Name: transfer function gain and origin effects
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 19 s
Channel: JoVE

Consider the transfer function of a system expressed in standard form.
The transfer function has a constant gain, a zero and a pole at the origin, a simple zero, a simple pole, a quadratic pole, and a quadratic zero.
Bode plots for individual factors are plotted separately and combined graphically.
For the positive constant gain term, the magnitude gain is constant, and the phase angle is zero and frequency-independent.
For the negative constant gain term, the magnitude gain remains the same, and the phase angle is &#177;180&#176;.
For a transfer function with a zero at the origin, the magnitude plot represents a straight line with a positive slope of 20dB/decade. The phase plot depicts a constant phase angle of 90&#176;.
For a pole at the origin, the magnitude plot&#39;s slope is -20dB/decade, and the phase shift is -90&#176;.
If there is more than one pole or zero at the origin, the slope of the magnitude plot and phase shift are multiplied by the number of poles or zeros present at the origin.

transfer function and bode plots-i

Transfer function and Bode Plots-I

A transfer function presented in its standard form integrates elements' constant gain, the zeros, and poles at the origin, simple zeros and poles, and quadratic poles and zeros. The transfer function can be written as H(&#969;):
<img alt="Transfer function equation H(&#969;) diagram for control systems analysis, detailing frequency response." src="/files/ftp_upload/15141/15141_Equation_1.svg" style="text-align: center; height: 40px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
The transfer function, H(&#969;), often expressed in the standard form is derived by normalizing the polynomial coefficients of the transfer function. The poles (j&#969;) and zeros (j&#969;) are critical frequencies where the magnitude and phase of the system's output experience significant changes.
Gain, K:
The transfer function has a constant term K, with a magnitude of 20 log10K and a phase angle of 0&#176; for a positive value of K. Both magnitude and phase are constant with frequency. If K is negative, the magnitude remains unchanged, but the phase is &#177;180&#176;. When the gain K =1, the magnitude will become zero along with the phase angle. In Bode plot terms, the magnitude is expressed in decibels (dB) as 20log10K, and its phase remains at 0&#8728; or 180&#8728; depending on the sign of K.
Pole/Zero at the Origin:
A pole or zero at the origin has a defining impact on the plot. A zero (j&#969;)+1 at the origin has a magnitude of 20 log10 &#969; and the phase is 90&#176;. The slope of the magnitude plot is 20dB/decade, and the phase is constant with frequency. 
For a pole (j&#969;)-1 at origin, the magnitude is -20dB/decade, and the phase is -90&#176;. Generally, for (j&#969;)N, where N is an integer, the magnitude plot will have a slope of -20NdB/decade while the phase is 90N&#176;.
The constant magnitude K, poles/zeros at origins have the phase angle that does not change with the frequency. In the case of zeros/poles, the phase angle changes when the number of poles/zeros changes at the origin.

Consider the transfer function of a system expressed in standard form.
The transfer function has a constant gain, a zero and a pole at the origin, a ...

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Electrical Engineering

Unit 1

Frequency Response

KEY TERMS AND CONCEPTS

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 is 0° when K is positive. If K is negative, the magnitude stays the same, but the phase becomes ±180°. When K = 1, the magnitude is 0 dB and the phase angle is 0°.

A zero at the origin has a strong effect on the Bode plot. For a zero at the origin, the magnitude is 20 log 10 ω, and the phase is 90°. The magnitude rises with a slope of 20 dB/decade, and the phase stays constant as frequency changes. A pole at the origin acts in the opposite way. Its magnitude slope is -20 dB/decade, and its phase is -90°.

More generally, a factor of (jω)^N changes the plot by an integer amount. For a zero at the origin, the magnitude slope becomes 20N dB/decade and the phase is 90N°. For a pole at the origin, the magnitude slope becomes -20N dB/decade, and the phase is also 90N°. In both cases, the constant gain K and the poles or zeros at the origin keep a phase angle that does not change with frequency. The phase changes when the number of poles or zeros at the origin changes.

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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...

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key terms and concepts

Explore how constant gain, poles, and zeros at the origin affect transfer functions and Bode plots, including dB slope and phase changes.

Transfer Function Gain and Origin Effects | 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.

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Transfer Function Gain and Origin Effects

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