Bode Diagrams of Transfer Functions and Impedances

In the design of a signal processing network, control system, or other analog system, it is usually necessary to work with frequency-dependent transfer functions and impedances, and to construct Bode diagrams. The Bode diagram is a log-log plot of the magnitude and phase of an impedance, transfer function, or other frequency-dependent complex-valued quantity, as a function of the frequency of a sinusoidal excitation. The objective of this handout is to describe the basic rules for constructing Bode diagrams, to illustrate some useful approximations, and to develop some physical insight into the frequency response of linear circuits. Experimental measurement of transfer functions and impedances is briefly discussed. Real systems are complicated, and hence their analysis often leads to complicated derivations, long equations, and lots of algebra mistakes. And the long equations are not useful for design unless they can be inverted, so that the engineer can choose element values to obtain a given desired behavior. It is further desired that the engineer gain insight sufficient to design the system, often including synthesizing a new circuit, adding elements to an existing circuit, changing connections in an existing circuit, or changing existing element values. So design-oriented analysis is needed [1]. Some tools for approaching the design of a complicated converter system are described in these notes. Writing the transfer functions in normalized form directly exposes the important features of the response. Analytical expressions for these features, as well as for the asymptotes, lead to simple equations that are useful in design. Well-separated roots of transfer function polynomials can be approximated in a simple way. Section 3 describes a graphical method for constructing Bode plots of transfer functions and impedances, essentially by inspection. If you can do this, then: (1) it's less work and less algebra, with fewer algebra mistakes; (2) you have much greater insight into circuit behavior, which can be applied to design the circuit; (3) approximations become obvious.