Linear Circuit Transfer Functions
Linear Circuit Transfer Functions
Electrical Analysis - Terminology and Theorems
This first chapter is an introduction to some of the basic definitions and terms you must understand in order to perform electrical analysis with efficiency and speed. By electrical analysis, I imply finding the various relationships that characterize a particular electrical network. To excel in this field, as in any job, you need to master a few tools. Obviously, they are innumerable and I am sure you have learned a plethora of theorems during your student life. Some names now seem distant simply because you never had a chance to exercise them. Or you actually did but implementation was so obscure and complex that you left quite a few of them aside. This situation often happens in an engineer's life where real-case experience helps clean up what you have learned at school to only retain techniques that worked well for you. Sometimes, when what you know fails to deliver the result, it is a good opportunity to learn a new procedure, better suited to solve your current case. In this chapter, I will review some of the founding theorems that I extensively use in the examples throughout this book. However, before tackling definitions and examples, let us first understand what the term transfer function designates.
1.1 Transfer Functions, an Informal Approach
Assume you are in the laboratory testing a circuit encapsulated in a box featuring two connectors: one for the input, the second for the output. You do not know what is inside the box, despite the transparent case in the picture! You now inject a signal with a function generator to the input connector and observe the output waveform with an oscilloscope. Using the right terminology, you drive the circuit input and observe its response to the stimulus. The input waveform represents the excitation denoted u and it generates a response denoted y . In other words, the excitation variable propagates through the box, undergoes changes in phase, amplitude, perhaps induces distortion etc. and the oscilloscope reproduces the response on its screen.
The waveform displayed by the oscilloscope is a time-domain graph in which the horizontal axis x is graduated in seconds while the vertical axis y indicates the signal amplitude (positive or negative). Its dimension depends on the observed variable (volts, amperes and so on). The input waveform is denoted in lower case as it is an instantaneous signal, observed at a time - the instant t - u ( t ). A similar notation applies to the output signal, y ( t ). In Figure 1.1 , you see a low duty ratio square-wave injected in the box engendering a rather distorted waveform on the output.
Figure 1.1 A black box featuring an input and an output signal. What is the relationship linking output and input waveforms?
This ringing signal tells us that the box could associate resonant elements, probably capacitors and inductors but not much more than that. If we change the excitation, what type of shape will we obtain? Knowing what is inside the box will let us predict its response to various types of excitation signals.
There are several available ways to characterize an electrical linear circuit. One of them is called harmonic analysis . The input signal is replaced by a sinusoidal waveform and you observe how the stimulus propagates through the box to form the response. This is shown in Figure 1.2 :
Figure 1.2 The black box is now driven by a sinusoidal stimulus for a small-signal analysis.
The excitation level must be of reasonable amplitude - understand small - so that the response signal is not distorted. The input signal dc bias must also be set accounting for the physical constraints imposed by the activ