Such circuits can be analyzed in the time domain, but often the necessary information is more readily obtained by working in the frequency domain, which requires that we assign an impedance to inductors and capacitors. The operation of circuits that include capacitors or inductors is less straightforward than resistor-only circuits because capacitors and inductors introduce time dependency into the behavior of a circuit. The letter j is preferred in the context of electrical engineering because it eliminates the possibility of confusion between the imaginary unit and the letter used to denote electric current. Note: For this example we will use j instead of i for the imaginary unit. Here we have an electrical network consisting of resistors, an inductor, and a capacitor in addition to a current source and a voltage source. This is very useful as it reduces the computational burden almost to half, and the improvement becomes more pronounced as the length of the DFT/IDFT increases. That is, points $$ P_1^* $$ and $$ P_2^* $$ are the complex conjugate pairs of points $$ P_1 $$ and $$ P_2 $$, respectively.įigure 3: Representation of conjugate symmetry exhibited by DFTs and IDFTsįor example, let us assume that $$ (1, 0.25 0.3i, 0.5 - 0.2i, 0.3 - 0.6i) $$ are the first four points of the 7-point DFT of a real sequence. This is because the DFT and IDFT of real sequences exhibit conjugate symmetry about the midpoint.įigure 3 shows an example of a 5-point DFT of a real sequence that exhibits conjugate symmetry with respect to points $$ P_1 $$ and $$ P_2 $$. Thus, if we find that the root of a particular equation is $$ x iy $$, then its complex conjugate $$ x - iy $$ is also a root.Ĭonjugate pairs are also involved when we need to find a Discrete Fourier Transform (DFT) or Inverse Discrete Fourier Transform (IDFT). To simplify this process, we can turn to an axiom of mathematics which states that complex roots always appear in pairs. Next, suppose that we need to find the roots of a polynomial expression. ![]() Case III: Finding Roots of Polynomial Expressions The result has fractions for its real and imaginary parts, but because the imaginary unit does not appear in the denominator, the simplified expression is more mathematically standard and more convenient to work with than the original expression. This kind of simplification is referred to as rationalization. ![]() Complex NumbersĬomplex numbers are numbers which are represented in the form $$ z = x i y $$, where x and y are the real and imaginary parts (respectively) and $$ i =\sqrt i $$ This article provides insight into the importance of complex conjugates in electrical engineering.
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