Complex Number In Electrical Engineering

Complex Numbers:
Complex numbers are such type of numbers which contain a real part and an imaginary part. Every real number can be a complex number but every complex number can’t be a real number complex no.’s r very important for the applications of quantum mechanics, electrical engineering, conformal mapping, method of steepest descent and for method of stationary phase etc.
As we have to talk about only the importance of complex numbers in the electrical engineering so let us begin our discussion.
Significance in Electrical Engineering:
Before we start the discussion we should keep in mind that every real quantity can be expressed in terms of complex numbers easily, rather than explaining it in terms of real numbers.
Voltage and Current:

We can describe the electric field strength and magnetic field strength by just one complex number.
In Electromagnetism you often deal with sinusoidal waveforms. The link between sinusoidal and complex numbers is done by Euler 's formula. We can write the sinusoidal as complex exponentials which helps a lot as they are then easily understandable.
Complex number can also be used to measure the strength of an electromagnetic field as it is a directly measurable quantity. If the field will be all electric means that there will be no magnetic component then complex number will be purely real and it will be pure imaginary if there will be no electric component, only containing magnetic ones. And for cases like in which both are present then the complex number will consist of a non-zero real part and a non-zero imaginary part, basically they will have some value instead of being zero.
If an attenuating medium is placed in the path of an electromagnetic wave than it can also be measured by using a complex number. The number will be purely real if the medium affects the magnitude of the wave but leaves the phase unchanged, and purely imaginary if the medium shifts the phase by 90 degrees. In general, a medium which scales the magnitude by a factor of R and shifts the phase by an angle t is described by the complex number as R*exp(

From Figure 4, it is easy to see that this definition of the complex exponential agrees with Euler’s equation: θ θ θ e cos j sin j = ± ± (23) With the definition of equation (22), we can define any arbitrary complex number in terms of complex numbers. For example, our previous complex number A can be represented as: A j A Ae θ = (24) We can generalize our definition of the complex exponential to time-varying signals. If we define a time varying signal j t e ω , we can use equation (23) to write: e cos t j sin t j t ω ω ω = ± ± (25) The signal j t e ω can be visualized as a unit vector rotating around the origin in the complex plane; the tip of the vector scribes a unit circle with its center at the origin of the complex plane. This is illustrated in Figure 3. The vector rotates at a rate defined by the quantity ω -- the vector makes one complete revolution every ω 2π seconds. The projection of this rotating vector on the real axis traces out the signal cosωt , as shown in Figure 3, while the projection of the rotating vector on the imaginary axis traces out the signal sinωt , also shown in Figure 5. Thus, we interpret the complex exponential function j t e ω as an alternate “type” of sinusoidal signal. The real part of this function is cosωt while the imaginary part of this function is