Ham radio: Impedance-5: Imaginary Numbers

[last updated: 2021-01-19]
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impedance-1: Why
impedance-3: Reactance
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• The Deal with Imaginary Numbers:     i   and   j
  They're kind of a trick, a smoke and mirrors. They're 'imaginary' from the standpoint of mathematics, but also meaningless in terms of any physical parameter or property of impedance and electronics. The imaginary number itself does not refer to anything in the real world of electronics.
  All they are is a mechanical device, a trick, a shortcut, a convenient way to plot a complex quantity, that is, a quantity that is really two quantities.


Explanation:

• Resistance is a pure, single-valued quantity. You can plot resistance on a number line, on a straight line. One single number is sufficient to fully describe/specify a resistance. A resistance might be 35 ohms, or 2 kohms, but it's a single number/quantity.

• But impedance is really two quantities - a resistance and a reactance. To fully specify an impedance requires two numbers, which you can't put onto a straight number line.
• So instead you do it on a graph, on a rectangular coordinate system with an x-y plot, where you can put the resistive component on the x-axis, the straight, number line, and plot the reactive component on the y-axis. (See below for the alternative polar coordinate system).

• In mathematics, quantities that are really two numbers are called "complex" numbers, and mathematicians have developed the shorthand trick to display them on a plot by using the "imaginary number" i or j to denote the component of the complex number that must be plotted in the y direction, ie. not on the same number line as the "normal" component.
• The imaginary number doesn't enter into the value in any mathematical way, it's just a signal, a flag, that says you must plot the value following it on the y-axis.

• In summary:
  Impedance is not a single number. It's really two numbers: resistance and reactance. In order to display an impedance, and capture both components, we use the mathematical trick of putting it into the form of a "complex number", and we denote an impedance as:
        Z = R + jX (rectangular coordinates)
  The R and X are the resistive and reactive components of impedance. The "j" is the imaginary number flag and just tells you that this component, the X, cannot simply be added to and plotted on the same line as the R, but must be handled independently and plotted on the y-axis.
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• Another way to plot impedances, that captures both resistive and reactive components, is to do it on a graph with a polar coordinate system. Then you can plot the resistive component on the x-axis as before, but the reactive you plot directly as an angle of phase shift off of horizontal (CCW for inductive and CW for capacitive phase shift).
• In this way, you are still specifying two components for an impedance, but now instead of listing the resistive and the reactive, you state the resistive and the phase shift angle. This fully specifies an impedance, because you can calculate/derive the reactive component from these two values.
• In polar form, we denote an impedance as:
        Z = R ∠ θ ("theta")
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• Convert between rectangular and polar coordinate systems:

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