The Complex Plane

The complex plane is a plane whose coordinates are represented by Z.

Complex PlaneFigure 2.14 illustrates two circles. Consider the point p on both circles. In figure 2.14a, the point p has retangular coordinate given by x,y. Where x is its horizontal distance along the X axis and y is its vertical distance along the Y axis. The polar coordinate of point p is (r,θ). Where r is called its position vector and θ its direction. Consequently, the magnitude |r| of the position vector r, called its absolute value is given by: |r|2 = x2 + y2. But x = |r|cosθ and y = |r|sinθ. So, |r|2 = |r|2(cos2θ + sin2θ). If |r| = 1, we have cos2θ + sin2θ = 1. The absolute value, |r| which represents the magnitude of the position vector r can be represented by r.

In figure 2.14b, the point p has complex coordinate given by Z = x + iy. Where x is called the real component (Re(Z)) of Z and iy is called the imaginary component (Im(Z)) of Z. The entity i is defined as √-1. The conjugate of Z is defined as Z' = x - iy. The polar form of Z when r = 1 is given by Z = cosθ + isinθ. A plane with Z coordinates is a complex plane.

The mathematical entity, e, known as Euler Identity is related to the polar form of Z as follows: e = cosθ + isinθ. For a unit-radius circular complex plane. The Euler Identity can be proved by equating the Maclaurin series of e to the Maclaurin series of cosθ + isinθ. In general, the Euler Identity is expressed as : re = r(cosθ + isinθ)

Complex numbers can be used to represent sinusoidal signals as follows:
Acos(ωt + θ) = Re(Aei(ωt + θ))= Re[(Ae)(Aeiωt)].
Essentially, the equation says that a generalized sinusoid is the real part of a complex position vector having a magnitude equal to the peak amplitude of the sinusoid and an angle given by (ωt + θ).

The mathematical entity Ae is defined as the complex phasor corresponding to the sinusoidal signal, Acos(ωt + θ).
This phasor is represented as A<θ,
that is, Ae = complex phasor notation for Acos(ωt + θ) = A<θ

The frequency ω of the sinusoidal signal is not explicitly stated in the phasor representation of the sinusoid. In general a sinusoidal signal can be mathematically represented either in its time domain form:
v(t) = Acos(ωt + θ)

Or its phasor or frequency-domain form:
V(iω) = Ae = A<θ, where indicates the eiωt dependence of the phasor.

Peter Oye Sagay

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