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.

frequentierespons

Een overdrachtsfunctie die in zijn standaardvorm wordt gepresenteerd, integreert de constante versterking van de elementen, de nullen en polen aan de oorsprong, eenvoudige nullen en polen, en kwadratische polen en nullen. De overdrachtsfunctie kan worden geschreven als H(ω):

<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="text-align: center; height: 40px;">

De overdrachtsfunctie, H(ω), vaak uitgedrukt in de standaardvorm, wordt afgeleid door de polynoomcoëfficiënten van de overdrachtsfunctie te normaliseren. De polen (jω) en nullen (jω) zijn kritische frequenties waarbij de grootte en fase van de output van het systeem aanzienlijke veranderingen ondergaan.

Winst, K:

De overdrachtsfunctie heeft een constante term K, met een grootte van 20 log_10K en een fasehoek van 0° voor een positieve waarde van K. Zowel de grootte als de fase zijn constant met de frequentie. Als K negatief is, blijft de grootte ongewijzigd, maar is de fase ±180°. Wanneer de versterking K = 1, wordt de magnitude samen met de fasehoek nul. In termen van de Bode-grafiek wordt de magnitude uitgedrukt in decibel (dB) als 20log_10K, en de fase blijft op 0∘ of 180∘, afhankelijk van het teken van K.

Pool/nul bij de oorsprong:

Een pool of nul aan de oorsprong heeft een bepalende invloed op de plot. Een nul (jω)^+1 aan de oorsprong heeft een grootte van 20 log_10 ω en de fase is 90°. De helling van de magnitudegrafiek is 20 dB/decade, en de fase is constant met de frequentie.

Voor een pool (jω)^-1 bij de oorsprong is de magnitude -20dB/decade, en de fase -90°. Over het algemeen zal voor (jω)^N, waarbij N een geheel getal is, de magnitudegrafiek een helling hebben van -20NdB/decade, terwijl de fase 90N° is.

De constante magnitude K, polen/nullen aan de oorsprong hebben een fasehoek die niet verandert met de frequentie. In het geval van nullen/polen verandert de fasehoek wanneer het aantal polen/nullen bij de oorsprong verandert.

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

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Transfer Function Gain and Origin Effects

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