Using Phasors in AC Circuits

Dealing with sines and cosines is mathematically a pain in the ass. Which is unfortunate because in AC it's mostly what we do. Luckily it's possible to use a new mathematical tool called "phasors" to remove the trigonometry from our problems and replace it with normal multiplication, division, addition and subtraction. Phasors are a way of representing sine waves as well as complex numbers and operating on them without needing to do any complex algebra or trigonometry. Before we go into how to use phasors we'll need to do a quick refresher on complex numbers.

A complex number is a point on a 2D graph written \(x + iy\) where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate. \(x + iy\) is the same as point (x, y) in geometry.

The \(i\) in complex numbers happens to be: \(i = \sqrt{-1}\)

In a complex number the \(x\) is called the "Real" part and the \(y\) is called the "Imaginary" part (hence the use of the letter \(i\) to denote the imaginary part).

Two complex numbers can be added together by adding their real and imaginary parts separately.

If \(Z_1 = 1 + 2i\) and \(Z_2 = -3 + i\) find \(Z_1 + Z_2\)

The real part of \(Z_1 + Z_2\) is going to be the real part of \(Z_1\) added to the real part of \(Z_2\). So the real part of \(Z_1 + Z_2\) is \(-2\). Likewise the imaginary part of \(Z_1 + Z_2\) is going to be \(2 + 1 = 3\) So \(Z_1 + Z_2 = -2 + 3i\)

Polar form is given by \(Z = re^{i\theta}\) To convert \(Z = x + iy\) into polar form we use the formulas: \(r = \sqrt{x^2 + y^2}\) \(\theta = \tan^{-1}(\frac{y}{x})\)

Convert \(1 + i\) into polar form

We just use the above formulas to get: \(r = \sqrt{2}\) \(\theta = \tan^{-1}(1) = \frac{\pi}{4}\) So \(i + i = \sqrt{2}e^{i\frac{\pi}{4}}\)

To convert a polar form to a rectangular form use the following formulas: For \(Z = Ae^{i\theta}\): \(Z = A\cos(\theta) + iA\sin(\theta)\)

To multiply two complex numbers use their polar forms and the following formulas: \(Z_3 = Z_1Z_2\) \(r_{Z3} = r_{Z1}\cdot r_{Z2}\) \(\theta_{Z3} = \theta_{Z1}+\theta_{Z2}\)

To divide two complex numbers use their polar forms and the following formulas: \(Z_3 = \frac{Z_1}{Z_2}\) \(r_{Z3} = \frac{r_{Z1}}{r_{Z2}}\) \(\theta_{Z3} = \theta_{Z1}-\theta_{Z2}\)

To write the polar complex number \(re^{i\theta}\) as a phasor we use the slightly different form: \(r \angle \theta\)

If all of the AC sources in a circuit have the same frequency then the voltage and current everywhere in the circuit has that one frequency.

If all of the AC frequencies in a circuit are the same we can write all the voltages and currents as phasors.

To write a cosine wave \(A\cos(\omega t + \theta)\) as a phasor it's simply: \(A \angle \theta\)

To write a sine wave \(A\sin(\omega t + \theta)\) as a phasor we write: \(A \angle (\theta - 90^\circ)\)

Find \(12\angle 30^\circ \cdot 2\angle 11^\circ\)

\(12 \cdot 2 = 24\) \(30 + 11 = 41\) So \(12\angle 30^\circ \cdot 2\angle 11^\circ = 24\angle 41^\circ\)

Find \(\frac{12\angle 30^\circ}{2\angle 11^\circ}\)

\(\frac{12}{2} = 6\) \(30 - 11 = 19\) so \(\frac{12\angle 30^\circ}{2\angle 11^\circ} = 6\angle 19^\circ\)

To add two phasors we must first change them into rectangular form, add them like we normally add two complex numbers, then turn the result back into phasor form.

Find \(5\angle 30^\circ + 2\angle 45^\circ\)

\(5\angle 30^\circ = 5\cos(30^\circ) + 5i\sin(30^\circ) \approx 4.33 + 2.5i\) \(2\angle 45^\circ = 2\cos(45^\circ) + 2i\sin(45^\circ) \approx 1.41 + 1.41i\) So \(5\angle 30^\circ + 2\angle 45^\circ \approx 5.74 + 3.91i\) Now we return it to phasor form: \(5.74 + 3.91i = 6.95\angle 34.26^\circ\)