# Circuit Theory/Phasor Arithmetic

## Forms

Phasors have two components, the magnitude (M) and the phase angle (φ). Phasors are related to sinusoids through our cosine convention:

$\mathbb{C} = |M| \angle \phi = |M| \cos (t\omega + \phi)$

Remember, there are 3 forms to phasors:

#### Phasor Form

• $\mathbb{C} = |M| \angle \phi$

#### Rectangular Form

• $\mathbb{C} = A + jB$

#### Exponential Form

• $\mathbb{C} = |M|e^{j\phi}$

Phasor and Exponential forms are identical and are also referred to as polar form.

## Converting between Forms

When working with phasors it is often necessary to convert between rectangular and polar form. To convert from rectangular form to polar form:

$|M| = \sqrt{A^2 + B^2}$
$\phi = \arctan \left( \frac{B}{A} \right)$

To convert from polar to rectangular form:

A is the part of the phasor along the real axis

$A = |M|\cos \left( \phi \right)$

B is the part of the phasor along the imaginary axis

$B = |M|\sin \left( \phi \right)$

To add two phasors together, we must convert them into rectangular form:

$\mathbb{C}_1 = A_1 + jB_1$
$\mathbb{C}_2 = A_2 + jB_2$
$\mathbb{C}_1 + \mathbb{C}_2 = (A_1 + A_2) + j(B_1 + B_2)$

This is a well-known property of complex arithmetic.

## Subtraction

Subtraction is similar to addition, except now we subtract

$\mathbb{C}_1 = A_1 + jB_1$
$\mathbb{C}_2 = A_2 + jB_2$
$\mathbb{C}_1 - \mathbb{C}_2 = (A_1 - A_2) + j(B_1 - B_2)$

## Multiplication

To multiply two phasors, we should first convert them to polar form to make things simpler. The product in polar form is simply the product of their magnitudes, and the phase is the sum of their phases.

$\mathbb{C}_1 = M_1 \angle \phi_1$
$\mathbb{C}_2 = M_2 \angle \phi_2$
$\mathbb{C}_1 \times \mathbb{C}_2 = M_1 \times M_2 \angle {\phi_1+\phi_2}$

Keep in mind that in polar form, phasors are exponential quantities with a magnitude (M), and an argument (φ). Multiplying two exponentials together forces us to multiply the magnitudes, and add the exponents.

## Division

Division is similar to multiplication, except now we divide the magnitudes, and subtract the phases

$\mathbb{C}_1 = |M_1| \angle \phi_1$
$\mathbb{C}_2 = |M_2| \angle \phi_2$
${\mathbb{C}_1 \over \mathbb{C}_2} = {|M_1| \over |M_2|} \angle {\phi_1-\phi_2}$

## Inversion

An important relationship that is worth understanding is the inversion property of phasors:

$\mathbb{C} = M_C\angle 0 = -M_C \angle \pi$

Or, in degrees,

$\mathbb{C} = M_C\angle 0^\circ = -M_C \angle 180^\circ$

On the normal cartesian plane, for instance, the negative X axis is 180 degrees around from the positive X axis. By using that fact on an imaginary axis, we can see that the Negative Real axis is facing in the exact opposite direction from the Positive Real axis, and therefore is 180 degrees apart.

## Complex Conjugation

Similar to the inversion property is the complex conjugation property of phasors. Complex conjugation is denoted with an asterisk above the phasor to be conjugated. Since phasors can be graphed on the Real-Imaginary plane, a 90 degree phasor is a purely imaginary number, and a -90 degree phasor is its complex conjugate:

$\mathbb{C} = M \angle 90^\circ$
$\mathbb{C}^* = M \angle -90^\circ = M \angle 270^\circ$

Essentially, this holds true for phasors with all angles: the sign of the angle is reversed to produce the complex conjugate of the phasor in polar notation. In general, for polar notation, we have:

$\mathbb{C} = M \angle \phi$
$\mathbb{C}^* = M \angle -\phi$

In rectangular form, we can express complex conjugation as:

$\mathbb{C} = A + jB$
$\mathbb{C}^* = A - jB$

Notice the only difference in the complex conjugate of C is the sign change of the imaginary part.