Name: bode plot slope and phase changes
Uploaded: 2025-05-22T09:06:14.000Z
Duration: 1 min 23 s
Channel: JoVE

Consider the transfer function in its standard form, with poles and zeros
For a transfer function with a simple zero, the magnitude gain at small frequency values is a straight line with zero slope, and the phase approaches zero.
At the corner frequency, the asymptotic magnitude deviates from the zero-slope line, and the phase approaches 45 degrees.
At higher frequencies, the magnitude plot forms a +20 dB/decade line, and the phase is 90 degrees.
A simple pole is the reciprocal of a simple zero. This means that pole based Bode plots mirror the simple zero plot, reflected about the horizontal axis.
Consider a quadratic pole transfer function.
At lower frequencies, the gain and phase angle approaches zero.
At the corner frequency, the asymptotic magnitude deviation depends on the damping factor, and the phase angle is nearly -90 degrees.
At higher frequencies, the magnitude plot forms a straight line with a slope of -40 dB/decade and a phase of -180 degrees.
For more than one quadratic pole, the slope of the line and phase shift are multiplied by the number of poles.

transfer function and bode plots-ii

In the standard form, the transfer function is shown in constant gain, poles/zeros at origin, simple poles/zeros, and quadratic poles/zeros; each contributing uniquely to the system's overall response. The term represents the magnitude of the simple zero:
<img alt="Transfer function equation \( H_{dB} = 20 \log_{10} \left| 1 + \frac{j\omega}{\omega_1} \right| \)." src="/files/ftp_upload/15142/15142_Equation_1.svg" style="text-align: center; height: 36px;" data-altupdatedat="1754904545000" data-alt="Frequency response equation \(H_{dB}=20\log_{10}(1+\frac{j\omega}{\omega_1})\) in signal processing." />
The Bode magnitude plot remains flat at low frequencies (approaching 0 dB) and begins to ascend at 20 dB/decade after a specific frequency known as the corner or break frequency, &#969;1. This is the frequency where the magnitude plot's slope changes and the actual response begins to deviate from the straight-line approximation. This deviation is quantified as 3 dB at &#969;=&#969;1.
The phase angle &#981;, expressed as:
<img alt="Phase angle formula: &#966;=tan&#8315;&#185;(&#969;/Z&#8321;) values chart; &#969;=0, Z&#8321;, &#8734; in circuit analysis equations." src="/files/ftp_upload/15142/15142_Equation_2.svg" style="text-align: center; height: 60px;" data-altupdatedat="1754904545000" data-alt="Phase angle equation diagram, &#966;=tan&#8315;&#185;(&#969;/Z&#8321;) with angular frequency conditions " />
Phase angle starts at 0&#176; and approaches 90&#176; asymptotically as the frequency increases. For frequencies much lower than the corner frequency (&#969;&#8810;&#969;1), the term j&#969;/&#969;1 is very small, so the magnitude is negligible, and the phase is essentially zero. As the frequency approaches &#969;1, leading to a -3 dB point in magnitude and a phase angle of 45&#176;. For frequencies much higher than &#969;1 (&#969;&#8811;&#969;1), the magnitude's slope changes to 20 dB/decade, and the phase settles at 90&#176;.
<img alt="Bode plot diagrams of magnitude and phase, illustrating frequency response and gain, 3dB point." src="/files/ftp_upload/15142/15142_Figure_1.svg" style="text-align: center; height: 160px;" data-altupdatedat="1754904545000" data-alt="Bode plot charts; frequency response; gain and phase shift; logarithmic scale analysis." />
Quadratic pole/zero:
The magnitude and phase angle of a quadratic pole is:
<img alt="Transfer function magnitude equation \(H_{dB}\) for frequency response analysis." src="/files/ftp_upload/15142/15142_Equation_3.svg" style="text-align: center; height: 42px;" data-altupdatedat="1754904545000" data-alt="Frequency response formula, transfer function equation, mathematical representation, H_dB calculation." />
<img alt="Phase angle equation in mechanical systems; formula, &#966;=-tan&#8315;&#185;(2&#958;&#969;/&#969;&#8345;)/(1-(&#969;/&#969;&#8345;)&#178;)." src="/files/ftp_upload/15142/15142_Equation_4.svg" style="text-align: center; height: 60px;" data-altupdatedat="1754904545000" data-alt="Phase lag formula and values; equation diagram; mechanical dynamics analysis." />
The amplitude plot for a quadratic pole has two parts: a flat response below the natural frequency &#969;n, and a -40 dB/decade slope above &#969;n, with the actual plot's peak varying with the damping factor &#950;2. The phase plot for a quadratic pole decreases linearly with a slope of -90&#176; per decade, starting at one-tenth of the natural frequency and ending at ten times that, influenced by the damping factor &#950;2.

Transfer function and Bode Plots-II

In the standard form, the transfer function is shown in constant gain, poles/zeros at origin, simple poles/zeros, and quadratic poles/zeros; each ...

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Electrical Engineering

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Frequency Response

KEY TERMS AND CONCEPTS

bode plot slope and phase changes

Bode plots show how a transfer function changes with frequency. The standard form includes constant gain, poles and zeros at the origin, simple poles and zeros, and quadratic poles and zeros. Each part shapes the system response in a different way.

A simple zero sets the magnitude trend first. The Bode magnitude plot stays flat at low frequencies, near 0 dB, and then rises at 20 dB per decade after the corner or break frequency, ω1. That corner frequency is where the straight-line approximation starts to differ from the actual curve, and the error is 3 dB at ω = ω1.

The phase angle for a simple zero also changes with frequency. It starts at 0° and rises toward 90° as frequency increases. When ω is much smaller than ω1, the term jω/ω1 is very small, so the magnitude is negligible and the phase is close to zero. At ω = ω1, the magnitude is at the -3 dB point and the phase is 45°. When ω is much larger than ω1, the magnitude slope is 20 dB per decade and the phase settles near 90°.

Quadratic poles have a different shape on Bode plots. Their magnitude is flat below the natural frequency, ωn, and drops at -40 dB per decade above ωn. The actual peak depends on the damping factor, ζ2. Their phase plot decreases linearly at -90° per decade from one-tenth of ωn to ten times ωn, and that shape is also affected by ζ2.

Bode Plot Slope and Phase Changes

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Bode Plot Slope and Phase Changes

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