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Using Complex Numbers In Circuit Analysis

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Complex Analysis:

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
Murray R. Spiegel described complex analysis as "one of the most beautiful as well as useful branches of Mathematics".
Using Complex Numbers in Circuit Analysis:

The purpose of this note is to review the algebra of concept numbers and show how they can be used to simplify analyses …show more content…

Take a look at the equations in the previous section. The addition and subtraction equations do not mix up the real and imaginary parts, but the equations for multiplication and division do. Multiplying a complex number by a real constant also obviously does not mix up the real and imaginary parts. Essentially, a linear equation is one that will not mix up the real and imaginary parts of the voltages and currents. From a practical standpoint, a linear circuit is one that includes only passive components (resistors, capacitors, and inductors) plus voltage and/or current sources. No diodes, transistors, vacuum tubes, etc. are allowed. It is perhaps worth mentioning here that the same formalism, with the same advantages of using complex numbers, works in mechanics when dealing with small, harmonic oscillations of mechanical systems. voe^(j(wt)) or Io e^(j(wt)) where the phase can be taken to be zero if there is only one source. Otherwise the relative phases of the sources must be taken into account. Then treat each passive component as an impedance

Resistor: Z = R
Capacitor

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