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

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.

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

Education

JoVE Core

Electrical Engineering

Unit 1

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

network function of a circuit

How Transfer Functions Shape Circuit Response

Transfer functions show how a circuit responds as signal frequency changes. They describe the link between input and output in frequency response analysis. This makes them useful for studying how circuits behave in electronics and engineering. A transfer function is a mathematical tool that relates phasor output to phasor input. It can be written as voltage gain, current gain, transfer impedance, or transfer admittance. The poles and zeros are the most important parts of the function. Zeros...

frequency response of a circuit

Inductive Circuit Frequency Response

Inductive circuit frequency response shows how a circuit with an inductor reacts to different frequencies. The analysis starts in the time domain and then moves into the frequency domain. In that step, the inductor is replaced by an impedance, and the circuit is written with phasors, which are sinusoidal signals shown with magnitude and phase. The transfer function describes how the output changes compared with the input at each frequency. A key parameter is the time constant, which is the...

gain

Circuit Gain and Phase Shift

Circuit gain and phase shift describe how a linear circuit changes a sinusoidal input voltage or current. These properties matter in circuits with reactive elements, because the response depends on the input frequency. As the frequency changes, the gain and phase shift also change. Gain compares the size of the output sinusoid to the size of the input sinusoid. If Vin is the input and Vout is the output, the gain K is the ratio of the output amplitude to the input amplitude. A gain greater...

bode plots

Reading Frequency Response on Bode Plots

Bode plots show how a circuit or system responds across frequency. They use a logarithmic frequency scale on the x-axis and gain in decibels on the y-axis. This format makes it easier to display a wide range of frequencies and to see how circuit parts affect behavior over that range. A network function gives the ratio of output to input for a system. Its complex form provides both magnitude and phase angle. The logarithmic gain on a Bode plot comes from the magnitude, calculated by multiplying...

transfer function and bode plots-i

Transfer Function Gain and Origin Effects

Transfer function gain and poles or zeros at the origin shape a system's Bode plot. The transfer function is often written in standard form after the polynomial coefficients are normalized. In that form, the constant gain, zeros, poles, and quadratic terms all appear clearly. The poles and zeros mark critical frequencies where the output magnitude and phase change noticeably. The constant gain K creates a fixed magnitude and phase term. Its magnitude is 20 log 10 K in decibels, and its phase...

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...

bode plots construction

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...

series resonance

Series Resonance in RLC Circuits

Series resonance in an RLC circuit happens when the inductive reactance and capacitive reactance are equal. An RLC circuit contains a resistor, an inductor, and a capacitor. The impedance of the circuit is the ratio of the supply voltage to the circuit current. At resonance, the imaginary part of the impedance becomes zero. The frequency at which this occurs is called the resonant frequency. It depends on the inductance (L) and capacitance (C) of the circuit. Mathematically, the resonant...

characteristics of series resonant circuit

characteristics-of-series-resonant-circuit

Series Resonance and RLC Filter Behavior

Series resonance in an RLC circuit happens when the inductor, capacitor, and resistor are connected in series. At the resonant frequency, the inductive reactance and capacitive reactance are equal in size but opposite in sign, so they cancel each other. As a result, the circuit impedance becomes very small and is mainly set by the resistance. The resonant frequency of a series RLC circuit can be calculated from its inductance and capacitance. At this frequency, the current reaches its maximum...

parallel resonance

Parallel RLC Tuning at Resonance

A parallel RLC circuit uses a resistor, inductor, and capacitor connected to the same nodes. Because they share the same voltage, the circuit is analyzed with admittance, which describes how easily current can flow. In a parallel RLC circuit, admittance helps show how the circuit responds as the frequency changes. Resonance happens when the net reactance is zero. At that point, the capacitive and inductive effects cancel each other out. The resonant frequency is found from this condition. At...

frequency response of op amp circuits

Op Amp Gain and Bode Plot Basics

Operational amplifiers are used for signal conditioning, filtering, and mathematical tasks such as addition, subtraction, integration, and differentiation. Their frequency response shows how amplifier gain changes as the input frequency changes. This makes the gain of an op-amp a key idea in circuit design. At low frequencies, an op-amp can hold a constant gain called the dc gain, A0. As frequency rises, the gain starts to drop after the corner frequency, or break frequency, ω1. In other...

passive filters

Audio Frequency Control with Passive Filters

Passive filters shape the frequency spectrum of electrical signals. They use only passive parts such as resistors, inductors, and capacitors. Because they do not need an external power source, they can let selected frequency ranges pass or block others in many applications. Low-pass filters pass signals below the cutoff frequency, ωc, and reduce signals above it. In audio systems, they can send bass frequencies to woofers or subwoofers. For an RC low-pass filter, the transfer function H(s)...

active filters

Active Filter Design and Frequency Response

Active filters use op-amps, resistors, and capacitors to shape a signal's frequency response. They remove unwanted frequency components while keeping the parts of the signal that are needed. The design depends on the choice of resistor and capacitor values, which set the cutoff points and help define the overall response. A first-order low-pass active filter passes signals below a cutoff frequency and reduces signals above it. Its transfer function includes the low-frequency gain, also called...

scaling

Circuit Component Scaling for Filters

Circuit component scaling helps turn simple circuit values into practical ones for filter design and circuit analysis. Engineers often start with standard values like 1 ohm, 1 henry, or 1 farad because they make examples and problems easier to work through. Later, those values can be scaled to more realistic levels. This method is useful in filters, resonant circuits, and many other circuit analysis problems. It reduces calculation complexity while keeping the circuit behavior easy to study.

design example

Touch-Tone Filter Design Example

Touch-tone telephony uses dual-tone multi-frequency (DTMF) signaling and a filter design example to show how keypad tones are separated. It replaced the old rotary dial with a matrix-style keypad arranged in four rows and three columns. Each of the 12 buttons produces a unique signal pair. Every key press creates two sinusoidal tones at the same time. One tone comes from a low-frequency group between 697 and 941 Hz. The other comes from a high-frequency group between 1209 and 1477 Hz. This...

key terms and concepts

Learn how simple zeros and quadratic poles shape Bode plot magnitude and phase, including corner frequency, 3 dB shift, and damping effects.

Bode Plot Slope and Phase Changes | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Bode Plot Slope and Phase Changes

Description

Transcript

Tags