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6. Useful impedance identities Consider the following series and parallel combinations of a pure resistance and a pure reactance: Ll R Using the series and parallel combination rules for the complex impedance ( = R + jX), derive expressions for R, and X, in terms of R, and X, such that these two circuits exhibit exactly the same impedance (that is, Z; = Z;) at all frequencies, and show that they are: [10 marks] Zz 2 Ry =Ry (1 +) and X, =xu(1+ [Recall: Z,, = jX for any purely reactive impedance. ] [Recall: For any two impedances to be exactly the same, Z; = Z,, their real and imaginary parts must also be equal: Re[Z,] = Re[Z,] and Im[Z,] = Im[Z;]] Note: The following algebraic facts may be useful: Zy =73 = |22 = |Z,)? Z,=Z,= S [Hint: There are many approaches to solving this problem. Some are really quite straightforward. Some can become surprisingly complicated quite quickly. Consider carefully what you are trying to solve for and how best to target that most directly. However, if you find yourself mired in a swamp of complicated polynomials of resistances and reactances, consider carefully the facts provided above. And if you are finding it hard to eliminate a variable individually, it may be easier to eliminate it as part of eliminating a whole expression.]