2.1 Complex numbers and phasor notation

Complex Numbers

Understanding Complex Numbers and Their Components

A complex number has the form $a + jb$, where $a$ is the real part and $b$ is the imaginary part. In electrical engineering, we use $j$ instead of $i$ for the imaginary unit (since $i$ already means current). The defining property is $j = \sqrt{-1}$, or equivalently, $j^2 = -1$.

You can visualize any complex number as a point on the complex plane: the horizontal axis is the real part, and the vertical axis is the imaginary part. This geometric picture becomes critical once we start working with phasors.

Why do we need complex numbers at all? Real numbers can't capture phase relationships between signals. When you're analyzing AC circuits, voltage and current are sinusoids that may be shifted in time relative to each other. Complex numbers let you encode both magnitude and phase in a single quantity, which makes the math far more manageable.

Applications of Complex Numbers in Electrical Engineering

• Model AC circuits where voltages and currents are sinusoidal
• Represent impedance as a single complex quantity that captures both resistance (real part) and reactance (imaginary part)
• Encode magnitude and phase of sinusoidal signals compactly
• Enable frequency-domain analysis techniques like Fourier transforms
• Simplify circuit calculations that would otherwise require solving differential equations in the time domain

Phasor Notation

Fundamentals of Phasors

A phasor is a complex number that represents a sinusoidal signal at a single frequency. It captures two pieces of information: the signal's amplitude (magnitude) and its phase angle relative to some reference.

Consider a time-domain signal $v(t) = V_m \cos(\omega t + \phi)$. The corresponding phasor is $\mathbf{V} = V_m \angle \phi$. Notice that the angular frequency $\omega$ drops out of the phasor representation. That's the whole point: since every signal in a linear AC circuit shares the same frequency, you don't need to carry $\omega$ through every calculation. You just track amplitude and phase, then reinsert $\omega$ at the end when converting back to the time domain.

Geometrically, you can think of a phasor as a vector in the complex plane that rotates counterclockwise at angular frequency $\omega$. The phasor itself is a snapshot of that vector at $t = 0$.

Phasor Representations

Phasors can be written in two equivalent forms, and converting between them is a skill you'll use constantly.

Polar form: $A \angle \theta$, where $A$ is the magnitude and $\theta$ is the phase angle.

Rectangular form: $a + jb$, where $a$ is the real component and $b$ is the imaginary component.

To convert between them:

• Polar to rectangular: $a = A \cos\theta$, $b = A \sin\theta$
• Rectangular to polar: $A = \sqrt{a^2 + b^2}$, $\theta = \tan^{-1}(b/a)$

Watch the quadrant when computing $\theta$. The $\tan^{-1}$ function on most calculators returns values in $(-90°, 90°)$, so if the complex number lies in the second or third quadrant (negative real part), you need to add or subtract $180°$ to get the correct angle.

Complex Number Operations

Manipulating Complex Numbers

Each operation has a form where it's easiest to perform:

• Addition/Subtraction — Use rectangular form. Add or subtract real and imaginary parts separately: $(a + jb) + (c + jd) = (a+c) + j(b+d)$
• Multiplication/Division — Use polar form. For multiplication, multiply magnitudes and add angles: $(A_1 \angle \theta_1)(A_2 \angle \theta_2) = A_1 A_2 \angle (\theta_1 + \theta_2)$. For division, divide magnitudes and subtract angles.
• Complex conjugate — Flip the sign of the imaginary part: $(a + jb)^* = a - jb$. In polar form, this negates the angle: $(A \angle \theta)^* = A \angle (-\theta)$. Conjugates are useful for rationalizing denominators: multiply numerator and denominator by the conjugate of the denominator.

The powers of $j$ cycle every four:

$j^1 = j, \quad j^2 = -1, \quad j^3 = -j, \quad j^4 = 1$

This pattern repeats, so for any power of $j$, just find the remainder when you divide the exponent by 4.

Euler's Formula and Exponential Form

Euler's formula is the bridge connecting exponentials to trig functions:

$e^{j\theta} = \cos\theta + j\sin\theta$

This gives us a third way to write a complex number, the exponential form: $Ae^{j\theta}$. It's equivalent to polar form ($A \angle \theta$) but is especially powerful for multiplication and division, since the rules of exponents apply directly.

A rotating phasor in the time domain is written as $Ae^{j(\omega t + \phi)}$. The real part of this expression, $A\cos(\omega t + \phi)$, recovers the physical sinusoidal signal. This is why the phasor transform works: operating on complex exponentials with linear circuit equations is algebraically simpler than working with cosines and sines directly, and you can always extract the real part at the end to get your actual signal.