Linear Circuit Transfer Functions An Introduction to Fast Analytical Techniques by Christophe P. Basso
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Linear Circuit Transfer Functions An Introduction to Fast Analytical Techniques by Christophe P. Basso


First as a student and later as an engineer, I have always been involved in the calculation of transfer functions. When designing power electronics circuits and switch mode power supplies, I had to apply my analytical skills on passive filters. I also had to linearize active networks when I needed the control-to-output dynamic response of my converter. Methods to determine transfer functions abounded and there are numerous textbooks on the subject. I started in college with mesh-node analysis, and at some point ended up using state variables. If all paths led to the correct result, I often struggled rearranging equations to make them fit a friendly format.

Matrices were useful for immediate numerical results but, when trying to extract a meaningful symbolic transfer function, I was often stuck with an intractable result. What matters with a transfer function formula is that you can immediately distinguish poles, zeros and gains without having to rework the expression. This is the idea behind the term low-entropy, a concept forged by Dr. Middlebrook. Simulation gives you an idea where poles and zeros hide by interpreting the phase and magnitude plots with minimum-phase functions.

However, inferring which terms really affect a pole or a zero position from a Bode plot is a different story. Fortunately, if the transfer function is written the right way, then you can immediately identify which elements contribute to the roots and assess how they impact the dynamic response. As some of these parasitics vary in production or drift with temperature, you have to counteract their effects so that reliability is preserved during the circuit’s life. The typical example is when you are asked to assess the impact of a parasitic term variation on a product you have designed: if a new capacitor or a less expensive inductor is selected by the buyers, will production be affected?

Is there a chance that stability will be jeopardized in some operating conditions? Implementing the classical analysis method will surely deliver a result describing the considered circuit, but extracting the information you need from the final expression is unlikely to happen if the equations you have are disorganized or in a high-entropy form. This is where Fast Analytical Circuit Techniques (FACTs) come into play. The acronym was formed by Dr. Vatché Vorpérian, who formalized the technique you are about to discover here. Before him, Dr. Middlebrook published numerous papers and lectured on his Extra-Element Theorem (EET), later generalized to the N extra-element theorem by one of his alumni. Since Hendrik Bode in the 40’s, authors have come up with techniques aiming to simplify linear circuit analysis through various approaches.

All of them were geared towards determining the transfer function at a pace quicker than what traditional methods could provide. Unfortunately, while traveling and visiting customers world-wide, I have found that, despite all the available documentation, FACTs were rarely adopted by engineers or students. When describing examples in my seminars and showing the method at work in small-signal analysis, I could sense interest from the audience through questions and comments.

However, during the discussions I had later on with some of the engineers or students, they confessed that they tried to acquire the skill but gave up because of the intimidating mathematical formalism and the complexity of the examples. If one needs to be rigorous when tackling electrical analysis, perhaps a different approach and pace could make people feel at ease when learning the method. This is what I strived to do with this new book, modestly shedding a different light on the subject by progressing with simple-to-understand examples and clear explanations. As a student, I too struggled to apply these fast analytical circuits techniques to real-world problems; as such, I identified the obstacles and worked around them with success. Thus, the seeds for this book were sown.

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