Complex Plane
Peter Oye Sagay
The complex plane is a plane whose coordinates are represented by Z.
Figure 2.14 illustrates two circles. Consider the point p on both circles. In figure 2.14a, the point p has rectangular 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, eiθ, known as Euler Identity is related to the polar form of Z as follows: eiθ = cosθ + isinθ. For a unit-radius circular complex plane. The Euler Identity can be proved by equating the Maclaurin series of eiθ to the Maclaurin series of cosθ + isinθ. In general, the Euler Identity is expressed as : reiθ = r(cosθ + isinθ)
Complex numbers can be used to represent sinusoidal signals as follows:
Acos(ωt + θ) = Re(Aei(ωt + θ))= Re[(Aeiθ)(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 Aeiθ is defined as the complex phasor corresponding to the sinusoidal signal, Acos(ωt + θ).
This phasor is represented as A<θ,
that is,
Aeiθ = 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) = Aeiθ = A<θ,
where iω indicates the eiωt dependence of the phasor.