Signals and Systems/Second Order Transfer Function
The second order transfer function is the simplest one having complex poles. Its analysis allows to recapitulate the information gathered about analog filter design and serves as a good starting point for the realization of chain of second order sections filters.
- 1 All-Pole Second Order Transfer Function
- 2 Biquadratic Second Order Transfer Function
- 3 Example
All-Pole Second Order Transfer Function
The transfer function of a continuous-time all-pole second order system is:
Note that the coefficient of has been set to 1. This simplifies the writing without any loss of generality, as numerator and denominator can be multiplied or divided by the same factor.
The frequency response, taken for , has a DC amplitude of:
For very high frequencies, the most important term of the denominator is and the frequency response gets closer and closer to:
At high frequencies, the amplitude response looks like a (squared) hyperbol in a linear plot and like a straight line with a negative slope in a log-log plot. Plotting the frequencies in decades and the amplitude in decibels reveals a slope of -40 [dB/decade].
Having a given amplitude at DC and an amplitude nearing zero at high frequencies indicates that the transfer function is of lowpass type.
The poles of the system are given by the roots of the denominator polynomial:
If the term inside the square root is negative, then the poles are complex conjugates. This is the general case in filter design: there is poor interest in a second order transfer function having two real poles.
From the location of the poles, the transfer function can be rewritten as:
The amplitude of the poles gives the corner frequency of the filter. The corner frequency is defined as the abscissa of the point where the horizontal and the -40 [dB/decade] lines meet in the log-log magnitude response plot. Note that this is not necessarily the -3 [dB] attenuation frequency of the filter.
The closer the poles are to the imaginary axis, the more a resonance will appear at a frequency smaller but close to the corner frequency of the system.
Transfer Function Rewritten
With this, the transfer function with unity gain at DC can be rewritten as a function of the corner frequency and the damping in the form:
Both and have a unit of [s-1]. has a unit of  and so does the total transfer function.
Biquadratic Second Order Transfer Function
The name biquadratic stems from the fact that the functions has two second order polynomials:
The poles are analysed in the same way as for an all-pole second order transfer function. They determine the corner frequency and the quality factor of the system.
The zeroes are used to affect the shape of the amplitude response:
- Placing zeroes on the imaginary axis at frequencies a little higher than the corner frequency gives more attenuation in the stopband and allows a faster transition from passband to stopband. This is what happens with Chebyshev type 2 and elliptic filter functions.
- Placing the zeroes on the imaginary axis precisely at the corner frequency forces the amplitude to zero at that specific point. This corresponds to a bandstop (or notch) function.
- Placing a single zero at the (0, 0) coordinate of the s-plane transforms the function into a bandpass one. This is done by setting coefficients and .
- Placing both zeroes at the (0, 0) coordinate transforms the function into a highpass one. This is done by setting coefficients and .
- Placing the zeroes on the right half plane, symetrically to the poles gives an allpass function: any point on the imaginary axis is at the same distance from a zero and from the associated pole. This allpass function is used to shape the phase response of a transfer function.