function, identify DC gain, mark corner frequencies from poles/zeros, draw asymptotic magnitude lines (−20 dB/dec per pole, +20 dB/dec per zero), add individual phase contributions, then smooth the curves near corner frequencies with ±3 dB magnitude corrections.

Quick Answer

A Bode plot displays a system's frequency response as two graphs: magnitude (dB) vs. log frequency and phase (degrees) vs. log frequency. To plot one, express the transfer function H(s) in factored form, identify corner frequencies from poles and zeros, sketch asymptotic approximations at ±20 dB/decade per pole or zero, then refine near corner frequencies. For H(s) = 10/(s+1)(s+10), the magnitude starts at 20 dB, breaks down at ω=1 rad/s, and again at ω=10 rad/s, rolling off at −40 dB/decade at high frequencies.

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Step-by-Step: How to Plot a Bode Diagram

Start by writing your transfer function in standard form with the DC gain factored out. For H(s) = 100/[(s+2)(s+50)], rewrite as H(s) = (100/100) · 1/[(s/2 + 1)(s/50 + 1)] = 1/[(s/2 + 1)(s/50 + 1)]. The DC gain is 1 (0 dB). Mark corner frequencies at ω₁ = 2 rad/s and ω₂ = 50 rad/s on the frequency axis. Below ω₁, the magnitude is flat at 0 dB. Between ω₁ and ω₂, it slopes down at −20 dB/decade (one pole contributing). Above ω₂, both poles contribute, giving −40 dB/decade. The actual curve rounds off by about 3 dB at each corner frequency.

Key Formulas

Asymptotic Approximations: The Straight-Line Method

Bode's key insight was that each pole and zero contributes a simple straight-line segment to the overall plot. A real pole at s = −a contributes 0 dB below ω = a and −20 dB/decade above it. A real zero at s = −b contributes 0 dB below ω = b and +20 dB/decade above it. Complex conjugate poles create a resonance peak whose height depends on the damping ratio ζ — underdamped systems (ζ < 0.707) show a visible peak. For phase, each pole transitions from 0° to −90° over roughly one decade centered on its corner frequency. You simply add all these individual contributions together.

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Plotting Phase: The Key Details Engineers Miss

Phase plotting requires more care than magnitude. Each real pole contributes a phase shift that begins one decade before the corner frequency and ends one decade after, transitioning from 0° to −90°. At exactly the corner frequency, the phase contribution is −45°. For a system with multiple poles, these transitions overlap and add. A common mistake is forgetting the phase contribution of the DC gain: a negative gain (K < 0) adds −180° of phase at all frequencies. Pure time delays add phase that decreases linearly with frequency: ∠e^{−sT} = −ωT radians, which can devastate phase margin at high frequencies.

Bode Plots for Common Transfer Function Types

Integrators (1/s) appear as a straight line at −20 dB/decade passing through 0 dB at ω = 1, with constant −90° phase. Differentiators (s) are the mirror image: +20 dB/decade with +90° phase. First-order systems are the building blocks described above. Second-order systems with transfer function ω²ₙ/(s² + 2ζωₙs + ω²ₙ) have flat magnitude below ωₙ, a peak of 1/(2ζ) at ωₙ (when ζ < 0.707), and −40 dB/decade rolloff above. The phase transitions from 0° to −180° with the steepest slope at ωₙ. PID controllers combine all of these: integral action for low-frequency boost, proportional gain for midband, and derivative action for high-frequency phase lead.

Using Software vs. Hand-Sketching Bode Plots

While software like MATLAB, Python's control library, and the LAPLACE Calculator generate exact Bode plots instantly, hand-sketching remains an essential skill. Sketching forces you to understand what each pole, zero, and gain does to the frequency response — knowledge that is critical when designing compensators or debugging unexpected system behavior. The recommended workflow is: sketch by hand first to build intuition, then verify with software. When your hand sketch and the software disagree, investigating why deepens your understanding of the system.

Frequently Asked Questions

Factor the transfer function, identify DC gain, mark corner frequencies from poles/zeros, draw asymptotic magnitude lines (−20 dB/dec per pole, +20 dB/dec per zero), add individual phase contributions, then smooth the curves near corner frequencies with ±3 dB magnitude corrections.

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