An Indepth Look at Duals and Their Circuits Jeffrey Phillips Freeman 2020-09-28 Duality is an approach that has been applied across countless disciplines where one takes an existing structure and transforms it into an equivalent structure, often with the intention of making it more useful for a particular context. In electronic circuits this usually means we take an existing circuit schematic and transform it in such a way that it serves a similar purpose but suited to our specific use case. One extremely trivial example of this would be converting two resistors in series, which act as a voltage divider, into two resistors in parallel producing a current divider. Another similarly trivial example would be to take a voltage divider and double the values of its resistors such that it divides the voltage by the same ratio but uses half the current to do so. In both cases the fundamental idea of dividing a value by a given ratio is the same, we just transform the circuit in different ways that are suited to our needs. One of the key advantages of duality is that it is feature-preserving, as such if the original circuit has particular desirable features but otherwise may not be well suited for our application we can transform the circuit into a dual in such a way as to preserve the desirable features but transform the undesirable features. For example a resistor based voltage divider has the advantageous feature of being relatively stable across various frequencies where other types of voltage dividers may have a very limited frequency range, so a resistor based voltage divider may be best suited for an extremely broadband application (large range of frequencies). However our application may require low-power consumption and the voltage divider may not need to drive a load and only need to be provided as an input to an IC with high impedance. As such if our reference circuit is for a voltage divider that uses relatively small values for resistors, and thus draws too much power, we can transform the voltage divider into its dual that preserves the frequency-stability of the original but reduces the current draw. This of course is a very trivial example, the concept of duality can, and often is, applied to much larger complex circuits as well. For this reason understanding circuit duality, and duality in general, specifically how to recognize it, apply it, and common circuit applications is a vital tool in anyone’s mental toolbox. 1 What is a Dual? Duality is the transformation of a mathematical model or structure into an equivalent mathematical model or structure such that each element of the original has a one-to-one relationship with an element in the result. The transformation between its elements usually represents an involution, which means if the same transformation is applied twice you wind up with the original; at the very least the transformation must be reversible (invertible) back to its original form. This implies that the transformation must be unique for any given input. Both the overall structure once converted is said to be the dual of the original, but so too are the individual elements of the structure considered the dual of their counterpart in the dual structure. Taking the reciprocal is an example of an involution, and thus a trivial example of duality. 1 f(x) = x Since f(x) is an involution the following must be true for any involution function. f(f(x)) = x Of course for the reciprocal this holds true. x = 5 1 f(5) = 5 1 f( ) = 5 5 1 Therefore we can say 5 is the dual of 5 under the reciprocal transformation. An involution can sometimes be an identity function, and thus create fixed points where the involution of one element is unchanged. This of course still holds true to the rules of a duality transformation whereby if the involution is applied twice you still wind up at the original, it is invertible. The identity transformation would be: f(x) = x Therefore it is trivial to see this holds true as an involution since the following is true. 2 f(f(x)) = x In this case any value is the dual of itself under the identity transformation. Which isn’t really saying much, but it is important to understand a fixed point is still a dual caused by an involution. As stated earlier duals are usually transformed between each other through an involution function, but this does not need to be the case. Any invertible (reversible) function can be a valid way to express duality. The technical term for the property of a function to be reversible is to say it is bijective But this is just a fancy way of saying you can reverse the function to get to where you started. For example adding one to a value is a bijective function since you can also subtract one and always get to where you started and there is no ambiguity in doing so. However multiplying by 0 is not bijective (invertible) because once you multiply by 0 you have no way of getting back to where you started, all numbers would transform into 0. In other words in order for a function to be invertible the output of the function must be a unique value for any given input of the function, otherwise ambiguity is introduced and there would be no way to reverse the process. f(x) = x + 1 As stated equation () is invertible as you can always subtract 1 and get back to where you started. f −1(x) = x − 1 f −1(f(x)) = x In this case f −1(x) is called the inverse function to f(x) and the notation used in equation () is the typical notation used to represent an inverse function. It should be trivially obvious but keep in mind an inverse function must work in both directions, in other words. f(f −1(x)) = f −1(f(x)) = x It should be noted that an involution function is closely related to a function and its inverse. All an involution function really is is a function where its inverse is itself. Also bear in mind not all values will have a dual, consider the reciprocal function, which is an involution (its own inverse) and thus a valid transformation. 3 1 f(x) = f −1(x) = x In the case of the reciprocal function a value of 0 for x is undefined since you can not divide by 0, any other real value other than 0, however, is valid. Therefore we can say that under the reciprocal transformation all real number values have a dual except for 0, which does not have a dual. Examples of Duals There are many common examples of duals in almost every subject from philos- ophy, to mechanical engineering, it is a pervasive idea that can often be useful in many fields. Here are a few example values and their duals under different inversion transformations. • True is the dual of False under the negation transformation • 10 is the dual of 0.1 under the reciprocal transformation • 5 is the dual of -5 under the negation transformation. • A current divider circuit is the dual of a voltage divider circuit under series-parallel transformation • A capacitor based high-pass filter is the dual of an inductor based high-pass filter under reciprocal impedance transformation • A bandpass filter is the dual of a band-stop filter under series-parallel transformation • Position is the dual of velocity under the derivative/integral transformation • Up is the dual of Down under vertical flip transformation • In philosophy the mind is the dual of the physical world under dual-aspect theory Similarly here are some examples of transformations and their inverse that are therefore capable of producing duals • Reciprocal transformation is its own inverse, an involution. • Negation transformation is its own inverse, an involution. • derivative transformation is the inverse of an integral transformation • A geometric flip transformation is its own inverse, an involution • doubling a value is the inverse of halving a value The Dual of a function Just as we have shown above that individual variables and values have a dual under an invertible function, likewise functions can also have duals in the same manner. Imagine we have an invertible function T (x) which will convert something to its dual, and we have some function f(x) we wish to find the dual of, then simply by passing the function into T we can produce its dual. Specifically T (f(x)) = f T (x) where the functions f(x) and f T (x) are duals of each other. 4 It is important to note here only T (x) needs to be invertible; neither f(x) nor f T (x) needs to have this property. For example; say the transformation under which we create the duals is the reciprocal function, which is invertible, but f(x) is the square function, which is not invertible. We know it isn’t invertible because 10 squared is 100 and -10 squared is also 100. So there is no way to reverse the value of 100 and get the original value since some information was lost, we no longer know if the original value was positive or negative. 1 T (x) = X f(x) = x2 1 f T (x) = T (f(x)) = x2 2 T 1 We can now see the function f(x) = x is the dual of f (x) = x2 under the reciprocal transformation. Manipulating A System of Equations Things get slightly more complicated when we start talking about systems rather than single variables or functions.