Name: reading frequency response with bode plots
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 24 s
Channel: JoVE

Tissue electrodes in electrocardiograms (E-C-G) establish a conductive pathway for electrical currents between the tissues and the measuring electrodes, enabling the observation of heart activity.
The electrode-tissue interface dynamics has a circuit model encompassing electrode resistance, the capacitance at the electrode-tissue interface, and tissue resistance. The potential difference represents the voltage difference between the electrode and the tissue.
Here, the input impedance equals the tissue resistance. The output impedance is the addition of the tissue resistance to the parallel combination of resistance and capacitance at the electrode-tissue interface.
The ratio of the output phasor to the input phasor, calculated using the known resistance and capacitance values is the transfer function.
This can be approximated across three distinct frequency ranges.
The Bode magnitude plot on a semilog graph depicts the calculated logarithmic gain in decibels against frequency in radians per second.
The low and high-frequency asymptotes are horizontal lines with a constant gain. In the intermediate frequency range, the asymptotic magnitude plot is linear with a 20-decibel-per-decade slope.

bode plots construction

The Bode plot is an essential tool in control system analysis, mapping the frequency response of a system through a magnitude plot and a phase plot, both against a logarithmic frequency axis. To construct a Bode plot, consider the transfer function H(&#969;):
<img alt="Transfer function equation H(&#969;)=200j&#969;/((j&#969;+2)(j&#969;+10)); control systems analysis." src="/files/ftp_upload/15102/15102_Equation_1.svg" style="text-align: center; height: 32px;" data-altupdatedat="1761848533000" data-alt="Equation 1" />
It has constant gain, zeros, and poles. After normalizing the transfer function is written as:
<img alt="Transfer function equation: \( H(\omega) = \frac{10j\omega}{1+j\omega/2} \times \frac{1}{1+j\omega/10} \); formula." src="/files/ftp_upload/15102/15102_Equation_2.svg" style="text-align: center; height: 32px;" data-altupdatedat="1761848533000" data-alt="Equation 2" />
Each term in this normalized function affects the Bode plot distinctly. The term 10j&#969; has a constant gain of 10 and zero at origin j&#969;. Each simple pole term (1+j&#969;/&#969;i) introduces a breakpoint or corner frequency &#969;i. The zero's positive slope is evident from the origin, while the poles' negative slopes commence at their corner frequencies, &#969;=2 and &#969;=10 respectively.
<img alt="Frequency response equation, \( H_{dB} = 20\log_{10}10 + 20\log_{10}|j&#969;| - 20\log_{10}|1+\frac{j&#969;}{2}| \)." src="/files/ftp_upload/15102/15102_Equation_3.svg" style="text-align: center; height: 36px;" data-altupdatedat="1761848533000" data-alt="Equation 3" />
<img alt="Static equilibrium equation: &#966; = 90&#176; - tan&#8315;&#185;(&#969;/2) - tan&#8315;&#185;(&#969;/10), mathematical concept." src="/files/ftp_upload/15102/15102_Equation_4.svg" style="text-align: center; height: 28px;" data-altupdatedat="1761848533000" data-alt="Equation 4" />
 Once the individual contribution of each term is identified, the overall Bode plot can be constructed. This involves superposing the slopes and phase changes of each factor. At low frequencies, the magnitude plot starts at 20 dB (since 20log10(10) =20) and maintains a flat response up to the first corner frequency. The phase plot starts at 90&#176; due to the zero at the origin and begins its descent before the first corner frequency.
The magnitude plot's slope will change at each corner frequency, decreasing by 20 dB/decade at &#969;=2 and again at &#969;=10. Correspondingly, the phase plot will bend downwards, approaching -90&#176; at a frequency much higher than &#969;=10.
Finally, the asymptotic Bode plot, which consists of these straight lines, can be adjusted to approximate the actual frequency response more closely. This involves adding a smooth curve intersecting the asymptotic plot at each corner frequency, typically resulting in a slight overshoot near the corner frequencies, known as peaking.

Bode Plots Construction

Tissue electrodes in electrocardiograms (E-C-G) establish a conductive pathway for electrical currents between the tissues and the measuring electrodes, ...

Education

JoVE Core

Electrical Engineering

Unit 1

Frequency Response

KEY TERMS AND CONCEPTS

reading frequency response with bode plots

Bode plots show the frequency response of a system with a magnitude plot and a phase plot. Both plots use a logarithmic frequency axis. In control system analysis, this makes it easier to see how a transfer function changes with frequency.

To build a Bode plot, start with the transfer function H(ω). After normalization, the function includes constant gain, zeros, and poles. Each term affects the plot in a different way.

A constant gain of 10 gives a starting magnitude of 20 dB, since 20 log10(10) = 20. The term 10jω also creates a zero at the origin, so the magnitude has a positive slope from the start. Each simple pole term, such as 1 + jω/ωi, adds a breakpoint or corner frequency ωi.

The zero at the origin gives the phase plot an initial value of 90°. The phase then begins to fall before the first corner frequency. The poles at ω = 2 and ω = 10 add negative slope changes to the magnitude plot, each reducing the slope by 20 dB per decade at its corner frequency.

The full Bode plot comes from adding the effects of each factor together. At low frequencies, the magnitude stays flat until the first corner frequency. The phase curve bends downward and approaches -90° at a frequency much higher than ω = 10.

The first result is an asymptotic Bode plot made of straight-line segments. A smoother curve can then be added to match the actual frequency response more closely. This adjustment often crosses the asymptotic lines near each corner frequency and can show a small peak, called peaking.

Reading Frequency Response with Bode Plots

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Learn how Bode plots track magnitude and phase on a log frequency axis, including gain, zeros, poles, corner frequencies, and peaking.

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electrical engineering

Reading Frequency Response with Bode Plots

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