The transfer function is widely used in circuits, many calculation formulas in the circuit are obtained through the derivation of the transfer function, and the amplitude-frequency characteristics and phase frequency characteristics of the circuit are also obtained through the transfer function, so the transfer function is of great help in circuit design and understanding of the circuit.
The expression for the zero pole is:
Pi is called the pole of g(s) and zi is called the zero point of g(s).
Suppose we have a transfer function where the variable s appears in the numerator and denominator. In this case, at least one s-value will make the numerator zero, and at least one s-value will make the denominator zero. The value that makes the numerator zero is the zero point of the transfer function, and the value that makes the denominator zero is the pole of the transfer function.
The pole frequency corresponds to the angular frequency, at which the slope of the amplitude curve is -20db decade (ten octaves) and the slope of the phase curve is -45° decade.
The zero point corresponds to an angular slope, the slope of the amplitude curve is 20dB decade, and the slope of the phase curve is 45° decade.
In fact, the transfer function is an algebraic operation in which the Ralph transform obtains the s-domain, where s is a complex number, the actual part represents the time delay, and the imaginary part represents the frequency. By analyzing the functions of the s-domain and processing the zero pole mentioned above, it is also possible to obtain information such as the frequency characteristics and phase characteristics of the signal, so as to better understand and process the signal.
So how does it get the frequency characteristics, the phase characteristics?Let's start with the concept of plural. For any two real numbers x and y, z=x+jy is called a complex number, where x is the real part, y is the imaginary part, and j is a distinction made to distinguish the electric current i in the circuit.
z=r(cos +jsin) is a trigonometric form of z, where r is called the absolute value or modulus of the complex number z, called the amplitude angle or phase angle, and they are calculated as follows:
where r and s reflect the amplitude-frequency characteristics and phase-frequency characteristics. The zero point of the pole is a turning point in the waveform of the amplitude-frequency characteristic and the phase-frequency characteristic.
The characteristics of the transfer function have been described above. The characteristics of the transfer function can give us a better understanding of the circuit, but how to derive the transfer function of a circuit has become a problem. The method used to write the transfer function of a circuit is the complex impedance method. Complex impedance is the ratio of the phasor of voltage to the current phasor of a port in a circuit, usually expressed as z(s). For a linear time-invariant system, where the complex impedances of the input and output ports are and zin(s) and zout(s), respectively, then the transfer function of the system can be expressed as:
Complex impedance can reduce the complexity of calculations, make mistakes less susceptible, and make it easier for us to analyze circuits. The complex impedances of resistors, capacitors, and inductors are shown in the table below.
The Bode plot is a semi-logarithmic coordinate plot of the transfer function of a linear non-time-varying system with a frequency (one axis is the ordinary coordinate axis with uniform indexing, and the other axis is the logarithmic coordinate axis with uneven indexing), the horizontal axis is the frequency, and the vertical axis is represented by the logscale, and the frequency response of the system can be seen by using the Bode plot. A Bode plot is usually a combination of two plots, one representing the change in the decibel value of the frequency response gain versus the frequency, and the other phase plot representing the change in the phase of the frequency response versus the frequency. In fact, it is the amplitude-frequency characteristic and phase-frequency characteristic of the complex number mentioned earlier.
Bode plots can be drawn using computer software (e.g. multisim) or instruments, or you can draw them yourself. The Bode plot can be used to see the magnitude and phase of the system gain at different frequencies, and the trend of magnitude and phase with frequency can also be seen.
The pole causes the amplitude-frequency response to decrease at a rate of -20dB dec after the cut-off frequency fp, and the pole also causes a phase shift before and after the cut-off fp, resulting in a phase shift of up to -90 degrees, and at the cut-off frequency fp, the amplitude is attenuated by 3dB, and the phase is shifted by -45 degrees.
The zero point causes the amplitude frequency response to rise at a rate of +20dB dec after the cut-off frequency fc, and the zero point also causes a phase shift before and after the cut-off fqu, causing a maximum phase shift of +90 degrees, which is at the cut-off frequency fc, and the amplitude increases by 3dB and the phase shifts by +45 degrees.
In the case of ignoring the load, from the input terminal, r and c are connected in series, so the compound impedance of the input terminal is:
From the output, the complex impedance is:
So the transfer function of the RC low-pass filter circuit is:
According to the previous zero pole definition of the transfer function, this function has a pole of rcs+1=0 s=-1 rc, then the frequency of the pole is j=-1 rc, and the modulo of j can obtain f=1 2 rc.
Its amplitude and frequency characteristics are:
Rule. 
The cut-off frequency is at -3db, so. So. Rule.
Therefore, f=1 2 rc is called the cut-off frequency of the circuit.
The phase frequency characteristics of this circuit are:
At the cut-off frequency, so we can draw its Bode plot as: