9.5:
Transfer function and Bode Plots-I
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 ±180°.
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°.
For a pole at the origin, the magnitude plot's slope is -20dB/decade, and the phase shift is -90°.
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.
9.5:
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(ω):
The transfer function, H(ω), often expressed in the standard form is derived by normalizing the polynomial coefficients of the transfer function. The poles (jω) and zeros (jω) 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° 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 ±180°. 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∘ or 180∘ 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 (jω)+1 at the origin has a magnitude of 20 log10 ω and the phase is 90°. The slope of the magnitude plot is 20dB/decade, and the phase is constant with frequency.
For a pole (jω)-1 at origin, the magnitude is -20dB/decade, and the phase is -90°. Generally, for (jω)N, where N is an integer, the magnitude plot will have a slope of -20NdB/decade while the phase is 90N°.
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.