Argand plane

The point z can also be located by establishing polar coordinates in the complex plane where r is the radius vector and 0 is the phase angle. Draw suitable polar coordinates for the Argand plane. What is r for the point 7 = 3 + 4i7 What is 0 in degrees and radians ... 

A mathematician would say that a plot of Z" (as y ) against Z (as jc ) forms an Argand diagram (or Argand plane ). As electroanalysts, we will call such a set of axes a Nyquist plot or simply an impedance plot (see Figure 8.9). 

The representation of the sampling by a unipolar, single-rotation-axis, U(l) sampler of a SU(2) continuous wave that is polarization/rotation-modulated is shown in Fig. 2, which shows the correspondence between the output space sphere and an Argand plane [28]. The Argand plane, S, is drawn in two dimensions, x and v, with z = 0, and for a set snapshot in time. A point on the Poincare sphere is represented as P(t,x,y,z), and as in this representation t = 1 (or one step in the future), specifically as P(l,x,y,z). The Poincare sphere is also identified as a 3-sphere, S 1, which is defined in Euclidean space as follows ... 

Figure 2. Correspondence between the output space sphere and an Argand plane. (After Penrose and Rindler [28].)...

Figure 19 Vector diagram in the Argand plane of FPH = FP + FH. As can be seen, the maximum angular difference between Fph and FP occurs when FH is perpendicular to FP. When FH is much smaller than FP, this angular difference is small. However, not all FP s are large not all FH will be much smaller than FP even when the averages are very different.

Drawing these on a diagram in the Argand plane, (Figure A6.2) shows that the total displacement along the real axis, X, is ... 

This kind of a figure is called an Argand diagram, and the plane of the figure is called the Argand plane or the complex plane. The horizontal coordinate represents the real part of the number and the vertical coordinate represents the imaginary part. The horizontal axis, labeled R, is called the real axis, and the vertical axis, labeled I, is called the imaginary axis. 

Figure 2.13 shows the location of a complex number and of its complex conjugate in the Argand plane. 

Figure 2.13 A complex number, z x + iy, and its complex conjugate, z = x — iy, in the Argand plane.

The other square root is obtained by realizing that if 0 is increased by In, the same point in the Argand plane is represented. Therefore, the square root of is... 

This implies that the complex impedance vector can be represented in the Argand plane, where we plot the imaginary component Im (Z) = Z" versus the real component Re (Z) = Z. Hence we can also write... 

The HFR corresponds to the extrapolated real-axis intercept for impedance spectra plotted on the Argand plane. The reported frequency ranges vary, but they usually include measurements at 1 kHz or larger frequencies. 

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