Angle notation.html

Angle notation or phasor notation is a notation used in electronics using the $\ang \!\$ sign. It is an abbreviation that arose due to the frequent occurrence of the expression e^{j θ}. There is also usually an implied conversion from degrees to radians. For example:

$\ang 90^\circ \!\ \quad \stackrel{\mathrm{represents}}{\longrightarrow}\quad e^{j\pi/2}\,$

The quantity A•e^{j θ} represents a phasor, with amplitude A and phase angle θ, which can be interpreted as the polar coordinates of a vector. The corresponding rectangular coordinates of the 90° example above are: (0,1).

Phasor notation is useful for multiplication or division of complex data.

Since

$Ae^{i \theta} = A\cos \theta + iA \sin \theta\ = X + iY\,\!$

and

$Ae^{i \theta} = A\ang \theta\,\!$

we can readily convert back and forth between complex numbers and phasor notation. When adding or subtracting, complex numbers have the advantage over phasor notation. When multiplying or dividing, however, phasor notation has advantages over complex numbers. For example, given

$Ae^{i \theta_1} = A\ang \theta_1 = A\cos\theta_1 + iA\sin\theta_1 = X_1 + iY_1\,\!$

$Be^{i \theta_2} = B\ang \theta_2 = B\cos\theta_2 + iB\sin\theta_2 = X_2 + iY_2\,\!$

When using phasors to multiply or divide we have

$Ae^{i \theta_1}\times Be^{i \theta_2} = AB\ang (\theta_1+\theta_2)\,\!$

$(Ae^{i \theta_1})/(Be^{i \theta_2}) = (A/B)\ang (\theta_1-\theta_2)\,\!$

When using complex numbers to multiply we have

$Ae^{i \theta_1}\times Be^{i \theta_2} = X_1 X_2 - Y_1 Y_2 + i(X_1 Y_2 + X_2 Y_1) \,\!$

but when dividing it becomes more difficult. First one must apply the complex inverse identity to the denominator, followed by the complex multiply.

${1\over{X_2 + iY_2}}={{X_2 - iY_2}\over{(X_2 + iY_2)(X_2 - iY_2)}}={{(X_2 - iY_2)}\over {({X_2}^2 + {Y_2}^2)}}=X_3+iY_3\,\!$

giving

$Ae^{i \theta_1}\times Be^{i \theta_2} = X_1 X_3 - Y_1 Y_3 + i(X_1 Y_3 + X_3 Y_1)\,\!$

with significantly more math operations than the division using phasor notation.

Phasor multiplication or division takes one multiply/divide and one add/subtract.

Complex multiplication takes four multiplies and two adds, while complex division takes an additional two multiplies and an add to compute (X₂²+Y₂²) followed by either two divides or one divide and two multiplies to compute $X 3$ and $Y 3$ , for a total of either six multiplies, three adds, and two divides, or eight multiplies, three adds, and one divide.

In the field of signal processing, much of the math involves multiplying or dividing sometimes large matrices of complex numbers. In such cases, phasor notation can be applied for significant computational speedup.

References

Nilsson, James W.f; Riedel, Susan A. (2005). Electric Circuits. Upper Saddle River, New Jersey: Pearson Prentice Hall. ISBN 0-13-146592-9.