Download COMPLEX NUMBERS: Polar to Rectangular Number Conversion

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COMPLEX NUMBERS: Polar to Rectangular Number Conversion.
Equations used in electronics are frequently written in 'polar form' to
mathematically represent the amplitude of an AC signal with its
corresponding phase angle.
A complementary method is in 'rectangular form', which then enables
these values to be drawn as phasors. This is often referred to as j
notation.
Real numbers (± r) appear on the horizontal axis. Imaginary numbers (± j)
on the vertical axis. Positive phase angles are anti-clockwise. Negative
phase angles clockwise. Beginning at the +r or 3 O'clock position as 0°.
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COMPLEX NUMBERS: Rectangular to Polar Form.
When complex numbers are written in rectangular form, the values they
actually represent are not always immediately obvious. Whereas in polar
form it is easy to visualise the amplitude and the phase angle. Which is
one reason why its useful to be able to convert from one to the other.
The length of the vector line can be made to represent voltage, current
or impedance within an AC circuit, along with the resultant phase angle.
Conversion is based on the trigonometric relationships for the right angle
triangle.
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COMPLEX NUMBERS: Rectangular Addition.
Rectangular addition may be used to combine two out-of-phase voltage or
current amplitudes.
The resultant value and its phase angle can be produced simply by finding
the sum of the 'real' numbers and the sum of the 'imaginary' numbers
and plotting the result as new values on the r and j axis.
To find the answer in polar form, use the rectangular to polar conversion
formulae, previously shown.
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COMPLEX NUMBERS: Rectangular Subtraction.
Complex number subtraction is carried out by subtracting the 'real' parts
and the 'imaginary' parts to produce a new resultant and phase angle.
Complex numbers are used for calculating, out-of-phase voltages and
currents which cannot be added or subtracted directly.
Impedance can also be shown as a complex number, where resultant RC or
RL voltages are at opposing phase angles.
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COMPLEX NUMBERS: Complex Number Multiplication.
Multiplication of complex numbers can be made easier by first converting
the two rectangular numbers into their equivalent polar form.
Here the two amplitudes A and B are multiplied together and their phase
angles added.
Conversion of the resultant back to rectangular form can then be carried
out.
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COMPLEX NUMBERS: Complex Number Division.
To divide two complex numbers convert each into its polar form and
divide the amplitudes A and B, then subtract the phase angles.
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