AC Electrical Theory an Introduction to Phasors, Impedance and Admittance, with Emphasis on Radio Frequencies

1 AC electrical theory An introduction to phasors, impedance and admittance, with emphasis on radio frequencies. By David W. Knight* Version 0.12. 18th Dec. 2015. © D. W. Knight, 2005 - 2015. * Ottery St Mary, Devon, England. Please check the author's website to ensure that you have the most recent versions of this article and its associated documents: http://www.g3ynh.info/ Recent changes (0.11→0.12): Grammatical corrections. More whitespace. Minimum linewidth in diagrams increased to 2px to improve rendering in online pdf viewers. Adoption of SI notation guidelines. Table of Contents Preface.............................................................2 24. Phasor theorems.......................................70 1. Field electricity............................................3 25. Generalisation of Ohm's law...................76 2. Circuit analysis overview..........................10 26. General statement of Joule's law.............78 3. Basic electrical formulae...........................14 27. Bandwidth................................................81 4. Resonance..................................................19 28. decibels & logarithms..............................82 5. Impedance, resistance, reactance...............21 29. Bandwidth of a series resonator..............86 6. Vectors & scalars.......................................22 30. Logarithmic frequency............................92 7. Balanced vector equations.........................25 31. A proper definition for resonant Q...........94 8. Phasors.......................................................29 32. Bandwidth in terms of Q.........................95 9. Voltage magnification & Q........................32 33. Lorentzian line-shape function................96 10. Power factor & scalar product ................36 34. Maximum power transfer........................99 11. Phasor dot product...................................39 35. The potential divider..............................105 12. Complex numbers....................................40 36. Output impedance of potential divider..106 13. Complex arithmetic.................................45 37. Thévenin's theorem................................107 14. Impedances in parallel.............................46 38. Measuring source resistance..................108 15. Dimensional consistency.........................47 39. Error analysis.........................................109 16. Parallel resonance....................................49 40. Antenna system Q..................................116 17. Dynamic resistance..................................51 41. Basic impedance transformer................118 18. Double-slash notation..............................53 42. Auto transformers..................................123 19a. Parallel-to-series transformation............55 43. Prototype Z-matching network..............126 19b. Series-to-parallel transformation...........57 44. Admittance, conductance, susceptance..128 20. Parallel resonator in parallel form...........58 45. Parallel resonator BPF...........................132 21. Imaginary resonance................................62 46. Unloaded Q of parallel resonator..........137 22. Phase analysis..........................................64 47. Current magnification............................140 23. Resistance tuned LC resonator?..............67 48. Controlling loaded Q.............................142 2 Preface This document provides an introduction to the subject of AC circuit analysis, with particular emphasis on radio-frequency applications. It was developed as part of a collection of writings on the subject of radio-frequency impedance matching and measurement; which were first made available via the Internet in 2005, and were grouped under the working title 'From Transmitter to Antenna'. Its purpose was (and still is) to widen the audience for the other articles by providing essential background material, but it can just as well be read by those who have a more general interest. The approach adopted is that of starting with the basic laws of DC electricity and expanding them to deal with AC. The modified laws are then used to derive and explore results that are normally accepted without proof, thereby explaining the origins of various standard formulae and demonstrating the general method by which linear circuit design equations are obtained. The level of treatment is one that does not demand a high level of mathematical skill at the outset; because the required techniques are introduced as the narrative progresses. Hence the discussion should be accessible to anyone who has some knowledge of basic algebra and is reasonably familiar with circuit diagrams and electrical terminology. Apart from providing a conventional introduction to AC theory however, there is a subtext. This relates to the author's concerns as a scientist: one being that there appears to be an almost universal public misconception regarding the nature of electricity; and the other being a lack of mathematical rigour in the way in which phasor techniques are commonly used. Both of these issues can (and should) be addressed at this stage in the development of working knowledge, and so the accompanying discussion attempts to do that. As all experienced engineers and physicists know, our understanding of electricity comes from Maxwell's equations. The problem for those who wish to teach electrical subjects however, is that the electromagnetic field approach requires advanced mathematics and does not lead directly to the practicalities of circuit design. Hence it is sensible to hold back on the more abstract ideas until they become unavoidable; but that leaves the problem of how to dispel the notion that electricity is synonymous with electrons flowing through wires. The author's solution is to provide an extended preamble; which gives a purely qualitative explanation of electricity in terms of fields; and is intended to leave the reader with the same mental picture as will be held by those who are familiar with Maxwell's theory. On the matter of mathematical rigour; it is not the intention to wrap the subject in formalism, but merely to eliminate certain bad practices. To this end, we pay particular attention to the definitions and properties of the mathematical objects involved, and develop a way of working that identifies and preserves the algebraic signs of the circuit parameters. The outcome is an internally-consistent theory of circuits, which produces results that are correct in both magnitude and phase. In this way, we eliminate the need for the so-called 'physical considerations' traditionally used to resolve ambiguities; and we discover which of the two commonly used definitions of admittance is actually the correct one. Some, of course, will ask: 'why should we bother to learn phasor analysis when we can model circuits using SPICE?' In fact, the use of SPICE is highly recommended; but simulation is essentially a way of checking existing design work. It does not offer a systematic approach to the business of optimising circuits or inventing new ones. The techniques discussed here, on the other hand, allow the equations describing the behaviour of a circuit to be written down explicitly. We might then, for example, separate out the terms describing unwanted behaviour with a view to making alterations that will eliminate them. Such is the basis for the development of precision measuring instruments and all manner of other high-performance circuitry. David Knight. June 2012. 3 1. Field electricity When AC theory is introduced, and especially when there is a bias towards radio frequencies, the very first new idea required (by many people at least) is a correct understanding of the word 'electricity'. The teaching of basic science often involves what are known as 'lies to children', and the one about electricity being "electrons flowing through wires" is an intellectual dead-end. Electricity is actually an invisible form of light. Specifically, it is electromagnetic energy of very long wavelength (in comparison to visible light); which is why we can build devices called 'radio transmitters' that cause electricity to propagate off into space. Hence we will never understand AC electricity by counting electrons; and we must first refine our ideas of voltage and current by thinking about certain mysterious entities known as 'fields'. The term 'field' has a vernacular meaning: "region of influence", and this is the route by which it came into the language of physics. In scientific parlance however, a field is more rigorously defined as 'a quantity that can take on different values, and possibly also different directions of maximum action, at different points in space and time'. The geometric field idea can be used (say) to describe the 3D temperature gradient around a hot object, or the average velocities of molecules in a flowing liquid; but the fields that are the most perplexing, and that ultimately reveal the deepest secrets of the Universe, are those that appear to produce action at a distance. Of these so-called 'force fields'; the gravitational, electric and magnetic are the most familiar; and of course, it is the latter two that concern us here. The beginning of what is loosely called 'modern physics' can be traced to a single deduction made by James Clerk Maxwell in the latter part of the 19th Century. Maxwell collected the details of every known scientific result concerning electricity and magnetism, and lent his phenomenal mathematical skill to the problem of finding a single

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