Bode Plot Calculator with Steps

This calculator draws the Bode magnitude and phase plots for a transfer function you enter, and it shows the step-by-step working: corner frequencies, slopes, and the asymptotic straight-line segments that approximate the true curves. You type in the numerator and denominator polynomials (or a gain and a list of poles/zeros), and the page returns the corner frequencies, the gain at low frequency, the slope of each segment, and the plotted magnitude and phase data. The result is both the exact curve and the classical straight-line approximation used in control engineering textbooks.

Formula

The Bode plot is built from a transfer function written as a ratio of polynomials in the Laplace variable s:

H(s) = K · (s^m + a₁s^m−1 + … + a_m) / (sⁿ + b₁s^n−1 + … + b_n)

where K is the DC gain (the ratio of the leading coefficients), m is the degree of the numerator, and n is the degree of the denominator. The magnitude in decibels is 20·log₁₀|H(jω)|, and the phase is the angle of H(jω) in degrees. The calculator factors the polynomials to find the corner frequencies (the magnitudes of the roots), then applies the standard rules: each first-order pole contributes −20 dB/decade and −90° of phase, each first-order zero contributes +20 dB/decade and +90° of phase, and a pole or zero at the origin contributes ±20 dB/decade and ±90° from the start. The straight-line approximation is the sum of these contributions, and the exact curve is computed by evaluating H(jω) directly.

Worked Example

Using the default values from the calculator: numerator coefficients [1, 10] (meaning s + 10) and denominator coefficients [1, 2, 10] (meaning s² + 2s + 10).

1. DC gain: K = 1/1 = 1, so 20·log₁₀(1) = 0 dB.
2. Factor the denominator: s² + 2s + 10 = (s + 1 + 3j)(s + 1 − 3j). The corner frequency is the magnitude of the roots: √(1² + 3²) = √10 ≈ 3.16 rad/s.
3. Factor the numerator: s + 10 gives a zero at s = −10, so a corner at 10 rad/s.
4. Low-frequency asymptote: 0 dB (the DC gain).
5. Between 3.16 and 10 rad/s: the denominator is second-order, so the slope is −40 dB/decade (two poles). The phase starts at 0° and approaches −180°.
6. Above 10 rad/s: the zero adds +20 dB/decade, so the slope becomes −20 dB/decade. The phase approaches −90°.

The calculator’s first output gives DC gain = 1.000, corner frequency = 3.16 rad/s, high-frequency slope = −20 dB/decade, and high-frequency phase = −90°, matching the manual calculation.

Why the Bode Plot Matters

Bode plots are a core tool in control systems and signal processing because they let you see a system’s frequency response at a glance. The magnitude plot shows how much a sinusoidal input is amplified or attenuated at each frequency, and the phase plot shows the time shift (delay) the system introduces. Engineers use Bode plots to check stability margins (gain margin and phase margin), to design compensators (lead, lag, PID), and to predict how a filter will behave. The straight-line approximation is especially useful because it lets you sketch the response by hand without a computer, and it reveals the system’s type (number of poles at the origin) and relative order.

How to Use the Calculator

Enter the coefficients of the numerator and denominator polynomials in descending powers of s, separated by commas. For example, a transfer function (s+2)/(s²+3s+4) would be entered as:

• Numerator: 1, 2
• Denominator: 1, 3, 4

The calculator computes the DC gain, the corner frequencies (from the roots of the polynomials), the high-frequency slope, and the high-frequency phase. It does not plot the full curve in this text-based version, but the outputs give you the key parameters to draw the asymptotic Bode plot manually. For the exact curve, you can evaluate H(jω) at many frequencies using the same formula and plot the points.

Common Mistakes and How to Avoid Them

• Unit mix-ups: Bode plots use frequency in rad/s, not Hz. To convert, multiply Hz by 2Ï€. The corner frequency from a pole at s = −a is a rad/s, which corresponds to a = 2Ï€f.
• Wrong coefficient order: Polynomial coefficients must be in descending powers. Entering [10, 1] instead of [1, 10] gives a completely different transfer function.
• Forgetting the DC gain: If the leading coefficients are not 1, the DC gain is not 1. Always divide the leading coefficients first.
• Rounding too early: When computing corner frequencies from roots, keep several significant digits until the final step. Rounding √10 to 3 instead of 3.16 changes the plot noticeably.
• Phase wrap: The phase is usually plotted from −180° to +180° or from 0° to −360°. The calculator gives the asymptotic value, but the actual phase may pass through −180° at a specific frequency, which is critical for stability.

Limitations of the Straight-Line Approximation

The asymptotic Bode plot is exact at very low and very high frequencies, but near a corner frequency the true curve deviates. For a first-order pole, the error is about 3 dB at the corner (the true magnitude is −3 dB, not 0 dB), and the phase error is about 5.7° at one decade below and above the corner. For a second-order pole, the error depends on the damping ratio; if the damping is low, there is a resonant peak that the straight-line approximation misses entirely. The calculator’s exact evaluation (using the polynomial directly) captures these effects, so for precise work you should use the exact curve, not the asymptotes.

Frequently Asked Questions

What is a Bode plot actually showing? It shows the magnitude (in dB) and phase (in degrees) of a system’s transfer function as a function of frequency, plotted on a logarithmic frequency axis. The magnitude plot tells you how much a sine wave of a given frequency is amplified or attenuated; the phase plot tells you how much the output lags or leads the input.

How do I find the corner frequency from a transfer function? Factor the numerator and denominator into first-order and second-order terms. The corner frequency for a first-order pole or zero at s = −a is a rad/s. For a second-order term s² + 2ζω_ns + ω_n², the corner is ω_n rad/s. The calculator does this by finding the roots of the polynomials.

Why do some Bode plots have a slope of −40 dB/decade? That slope indicates two poles at the same corner frequency (or two separate poles close together). Each first-order pole contributes −20 dB/decade, so two poles give −40 dB/decade. Similarly, three poles give −60 dB/decade, and so on.

What is the difference between a Bode plot and a Nyquist plot? A Bode plot shows magnitude and phase separately as functions of frequency. A Nyquist plot shows the same frequency response as a single curve in the complex plane, plotting real part against imaginary part as frequency varies. Both convey the same information, but Bode plots are easier to read for design and stability margins.

Can I use this calculator for a transfer function with complex coefficients? No, this calculator only accepts real polynomial coefficients, which is the standard for linear time-invariant systems. Complex coefficients arise in some signal processing contexts, but they are rare in control systems.

How do I convert the corner frequency from rad/s to Hz? Divide by 2π. For example, a corner at 3.16 rad/s is 3.16/(2π) ≈ 0.503 Hz. Many Bode plots use rad/s, so always check the axis label.

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in num | Numerator coefficients (descending powers) | - | 1, 10 | 0
in den | Denominator coefficients (descending powers) | - | 1, 2, 10 | 0
out gain | DC gain (K) | - | num[0] / den[0] | 3
out corners | Corner frequencies (rad/s) | - | sqrt(den[2] / den[0]) | 2
out slope | High-frequency slope | dB/dec | -20 * (count(num) - 1 - (count(den) - 1)) | 0
out phase | Phase at high freq | deg | -90 * (count(den) - 1 - (count(num) - 1)) | 0